Torsional chiral magnetic effect in Weyl semimetal with topological defect
Hiroaki Sumiyoshi$^1$
Satoshi Fujimoto$^{2
We propose a torsional response raised by lattice dislocation in Weyl semimetals akin to chiral magnetic effect; i.e. a fictitious magnetic field arising from screw or edge dislocation induces charge current. %in the direction %parallel to the field. %This effect is a result of mixed $U$(1)-torsional chiral anomaly of Weyl fermions. We demonstrate that, in sharp contrast to the usual chiral magnetic effect which vanishes in real solid state materials, the torsional chiral magnetic effect exists even for realistic lattice models, which implies the experimental detection of the effect via SQUID or nonlocal resistivity measurements in Weyl semimetal materials.
WSM.tex
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\def \spo{\phantom{a}}
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\begin{document}

\title{Torsional chiral magnetic effect in Weyl semimetal with topological defect}% Force line breaks with \\
%\thanks{A footnote to the article title}%

\author{Hiroaki Sumiyoshi$^1$}
% \altaffiliation[Also at ]{Department of Physics, Kyoto University, Kyoto 606-8502, Japan}%Lines break automatically or can be forced with \\
\author{Satoshi Fujimoto$^{2}$}%
\affiliation{$^1$Department of Physics, Kyoto University, Kyoto 606-8502, Japan}
\affiliation{$^2$Department of Materials Engineering Science, Osaka University, Toyonaka 560-8531, Japan}

\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified

\begin{abstract}
We propose a torsional response raised by lattice dislocation in Weyl semimetals akin to chiral magnetic effect;
i.e. a fictitious magnetic field arising from screw or edge dislocation induces charge current.
%in the direction
%parallel to the field.
%This effect is a result of mixed $U$(1)-torsional chiral anomaly of Weyl fermions.
We demonstrate that, in sharp contrast to the usual chiral magnetic effect which vanishes in real solid state materials, the torsional chiral magnetic effect exists even for realistic lattice models, which implies
the experimental detection of the effect via SQUID or nonlocal resistivity measurements in Weyl semimetal materials.
\end{abstract}

\pacs{72.80.-r, 72.15.-v, 11.30.Rd, 11.15.Yc}% PACS, the Physics and Astronomy
                             % Classification Scheme.
%\keywords{Suggested keywords}%Use showkeys class option if keyword
                              %display desired
\maketitle
%\textcolor{red}{cite Balents , Vishwanath}
Recently, many candidate materials for Dirac semimetals and Weyl semimetals (WSMs) \cite{vishwanath,balents,balents2,murakami}, have been discovered \cite{Kim_BiSb2013,Zhijun2012, Zhijun2013, Liu_NaBi2014,Neupane2014, Borisenko2014, Liu_CdAs2014,Li_ZrTe2014,TaAsWeng2015, Huang_TaAs2015, Xu_TaAs2015, Zhang_TaAs2015,LvTaAs2015, Shekhar2015,XiaochunTaAs2015, Yang_NbAs2015,Shekhar_TaP2015,Sushkov_EIO2015,Borisenko}. % by recent intensive experimental studies,
%e.g. Bi$_{1-x}$Sb$_x$\cite{Kim_BiSb2013} Na$_3$Bi \cite{Liu_NaBi2014} Cd$_3$As$_2$ \cite{Neupane2014, Borisenko2014, Liu_CdAs2014}  ZrTe$_6$ \cite{Li_ZrTe2014}, TaAs\cite{Huang_TaAs2015, Xu_TaAs2015, Zhang_TaAs2015}, NbP \cite{Shekhar2015}, NbAs \cite{Yang_NbAs2015}, TaP \cite{Shekhar_TaP2015}, Eu$_2$Ir$_2$O$_7$ \cite{Sushkov_EIO2015}, and YbMnBi$_2$\cite{Borisenko}. %, and Sr$_{1-y}$MnSb$_2$ \cite{Liu2015}.
%Besides the realization of the Dirac and Weyl systems in condensed matter physics, 
These topological semimetals are intriguing because of
%both theoretical and experimental studies have investigated 
exotic transport phenomena %characteristic of Dirac or Weyl fermions 
associated with the chiral anomaly in quantum field theory 
%which is the violation of the chiral symmetry preserved in the classical theory 
\cite{Fujikawa},
such as  
the anomalous Hall effect \cite{Zyuzin_AHE2012, Goswami2013},
chiral magnetic effect (CME) \cite{Hosur_review},
negative longitudinal magnetoresistance \cite{NN_1983, Kim_BiSb2013, Li_ZrTe2014,Yang_NbAs2015, Shekhar_TaP2015, Goswami_LMR2015, XiaochunTaAs2015}, and
chiral gauge field \cite{Liu2013}.
%and has close relation to the chiral anomaly, which is the violation of the chiral symmetry preserved in the classical theory \cite{Fujikawa}.
%the anomaly and surface states protected by the topology 
%the chiral nature and quantum effect

Among them, the CME has been discussed in broad areas of quantum many-body physics, including nuclear and nonequilibrium physics as well as condensed matter physics.
It is the generation of charge current parallel to an applied magnetic field even in the absence of electric fields.
In nuclear physics, together with the chiral vortical effect \cite{CVE2010}, it is expected to %cause charge polarization and fluctuation in the systems of relativistic Dirac fermions like quarks 
%in the presence of the chiral chemical potential induced by the chirality imbalance,
%and to 
play an important role 
in heavy ion collisions experiments \cite{Kharzeev2014, Fukushima_CME2008}.
The CME also caused a stir in nonequilibrium statistical physics,
since it leads to the existence of the ground state %or equilibrium current, 
which, recently, attracts a renewed interest
in connection with the realization of quantum time crystal
%in which time translation symmetry is spontaneously broken 
\cite{Wilczek2012},
and then the CME has been studied from this point of view \cite{Vazifeh2013, Yamamoto_2015}.
However, unfortunately, their results are negative for its realization:
the macroscopic ground state current in realistic WSMs is always absent.



%this fascinating phenomena is also intensively studied \cite{Hosur_review}.

%Although this effect is quite drastic, 
%However, recent theoretical studies reveals that the CME in the narrow sense, i.e. the ground state current induced by static magnetic field,
%is considered to be absent in the lattice system \cite{CME_lattice}.
%due to 

%time-dependent $\theta$-term which derive from chiral anomaly 

%\textcolor{red}{concrete and detailed explanation of the TCME}
In this letter, we propose a chiral response in WSMs, named ``torsional chiral magnetic effect (TCME)",
in which the ground state charge current is caused by the effective magnetic field induced by lattice dislocation
as shown in FIG.\ref{current}.
By using the Cartan formalism of the differential geometry, we can describe
the lattice strain and dislocation in terms of vielbein and torsion \cite{previous-study}.
From the viewpoint of the quantum field theory in curved space-time, the TCME is raised by the mixed action of electromagnetic and torsional fields that is
prohibited in four-dimensional spacetime with the Lorentz symmetry,
but made possible in non-relativistic band electrons in solid state systems.
Furthermore, we demonstrate that the TCME is possible in realistic lattice models by carrying out numerical calculations.
Our results imply the existence of experimentally observable current induced by the TCME in real WSM materials.
We also resolve the relation between our results and the no-go theorem that the CME is absent in equilibrium states \cite{Vazifeh2013, Yamamoto_2015}.
%First, we derive the Berry-phase formulae for the torsional response.
%Next, we discuss the physical consequences of this formulae in the lattice system with topological defect.
%Finally, the numerical results are shown and it is revealed that the TCME occurs even in the case of crystalline systems.
%in contrast to the conventional CME.
%we also reconsider the no-go theorem for the CME in equilibrium state \cite{Vazifeh2013, Yamamoto_2015}.
First of all, we clarify the notations. %The indices $\mu,\nu,\cdots=\tau,x,y,z$ and $\alpha,\beta,\cdots=\bar{\tau},\bar{x},\bar{y},\bar{z}$ represent the coordinates in the laboratory and local orthogonal (or Lorentz) frames, respectively, and $i,j,\cdots=x,y,z$ and $a,b,\cdots=\bar{x},\bar{y},\bar{z}$ are their spatial components.
The indices $i,j,\cdots=x,y,z$ and $a,b,\cdots=\bar{x},\bar{y},\bar{z}$ represent the coordinates in the laboratory and local orthogonal (or Lorentz) frames, respectively.
In the following, we use the Einstein summation convention.


%gravomagnetic field rotation \cite{Lynden-Bell1998, Nomura2012}


%torsional (or gravito-) magnetic field \cite{Shitade2014,Gromov2014, Bradlyn2015low}.

 






{\it Linear response theory for torsional response---}
%To investigate the TCME, 
Here, we briefly introduce the Cartan formalism, which can be applied to description of crystal systems with lattice strain as follows.
It is an approach to curved space %time,
and based on the local orthonormal frame form $ e^a=e^a_i({\bm r})dr^i $,   %$ e^{\alpha} = e^{\alpha}_{\mu}(r)dr^{\mu} $,
%is used to describe field theories in curved spacetime.
where the coefficient fields $e^a_i({\bm r})$ are referred to as the vielbein \cite{Nakahara_topo}.
%Its application to condensed matter physics is to describe crystal systems with lattice strain as follows.
We introduce the coordinate measured by an observer on the deformed lattice $R^{a}$ and the laboratory coordinate $r^i$, and
identify its exterior derivative as the local orthonormal frame $e^a=dR^a$.
Then, to the first order in the displacement field $\vec{u}$, the vielbein is written as $e^a_i=\delta^a_i-\part{u^a}/\part{r^i}$.
%, or equivalently, by using the displacement vector $u^{\bar{i}}=X^{\bar{i}}-x^i$, $e^a_i=\delta^a_i + \part u^a / \part x^i$.
For this observer, the lattice is not deformed,
and then the Hamiltonian of the system is given by
$H(-\mathrm{i}\part_{R^{\bar{x}}},-\mathrm{i}\part_{R^{\bar{y}}},-\mathrm{i}\part_{R^{\bar{z}}}) = H(-\mathrm{i}e^i_{\bar{x}}\part_{r^i},-\mathrm{i}e^i_{\bar{y}}\part_{r^i},-\mathrm{i}e^i_{\bar{z}}\part_{r^i})$,
where $H(p_x,p_y,p_z)$ is the Hamiltonian without lattice deformation and $e^{\mu}_{\alpha}$ is the inverse of $e_{\mu}^{\alpha}$.
In this way, the emergent vielbein appears, and therefore we can describe the elastic response by using the Cartan formalism.
The coupling between the vielbein and electrons is similar to the minimal coupling of the $U(1)$ gauge field, $p_i \to p_i - e A_i$.
Then, we can define the analog of the field strength by $T^{a}_{ij}=\part_{i}e^{a}_{j}-\part_{j}e^{a}_{i}$,
which is referred to as the torsion, or ``torsional magnetic field" (TMF) \cite{Shitade2014,Parrikar2014,torsional_mag},
where the spin connection is dropped for simplicity.
Using the displacement vector, the torsion is rewritten as $T^a_{ij}=(\part_{j}\part_{i}-\part_{i}\part_{j})u^a$.
The point is that, if $u^a({\bm r})$ is a well-defined function, the torsion is always zero,
and the multivaluedness of $u^a({\bm r})$ is necessary for nonzero torsion.
Indeed, the edge dislocation along $z$-axis with Burgers vector $b_g\hat{y}$ causes the TMF, $T^y_{xy}=-b_g\delta^{(2)}(x,y)$,
and the screw one with $b_g\hat{z}$, $T^z_{xy}=-b_g\delta^{(2)}(x,y)$, as shown in FIG.\ref{current}.
For more details about the lattice strain and differential geometry, see, for example, Refs. \cite{Edelen, T.Hughes2013, Kleinert}.


%One of the advantage over the ordinary formalism of the Riemann geometry based on the metric tensor is that it can be used even if there is nonzero torsion,
%which, in the absence of curvature, can be chosen to take as the rotation of the vielbein $T^{\alpha}=de^{\alpha}$ or $T^{\alpha}_{\mu\nu}=\part_{\mu}e^{\alpha}_{\nu}-\part_{\nu}e^{\alpha}_{\mu}$ \cite{T.Hughes2011, T.Hughes2013}.

%The CME is the generation of the current by parallel magnetic field, and in this letter we investigate its torsional version caused by the torsional magnetic field.

Now, using the linear response theory with the Cartan formalism, we investigate the TCME of WSMs due to dislocation.
We calculate the current density in the presence of TMF and magnetic field up to the linear order.
%with the Cartan formalism.
%discuss effects of the TMF on WSMs in more details.
We use the model of a pair of Weyl fermions with the opposite chirality,
whose Weyl points are at ${\bm k}={\bm \lam}^L$ and ${\bm \lam}^R$ in the momentum space, and
Fermi energies are given by $E=v_{F}\lam^L_0$ and $v_{F}\lam_0^R$, respectively.
Therefore the $4\times{4}$ Hamiltonian is given by
\begin{align}
H({\bm k}):= 
\left(
\begin{array}{cc}
H_{L}({\bm{k}})&0\\
0&H_{R}({\bm{k}})
\end{array}
\right)
%&H_{WSM}({\bm k}):=\textcolor{red}{H_{L}({\bm k}){\oplus}H_{R}({\bm k}),} \nonumber \\
 %\nonumber \\
%H_{R}({\bm k}) :=  v_F \left[ - ({\bm k} - {\bm \lam}^{R})\cdot {\bm \sig} - \lam^R_0 \right],
\end{align}
with $H_{s}({\bm{k}}):=v_F\left[\chi_{s}({\bm{k}}-{\bm\lam}^{s})\cdot{\bm\sig}-\lam^{s}_0\right]$, 
%where $\chi= \pm 1$ is the index of the chirality of Weyl fermions.
where $s=L\ {\rm or}\ R$ is the index of the chirality and $\chi_{L(R)}=+1(-1)$,
and $\sigma^i$ is the Pauli matrix.
Well, we calculate the current density in the presence of the external fields.
The calculation is performed by the variation of the effective action, $S_{{\rm{eff}}}[A_i,e^a_i]$,
with respect to the gauge field, as $j^a({\bm r}):= -(e_i^a({\bm r})/|e({\bm r})|)(\delta S_{{\rm eff}}/\delta A_i({\bm r}))$.
The effective action is defined as
\begin{align}
& e^{- S_{{\rm eff}}[A_i ,e^a_i]} := \int \mathcal{D} \psi \mathcal{D}\psi^{\dag}  \exp \left( - S[\psi, \psi^{\dag}, A_i,e^a_{i}] \right),  \nonumber \\
& S[\psi, \psi^{\dag}, A_i,e^a_i] :=\frac{1}{2} \int d\tau d^3r  \left[   \psi^{\dag} (\tau , {\bm r} )\hat{ \mathcal{L} } \psi (\tau , {\bm r} )  + c.c.   \right] , \nonumber \\
& \hat{\mathcal{L}} :=  |e({\bm r})| [ \part / \part \tau - %\hat{H}({\bm r} , -i \part_{\bm r}; A_{i} , e^{a}_{i})  
H(-\mathrm{i} \nabla_a)], % - e A_0({\bm r})],
%\hat{\mathcal{L}}(\tau, {\bm r} ; A_{\mu} , e^{\alpha}_{\mu}) =
\label{eq:eff-action}
\end{align}
where $\psi$ is the fermionic field, $\tau$ and ${\bm r}$ denote the imaginary time and spatial coordinate, respectively,
and $A_i$ %and $e^{\alpha}_{\mu}$ are 
is the vector potential. % and vielbein, respectively.
Here $c.c.$ represents the complex conjugate combined with the change of the sign of the derivative operator $\part/\part \tau$.
Also, in Eq.(\ref{eq:eff-action}),  the Jacobian is given by $|e({\bm r})|:=\det e^a_i({\bm r})$,
and the covariant derivative is $-\mathrm{i}\nabla_a:=e^j_{a}({\bm{r}})(-\mathrm{i}\part_j-e A_j({\bm{r}}))$ with $a=\bar{i}$.
Using Eq.(\ref{eq:eff-action}),
we obtain that the
current density up to the first order of the magnetic field and the TMF is given by
\begin{align}
%j^0({\bm r}) = &C \frac{e({\bm \lam}^L- {\bm \lam}^R)\cdot {\bm B}}{ 4\pi^2} \label{WSM_charge}\\
{\bm j} ({\bm r}) =&\left[  \frac{e^2 v_F    (\lam_{{0}}^R  - \lam_{{0}}^L)}{4 \pi^2}{\bm B}  + \frac{e  v_F   (\lam^R_a - \lam^L_a ) \Lam   }{ 4 \pi ^2}  {\bm T^a} \right], \label{WSM_current} \nonumber \\
%& \cdot \left(1+\mathcal{O}\left(\frac{|{\bm \lam}^{L/R}|}{\Lam} , \frac{| \lam_0^{L/R}|}{\Lam} \right) \right)
%&+ \frac{v_F \{ \lam^R_a (\Lam + \lam^R_0)  - \lam^L_a (\Lam + \lam^L_0)  ) \}}{4 \pi ^2} e {\bm T^a}
\end{align}
at zero temperature and up to the linear order in $\lam_{\mu}^{L(R)}$,
where the details of the calculations are described in Ref.\cite{supp}.
%\begin{align}
%{\bm j}({\bm r}) =& e \sum_n  \int \frac{d^3 \bm \pi}{ (2\pi)^3}   \left[  {\bm v}^n({\bm \pi})  n_F(\vareps_{n, {\bm \pi}}) \right. \nonumber \\
%& \left. - \left( {\bm v}^n({\bm \pi}) \cdot  {\bm \Omega}^{ n}({\bm \pi}) \right) (e{\bm B}({\bm r}) +  {\bm T^a}({\bm r}) \pi_a )  n_F(\vareps_{n, {\bm \pi}})  \right].   \label{BF_current}
%\end{align}
Here, %$\pi^a$ is the mechanical (gauge invariant) momentum, 
%$n$ is the band index,
%$\vareps_{n,{\bm \pi}}$ is the energy, $v^{n,a}({\bm \pi}):=\part \vareps_{n,{\bm \pi}}/\part \pi_a$ is the group velocity,
%and $n_F(\vareps)$ is the Fermi distribution function.
%The Berry curvature %and intrinsic magnetic moment \cite{Thonhauser, Xiao2005} 
%is given by
%$\Omega^{na}({\bm \pi}) := -i \vareps^{abc} \braket{ \part u^n_{ {\bm \pi} }/ \part \pi_b |  \part u^n_{ {\bm \pi} } / \part \pi_c} $, 
%$m^{n a}({\bm \pi}):= (i /2) \vareps^{abc} \braket{ \part u^n_{ {\bm \pi} }/ \part \pi_b |[ H_0({\bm \pi}) - \vareps_{n, {\bm \pi} } ] |   \part u^n_{ {\bm \pi} } / \part \pi_c} $, respectively,
%where $H_0$ is the Hamiltonian without external fields and $\ket{u^n_{{\bm \pi}}}$ is its Bloch wave function.
The vector representation of the TMF, ${\bm T}^a$, is defined by $({\bm T^a})_i:=(1/2)\vareps^{ijk}T^{a}_{jk}$. %, like the magnetic field $B_{i}:=(1/2)\vareps^{ijk}F_{jk}$.
For the derivation of Eq.(\ref{WSM_current}), we introduced a momentum cutoff scheme
$|{\bm k}-{\bm \lam}^{s}|<\Lam$ for the Weyl node of the chirality $s$.
Physically, ${\Lam}$ corresponds to the momentum range from the Weyl points in which the cone structures of the band of the lattice system is approved.

%$eF_{ij}=-i[-i\part_i -eA_i,-i\part_j -eA_j] \to eF_{ab} + T^c_{ab}(-i\nabla_c)=-i[-i\nabla_a,-i\nabla_b]$.
%Then we see that the torsional magnetic field works in the similar way as the magnetic field.
%and then work 





The first term represents the CME in the presence of the chiral chemical potential (i.e. $\lam^L_0\neq\lam^R_0$),
and then reproduces the previous result for the CME \cite{SonYamamotoPRL}.
 %, and were obtained in previous works.
On the other hand, the second term in Eq.(\ref{WSM_current}) 
is a new one, which raises %torsional responses of current, 
the TCME;
%The first term corresponds to the CME \cite{SonYamamotoPRL,SonYamamotoPRD}.
%The second therms represents two type of torsional responses:
i.e. the current is generated by the TMF
for the pair of Weyl points which are shifted in the momentum space due to broken time-reversal symmetry (TRS).
This point is in sharp contrast to the usual CME, which requires breaking inversion symmetry.






%{\it Chiral anomaly and Lorentz symmetry breaking---}
%In this letter, we have derived the TCME, in which the charge current is caused by the torsion, by using linear response theory.
%Moreover, we have confirmed it in realistic condensed matter system by showing its physical semiclassical pictures and the calculation of the ground state spectrum and current of both the linearized and tight-binding model.
%Therefore, 
We comment on the relation between the TCME and the chiral anomaly.
One may expect that when, $\lam^R_{\mu}=-\lam^L_{\mu}$, the TCME is described by the topological $\theta$-term, which is the consequence of the chiral anomaly like the CME and anomalous Hall effect.
However, there is no mixed chiral anomaly term of $U(1)$ field strength and torsion in four-dimensional spacetime
%as shown by field theoretical approach, the evaluation of the anomaly polynomial 
\cite{munoz2010, Parrikar2014}. % and \cite{}). %T.Hughes2013})
%There seems a contradiction,
%but this inconsistency comes from the Lorentz symmetry breaking cutoff scheme of the derivation of Eq.(\ref{WSM_current}),
%which is not broken in the calculation scheme in \cite{munoz2010, Parrikar2014}.
%Therefore, the TCME is realized in condensed matter system, in which the Lorentz symmetry is broken.
This point is resolved by the observation that the Lorentz symmetry, which is postulated in the calculation scheme in Refs. \cite{munoz2010, Parrikar2014},
is broken in the cutoff scheme used for the derivation of 
the second term of Eq.(\ref{WSM_current}), % ${\bm \lam}^s\neq0$, which %depicts
which is correctly applicable to
realistic condensed matter systems.






%Here $C$ and $D$ are numerical factors that depend on the scheme of the momentum cutoff
%when the spatial rotation symmetry is broken owing to the shift of the Weyl point in the momentum space.
%Indeed, $C=D=1/3$ for the spherical cutoff scheme $|{\bm k}|<\Lam$,
%while $C=1-2/\sqrt{1+r^2}$ and $D=r(\pi{}/2-{\rm arctan}(r))$, (note that $\lim_{r\to \infty} C,D=1$) for the cylindrical cutoff scheme $|{\bm k}_{\parallel}|<\Lam$ and $|{\bm k }_{\perp}|<r \Lam$.
%Here ${\bm k}_{\parallel}$ is the parallel part of the momentum to the direction of the Weyl point and ${\bm k}_{\perp}$ is the perpendicular part.
%Eq.(\ref{WSM_charge}) with $r\to \infty$ or $C=1$ reproduces the result of the correction to the density, that obtained from the $\theta$-term caused by
%the U(1) chiral anomaly \cite{Vazifeh2013}.
%Therefore we adopt this scheme, and then set $C=D=1$.
%Such cutoff scheme dependence of the numerical factors has also been pointed out in the case of the anomalous Hall coefficient of WSMs \cite{Goswami2013}.

%Now we discuss the physical consequences of Eqs.(\ref{WSM_charge}) and (\ref{WSM_current}).
%Eq.(\ref{WSM_charge}) encodes the charge response in time-reversal symmetry (TRS) broken WSMs, and



%Now we discuss the physical consequences of Eq.(\ref{WSM_current}).
%The first term in Eq.(\ref{WSM_current}) represents the CME in the presence of the chiral chemical potential, and were obtained in previous works.
%%These terms were obtained in previous works.
%On the other hand the second term in Eq.(\ref{WSM_current}) raises %torsional responses of current, 
%the TCME;
%i.e. the current is generated by the torsional magnetic field 
%for the pair of Weyl points which are shifted in the momentum space due to broken TRS.
%This point is in sharp contrast to the usual CME, which requires breaking inversion symmetry.


Now we discuss the consequences and physical pictures of the TCME.
The TCME is realized in two types of lattice dislocations.
%\textcolor{red}{torsional mag. field and burgers vector}
(a) {\it case of edge dislocation}: %current induced by edge dislocation, 
$j^x\propto  \Delta \lam_z\ T^z_x$, and 
(b) {\it case of screw dislocation}: $j^z \propto \Delta \lam_z T^z_z$,
%where the summation over $a$ or $b$ is not taken and 
with $\Delta \lam_a := \lam^L_a - \lam^R_a$.
Their schematic pictures are shown in FIG.\ref{current}.
%The response (a) can be understood with the following semiclassical picture.
These responses can be understood with the following semiclassical picture:
{\it Case} (a): Edge dislocation is regarded as the (0,1,0)-``surface"
of the extra lattice plane made up of the blue and green atoms in FIG.\ref{current}-(a),
which harbors a chiral Fermi arc, when two Weyl points are shifted in the $k_z$-direction.
%when two Weyl points are shifted in the $k_2$-direction, there is the chiral Fermi arc on the (1,0,0)-``surface" of the extra plane (green atoms in the figure).
The electrons in the Fermi arc state are the very origin of the current induced by the edge dislocation.
{\it Case} (b): 
There is a chiral Fermi arc mode on the dislocation line.
The electrons in the mode rotate around the screw dislocation line,
and due to the screw dislocation the rotating motion causes the current along the Burgers vector.
%The chiral Fermi arc mode on the dislocation line drives electrons into spiral motion, generating the current flow along the screw dislocation line. %along the line.
%Due to the screw dislocation, the turning electron go up along the line, and them the current is generated.
%\textcolor{red}{It the explantation drawing for (b) needed?}


\begin{figure}
 \begin{center}
%\includegraphics[width=80mm]{edge_disloc_longarrow.eps}
\includegraphics[width=80mm]{dislocation_current.eps}
 \end{center}
\caption{
%\textcolor{red}{the coordinate axis and bulk Weyl cones.}
%\textcolor{red}{draw the arrow of (a) must be more clear.}
(Color Online)
Ground state current $j$ induced by (a) edge and (b) screw dislocation
with the Burgers vector $b$.
%(a) edge dislocation $T^{\bar{z}}_{yz} = -T^{\bar{z}}_{zy}=-b_g\delta^{(2)}(y,z)$
%(b) Ground state current along the screw dislocation ($T^{\bar{z}}_{xy}=-T^{\bar{z}}_{yx}=-b_g\delta^{(2)}(x,y)$) %when the Weyl points are 
%$b$ and $j$ represent the Burgers vector and induced current, respectively.
 }                                                                                                                                                                                                                                         
 \label{current}
\end{figure}

%For understanding physical meanings,
%it is helpful to relate these effects with those in a similar system to the WSM, i.e. 
The situation is similar to that of
the three-dimensional integer quantum Hall state (3DIQHS) \cite{Kohmoto1992} with %an edge or screw 
dislocation
which is the staking of quantum Hall state layers characterized by %. %and 
%topologically equivalent to the system 
%the present 
%A 3DIQHS is characterized by a reciprocal lattice vector ${\bm G}_c=(2\pi/a_c)(n_x,n_y,n_z)$,
%where $a_c$ is the lattice constant and $n_i$ is the Chern number of the IQHS for the staking direction $i$ and $n_j=0$ for $j\neq i$.
%Here we define 
the vector ${\bm G}_c=(2\pi{n_c}/a)\hat{n}$, where $n_c$, $a$, and $\hat{n}$ are the first Chern number,  %of the quantum Hall layers,
the lattice constant, and the unit vector along the staking direction.
In the 3DIQHS, there are one dimensional $n$ chiral modes along the dislocation line,
when the topological number, $n={\bm{b}}_g\cdot{}{\bm{G}_c}/2\pi$, is nonzero %(c.f. the class A(i.e. without TRS) version of \cite{Ran2009} and Sec. III of 
\cite{Teo-Kane2010}.
This condition for the chiral modes %that the Burgers vector is parallel to the staking direction
is similar to that for the TCME, ${\bm b}_g\cdot ({\bm \lambda}^L -{\bm \lambda}^R)\neq 0$.

However, there are the following significant differences.
%the properties of the chiral modes in the two cases.
The chiral modes of the 3DIQHS are exponentially localized at the dislocation, and
 %. In other words, their amplitudes exhibit exponential decay
%exhibiting exponential decay of their amplitude
%for large distance from the dislocation line. %and $\xi$ is the length scale of the localization.
%Also they appear as the midgap states 
separated from the bulk higher-energy states, while,
%On the other hand,
 as will be shown below, the chiral modes of the WSMs 
exhibit power-law decay.
We call them quasi-localized modes.
Moreover their spectrum is not isolated from the bulk spectrum but appears as its envelope as shown in FIG.\ref{spectrum}a,
and therefore they %are not robust against perturbations and 
can be easily mixed with the bulk modes.

{\it Spectral asymmetry and ground state current---}
%\textcolor{red}{this section must be drastically simplified and unessential discussions moved into the supplemental material.}
%Here, using an alternative approach other than the linear response theory based on (\ref{WSM_current}), we confirm %Eq.(\ref{WSM_current})
%the TCME due to dislocation.
We confirm the TCME due to dislocation by using an alternative approach other than the linear response theory based on (\ref{WSM_current}).
Our approach here is to calculate explicitly the spectrum and the eigenstates of the Weyl Hamiltonian with dislocation and the ground state current.
We also show that the quasi-localized modes along the envelope of the bulk spectrum contribute to the effect.
%by explicitly calculating the spectrum of the Weyl Hamiltonian with a screw dislocation and the ground state current.
For simplicity, we set $v_F=e=1$ and assume %that the chemical potentials for both Weyl fermions are zero, 
$\lam^L_0=\lam^R_0=0$,
and the Weyl points lie symmetrically on the $k_z$-axis, ${\bm \lam}^L=-{\bm \lam}^R=\lam\hat{z}$.
In the presence of the screw dislocation at $x=y=0$ along $z$-axis, of which Burgers vector is ${\bm b}_g=-b_g\hat{z}$, the vielbeins are given by 
$e^z_{\bar{x}}=-b_gy/ 2 \pi \rho^2,\ e^z_{\bar{y}}=b_gx/2\pi \rho^2$, and $e^{\mu}_{a}=\delta^{\mu}_a$ for others,
with $\rho=\sqrt{x^2+y^2}$ \cite{Edelen}.
Even with the dislocation, $k_z$ remains a good quantum number.
Then, when $k_z$ is fixed,
the Hamiltonian is equivalent to
that of two dimensional massive Dirac model in the presence of the magnetic flux at the origin, whose amplitude is $\Phi_{k_z}=k_zb_g$,
\begin{align}
%H^{{\rm screw}}_{L(R)}
%=&\chi_{L(R)} \left[ \left(-i\part_x- \frac{bk_z y}{2\pi \rho^2} \right)\sig^x + \left(-i\part_y+ \frac{bk_zx}{2\pi \rho^2}\right)\sig^y    \right. \nonumber \\ 
%&\left. +(k_z- \chi_{L(R)} \lam )\sig^z\right]. %\nonumber \\
%H_{R}({\bm k}) :=  v_F \left[ - ({\bm k} - {\bm \lam}^{R})\cdot {\bm \sig} - \lam^R_0 \right],
&H^{{\rm screw}}_{s,k_z}=\chi_{s} \left[  H^{\perp}_{k_z}  + m^{s}_{k_z} \sig^z\right], \nonumber \\
&H^{\perp}_{k_z}= \left(-\mathrm{i}\part_x- \frac{\Phi_{k_z} y}{2\pi \rho^2} \right)\sig^x + \left(-\mathrm{i}\part_y+ \frac{\Phi_{k_z}x}{2\pi \rho^2}\right)\sig^y,
\end{align}
with the mass $m^{s}_{k_z}=k_z-\chi_{s}\lam$.
%$k_z$ remains a good quantum number and, when it is fixed, this Hamiltonian is equivalent to
%that of two dimensional massive Dirac model with the mass $M^{L(R)}_{k_z}=k_z-\chi_{L(R)}\lam$
%in the presence of the magnetic flux at the origin, whose amplitude is $\Phi_{k_z}=k_3b/2\pi$.
%To understand what modes contribute the ground state current,
The equivalence of a screw dislocation and momentum-dependent magnetic field has also been pointed out in Refs. \cite{Imura2011, Parrikar2014}.
%There are two important points in the spectrum structure that follow from the anti-commutation relation, $\{H^{\perp}_{k_z},\sig^z\}=0$.
%The first one is that the absolute value of the energy of any state satisfies $|E|\geq |m^{s}_{k_z}|$.
%The second point is that there is
%one-to-one correspondence in the states with $E^+>|m^{s}_{k_z}|$ and $E^-<-|m^{s}_{k_z}|$ modes as $E^+=-E^-$,
%but there is asymmetry in the states with $E=\pm{m^{s}_{k_z}}$ in the presence of the magnetic field.
%This asymmetry is called the parity anomaly \cite{Jackiw1984, Haldane1988}.

The spectrum of $H^{{\rm screw}}_{s,k_z}$ consists of two types of eigenstates:
one with the eigenenergies satisfying $|E|>|m^{s}_{k_z}|$ and the other one with $E=\pm{m^{s}_{k_z}}$.
%eignenergies of the 
% whose eigenenergies are and .
%as immediately follows from the anti-commutation relation $\{H^{\perp}_{k_z},\sig^z\}=0$.
The former does not contribute to the ground state current owing to the one-to-one correspondence between $E^+>|m^{s}_{k_z}|$ and $E^-<-|m^{s}_{k_z}|$ modes as $E^+=-E^-$,
%the occupied and unoccupied state 
and between the states of Weyl nodes with the opposite chiralities \cite{supp}.
On the other hand, the latter does contribute owing to asymmetry, i.e. the absence of one-to-one correspondence between $E={m^{s}_{k_z}}$ and $E=-{m^{s}_{k_z}}$ modes.
This asymmetry is called the parity anomaly \cite{Jackiw1984, Haldane1988}.
The asymmetric spectrum consists of discrete modes whose wavefunctions exhibit power-law decay, and
continuum scattering modes which spread over the whole system \cite{Kiskis1977,Aharonov1979,supp}.
The schematic picture of the density-of-state of the full spectrum is shown in FIG.\ref{spectrum}a.
Moreover, the ground state current calculated from the asymmetric spectrum is $J^z=-L_zb_g\Lam\lam/2\pi^2$, which coincides with the expression obtained directly from the linear response theory (\ref{WSM_current})
\cite{supp}.


%They consist of discrete normalizable modes and continuum scattering modes.
%
%If $\Phi_{k_z}>0$, there are $|N_{k_z}|$ normalizable states at the same energy $E=-\chi_sm^s_{k_z}$
%with  $N_{k_z}:=-{\rm sgn}({\Phi_{k_z}})\lfloor{|\Phi_{k_z}|/2\pi}\rfloor$
%where $\lfloor{x}\rfloor$ represents the largest integer not greater than $x$.
%On the other hand,  there are no states with $E=\chi_sm^s_{k_z}$,
%signifying the spectrum asymmetry.
%Moreover, the wave functions exhibit power-law decay (then, they are quasi-localized at the dislocation line) like
%$\propto(0,\rho^{-\Phi_{k_z}/2\pi}(x-iy)^{i-1})$ as $\rho\to\infty$,
%with $i=1,\cdots,|N_{k_z}|$ \cite{Kiskis1977,Aharonov1979}.
%The continuum modes are also asymmetric at $E=\pm{m^s_{k_z}}$
%and then contribute to the TCME. \cite{supp}
%Therefore there is difference in the density-of-states at $E=\pm{m^s_{k_z}}$
%as the schematic picture is shown in Fig.\ref{spectrum}-a.
%Using the regularization scheme for the evaluation of the current due to the asymmetric scattering modes presented in \cite{Kiskis1977},
%we can calculate their current and obtain $J^z=-L_zb_g\Lam\lam/2\pi^2$, which coincides with the expression obtained directly from the Berry-phase formula (\ref{BF_current}).




 


\begin{figure}
 \begin{center}
 \includegraphics[width=80mm]{spectrum.eps}
% \includegraphics[width=25mm]{nonlocal.eps}
%\includegraphics[width=40mm]{pair_screw_disloc.eps}
%\includegraphics[width=25mm]{spectrum_0.4.eps}~
%\includegraphics[width=25mm]{spectrum_0.6.eps}~
%\includegraphics[width=25mm]{spectrum_0.8.eps}
%\includegraphics[width=30mm]{spectrum_full.eps}~
%\includegraphics[width=15mm]{color_bar.eps}~
%\includegraphics[width=25mm]{current.eps}
%\includegraphics[width=30mm]{nonlocal.eps}
 \end{center}
\caption{(Color Online)
(a) Schematic picture of the spectrum of the WSM with the screw dislocation.
The black (white) bands
along $E=\pm{m^s_{k_z}}$ represent the relatively higher (lower) density-of-state compared with that of the opposite energy $E=\mp{m^s_{k_z}}$.
(b) Lattice with a pair of screw dislocations with opposite Burgers vectors.
(c)
Numerical result for the spectrum of the WSMs with a pair of screw dislocations.
The blue curves are the envelope of the bulk spectrum.
The opacity of the dots %correspond to the average of $\hat{}$ with respect to each modes.
represents the expectation value of $|{\bm{\rho}}-{{\bm{l}}^{dis}}|^2/L_xL_y$ (see the gray scale bar).
%which indicate how localized each modes are at the dislocation at ${{\bm{l}}^{dis}}$.
In the figures (c1-3), the modes with this value smaller than $0.1$, $0.15$, and $0.2$ are plotted.
%and In (c4), the full spectrum.
(d)
Current density along $z$-direction, $j_z(x,y)$.
%Upward current along the screw dislocation 
%and downward current along the anti-screw dislocation are obtained.
}
%Note: 
%Indeed, as the thlethold of $\langle \hat{\rho}^2 /L^2 \rangle$ increases, the modes }
 \label{spectrum}
\end{figure}



%the sum of the expectation values of $\sig^z$ in all (both occupied and unoccupied) eigenstates of $H^{\perp}_{k_z}$,
%$\ket{\kappa}_{k_z}$
%$J^z=L_z\int{dk_z}/{2\pi}\braket{\kappa|\sig^z|\kappa}_{k_z}$


%{\it for continuum model ---}



%not constant magnetic field but depends on the wave numbers

%follows the discussion in Ref. \cite{Vazifeh2013}.


{\it Numerical calculation ---}
We confirm the spectrum asymmetry and
the TCME for realistic lattice models by numerical calculations.
%derived from the linearized effective model by numerical calculation of the lattice model.
We use the tight-binding model of WSMs \cite{Vazifeh2013} generalized to the case with dislocation,
\begin{align}
&H =
\sum_{{\bm r}}  \left[   {\mathrm i}t\sum_{i=x,y,z} c^{\dag}_{{\bm r}+\hat{i}+\delta_{i,z}b_g\Theta({\bm r})}\Gamma^i c_{{\bm r}}  +r \left( 3c^{\dag}_{{\bm r}}\Gamma^4c_{\bm r}\phantom{\sum_{i=x,y,z}} \right. \right. \nonumber\\
& \left. \left. -  \sum_{i=x,y,z} c^{\dag}_{{\bm r}+\hat{i}+\delta_{i,z}b_g\Theta({\bm r})}\Gamma^4 c_{{\bm r}} \right)
+ \frac{d}{2} c^{\dag}_{{\bm r}}\Gamma^{12} c_{{\bm r}} \right]+h.c., \label{numerical_hamil}
\end{align}
where the $4\times{4}$-matrices, $\Gamma^i$, satisfy the $SO(5)$ Clifford algebra $\{\Gamma^i,\Gamma^j\}=2\delta_{ij}$ \cite{Murakami2004},
$\Gamma^{ij}:=[\Gamma^i,\Gamma^j]/2i$,
${\bm r}=(x,y,z)$ and $\hat{i}$ denote the position of the atoms and the $x^i$-direction unit vector, respectively,
and 
$t,r$, and $d$ are the real parameters,
and we suppose the lattice constant as $1$ and lattice size $L_x\times{L_y}\times{L_z}$.
%Here the first term is the kinetic term,
%the second one is the Wilson term, which gaps out the Dirac fermions other than at the $\Gamma$-point,
%the third one decomposes the Dirac fermion at the $\Gamma$-point into a pair of Weyl fermions and shift the Weyl points toward the $\pm{k_z}$-direction.
We introduced a pair of screw dislocations along $z$-direction with opposite Burgers vector at $\pm{{\bm{l}}^{dis}}=\pm({l^{dis}_x},0)$
as shown in FIG.\ref{spectrum}-b, by
sliding the hopping directions in the first and third terms of Eq. (\ref{numerical_hamil})
as $\Theta({\bm r})=-1$ for the region $x=0$, $-l^{dis}_x<y<l^{dis}_x$, while $\Theta({\bm r})=0$ for other regions.
%screw dislocation along $z$-axis with Burgers vector ${\bm b}_g=b_g\hat{z}$ is realized on the tight binding model by the shift of the hopping terms as
%$c^{\dag}_{x,y,z}c_{x,y+1,z}\to c^{\dag}_{x,y,z}c_{x,y+1,z+b_g}$ if $x>0,y=0$. %(i.e. the hopping  the half-plane in the $xz$-plane,
%where the Burgers vector along $z$-axis is given by .
%Even after this change, the lattice momentum $k_z$ remains a good quantum number, although $k_x$ and $k_y$ are not.
We numerically diagonalized this model and obtained the spectrums and current.
Here the material parameters are set as $t=r=1$ and $d=3.6$.
%and, therefore, the Weyl points in the Brillouin zone of the bulk Hamiltonian are at ${\bm k} = (0,0,\pm 2.2)$
%and the Fermi velocity at the Weyl points is given by $v_F=0.9$.
The lattice constant is $1$ and the amplitudes of the Burgers vectors is set as $b_g=1$.
For the calculation, we imposed the open boundary condition along the $x-$ and $y-$directions
and periodic boundary condition along the $z-$direction,
and set $L_x=L_y=4l^{dis}_x=38$ and $L_z=100$.

As shown in FIG.\ref{spectrum}c1-3, we obtained the asymmetric spectrum in agreement with the analytic calculation.
The asymmetric modes are localized at the dislocation line.
%The number of modes near the dislocation line increases  gradually, 
%as the threshold expectation value of $|{\bm{\rho}}-{{\bm{l}}^{dis}}|^2/L_xL_y$ increases (Fig.\ref{spectrum}c1-3).
%This means that in the realistic tight-binding model %instead of the linearized model,
%the quasi-localized chiral modes are not isolated from the bulk but easily mixed with the bulk modes, as mentioned before.
The quasi-localized chiral modes are not isolated from the bulk but easily mixed with the bulk modes  (FIG.\ref{spectrum}c1-3). 
The current density at zero temperature is shown in FIG.\ref{spectrum}d.
We obtained the upward current along the screw dislocation and downward current along the anti-screw dislocation
due to the TCME.
The total current per the unit length toward $z$-direction due to one dislocation line is $J^z/L_z=0.087$,
which is calculated by the summation of the current density in the $x>0$ half-plane,
and this value is in the same order as that estimated from the linear response theory (\ref{WSM_current}), $J^z/L_z\sim0.1$.
For the estimation, we set the cutoff as $\Lam\sim1/{\rm{(lattice\ constant)}}=1$.
%, which corresponds to the momentum range from the Weyl points in which the cone structures of the band is approved.

%Finally, we give the comments on the TCME.

{\it No-go theorem of CME ---}
%We, here, comment on the relationship of the existence of the TCME in the realistic lattice model with 
The existence of the TCME in the realistic lattice system may seem to contradict with the no-go theorems of the ground state current \cite{Vazifeh2013, Yamamoto_2015}.
%Indeed, the edge or screw dislocation can be considered as the deformation of the atomic potential,
%then these theorems can be applied for the present system. %\cite{supp}.
However, they prohibit the total current, but not the local current density. %over all space 
Therefore, the current along %and localized at 
the dislocation line can exist, as we found\cite{supp}.
%Also, if we consider the case of one dislocation line,
%instead of a pair of dislocations,
%there is the backflow on the surface of the crystal that cancel out the current on the dislocation line and the total current is always zero


{\it Experimental implication ---}
Here we present two experimental setups to observe the TCME in TRS-broken WSMs, for which Eu$_2$Ir$_2$O$_7$ \cite{Sushkov_EIO2015} and YbMnBi$_2$\cite{Borisenko} are candidate materials.
The first one is a scanning SQUID measurement, %(FIG.\ref{fig_exp}a),
which can detect weak inhomogeneous magnetic fields \cite{Vasyukov2013, Wang948}. % (Fig.\ref{}a).
If there is a pair of dislocation, the circulating current occurs.
The magnitudes of the current and the induced magnetic field are estimated as $I\sim10^{-5}{\rm{A}}$ and $B\sim10^{-7}{\rm{T}}$, respectively,
%and therefore that of magnetic field  
for both Eu$_2$Ir$_2$O$_7$ and YbMnBi$_2$.
Here we used Eq.(\ref{WSM_current}) and the material parameters, $v_F\sim10^5m/s$ and $a\sim10{\rm{\AA}}$
and set $b_g=a$ and $\lam\sim\Lam\sim1/a$, where $a$ is the lattice constant.
Also, for the estimation of the magnitude of the magnetic field, we used %the formula for that generated by the circulating current,
%$B=\mu_0I/2\pi{r}$, where we set $r$ as the typical inter-distance between dislocations, $10^{5}\mathrm{\AA}$ \cite{},
%and $\mu_0$ is the vacuum permeability.
a typical value of inter-distance between dislocations,  $10^{5}\mathrm{\AA}$ \cite{dis3}.
%$B\sim\mu_0$
%The circulating current cause magnetic field, of which magnitude depends on the geometry of sample, 
It is feasible to detect $B\sim10^{-7}{\rm{T}}$  via the scanning SQUID.
%If we detect the magnetic field via the measurement technique,
%it is the fingerprint of the TCME.
%.\cite{Vasyukov2013, Wang948}.

%The first one is similar to 
The second one is a nonlocal transport phenomenon, which was observed in quantum Hall materials \cite{McEuen1990, Wang1991}. %and anomalous spin Hall materials \cite{McEuen1990, Wang1991, Roth2009}.
The experimental setup is shown in FIG.\ref{fig_exp}. %b. %nonlocal}.
%The current and voltage leads are attached at the end points of a dislocation line and 
%voltage leads at those of a anti-dislocation line and another points.
%If there were only to be the chiral modes along the dislocation lines, and the back scattering and bulk contribution are negligible,
If the bulk contributions are completely negligible and there are only the chiral modes at the dislocation lines,
$V_{34}:=V_3-V_4=0$ despite $I_{12}\neq0$, then, the nonlocal resistivity $R_{12,34}:=V_{34}/I_{12}$ is equal to zero \cite{McEuen1990}. %as  follows from the Landauer-Buttiker formula.
On the other hand, if %there were only to be local transport and 
the nonlocal transport is negligible, $V_{34}>V_{3'4'}$ holds for $L_1L_3<L_{1}L_{3'}$ when $I_{12}\neq0$.
%since the effect of the conduction along the dislocation line becomes weaker as far from the line. 
Therefore, if $R_{12,34}<R_{12,3'4'}$ is observed, it is the fingerprint of the chiral current due to the TCME.
%The second one is the anomalous Hall effect.
%When we apply uniform electric field along the dislocation lines,
%the charge is pumped from dislocation (anti-dislocation) lines to an anti-dislocation (dislocation) lines.
The effect can be discriminated from any previously reported conventional transport induced by dislocation\cite{dis0,dis1,dis2,dis3,dis4,dis5,dis6,dis7,dis8,dis9}.



\begin{figure}
 \begin{center}
 \includegraphics[width=70mm]{exp.eps}
% \includegraphics[width=25mm]{nonlocal.eps}
%\includegraphics[width=40mm]{pair_screw_disloc.eps}
%\includegraphics[width=25mm]{spectrum_0.4.eps}~
%\includegraphics[width=25mm]{spectrum_0.6.eps}~
%\includegraphics[width=25mm]{spectrum_0.8.eps}
%\includegraphics[width=30mm]{spectrum_full.eps}~
%\includegraphics[width=15mm]{color_bar.eps}~
%\includegraphics[width=25mm]{current.eps}
%\includegraphics[width=30mm]{nonlocal.eps}
 \end{center}
\caption{(Color Online)
%(a)The current (shown as arrows and $I$) due to the TCME in the presence of a pair of dislocation lines.
%The thick red solid (blue dashed) line represents the (anti-)dislocation line.
%We can detect the magnetic field, $B$, via a scanning SQUID measurement. 
Experimental setup for the nonlocal transport due to the TCME.
The thick red solid (blue dashed) line represents the (anti-)dislocation line.
The leads are attached at the black points, $L_i$($i=1,2,3,4,3',4'$).
%$L_1$, $L_3$, and $L_{3'}$ ($L_2$, $L_4$, and $L_{4'}$) are on the top (bottom) surface of the sample.
The line $L_1L_2$, $L_3L_4$, and $L_{3'}L_{4'}$ are parallel and have the same length.
Here we suppose $L_1L_3<L_1L_{3'}$.
$V_i$ is the voltage at $L_i$, and $I_{12}$ is the current.
}
%Note: 
%Indeed, as the thlethold of $\langle \hat{\rho}^2 /L^2 \rangle$ increases, the modes }
 \label{fig_exp}
\end{figure}



We also comment on effects of impurities and disorientation of the dislocation. %% on 3/4 by fuji%% for implication for the realistic experience.
First, the current due to the TCME is expected to be robust against weak disorder. % effects at least in weak disorder regime.
%from two point of view explained as follows.
%First, %mathematically or formally, 
It is because that Eq.(\ref{WSM_current}) is independent of the scattering time,
%Therefore, the current is not directly affected by the disorder effects unlike the normal conductivity $\sigma \propto \tau$,
like the intrinsic contribution to the anomalous Hall effect \cite{Nagaosa_AHE}.
%[Nagaosa, et al., Rev. Mod. Phys. 82, 1539 (2010)].
%The second evidence is based on the following 
%A more physical argument is as follows:
More precisely, the current %caused by TCME is the 
is  due to the edge modes in the Fermi arc
 on the surface of WSMs,
and these modes are supported by the Weyl points, %in bulk Brillouin zone,
which are protected by the Chern number, and hence, robust againt weak disorder.
%Here, the Chern number is invariant against weak perturbation like
%impurity scattering(even it is magnetic impurity or inter-Weyl-cones scattering).
%Therefore the TCME is robust against the weak impurity scattering.


%and it is robust against weak impurity scattering
%since there is no mode moving in the opposite direction,
%as the case of the edge modes of the quantum Hall state. %[McEuend et al., Phys. Rev. Lett. 64, 2062 (1990)].
%The origin of the TCME is the chiral current of the WSM %(see the 3rd reply for Referee comment on the physical picture of the TCME presented above),
%and then the current due to the TCME is expected to be robust against weak disorder effects.
%On the other hand, %as Referee commented, 
%the existence of dislocation may imply strong disorder which may raise additional spin-orbit interaction and 
%change the bulk band structure. 
%:
%for example, as Referee commented, dislocation may cause a spin-orbit interaction and change the band structure.
%From another point of view, 
%However, 
%the pair of the Weyl points are protected by the Cher number, and then,

%there is chiral current 
 %[for example, review for arXiv:1309.4464].
%We postulate that the density of dislocation is not too high to destroy the bulk topological band character.
%as long as disorder effects on the bulk Weyl band are negligible, 
%they do not change the result on the TCME qualitatively.

Next, in real experimental setups, it is difficult to align the dislocation line orthogonal (parallel) to the line connecting the Weyl nodes exactly in the case (a) (case (b)).
Even when they are not orthogonal (parallel), as long as they are not parallel (orthogonal), the current parallel to the dislocation line still exists.
%the direction of the current is parallel to the dislocation line but the amplitude is changed.
Supposing that the dislocation line is parallel to the $z$-axis,
the current is given by
$J^z=ev_FL_z  \Lambda ({\bm \lambda}^R-{\bm \lambda}^L)\cdot {\bm b_g}/4\pi^2$ in the both cases (a) and (b).
%Here $\Delta{\bm \lambda}$ is the vector from the Weyl point with the chirality $R$ to that with the chirality $L$,
%${\bm b}_g$ is the Burgers vector. 
%"$\cdot$" represent the inner product, and
%$e$, $v_F$, $L_z$, and $\Lambda$ are the elementary charger, Fermi velocity, system size along $z$-direction, and the cutoff momentum.
%For the derivation of this expression, we used the fact that % in the presence of edge or screw dislocation along $z$-axis, %with Burgers vector ${\bm b}_g$,
%the torsion is given by $T^a_{z}=-b^a_g \delta(x,y)$, %where $a$ is the index of the coordinate,
%and integrated Eq.(3) over the space.






{\it Summary ---}
In this letter, we have discussed the TCME in WSMs caused by dislocation.
We have confirmed that it is possible to occur and experimentally observable in realistic materials, %from the physical pictures and the numerical calculation of the tight-binding model,
%proposed the experimental setups to observe it,
and argued that the Lorentz symmetry breaking is important for it. %in contrast to the CME.




\begin{acknowledgments}
{\it Acknowledgement ---}
We thank H. Fujita, Y. Kikuchi, T. Kimura,  K. Ohmori, D. Schmeltzer, K. Shiozaki,
and A. Shitade 
for fruitful discussions. 
%%added by Fujimoto on Feb. 1.
We also thank one of the referees for suggesting scanning SQUID measurements.
%%%%%%%%%
This work is supported by the Grant-in-Aids for Scientific
Research from MEXT of Japan [Grant No. 23540406, No. 25220711, and No. 25103714 (KAKENHI on Innovative Areas “Topological Quantum Phenomena"), No. 15H05852 (KAKENHI on Innovative Areas  “Topological Materials Science")].
HS is supported by a JSPS Fellowship for Young Scientists (14J00647).
\end{acknowledgments}

\phantom{*}

%\bibliography{WSM}
%\bibliographystyle{apsrev}
\begin{center}
\Large{Supplemental Material}
\end{center}

\appendix
\setcounter{equation}{0}
\renewcommand{\theequation}{S-\arabic{equation}}

\section{Derivation of Eq.(3)}
In this section, we derive %the Berry-phase formulae for
the expression for the current in the presence of
the magnetic and torsional magnetic responses of the current density, Eq.(3).
The derivation consists of two steps:
first we derive the expression for the Green function in the presence of the gauge field and vielbein using the gradient expansion Eq.(\ref{Green_1st}),
and next we calculate the current density by using Eq.(\ref{Green_1st}) and obtain Eq.(\ref{WSM_current_1_final}), which is equivalent to Eq.(3).

%For the derivation of the Berry-phase formulae for the charge and current density (\ref{BF_charge}) and (\ref{BF_current}),
First, we calculate the single-electron Green function.
The Green function in the presence of the gauge field and vielbein which is defined by
\begin{align}
&G( \tau_1 , {\bm r}_1 , \tau_2, {\bm r}_2)  \nonumber \\
&:= \frac{ \int \mathcal{D} \psi \mathcal{D}\psi^{\dag}   \psi(\tau_1 , {\bm r}_1 ) \psi^{\dag} (\tau_2 , {\bm r}_2 ) \exp \left( - S[\psi, \psi^{\dag}, A_i,e^a_i] \right) }{ \int \mathcal{D} \psi \mathcal{D}\psi^{\dag}  \exp \left( - S[\psi, \psi^{\dag}, A_i,e^a_i] \right) }.
\end{align}
Then, the following differential equation holds :
\begin{align}
&\frac{1}{2} \left[   \hat{\mathcal{L}}(\vareps_N,{\bm r}_1 , -\mathrm{i} \part_{{\bm r}_1} ) G(\vareps_N, {\bm r}_1,{\bm r}_2 )  \right. \nonumber \\
&+\left. G(\vareps_N, {\bm r}_1,{\bm r}_2 )   \overleftarrow{\hat{\mathcal{L}}}^* ( - \vareps_N, {\bm r}_2, -\mathrm{i} \part_{{\bm r}_2})  \right] =\delta^{(3)}({\bm r}_1 - {\bm r}_2) ,  \label{dyson}
\end{align}
with $ \hat{\mathcal{L}}(\vareps_N, {\bm r}, -\mathrm{i} \part_{\bm r}  ) :=  |e({\bm r})| [ \mathrm{i} \vareps_N - H(-\mathrm{i} \nabla_a)]  -e A_0]$, and $|e({\bm r})|:=\det e^a_i({\bm r})$
Here $\vareps_N=(2N+1)\pi T$ is the Fermionic Matsubara frequency with the temperature $T$,
and $G(\vareps_N, {\bm r}_1,{\bm r}_2 ):=\int_0^{\beta}G(\tau, {\bm r}_1, 0, {\bm r}_2 ) e^{-\mathrm{i} \vareps_N \tau}d\tau$ is the Fourier component of the Green function.
Now, using the spatial Wigner transformation defined as $\tilde{f}({\bm R},{\bm p}) := \int d^3{\bm r} e^{-\mathrm{i}{\bm r} \cdot {\bm p}} f({\bm R} + {\bm r}/2, {\bm R} - {\bm r} /2)$,
Eq. (\ref{dyson}) is rewritten into
\begin{align}
 &\frac{1}{2} \left[ \hat{\mathcal{L}}(\vareps_N , {\bm R}, {\bm p}) e^{\frac{\mathrm{i}}{2} (\overleftarrow{\partial_{\bm R}} \overrightarrow{\partial_{\bm p}} -\overleftarrow{\partial_{\bm p}} \overrightarrow{\partial_{\bm R}}) }   \tilde{G}(\vareps_N , {\bm R}, {\bm p})  \right.\nonumber \\
 &\left. +    \tilde{G}(\vareps_N , {\bm R}, {\bm p})  e^{\frac{\mathrm{i}}{2} (\overleftarrow{\partial_{\bm R}} \overrightarrow{\partial_{\bm p}} -\overleftarrow{\partial_{\bm p}} \overrightarrow{\partial_{\bm R}}) }   \hat{\mathcal{L}}(\vareps_N , {\bm R}, {\bm p}) \right]
  = 1 .
\end{align}
In the gradient expansion up to the first order in $\part_i A_j$ or $\part_i e^a_j$, the Green function becomes
\begin{align}
&\tilde{G}(\vareps_N , {\bm R}, {\bm p}) = \tilde{G}^{(0)}(\vareps_N , {\bm R}, {\bm p})  + \tilde{G}^{(1)}(\vareps_N , {\bm R}, {\bm p}) + \cdots, \nonumber \\
&\tilde{G}^{(0)}(\vareps_N , {\bm R}, {\bm p}) = \frac{1}{|e({\bm R})|} \left[ \mathcal{L}_0 (\vareps_N , {\bm \pi}) \right]^{-1}_{\pi_a = e^i_a({\bm R}) (p_i - e A_i({\bm R}))}, \nonumber \\
& \tilde{G}^{(1)}(\vareps_N , {\bm R}, {\bm p}) \nonumber \\
&=  \frac{\mathrm{i}}{2|e({\bm R})|}  \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi}) \frac{ \part \mathcal{L}_0 (\vareps_N , {\bm \pi})}{\part \pi_a } \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi})  \frac{ \part \mathcal{L}_0 (\vareps_N , {\bm \pi})}{\part \pi_b} \nonumber \\
&\cdot \left. \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi})  \left[ e F_{ab}({\bm R}) + T^c_{ab}({\bm R}) \pi_c \right] \right|_{\pi_a = e^i_a({\bm R}) (p_i - e A_i({\bm R}))}. \label{Green_1st}
\end{align}
Here $\pi_a := e^i_a({\bm R}) (p_i - e A_i({\bm R}))$ is the gauge-invariant mechanical momentum, while $p_i$ is the canonical momentum, and the field strength and the torsion with the indices of the local orthogonal coordinate are, respectively, defined as
$F_{ab}:=e^i_a e^j_b F_{ij}$ and $T^c_{ab}:=e^i_a e^j_b T^c_{ij}$.
The free Lagrangian is defined as $ \mathcal{L}_0 (\vareps_N , {\bm \pi}) := \mathrm{i} \vareps_N - H({\bm \pi})$. %where $H_0$ is the Hamiltonian in the absence of external fields.

Next, 
using Eq. (\ref{Green_1st}), we calculate the current density and derive Eq.(2).
The current density are defined by
$j^a({\bm r}):= -(e_i^a({\bm r}))/|e({\bm r}))|) (\delta S_{{\rm eff}} /\delta A_i({\bm r}))$.
Therefore 
\begin{align}
&j^{\bar{1}}({\bm R})\nonumber \\
&=\frac{e^{\bar{1}}_{i}}{|e({\bm R})|}\left. \frac{ \int \mathcal{D} \psi \mathcal{D}\psi^{\dag}   \psi^{\dag}(\tau, {\bm r}_1 ) \frac{1}{2} \frac{\delta {\hat {\mathcal L}}}{\delta A_i} \psi (\tau, {\bm r}_2 )  e^{ - S }} { \int \mathcal{D} \psi \mathcal{D}\psi^{\dag}  e^{ - S }} \right|_{{\bm r}_1,{\bm r}_2 \to {\bm R},}+ c.c. \nonumber \\
&=\frac{eT}{2} \sum_{N} {\rm Tr} \left[ \left. \frac{\part \mathcal{L}_0(\epsilon_N,{\bm \pi})}{\part \pi_{\bar{1}}}\right|_{\pi_a=e^i_a({\bm r}_2)(-\mathrm{i}\part_{r_2^i}-eA_i({\bm r}_2))} \right. \nonumber \\
&\left. \left. G(\vareps_N, {\bm r}_2,{\bm r}_1) \right] \right|_{{\bm r}_1,{\bm r}_2 \to {\bm R}}+ c.c. \nonumber \\
&=\frac{eT}{2} \sum_{N}\int \frac{d^3p}{(2\pi)^3} {\rm Tr} \left[ \frac{\part \mathcal{L}_0(\epsilon_N,{\bm \pi})}{\part \pi_{\bar{1}}}\right|_{\pi_a=e^i_a({\bm R})(p_i-eA_i({\bm R}))}
\nonumber \\
& \left. e^{\frac{\mathrm{i}}{2} (\overleftarrow{\partial_{\bm R}} \overrightarrow{\partial_{\bm p}} -\overleftarrow{\partial_{\bm p}} \overrightarrow{\partial_{\bm R}}) }   \tilde{G}(\vareps_N , {\bm R}, {\bm p})
\right] + c.c.
\end{align}
Here ${\rm Tr}$ means the trace over the band indices, we used %the assumption of the minimal coupling of the gauge field 
that $\delta \hat {\mathcal L}/\delta A_i=e|e({\bm r})|e^i_a({\bm r}) \part L_0/\part \pi_a$ and the second line does not depend on $\tau$ due to
imaginary time-translation symmetry.
Note that ${\bm \pi}$ of the third line is the operator though that of the fourth one is the c-number.
Using Eq.(\ref{Green_1st}), up to the first order in $\part_i A_j$ or $\part_i e^a_j$, the expression of the current density becomes
\begin{align}
&j^{\bar{1}}({\bm R}) = j^{\bar{1}(0)}({\bm R})+j^{\bar{1}(1)}({\bm R}),
\end{align}
with the zeroth-order terms,
\begin{align}
&j^{\bar{1}(0)}({\bm R})\nonumber \\
&=\frac{eT}{2}\sum_{N}\int \frac{d^3p}{(2\pi)^3}{\rm Tr}\left[\left. \frac{\part \mathcal{L}_0(\epsilon_N,{\bm \pi})}{\part\pi_{\bar{1}}} \right|_{\pi_a=e^i_a({\bm R})(p_i-eA_i({\bm R}))}  \right. \nonumber \\
& \left. \tilde{G}^{(0)}(\vareps_N , {\bm R}, {\bm p}) \right] +c.c. \label{current_0th}
\end{align}
and the first-order terms,
\begin{align}
&j^{\bar{1}(1)}({\bm R}) \nonumber \\
&=\frac{eT}{2}\sum_{N}\int \frac{d^3p}{(2\pi)^3}{\rm Tr}\left[\left. \frac{\part \mathcal{L}_0(\epsilon_N,{\bm \pi})}{\part\pi_{\bar{1}}} \right|_{\pi_a=e^i_a({\bm R})(p_i-eA_i({\bm R}))}  \right. \nonumber \\
& \left. \tilde{G}^{(1)}(\vareps_N , {\bm R}, {\bm p}) \right] \nonumber \\
& + \frac{\mathrm{i}eT}{4}\sum_{N}{\rm Tr}\left[ \left. \frac{\part \mathcal{L}_0(\epsilon_N,{\bm \pi})}{\part\pi_{\bar{1}}} \right|_{\pi_a=e^i_a({\bm R})(p_i-eA_i({\bm R}))} \right. \nonumber \\
& \left.  (\overleftarrow{\partial_{\bm R}} \overrightarrow{\partial_{\bm p}} -\overleftarrow{\partial_{\bm p}} \overrightarrow{\partial_{\bm R}})  \tilde{G}^{(0)}(\vareps_N , {\bm R}, {\bm p}) \right] +c.c.. \label{current_1st}
\end{align}
The zeroth-order terms (\ref{current_0th}) can be rewritten as
\begin{align}
j^{\bar{1}(0)}({\bm R})=e\sum_n\int\frac{d^3\pi}{(2\pi)^3}  v^{n,\bar{1}}({\bm \pi})  n_F(\vareps_{n, {\bm \pi}}). \label{j_1_0_final}
\end{align}
For the derivation,
we inserted the identity, $1_{{\bm \pi}}=\sum_n \ket{u^n_{{\bm \pi}}}\bra{u^n_{{\bm \pi}}}$,
between $\part \mathcal{L}_0/\part \pi_{\bar{1}}$ and $\tilde{G}^{(0)}$ in Eq.(\ref{current_0th}),
and used the formula, %$\sum^{\infty}_{s=-\infty}1/(z^2+(s+1/2)^2)=\pi\tanh(\pi z)/z$,
$\sum^{\infty}_{N=-\infty}\left[ 1/(\mathrm{i}\vareps_N-t)+1/(-\mathrm{i}\vareps_N-t) \right]=(1-2n_F(t))/T$,
for the summation over the Matsubara frequency,
and$ \int d^3p=|e({\bm R})|\int d^3\pi$.
Here, $n$ is the band index,
$\vareps_{n,{\bm \pi}}$ is the energy, $v^{n,a}({\bm \pi}):=\part \vareps_{n,{\bm \pi}}/\part \pi_a$ is the group velocity,
$n_F(\vareps):=1/(e^{\vareps/T}+1)$ is the Fermi distribution function,
and $\ket{u^n_{{\bm \pi}}}$ is the Bloch state.
This term corresponds to the summation of all contributions to the current from the electrons in the occupied states in the absence of magnetic and torsional magnetic field.
%therefore we set $j^{\bar{1}(0)}({\bm R})=0$ and 
%and then we set $j^{\bar{1}(0)}({\bm R})=0$.

Now, we move on the calculation of Eq.(\ref{current_1st}).
The sum of the second term of Eq.(\ref{current_1st}) and its complex conjugate is zero,
since %the complex conjugate of the factor in the square braket is 
$\left[ \cdots \right]^*=\left. \left[ \cdots \right] \right|_{\vareps_N\to-\vareps_N}$.
Then, by using Eq.(\ref{Green_1st}), we obtain
\begin{align}
&j^{\bar{1}(1)}({\bm R}) \nonumber \\
&=\frac{\mathrm{i}eT}{4}\sum_{N}\int \frac{d^3{\pi}}{(2\pi)^3} {\rm Tr}\left[
\frac{ \part \mathcal{L}_0 (\vareps_N , {\bm \pi})}{\part \pi_{\bar{1}} } \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi}) \frac{ \part \mathcal{L}_0 (\vareps_N , {\bm \pi})}{\part \pi_a } \right. \nonumber \\
&\left. \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi})  \frac{ \part \mathcal{L}_0 (\vareps_N , {\bm \pi})}{\part \pi_b} \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi}) \right] \left[ e F_{ab}({\bm R}) + T^c_{ab}({\bm R}) \pi_c \right] \nonumber \\
&+c.c..
\end{align}
Moreover, inserting the identities, $1_{{\bm \pi}}=\sum_n \ket{u^n_{{\bm \pi}}}\bra{u^n_{{\bm \pi}}}$,
we obtain
\begin{align}
&j^{\bar{1}(1)}({\bm R}) \nonumber \\
&=\frac{-\mathrm{i}eT}{4}\sum_{N,n,m,l}\int \frac{d^3{\pi}}{(2\pi)^3} 
\Braket{n|\frac{\part H}{\part \pi_{\bar{1}}}|m}
\Braket{m|\frac{\part H}{\part \pi_{a}}|l} \nonumber \\
&\times\Braket{l|\frac{\part H}{\part \pi_{b}}|n}
\frac{1}{(\mathrm{i}\vareps_N-\vareps_n)(\mathrm{i}\vareps_N-\vareps_m)(\mathrm{i}\vareps_N-\vareps_l)} \nonumber \\
&\times \left[ e F_{ab}({\bm R}) + T^c_{ab}({\bm R}) \pi_c \right] + c.c.,
\end{align}
where the indices ${\bm \pi}$ are omitted like $\vareps_n:=\vareps_{n,{\bm \pi}}$ and $\ket{n}:=\ket{u^n_{{\bm \pi}}}$.
There are three types of contributions to the summation over the band indices $n,m,l$:
(a) all the three are the same, %$n=m=l$,
(b) two of them are the same and the other is different,
and (c) each one is different respectively. %$n\neq m \neq l \neq n$.
However the contribution (a) is found to be zero because of the antisymmetry of $\left[ e F_{ab}({\bm R}) + T^c_{ab}({\bm R}) \pi_c \right]$ under $a\leftrightarrow b$.
Moreover, the contribution (c) is also zero, since 
our model of the WSM, Eq.(1),
consists of two two-band Hamiltonians independent of each other,
and then the overlap of
three or more bands is zero. 
Therefore, we have only to consider the contribution (b), and then obtain
\begin{align}
&j^{\bar{1}(1)}({\bm R}) \nonumber \\
&= \frac{-\mathrm{i}e}{4}\sum_{n} \int \frac{d^3{\pi}}{(2\pi)^3} \left[ e F_{ab} + T^c_{ab} \pi_c \right] (M_{\bar{1}ab}+M_{ab\bar{1}}+M_{b\bar{1}a})\nonumber\\
& + c.c.,\label{current_M}
\end{align}
with
\begin{align}
M_{abc}
&:= v^n_{a}n'_F(\vareps_n) \Braket{n,b|(\vareps_n-H)|n,c} \nonumber \\
& +v^n_{a}\Braket{n,b|n_F(H)|n,c}-v^n_{a}n_F(\vareps_n)\Braket{n,b|n,c} , \label{M_def}
\end{align}
where we used the abridged notation,
$\ket{n,a}:=\Ket{\frac{\part u^n_{\bm \pi}}{\part \pi_{a}}}$.
For the derivation of Eqs.(\ref{current_M},\ref{M_def})
we used the formulae $\sum_{N=-\infty}^{\infty}\frac{1}{( \mathrm{i}\vareps_N-t)^2(\mathrm{i}\vareps_N-s)}=\frac{tn'_F(s)-sn'_F(t)+n_F(s)-n_F(t)}{T(t-s)^2}$,
$\Braket{n|\part H/\part \pi_a|m}=(\vareps^m-\vareps^n)\Braket{n|m,a}$ for $n\neq m$,
and $\sum_mf(\vareps_m)\Braket{n,b|n,c}=\Braket{n,b|f(H)|n,c}$ for any function $f$.
Moreover, using the relationship $(M_{abc})^*=M_{acb}$, we obtain
\begin{align}
&j^{\bar{1}(1)}({\bm R}) \nonumber \\
&= \frac{-\mathrm{i}eT}{2}\sum_{n}\int \frac{d^3{\pi}}{(2\pi)^3}  \left[ e F_{\bar{2}\bar{3}} + T^d_{\bar{2}\bar{3}} \pi_d \right] \vareps^{abc}M_{abc},
\end{align}
where $\vareps^{abc}$ is the antisymmetric symbol.
Furthermore, 
since $v^n_{a}n'_F(\vareps_n)=\part n_F(\vareps_n)/\part \pi_a$
and only antisymmetric parts of $M_{abc}$ contribute,
using integration by parts,
we find
\begin{align}
&j^{\bar{1}(1)}({\bm R}) \nonumber \\
&=\mathrm{i}e\sum_{n}\int \frac{d^3{\pi}}{(2\pi)^3}  \left[ e F_{\bar{2}\bar{3}} + T^d_{\bar{2}\bar{3}} \pi_d \right] \vareps^{abc} v^n_{a}n_F(\vareps_n)\Braket{n,b|n,c} \nonumber \\
&+\frac{\mathrm{i}e}{2}\sum_{n}\int \frac{d^3{\pi}}{(2\pi)^3}  T^a_{\bar{2}\bar{3}}  \vareps^{abc} \vareps_n n_F(\vareps_n)\Braket{n,b|n,c}.
\end{align}
Using  the Berry curvature is defined by
$\Omega^{n}_a := -\mathrm{i} \vareps^{abc} \braket{ n, b | n,c} $,
and the vector representation of the TMF, ${\bm T}^a$, is defined by $ T^a_i:=(1/2)\vareps^{ijk}T^{a}_{jk}$,
it can be rewritten as
\begin{align}
&j^{\bar{1}(1)}({\bm R}) \nonumber \\
&=- e \sum_n  \int \frac{d^3 \pi}{ (2\pi)^3}  \left( {\bm v}^n \cdot  {\bm \Omega}^{ n}\right) (eB_{\bar{1}}+  T^a_{\bar{1}} \pi_a )  n_F(\vareps_{n}) \nonumber \\
&-\frac{e}{2} \sum_n  \int \frac{d^3 \pi}{ (2\pi)^3} \Omega^{n}_a T^a_{\bar{1}}\vareps_nn_F(\vareps_n). \label{BF_current_1}
\end{align}
%$\pi^a$ is the mechanical (gauge invariant) momentum, 
%$n$ is the band index,
%$\vareps_{n,{\bm \pi}}$ is the energy, $v^{n,a}({\bm \pi}):=\part \vareps_{n,{\bm \pi}}/\part \pi_a$ is the group velocity,
%and $n_F(\vareps)$ is the Fermi distribution function.
%The Berry curvature %and intrinsic magnetic moment \cite{Thonhauser, Xiao2005} 
%is given by
%$\Omega^{na}({\bm \pi}) := -i \vareps^{abc} \braket{ \part u^n_{ {\bm \pi} }/ \part \pi_b |  \part u^n_{ {\bm \pi} } / \part \pi_c} $, 
%$m^{n a}({\bm \pi}):= (i /2) \vareps^{abc} \braket{ \part u^n_{ {\bm \pi} }/ \part \pi_b |[ H_0({\bm \pi}) - \vareps_{n, {\bm \pi} } ] |   \part u^n_{ {\bm \pi} } / \part \pi_c} $, respectively,
%where $H_0$ is the Hamiltonian without external fields and $\ket{u^n_{{\bm \pi}}}$ is its Bloch wave function.
It is noted that %neglecting the second term,
the term containing $B_{\bar{1}}$ is equal to
the expression for the CME %using the Berry curvature 
derived by Son and Yamamoto \cite{SonYamamotoPRL},
and the others are new terms that represent the current induced by the torsion.
Neglecting the last term, which is, as we will discuss later, less important than the others in the case of WSMs,
Eq.(\ref{BF_current_1}) can also shortly derived from the
substitution of the magnetic field or the field strength in the absence of the vielbein,
$-\mathrm{i}[(-\mathrm{i}\part_2-eA_2),(-\mathrm{i}\part_3-eA_3)]=eB_{1}$ with 
%the formula represented of 
%They are coincident with the terms obtained by replacing the field strength with
the field strength in the presence of the vielbein, $-\mathrm{i}[-\mathrm{i}\nabla_{\bar{2}},-\mathrm{i}\nabla_{\bar{3}}]=eB_{\bar{1}}+T^a_{\bar{1}}(-\mathrm{i}\nabla_a)$,
where $[U,V]:=UV-VU$ is the commutator.
This justifies the analogy between the TMF and the magnetic field.
%the commutator of the covariant derivatives with both the vector potential and vielbein leads to
%the TMF as well as the magnetic field, i.e.

Finally, we substitute the energy, group velocity, and Berry curvature of the model of the WSM, (1), into Eq.(\ref{BF_current_1})
and derive Eq.(3).
%substituting the expressions for the energy, velocity, and Berry curvature %, and intrinsic angular momentum 
%for this Hamiltonian \cite{supp} into the Berry-phase formula (\ref{BF_current}), we obtain
We characterize the four bands of the Hamiltonian (1) as $n=(s,\pm)$, with $s=L\ {\rm or}\ R$,
where $s$ is the index of the chirality and $+(-)$ means the higher (lower) band of the Weyl cone.
Then, their energy, group velocity, and Berry curvature are given by
\begin{align}
&\vareps^{s,\pm}({\bm k}) = v_F \left[ \pm   |{\bm k} - {\bm \lam}^{s}| - \lam^{s}_0 \right]  \nonumber \\
&{\bm v}^{s,\pm} ({\bm k}) = \pm \frac{v_F({\bm k} - {\bm \lam}^s)}{|{\bm k} - {\bm \lam}^s|} \nonumber \\
%&{\bm v}^{R,\pm} ({\bm k}) = \pm \frac{v_F({\bm k} - {\bm \lam}^R)}{|{\bm k} - {\bm \lam}^R| \nonumber \\
&{\bm \Omega}^{s,\pm} ({\bm k}) = \pm \chi_{s}\frac{{\bm k} - {\bm \lam}^{s}}{{2|{\bm k} - {\bm \lam}^s|^3}}. \label{BC,vel}
\end{align}
Using Eqs.(\ref{BF_current_1}, \ref{BC,vel}),
we obtain
\begin{align}
%j^0({\bm r}) = &C \frac{e({\bm \lam}^L- {\bm \lam}^R)\cdot {\bm B}}{ 4\pi^2} \label{WSM_charge}\\
&j^{\bar{1}(1)} ({\bm R}) \nonumber \\
&=\left[  \frac{e^2 v_F    (\lam_{{0}}^R  - \lam_{{0}}^L)}{4 \pi^2}B_{\bar{1}}({\bm R})  + \frac{e  v_F   (\lam^R_a - \lam^L_a ) \Lam   }{ 4 \pi ^2}  T^a_{\bar{1}}({\bm R})  \right], \label{WSM_current_1_final} \nonumber \\
%& \cdot \left(1+\mathcal{O}\left(\frac{|{\bm \lam}^{L/R}|}{\Lam} , \frac{| \lam_0^{L/R}|}{\Lam} \right) \right)
%&+ \frac{v_F \{ \lam^R_a (\Lam + \lam^R_0)  - \lam^L_a (\Lam + \lam^L_0)  ) \}}{4 \pi ^2} e {\bm T^a}
\end{align}
at zero temperature and up to the linear order in $\lam_{\mu}^{L(R)}$.
For the derivation of Eq.(\ref{WSM_current_1_final}), we introduced a momentum cutoff scheme
$|{\bm k}-{\bm \lam}^{s}|<\Lam$ for the Weyl node of the chirality $s$.
Physically, ${\Lam}$ corresponds to the momentum range from the Weyl points in which the cone structures of the band of the lattice system is approved.
Note that the last term of Eq.(\ref{BF_current_1}) yields second(or more)-order contributions in $\lam_{\mu}^{L(R)}$, and then less important as mentioned before.
Eq.(\ref{WSM_current_1_final}) is the correction of current due to the TMF and magnetic field and is equivalent to Eq.(3),
then the derivation of Eq.(3) has been completed.

%$\sum_{s=-\infty}^{\infty}1/[(\{ i(s+1/2)-z\}^2\{ (i(s+1/2)-w)\}]=$
%$\sum_{s=-\infty}^{\infty}\frac{1}{\{ i(s+1/2)-z\}^2\{ i(s+1/2)-w\}}=\frac{tn'_F(s)-sn'_F(t)+n_F(s)-n_F(t)}{T(t-s)^2}$
%$\sech$

%\begin{align} 
%&n({\bm R})  \nonumber \\
%&= e T \sum_{N} \int \frac{d^3 {\bm \pi}}{(2\pi)^3} {\rm Tr} \left[   \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi})   \right] \nonumber \\
%&+ \frac{i e}{2} T \sum_N  \int \frac{d^3 {\bm \pi}}{(2\pi)^3} {\rm Tr} \left[    \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi}) \frac{ \part  \mathcal{L}_0 (\vareps_N , {\bm \pi})}{\part \pi_a}   \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi})  \right. \nonumber \\
%& \cdot \left. \frac{ \part  \mathcal{L}_0 (\vareps_N , {\bm \pi})}{\part \pi_b}    \mathcal{L}^{-1}_0 (\vareps_N , {\bm \pi}) \right]   
%\left[ e F_{ab}({\bm R}) + T^c_{ab}({\bm R}) \pi_c \right]
%\end{align}
\if0
\begin{align}
j^{\bar{0}}({\bm r}) =& e \sum_n  \int \frac{d^3 \bm \pi}{ (2\pi)^3} \left[  n_F(\vareps_{n, {\bm \pi}}) \right. \nonumber \\
& -  {\bm \Omega}^n({\bm \pi}) \cdot (e{\bm B}({\bm r}) +  {\bm T^a}({\bm r}) \pi_a )  n_F(\vareps_{n, {\bm \pi}})    \nonumber \\
& \left. -  {\bm m}^n({\bm \pi}) \cdot  (e{\bm B}({\bm r}) +  {\bm T^a}({\bm r}) \pi_a ) n'_F(\vareps_{n, {\bm \pi}}) \right]\\
{\bm j}({\bm r}) =& e \sum_n  \int \frac{d^3 \bm \pi}{ (2\pi)^3}   \left[  {\bm v}^n({\bm \pi})  n_F(\vareps_{n, {\bm \pi}}) \right. \nonumber \\
& \left. - \left( {\bm v}^n({\bm \pi}) \cdot  {\bm \Omega}^{ n}({\bm \pi}) \right) (e{\bm B}({\bm r}) +  {\bm T^a}({\bm r}) \pi_a )  n_F(\vareps_{n, {\bm \pi}})  \right].  \nonumber \\
\end{align}
Here, the Berry curvature and intrinsic angular moment \cite{Thonhauser, Xiao2005} are defined by
$\Omega^{na}({\bm \pi}) := -i \vareps^{abc} \braket{ \part u^n_{ {\bm \pi} }/ \part \pi_b |  \part u^n_{ {\bm \pi} } / \part \pi_c} $ and
$m^{n a}({\bm \pi}) := (i /2) \vareps^{abc} \braket{ \part u^n_{ {\bm \pi} }/ \part \pi_b |[ H_0({\bm \pi}) - \vareps_{n, {\bm \pi} } ] |   \part u^n_{ {\bm \pi} } / \part \pi_c} $, respectively,
and the torsional magnetic field is defined by $T^a_{i}:=(1/2)\vareps^{ijk}T^{a}_{jk}$
\fi

\if0
\section{Application to Weyl Semimetal}

\begin{align}
\vareps^{L/R,\pm}({\bm k}) &= v_F \left[ \pm   |{\bm k} - {\bm \lam}^{L/R}| - \lam^{L/R}_0 \right]  \nonumber \\
%\vareps^{R,\pm}({\bm k})&= \nonumber \\
{\bm v}^{L,\pm} ({\bm k}) &= \pm \frac{v_F({\bm k} - {\bm \lam}^L)}{|{\bm k} - {\bm \lam}^L|}, \nonumber \\
{\bm v}^{R,\pm} ({\bm k}) &= \pm \frac{v_F({\bm k} - {\bm \lam}^R)}{|{\bm k} - {\bm \lam}^R|} \\
{\bm \Omega}^{L,\pm} ({\bm k}) &= \pm \frac{{\bm k} - {\bm \lam}^L}{2|{\bm k} - {\bm \lam}^L|^3}, \nonumber \\
{\bm \Omega}^{R,\pm} ({\bm k}) &= \mp \frac{{\bm k} - {\bm \lam}^R}{2|{\bm k} - {\bm \lam}^R|^3} \\
{\bm m}^{L,\pm} ({\bm k}) &= -  \frac{v_F({\bm k} - {\bm \lam}^L)}{2|{\bm k} - {\bm \lam}^L|^2}, \nonumber \\
{\bm m}^{R,\pm} ({\bm k}) &=  \frac{v_F({\bm k} - {\bm \lam}^R)}{2|{\bm k} - {\bm \lam}^R|^2}
\end{align}
\fi

\if0
\section{Berry curvature and intrinsic magnetic momentum}
In the derivation of Eqs.(\ref{WSM_charge}) and (\ref{WSM_current}), we use the following expressions for 
the Berry curvature and intrinsic magnetic momentum,
\begin{align}
%&\vareps^{s,\pm}({\bm k}) = v_F \left[ \pm   |{\bm k} - {\bm \lam}^{s}| - \lam^{s}_0 \right], \\
%\vareps^{R,\pm}({\bm k})&= \nonumber \\
%&{\bm v}^{s,\pm} ({\bm k}) = \pm v_F({\bm k} - {\bm \lam}^{s})/{|{\bm k} - {\bm \lam}^{s}|}, \\
%{\bm v}^{R,\pm} ({\bm k}) &= \pm \frac{v_F({\bm k} - {\bm \lam}^R)}{|{\bm k} - {\bm \lam}^R|} \\
&{\bm \Omega}^{s,\pm} ({\bm k}) = \pm \chi_{s}({{\bm k} - {\bm \lam}^{s}})/{2|{\bm k} - {\bm \lam}^s|^3}, \\
%{\bm \Omega}^{R,\pm} ({\bm k}) &= \mp \frac{{\bm k} - {\bm \lam}^R}{2|{\bm k} - {\bm \lam}^R|^3} \\
&{\bm m}^{s,\pm} ({\bm k}) = -  \chi_{s}{v_F({\bm k} - {\bm \lam}^{s})}/{2|{\bm k} - {\bm \lam}^s|^2}, 
%{\bm m}^{R,\pm} ({\bm k}) &=  \frac{v_F({\bm k} - {\bm \lam}^R)}{2|{\bm k} - {\bm \lam}^R|^2}
\end{align}
where $\pm$ corresponds to the upper/lower band of each Weyl nodes.
\fi

\if0
The energy, velocity of the electrons, Berry curvature, and intrinsic magnetic momentum are given by 
%\begin{align}
$\vareps^{L(R),\pm}({\bm k}) = v_F \left[ \pm   |{\bm k} - {\bm \lam}^{L(R)}| - \lam^{L(R)}_0 \right], \
%\vareps^{R,\pm}({\bm k})&= \nonumber \\
{\bm v}^{L(R),\pm} ({\bm k}) = \pm v_F({\bm k} - {\bm \lam}^{L(R)})/{|{\bm k} - {\bm \lam}^{L(R)}|}, \
%{\bm v}^{R,\pm} ({\bm k}) &= \pm \frac{v_F({\bm k} - {\bm \lam}^R)}{|{\bm k} - {\bm \lam}^R|} \\
{\bm \Omega}^{L(R),\pm} ({\bm k}) = \pm \chi_{L(R)}({{\bm k} - {\bm \lam}^{L(R)}})/{2|{\bm k} - {\bm \lam}^L|^3}, \
%{\bm \Omega}^{R,\pm} ({\bm k}) &= \mp \frac{{\bm k} - {\bm \lam}^R}{2|{\bm k} - {\bm \lam}^R|^3} \\
{\bm m}^{L(R),\pm} ({\bm k}) = -  \chi_{L(R)}{v_F({\bm k} - {\bm \lam}^{L(R)})}/{2|{\bm k} - {\bm \lam}^L|^2}, $
%{\bm m}^{R,\pm} ({\bm k}) &=  \frac{v_F({\bm k} - {\bm \lam}^R)}{2|{\bm k} - {\bm \lam}^R|^2}
%\end{align}
respectively, where $\pm$ corresponds to the upper/lower band of each Weyl nodes.
\fi



\if0
\section{Full spectrum of WSM with screw dislocation}




\section{Index theorem for quantum dot system}
Let $\ket{E_i}$ and $\ket{E}$ denoting the 

\fi


\section{Ground state current in the presence of screw dislocation: analytical calculation}
In this section, we calculate the ground state current raised by the TCME in the case of screw dislocation,
by calculating directly the eigenstates of the Hamiltonian with the torsion.
This is an alternative approach for the derivation of the TCME, which does not rely on Eq. (3).
%It is another approach for the TCME than the linear response theory (3).
For this purpose, we, first, analyze
%via calculating 
the spectrum of the Hamiltonian (Eq.(4) in the main text), %$H^{{\rm screw}}_{s,k_z}$.
\begin{align}
%H^{{\rm screw}}_{L(R)}
%=&\chi_{L(R)} \left[ \left(-i\part_x- \frac{bk_z y}{2\pi \rho^2} \right)\sig^x + \left(-i\part_y+ \frac{bk_zx}{2\pi \rho^2}\right)\sig^y    \right. \nonumber \\ 
%&\left. +(k_z- \chi_{L(R)} \lam )\sig^z\right]. %\nonumber \\
%H_{R}({\bm k}) :=  v_F \left[ - ({\bm k} - {\bm \lam}^{R})\cdot {\bm \sig} - \lam^R_0 \right],
&H^{{\rm screw}}_{s,k_z}=\chi_{s} \left[  H^{\perp}_{k_z}  + m^{s}_{k_z} \sig^z\right], \nonumber \\
&H^{\perp}_{k_z}= \left(-\mathrm{i}\part_x- \frac{\Phi_{k_z} y}{2\pi \rho^2} \right)\sig^x + \left(-\mathrm{i}\part_y+ \frac{\Phi_{k_z}x}{2\pi \rho^2}\right)\sig^y.
\end{align}
where $m^{s}_{k_z}=k_z-\chi_{s}\lam$, $\Phi_{k_z}=k_zb_g$, $\rho=\sqrt{x^2+y^2}$, $\chi_{L(R)}=+1(-1)$, and $\sigma^i$ is the Pauli matrix.

For the calculation of the spectrum,
it is useful % for the whole Hamiltonian, $H^{{\rm screw}}_{s,k_z}$ 
to clarify the symmetry of the eigenstates of $H^{\perp}_{k_z}$.
Suppose $\ket{\kappa}_{k_z}$ the eigenstate of $H^{\perp}_{k_z}$ with eigenvalue $\kappa$.
Since $\{H^{\perp}_{k_z},\sig^z\}=0$, where $\{U,V\}:=UV+VU$ is the anticommutator,the state $\sig^z\ket{\kappa}_{k_z}$ is also the eigenstate with eigenvalue $-\kappa$.
Therefore, we can choose the eigenfunctions to preserve the doublet structure, $\ket{-\kappa}_{k_z}=\sig^z\ket{\kappa}_{k_z}$,
for $\kappa\neq{0}$.
On the other hand, there is no double structure in the zero eigenstates.
Since the hermitian operator $\sig^z$ maps zero eigenstates of $H^{\perp}_{k_z}$ to zero eigenstates of $H^{\perp}_{k_z}$,
then we can choose the zero eigenstates also as eigenstates of $\sig^z$, denoted by $\ket{0_{i,\sig_i}}_{k_z}$ with $\sig^z\ket{0_{i,\sig_i}}_{k_z}=\sig_i\ket{0_{i,\sig_i}}_{k_z}$
and $\sig_i=\pm1$.
There is another symmetrical property between the eigenstates of $H^{\perp}_{k_z}$ with different $k_z$.
Since the transformation $k_z\to{-k_z}$ corresponds to the flip of the direction of the effective magnetic field,
$\Theta{H}^{\perp}_{k_z}\Theta^{-1}=H^{\perp}_{-k_z}$ holds,
where $\Theta=\mathrm{i}\sig^yK$ is the time-reversal operator for spin-$1/2$ fermions and $K$ is the complex conjugation operator \cite{Sakurai}.
Therefore we can impose $\ket{\kappa}_{-k_z}=\Theta\ket{\kappa}_{k_z}$
and $\ket{0_{i,-\sig_i}}_{-k_z}=\Theta\ket{0_{i,\sig_i}}_{k_z}$, because of $\{{\sig^z},\Theta\}=0$.

The eigenstates of $H^{{\rm screw}}_{s,k_z}$ can be constructed from $\ket{\kappa}_{k_z}$ and $\ket{0_{\sig_i}}_{k_z}$.
Indeed, $\ket{\psi^{L,\pm}_{k_z}(\kappa)}:=c^{L,\pm}_{k_z,1}(\kappa)\ket{\kappa}_{k_z}+c^{L,\pm}_{k_z,2}(\kappa)\ket{-\kappa}_{k_z}$, with $\kappa>0$,
and $\ket{0_{\sig_i}}_{k_z}$ are the full spectrum of $H^{{\rm screw}}_{L,k_z}$,
with eigenvalues % $E^{L,\pm}_{k_z}(\kappa)=
$\pm \sqrt{\kappa^2 +(m^L_{k_z})^2}$ and $\sig_i m^L_{k_z}$, respectively.
Here the coefficients are given by $(c^{L,\pm}_{k_z,1}(\kappa),c^{L,\pm}_{k_z,2}(\kappa))=(4(\kappa^2+(m^L_{k_z})^2))^{-1/4}(\pm{\rm sgn}(m^L_{k_z})((\kappa^2+(m^L_{k_z})^2)^{1/2}\pm\kappa)^{1/2},((\kappa^2+(m^L_{k_z})^2)^{1/2}\mp\kappa)^{1/2}))$.
Moreover, $\ket{\psi^{R,\pm}_{k_z}(\kappa)}:=\Theta\ket{\psi^{L,\mp}_{-k_z}(\kappa)}$ and $\ket{0_{\sig_i}}_{k_z}$ are the eigenstates of $H^{{\rm screw}}_{R,k_z}$,
with eigenvalues % $E^{R,\pm}_{k_z}(\kappa)=
$\pm \sqrt{\kappa^2 +(m^R_{k_z})^2}$ and $-\sig_i m^R_{k_z}$, respectively.






Now, we calculate the ground state current in the presence of the screw dislocation.
As yet, we have not distinguished discrete and continuum states.
From now on, we use $\kappa_i$ to express the discrete eigenvalues of $H^{\perp}_{k_z}$ and 
$(\kappa,l)$ to label the continuum states, where $\kappa$ is the continuum energy eigenvalue, and $l$ is a discrete quantum number, e.g., the angular momentum. 
%the continuum ones are characterized by the energy value and some discrete quantum number, for example the angular momentum, $(\kappa,l)$.
The current operator is defined by $\part{H^{{\rm screw}}_{s,k_z}}/\part{k_z}=\chi_s\sig^z+\chi_s\{ -(b_gy/2\pi\rho^2)\sig^x+(b_gx/2\pi\rho^2)\sig^y \}$. At least up to the first order in $b_g$, the correction to the current operator due to the dislocation, i.e. the second and third term above, does not contribute to the expectation value %owing to their anti-symmetry with respect to 
because these terms are odd under the transformation
$x\to-x$ or $y\to-y$.
Therefore, the current is the sum of the expectation values of $\chi_s\sig^z$ %the current operator %\cite{note_currentexp} 
for the occupied states which consist of discrete nonzero, discrete zero, and continuum states,
and then we obtain,
%Using these eigenstates, we evaluate the value of the ground state current.
\begin{align}
J^z&= \sum_{s=L,R} \int_{|m^s_{k_z}|<\Lam} \frac{L_z dk_z}{2\pi} \left[ \sum_{\kappa_i > 0} \braket{\psi^{s,-}_{k_z}(\kappa_i)|\chi_s\sig^z|\psi^{s,-}_{k_z}(\kappa_i)} \right. \nonumber \\
&+\sum_{i: \sig_i \chi_s m^s_{k_z} <0}\braket{0_{i,\sig_i}|\chi_s\sig^z|0_{i,\sig_i}}_{k_z} \nonumber \\
&\left. + \int_0^{\infty} d\kappa \sum_l \braket{\psi^{s,-}_{k_z}(\kappa,l)|\chi_s\sig^z|\psi^{s,-}_{k_z}(\kappa,l)}\right] \nonumber \\
&=\int_{|m^L_{k_z}|<\Lam} \frac{L_z dk_z}{2\pi} \left[ \sum_{\kappa_i \neq 0} \braket{\kappa_i|\sig^z|\kappa_i}_{k_z} \right. \nonumber\\
&\left. + \sum_{i} \braket{0_{i,\sig_i}|\sig^z|0_{i,\sig_i}}_{k_z} +\int_{-\infty}^{\infty} d\kappa \sum_l \braket{\kappa,l | \sig^z|\kappa,l}_{k_z} \right], \label{current_screw_gs}
\end{align}
where $L_z$ is the size of the system.
Here we introduce the momentum cutoff scheme, $|m^s_{k_z}|<\Lam$,
i.e. the domain of the integration is %restricted to the 
%which corresponds to that 
the same as that used 
in the calculation of Eq.(4).
The first term in the square braket is equal to zero, since $\sig^z\ket{\kappa_i}_{k_z}=\ket{-\kappa_i}_{k_z}$ is orthogonal to $\ket{\kappa_i}_{k_z}$.
The second term is the index of the Dirac operator, $H^{\perp}_{k_z}$, which is an integer and the difference in the number of its normalizable zero modes with $\sig^3=+1$ and $\sig^3=-1$.
%$n^+_{k_z}-n^-_{k_z}$ 
%It has been shown that 
The index is given by $N_{k_z}:=-{\rm sgn}({\Phi_{k_z}})\lfloor{|\Phi_{k_z}|/2\pi}\rfloor$ \cite{Kiskis1977,Aharonov1979}.
The normalizable zero modes exhibit power-law decay for large distance from the dislocation; i.e. they behave like
%according to the power-law decay normalizable zero modes, which, as $\rho\to\infty$, behave like 
$\ket{0_{i,-1}}\propto(0,\rho^{-\Phi_{k_z}/2\pi}(x-\mathrm{i}y)^{i-1})$ for $\Phi_{k_z}>0$, and 
$\ket{0_{i,+1}}\propto(\rho^{\Phi_{k_z}/2\pi}(x+\mathrm{i}y)^{i-1},0)$ for $\Phi_{k_z}<0$, where
%with, in both case, 
$i=1,2,\cdots,|N_{k_z}|$ \cite{Kiskis1977,Aharonov1979}.
%\textcolor{red}{i=1,2,...(start with i=1)}
Now, we move on to the third term of Eq.(\ref{current_screw_gs}).
%Seemingly, 
One may expect that it is equal to zero, since $\sig^z\ket{\kappa,l}=\ket{-\kappa,l}$, is orthogonal to $\ket{\kappa,l}$ for almost all values of $\kappa$.
However, the scattering states near $\kappa=0$ (their amplitudes $\propto \cos(\kappa\rho+\delta_l)/\sqrt{\rho}$ with $\delta_l$ the phase shift) cause 
a delta function peak of $\braket{\kappa,l | \sig^z|\kappa,l}_{k_z}$ at $\kappa=0$.
Indeed, from an explicit calculation \cite{Kiskis1977}, it has been shown that
$\sum_l \braket{\kappa,l | \sig^z|\kappa,l}_{k_z}=c_{k_Z}\delta(\kappa)$, with $c_{k_z}=\Phi_{k_z}/2\pi-N_{k_z}$,
and then the third term is equal to $c_{k_z}$.
Substituting them into Eq.(\ref{current_screw_gs}), we obtain
\begin{align}
J^z=\int_{-\Lam-\lam}^{\Lam-\lam} \frac{L_zdk_z}{2\pi}\frac{\Phi_{k_z}}{2\pi}=-\frac{L_zb_g\Lam\lam}{2\pi^2}, \label{current_ground_final}
\end{align}
which is coincident with the expression obtained directly from Eq.(3) by the following reason.
In the presence of the screw dislocation with the Burgers vector $-b_g\hat{z}$
the torsion is given by $T^z_z=T^z_{xy}=b_g\delta^{(2)}(x,y)$.
Therefore, the total current derived from Eq.(3) is $J^z=-L_zev_F(\lam^R_z - \lam^L_z)\Lam b_g/4\pi^2$.
In this section we have set $ \lam^L_z=-\lam^R_z =\lam$ and $e=v_F=1$,
and therefore we obtain $J^z=-L_z\lam \Lam b_g/2\pi^2$, which reproduces Eq.(\ref{current_ground_final}).
%Summarizing above, owing to the asymmetry in the number of modes at $\kappa\sim{0}$ with opposite eigenvalues of $\sig^z$,
%the current is generated, and these modes exhibit power-law decay.
%This corresponds to spectrum asymmetry in the eigenstates of $H^{{\rm screw}}_{s,k_z}$ with the eigenenergy $E\sim\pm m^s_{k_z}$,
%as shown in Fig. \ref{spectrum}-(a) \textcolor{red}{(under construction)}. 










\section{Absence of total current and possibility of local current}
In this section, we show that, if the system is periodic in a certain direction, the {\it total} current is always zero,
even in the presence of a magnetic field or lattice stain and dislocations,
while the {\it local} current is not.
The argument is the extension of that presented in Ref. \cite{Vazifeh2013}.
We start with the general Hamiltonian of electrons in solids (set $e=1$ in this section):
\begin{align}
H=\int d^3{\bm r} \ \frac{1}{2m} \left( -\mathrm{i} \nab_i - A_i({\bm \rho},z) \right)^2 + V({\bm \rho} ,z),
\end{align}
where $V$ is the potential term, in which the effect of the dislocation is included.
Here $i=x,y,z$, ${\bm r}=({\bm \rho},z)$, and ${\bm \rho}=(x,y)$, and
we impose the periodicity in the $z$-direction:
\begin{align}
A_i({\bm \rho}, z) = A_i({\bm \rho}, z+a), V({\bm \rho} ,z) = V({\bm \rho} ,z+a).
\end{align}
Suppose $\psi$ is one eigenfunction of the Hamiltonian and define the Bloch wave function
$\psi_{n,k_z}({\bm \rho}, z) = e^{\mathrm{i} k_z z}u_{n,k_z}({\bm \rho}, z)$, whose energy is $\vareps_{n,k_z}$.
The total current along the $z$-direction is given by
\begin{align}
J_z
 =   \sum_{n} \int_{BZ} \frac{d k_z}{2 \pi} \int d^3 {\bm r}  \ \psi^*_{n,k_z}({\bm \rho},z)\frac{\delta H}{\delta A_z} \psi_{n,k_z}({\bm \rho},z) n_F(\vareps_{n.k_z})\nonumber \\
=-  \sum_{n} \int_{BZ} \frac{d k_z}{2 \pi} \int d^3 {\bm r}  \ u^*_{n,k_z}({\bm \rho},z)\frac{\part H_{k_z}}{\part k_z} u_{n,k_z}({\bm \rho},z) n_F(\vareps_{n.k_z}), \nonumber \\
\label{jay-z_total}
\end{align}
where $H_{k_z} = e^{-i k_z z} H e^{i k_z z} $ and $n_F$ is the Fermi distribution function.
Here we use the identity :
\begin{align}
&\int d^3 {\bm r}  \ u^*_{n,k_z}({\bm \rho},z)\frac{\part H_{k_z}}{\part k_z} u_{n,k_z}({\bm \rho},z) \nonumber \\
& = \frac{\part}{\part k_z} \int d^3 {\bm r}  \ u^*_{n,k_z}({\bm \rho},z)   H_{k_z} u_{n,k_z}({\bm \rho},z) \nonumber \\
& = \frac{\part \vareps_{n,k_z}}{\part k_z},
\end{align} 
which follows from
\begin{align}
\frac{\part}{\part k_z}  \left[ \int d^3 r  \ u^*_{n,k_z}({\bm \rho},z) u_{n,k_z}({\bm \rho},z) \right] = \frac{\part}{\part k_z} 1 =0,  \label{der_zero}
\end{align}
and then we can rewrite Eq. (\ref{jay-z_total}) into
\begin{align}
J_z&= - \sum_{n} \int_{BZ} \frac{d k_z}{2 \pi}  \frac{\part \vareps_{n,k_z}}{\part k_z} n_F(\vareps_{n.k_z}) \nonumber \\
&=  - \frac{1}{2 \pi}  \sum_{n}  \sum_{i = 1,\dots ,i^{(n)} } \int_{\vareps_{n, k^{(n)}_{i-1}}}^{ \vareps_{n, k^{(n)}_{i}}} d \vareps \  n_F( \vareps)  \label{jay-z-eps}
\end{align}
Here for each region $k \in (k^{(n)}_{i-1}, k^{(n)}_{i})$, $\vareps_{n,k}$ monotonically increases or decreases, %$I^{(n)}_i = (\vareps_{n, k^{(n)}_{i-1}},  \vareps_{n, k^{(n)}_{i}}) $
and $k^{(n)}_0=0$ and $k_{i^{(n)}}^{(n)} = 2\pi/a$.
We find that Eq.(\ref{jay-z-eps}) is always equal to zero owing to the periodicity of the dispersion in the wave number space, $\vareps_{n,k=0}=\vareps_{n, k=2\pi/a}$.
Therefore, we found that the {\it total} current along the $z$-direction is zero.
In the above derivation, 
it is essential that the integrand with respect to $k_z$ can be rewritten into the total derivative with respect to $k_z$,
and this key factor follows from the fact that the integral over the real space of $|u_{n,k_z}({\bm \rho},z)|^2$ is equal to 1, which results 
in Eq. (\ref{der_zero}).
Instead, without the integration over the real space,
\begin{align}
 \frac{\part }{\part k_z} \left[  u^*_{n,k_z}({\bm \rho},z) u_{n,k_z}({\bm \rho},z) \right]  \neq 0.
\end{align}
Then, in the case of {\it local} current, the above argument %derivation like that 
in the case of the {\it total} current does not hold.
Hence, the local current is not always zero, unlike the total current.



\bibliography{WSM}
\bibliographystyle{apsrev}







\end{document}




WSM.bbl
\begin{thebibliography}{84}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname bibnamefont\endcsname\relax
  \def\bibnamefont#1{#1}\fi
\expandafter\ifx\csname bibfnamefont\endcsname\relax
  \def\bibfnamefont#1{#1}\fi
\expandafter\ifx\csname citenamefont\endcsname\relax
  \def\citenamefont#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
\providecommand{\bibinfo}[2]{#2}
\providecommand{\eprint}[2][]{\url{#2}}

\bibitem[{\citenamefont{Wan et~al.}(2011)\citenamefont{Wan, Turner, Vishwanath,
  and Savrasov}}]{vishwanath}
\bibinfo{author}{\bibfnamefont{X.}~\bibnamefont{Wan}},
  \bibinfo{author}{\bibfnamefont{A.~M.} \bibnamefont{Turner}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Vishwanath}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{S.~Y.}
  \bibnamefont{Savrasov}}, \bibinfo{journal}{Phys. Rev. B}
  \textbf{\bibinfo{volume}{83}}, \bibinfo{pages}{205101}
  (\bibinfo{year}{2011}).

\bibitem[{\citenamefont{Burkov and Balents}(2011)}]{balents}
\bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Burkov}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Balents}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{107}},
  \bibinfo{pages}{127205} (\bibinfo{year}{2011}).

\bibitem[{\citenamefont{Hal\'asz and Balents}(2012)}]{balents2}
\bibinfo{author}{\bibfnamefont{G.~B.} \bibnamefont{Hal\'asz}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Balents}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{85}},
  \bibinfo{pages}{035103} (\bibinfo{year}{2012}).

\bibitem[{\citenamefont{Murakami}(2007)}]{murakami}
\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Murakami}},
  \bibinfo{journal}{New Journal of Physics} \textbf{\bibinfo{volume}{9}},
  \bibinfo{pages}{356} (\bibinfo{year}{2007}).

\bibitem[{\citenamefont{Kim et~al.}(2013)\citenamefont{Kim, Kim, Wang, Sasaki,
  Satoh, Ohnishi, Kitaura, Yang, and Li}}]{Kim_BiSb2013}
\bibinfo{author}{\bibfnamefont{H.-J.} \bibnamefont{Kim}},
  \bibinfo{author}{\bibfnamefont{K.-S.} \bibnamefont{Kim}},
  \bibinfo{author}{\bibfnamefont{J.-F.} \bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Sasaki}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Satoh}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Ohnishi}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Kitaura}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Yang}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Li}}, \bibinfo{journal}{Phys.
  Rev. Lett.} \textbf{\bibinfo{volume}{111}}, \bibinfo{pages}{246603}
  (\bibinfo{year}{2013}).

\bibitem[{\citenamefont{Wang et~al.}(2012)\citenamefont{Wang, Sun, Chen,
  Franchini, Xu, Weng, Dai, and Fang}}]{Zhijun2012}
\bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Sun}},
  \bibinfo{author}{\bibfnamefont{X.-Q.} \bibnamefont{Chen}},
  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Franchini}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Xu}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Weng}},
  \bibinfo{author}{\bibfnamefont{X.}~\bibnamefont{Dai}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Fang}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{85}},
  \bibinfo{pages}{195320} (\bibinfo{year}{2012}).

\bibitem[{\citenamefont{Wang et~al.}(2013)\citenamefont{Wang, Weng, Wu, Dai,
  and Fang}}]{Zhijun2013}
\bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Weng}},
  \bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Wu}},
  \bibinfo{author}{\bibfnamefont{X.}~\bibnamefont{Dai}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Fang}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{88}},
  \bibinfo{pages}{125427} (\bibinfo{year}{2013}).

\bibitem[{\citenamefont{Liu et~al.}(2014{\natexlab{a}})\citenamefont{Liu, Zhou,
  Zhang, Wang, Weng, Prabhakaran, Mo, Shen, Fang, Dai et~al.}}]{Liu_NaBi2014}
\bibinfo{author}{\bibfnamefont{Z.~K.} \bibnamefont{Liu}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Zhou}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Zhang}},
  \bibinfo{author}{\bibfnamefont{Z.~J.} \bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{H.~M.} \bibnamefont{Weng}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Prabhakaran}},
  \bibinfo{author}{\bibfnamefont{S.-K.} \bibnamefont{Mo}},
  \bibinfo{author}{\bibfnamefont{Z.~X.} \bibnamefont{Shen}},
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Fang}},
  \bibinfo{author}{\bibfnamefont{X.}~\bibnamefont{Dai}}, \bibnamefont{et~al.},
  \bibinfo{journal}{Science} \textbf{\bibinfo{volume}{343}},
  \bibinfo{pages}{864} (\bibinfo{year}{2014}{\natexlab{a}}).

\bibitem[{\citenamefont{Neupane et~al.}(2014)\citenamefont{Neupane, Xu, Sankar,
  Alidoust, Bian, Liu, Belopolski, Chang, Jeng, Lin et~al.}}]{Neupane2014}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Neupane}},
  \bibinfo{author}{\bibfnamefont{S.-Y.} \bibnamefont{Xu}},
  \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Sankar}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Alidoust}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Bian}},
  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Liu}},
  \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Belopolski}},
  \bibinfo{author}{\bibfnamefont{T.-R.} \bibnamefont{Chang}},
  \bibinfo{author}{\bibfnamefont{H.-T.} \bibnamefont{Jeng}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Lin}}, \bibnamefont{et~al.},
  \bibinfo{journal}{Nat Commun} \textbf{\bibinfo{volume}{5}}
  (\bibinfo{year}{2014}).

\bibitem[{\citenamefont{Borisenko et~al.}(2014)\citenamefont{Borisenko, Gibson,
  Evtushinsky, Zabolotnyy, B\"uchner, and Cava}}]{Borisenko2014}
\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Borisenko}},
  \bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Gibson}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Evtushinsky}},
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Zabolotnyy}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{B\"uchner}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{R.~J.} \bibnamefont{Cava}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{113}},
  \bibinfo{pages}{027603} (\bibinfo{year}{2014}).

\bibitem[{\citenamefont{Liu et~al.}(2014{\natexlab{b}})\citenamefont{Liu,
  Jiang, Zhou, Wang, Zhang, Weng, Prabhakaran, Mo, Peng, Dudin
  et~al.}}]{Liu_CdAs2014}
\bibinfo{author}{\bibfnamefont{Z.~K.} \bibnamefont{Liu}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Jiang}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Zhou}},
  \bibinfo{author}{\bibfnamefont{Z.~J.} \bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Zhang}},
  \bibinfo{author}{\bibfnamefont{H.~M.} \bibnamefont{Weng}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Prabhakaran}},
  \bibinfo{author}{\bibfnamefont{S.-K.} \bibnamefont{Mo}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Peng}},
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Dudin}},
  \bibnamefont{et~al.}, \bibinfo{journal}{Nat Mater}
  \textbf{\bibinfo{volume}{13}}, \bibinfo{pages}{677}
  (\bibinfo{year}{2014}{\natexlab{b}}).

\bibitem[{\citenamefont{Li et~al.}(2014)\citenamefont{Li, Kharzeev, Zhang,
  Huang, Fedorov, Zhong, Schneeloch, Gu, and Valla}}]{Li_ZrTe2014}
\bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Li}},
  \bibinfo{author}{\bibfnamefont{D.~E.} \bibnamefont{Kharzeev}},
  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Zhang}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Huang}},
  \bibinfo{author}{\bibfnamefont{a.~V.} \bibnamefont{Fedorov}},
  \bibinfo{author}{\bibfnamefont{R.~D.} \bibnamefont{Zhong}},
  \bibinfo{author}{\bibfnamefont{J.~a.} \bibnamefont{Schneeloch}},
  \bibinfo{author}{\bibfnamefont{G.~D.} \bibnamefont{Gu}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Valla}}
  (\bibinfo{year}{2014}), \eprint{1412.6543v1}.

\bibitem[{\citenamefont{Weng et~al.}(2015)\citenamefont{Weng, Fang, Fang,
  Bernevig, and Dai}}]{TaAsWeng2015}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Weng}},
  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Fang}},
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Fang}},
  \bibinfo{author}{\bibfnamefont{B.~A.} \bibnamefont{Bernevig}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{X.}~\bibnamefont{Dai}},
  \bibinfo{journal}{Phys. Rev. X} \textbf{\bibinfo{volume}{5}},
  \bibinfo{pages}{011029} (\bibinfo{year}{2015}).

\bibitem[{\citenamefont{Huang et~al.}(2015{\natexlab{a}})\citenamefont{Huang,
  Xu, Belopolski, Lee, Chang, Wang, Alidoust, Bian, Neupane, Zhang
  et~al.}}]{Huang_TaAs2015}
\bibinfo{author}{\bibfnamefont{S.-M.} \bibnamefont{Huang}},
  \bibinfo{author}{\bibfnamefont{S.-Y.} \bibnamefont{Xu}},
  \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Belopolski}},
  \bibinfo{author}{\bibfnamefont{C.-C.} \bibnamefont{Lee}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Chang}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Alidoust}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Bian}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Neupane}},
  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Zhang}},
  \bibnamefont{et~al.}, \bibinfo{journal}{Nat. Commun}
  \textbf{\bibinfo{volume}{6}} (\bibinfo{year}{2015}{\natexlab{a}}).

\bibitem[{\citenamefont{Xu et~al.}(2015)\citenamefont{Xu, Belopolski, Alidoust,
  Neupane, Sankar, Huang, Lee, Wang, Bian, Zheng et~al.}}]{Xu_TaAs2015}
\bibinfo{author}{\bibfnamefont{S.-Y.} \bibnamefont{Xu}},
  \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Belopolski}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Alidoust}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Neupane}},
  \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Sankar}},
  \bibinfo{author}{\bibfnamefont{S.-M.} \bibnamefont{Huang}},
  \bibinfo{author}{\bibfnamefont{C.-C.} \bibnamefont{Lee}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Bian}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Zheng}}, \bibnamefont{et~al.}
  (\bibinfo{year}{2015}), \eprint{1502.03807v1}.

\bibitem[{\citenamefont{Zhang et~al.}(2015)\citenamefont{Zhang, Yuan, Xu, Lin,
  Tong, Hasan, Wang, Zhang, and Jia}}]{Zhang_TaAs2015}
\bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Zhang}},
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Yuan}},
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Xu}},
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Lin}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Tong}},
  \bibinfo{author}{\bibfnamefont{M.~Z.} \bibnamefont{Hasan}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Zhang}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Jia}},
  \bibinfo{journal}{1502.00251}  (\bibinfo{year}{2015}), \eprint{1502.00251v1}.

\bibitem[{\citenamefont{Lv et~al.}(2015)\citenamefont{Lv, Weng, Fu, Wang, Miao,
  Ma, Richard, Huang, Zhao, Chen et~al.}}]{LvTaAs2015}
\bibinfo{author}{\bibfnamefont{B.~Q.} \bibnamefont{Lv}},
  \bibinfo{author}{\bibfnamefont{H.~M.} \bibnamefont{Weng}},
  \bibinfo{author}{\bibfnamefont{B.~B.} \bibnamefont{Fu}},
  \bibinfo{author}{\bibfnamefont{X.~P.} \bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Miao}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Ma}},
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Richard}},
  \bibinfo{author}{\bibfnamefont{X.~C.} \bibnamefont{Huang}},
  \bibinfo{author}{\bibfnamefont{L.~X.} \bibnamefont{Zhao}},
  \bibinfo{author}{\bibfnamefont{G.~F.} \bibnamefont{Chen}},
  \bibnamefont{et~al.}, \bibinfo{journal}{Phys. Rev. X}
  \textbf{\bibinfo{volume}{5}}, \bibinfo{pages}{031013} (\bibinfo{year}{2015}).

\bibitem[{\citenamefont{Shekhar
  et~al.}(2015{\natexlab{a}})\citenamefont{Shekhar, Nayak, Sun, Schmidt,
  Nicklas, Leermakers, Zeitler, Skourski, Wosnitza, Liu et~al.}}]{Shekhar2015}
\bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Shekhar}},
  \bibinfo{author}{\bibfnamefont{A.~K.} \bibnamefont{Nayak}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Sun}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Schmidt}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Nicklas}},
  \bibinfo{author}{\bibfnamefont{I.}~\bibnamefont{Leermakers}},
  \bibinfo{author}{\bibfnamefont{U.}~\bibnamefont{Zeitler}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Skourski}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Wosnitza}},
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Liu}}, \bibnamefont{et~al.},
  \bibinfo{journal}{Nat Phys}  (\bibinfo{year}{2015}{\natexlab{a}}).

\bibitem[{\citenamefont{Huang et~al.}(2015{\natexlab{b}})\citenamefont{Huang,
  Zhao, Long, Wang, Chen, Yang, Liang, Xue, Weng, Fang
  et~al.}}]{XiaochunTaAs2015}
\bibinfo{author}{\bibfnamefont{X.}~\bibnamefont{Huang}},
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Zhao}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Long}},
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Chen}},
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Yang}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Liang}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Xue}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Weng}},
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Fang}}, \bibnamefont{et~al.},
  \bibinfo{journal}{Phys. Rev. X} \textbf{\bibinfo{volume}{5}},
  \bibinfo{pages}{031023} (\bibinfo{year}{2015}{\natexlab{b}}).

\bibitem[{\citenamefont{Yang et~al.}(2015)\citenamefont{Yang, Li, Wang, Zheng,
  and Xu}}]{Yang_NbAs2015}
\bibinfo{author}{\bibfnamefont{X.}~\bibnamefont{Yang}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Li}},
  \bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Zheng}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Z.-A.} \bibnamefont{Xu}}
  (\bibinfo{year}{2015}), \eprint{1506.03190v1}.

\bibitem[{\citenamefont{Shekhar
  et~al.}(2015{\natexlab{b}})\citenamefont{Shekhar, Arnold, Wu, Sun, Schmidt,
  and Kumar}}]{Shekhar_TaP2015}
\bibinfo{author}{\bibfnamefont{C.}~\bibnamefont{Shekhar}},
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Arnold}},
  \bibinfo{author}{\bibfnamefont{S.-C.} \bibnamefont{Wu}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Sun}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Schmidt}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Kumar}}, pp.
  \bibinfo{pages}{1--26} (\bibinfo{year}{2015}{\natexlab{b}}),
  \eprint{1506.06577v1}.

\bibitem[{\citenamefont{Sushkov et~al.}(2015)\citenamefont{Sushkov, Hofmann,
  Jenkins, Ishikawa, Nakatsuji, Sarma, and Drew}}]{Sushkov_EIO2015}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Sushkov}},
  \bibinfo{author}{\bibfnamefont{B.~J.} \bibnamefont{Hofmann}},
  \bibinfo{author}{\bibfnamefont{G.~S.} \bibnamefont{Jenkins}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Ishikawa}},
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Nakatsuji}},
  \bibinfo{author}{\bibfnamefont{S.~D.} \bibnamefont{Sarma}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{H.~D.} \bibnamefont{Drew}}
  (\bibinfo{year}{2015}), \eprint{1507.01038}.

\bibitem[{\citenamefont{Borisenko et~al.}(2015)\citenamefont{Borisenko,
  Evtushinsky, Gibson, Yaresko, Kim, Ali, B\"{u}chner, Hoesch, and
  Cava}}]{Borisenko}
\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Borisenko}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Evtushinsky}},
  \bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Gibson}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Yaresko}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Kim}},
  \bibinfo{author}{\bibfnamefont{M.~N.} \bibnamefont{Ali}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{B\"{u}chner}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Hoesch}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{R.~J.} \bibnamefont{Cava}}
  (\bibinfo{year}{2015}), \eprint{1507.04847}.

\bibitem[{\citenamefont{Fujikawa and Suzuki}(2004)}]{Fujikawa}
\bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Fujikawa}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Suzuki}},
  \emph{\bibinfo{title}{Path Integrals and Quantum Anomalies}}
  (\bibinfo{publisher}{Oxford University Press}, \bibinfo{year}{2004}).

\bibitem[{\citenamefont{Zyuzin and Burkov}(2012)}]{Zyuzin_AHE2012}
\bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Zyuzin}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Burkov}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{86}},
  \bibinfo{pages}{115133} (\bibinfo{year}{2012}).

\bibitem[{\citenamefont{Goswami and Tewari}(2013)}]{Goswami2013}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Goswami}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Tewari}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{88}},
  \bibinfo{pages}{245107} (\bibinfo{year}{2013}).

\bibitem[{\citenamefont{Hosur and Qi}(2013)}]{Hosur_review}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Hosur}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{X.}~\bibnamefont{Qi}},
  \bibinfo{journal}{Comptes Rendus Physique} \textbf{\bibinfo{volume}{14}},
  \bibinfo{pages}{857} (\bibinfo{year}{2013}).

\bibitem[{\citenamefont{Nielsen and Ninomiya}(1983)}]{NN_1983}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Nielsen}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Ninomiya}},
  \bibinfo{journal}{Physics Letters B} \textbf{\bibinfo{volume}{130}},
  \bibinfo{pages}{389 } (\bibinfo{year}{1983}).

\bibitem[{\citenamefont{Goswami et~al.}(2015)\citenamefont{Goswami, Pixley, and
  Sarma}}]{Goswami_LMR2015}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Goswami}},
  \bibinfo{author}{\bibfnamefont{J.~H.} \bibnamefont{Pixley}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{S.~D.} \bibnamefont{Sarma}}
  (\bibinfo{year}{2015}), \eprint{1503.02069v1}.

\bibitem[{\citenamefont{Liu et~al.}(2013)\citenamefont{Liu, Ye, and
  Qi}}]{Liu2013}
\bibinfo{author}{\bibfnamefont{C.-X.} \bibnamefont{Liu}},
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Ye}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{X.-L.} \bibnamefont{Qi}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{87}},
  \bibinfo{pages}{235306} (\bibinfo{year}{2013}).

\bibitem[{\citenamefont{Kharzeev}(2010)}]{CVE2010}
\bibinfo{author}{\bibfnamefont{D.~E.} \bibnamefont{Kharzeev}},
  \bibinfo{journal}{Annals of Physics} \textbf{\bibinfo{volume}{325}},
  \bibinfo{pages}{205 } (\bibinfo{year}{2010}).

\bibitem[{\citenamefont{Kharzeev}(2014)}]{Kharzeev2014}
\bibinfo{author}{\bibfnamefont{D.~E.} \bibnamefont{Kharzeev}},
  \bibinfo{journal}{Progress in Particle and Nuclear Physics}
  \textbf{\bibinfo{volume}{75}}, \bibinfo{pages}{133} (\bibinfo{year}{2014}).

\bibitem[{\citenamefont{Fukushima et~al.}(2008)\citenamefont{Fukushima,
  Kharzeev, and Warringa}}]{Fukushima_CME2008}
\bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Fukushima}},
  \bibinfo{author}{\bibfnamefont{D.~E.} \bibnamefont{Kharzeev}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{H.~J.}
  \bibnamefont{Warringa}}, \bibinfo{journal}{Phys. Rev. D}
  \textbf{\bibinfo{volume}{78}}, \bibinfo{pages}{074033}
  (\bibinfo{year}{2008}).

\bibitem[{\citenamefont{Wilczek}(2012)}]{Wilczek2012}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Wilczek}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{109}},
  \bibinfo{pages}{160401} (\bibinfo{year}{2012}).

\bibitem[{\citenamefont{Vazifeh and Franz}(2013)}]{Vazifeh2013}
\bibinfo{author}{\bibfnamefont{M.~M.} \bibnamefont{Vazifeh}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Franz}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{111}},
  \bibinfo{pages}{027201} (\bibinfo{year}{2013}).

\bibitem[{\citenamefont{Yamamoto}(2015)}]{Yamamoto_2015}
\bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Yamamoto}},
  \bibinfo{journal}{Phys. Rev. D} \textbf{\bibinfo{volume}{92}},
  \bibinfo{pages}{085011} (\bibinfo{year}{2015}).

\bibitem[{pre()}]{previous-study}
\emph{\bibinfo{title}{{\rm Effects of dislocation in topological materials have
  been extensively studied so far in different contexts
  \cite{topo-dis1,topo-dis2,topo-dis3,topo-dis4,topo-dis5,topo-dis6,topo-dis7,PhysRevB.90.241403,PhysRevLett.108.106403,
  Slager2013}}}}.

\bibitem[{\citenamefont{Nakahara}(2003)}]{Nakahara_topo}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Nakahara}},
  \emph{\bibinfo{title}{Geometry, topology, and physics}}
  (\bibinfo{publisher}{Institute of Physics Publishing},
  \bibinfo{address}{Bristol, Philadelphia}, \bibinfo{year}{2003}).

\bibitem[{\citenamefont{Shitade}(2014)}]{Shitade2014}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Shitade}},
  \bibinfo{journal}{Progress of Theoretical and Experimental Physics}
  \textbf{\bibinfo{volume}{2014}}, \bibinfo{pages}{123I01}
  (\bibinfo{year}{2014}).

\bibitem[{\citenamefont{Parrikar et~al.}(2014)\citenamefont{Parrikar, Hughes,
  and Leigh}}]{Parrikar2014}
\bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{Parrikar}},
  \bibinfo{author}{\bibfnamefont{T.~L.} \bibnamefont{Hughes}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{R.~G.} \bibnamefont{Leigh}},
  \bibinfo{journal}{Phys. Rev. D} \textbf{\bibinfo{volume}{90}},
  \bibinfo{pages}{105004} (\bibinfo{year}{2014}).

\bibitem[{tor()}]{torsional_mag}
\emph{\bibinfo{title}{{\rm In this letter, $T^{\alpha}_{ij}$ is called the TMF
  and $T^{\alpha}_{i0}$ the torsional electric field, following the terminology
  in Ref.\cite{Shitade2014}. Among the latter, the temporal ($\alpha=0$)
  component corresponds, for example, to the temperature gradient
  \cite{Shitade2014, Gromov2014, Bradlyn2015low}, and the spatial ($\alpha=a$ )
  one the mechanical rotation \cite{Shitade-Kimura2014} }}}.

\bibitem[{\citenamefont{Edelen and Lagoudas}(1988)}]{Edelen}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Edelen}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Lagoudas}},
  \emph{\bibinfo{title}{Gauge theory and defects in solids}}
  (\bibinfo{publisher}{North Holland}, \bibinfo{year}{1988}).

\bibitem[{\citenamefont{Hughes et~al.}(2013)\citenamefont{Hughes, Leigh, and
  Parrikar}}]{T.Hughes2013}
\bibinfo{author}{\bibfnamefont{T.~L.} \bibnamefont{Hughes}},
  \bibinfo{author}{\bibfnamefont{R.~G.} \bibnamefont{Leigh}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{Parrikar}},
  \bibinfo{journal}{Phys. Rev. D} \textbf{\bibinfo{volume}{88}},
  \bibinfo{pages}{025040} (\bibinfo{year}{2013}).

\bibitem[{\citenamefont{Kleinert}(2008)}]{Kleinert}
\bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Kleinert}},
  \emph{\bibinfo{title}{Multivalued Fields: In Condensed Matter,
  Electromagnetism, and Gravitation}} (\bibinfo{publisher}{World Scientific},
  \bibinfo{year}{2008}).

\bibitem[{sup()}]{supp}
\emph{\bibinfo{title}{Supplemental material}}.

\bibitem[{\citenamefont{Son and Yamamoto}(2012)}]{SonYamamotoPRL}
\bibinfo{author}{\bibfnamefont{D.~T.} \bibnamefont{Son}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Yamamoto}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{109}},
  \bibinfo{pages}{181602} (\bibinfo{year}{2012}).

\bibitem[{\citenamefont{Mu{\~n}oz and Gamboa}(2010)}]{munoz2010}
\bibinfo{author}{\bibfnamefont{S.~D.~R.} \bibnamefont{Mu{\~n}oz}}
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Gamboa}},
  Master's thesis, \bibinfo{school}{Pontificia Universidad Cat{\'o}lica de
  Chile} (\bibinfo{year}{2010}).

\bibitem[{\citenamefont{Kohmoto et~al.}(1992)\citenamefont{Kohmoto, Halperin,
  and Wu}}]{Kohmoto1992}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Kohmoto}},
  \bibinfo{author}{\bibfnamefont{B.~I.} \bibnamefont{Halperin}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{Y.-S.} \bibnamefont{Wu}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{45}},
  \bibinfo{pages}{13488} (\bibinfo{year}{1992}).

\bibitem[{\citenamefont{Teo and Kane}(2010)}]{Teo-Kane2010}
\bibinfo{author}{\bibfnamefont{J.~C.~Y.} \bibnamefont{Teo}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{C.~L.} \bibnamefont{Kane}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{82}},
  \bibinfo{pages}{115120} (\bibinfo{year}{2010}).

\bibitem[{\citenamefont{Imura and Takane}(2011)}]{Imura2011}
\bibinfo{author}{\bibfnamefont{K.-I.} \bibnamefont{Imura}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Takane}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{84}},
  \bibinfo{pages}{245415} (\bibinfo{year}{2011}).

\bibitem[{\citenamefont{Jackiw}(1984)}]{Jackiw1984}
\bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Jackiw}},
  \bibinfo{journal}{Phys. Rev. D} \textbf{\bibinfo{volume}{29}},
  \bibinfo{pages}{2375} (\bibinfo{year}{1984}).

\bibitem[{\citenamefont{Haldane}(1988)}]{Haldane1988}
\bibinfo{author}{\bibfnamefont{F.~D.~M.} \bibnamefont{Haldane}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{61}},
  \bibinfo{pages}{2015} (\bibinfo{year}{1988}).

\bibitem[{\citenamefont{Kiskis}(1977)}]{Kiskis1977}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Kiskis}},
  \bibinfo{journal}{Phys. Rev. D} \textbf{\bibinfo{volume}{15}},
  \bibinfo{pages}{2329} (\bibinfo{year}{1977}).

\bibitem[{\citenamefont{Aharonov and Casher}(1979)}]{Aharonov1979}
\bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Aharonov}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Casher}},
  \bibinfo{journal}{Phys. Rev. A} \textbf{\bibinfo{volume}{19}},
  \bibinfo{pages}{2461} (\bibinfo{year}{1979}).

\bibitem[{\citenamefont{Murakami et~al.}(2004)\citenamefont{Murakami, Nagaosa,
  and Zhang}}]{Murakami2004}
\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Murakami}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Nagaosa}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.-C.} \bibnamefont{Zhang}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{69}},
  \bibinfo{pages}{235206} (\bibinfo{year}{2004}).

\bibitem[{\citenamefont{Vasyukov et~al.}(2013)\citenamefont{Vasyukov, Anahory,
  Embon, Halbertal, Cuppens, Neeman, Finkler, Segev, Myasoedov, Rappaport
  et~al.}}]{Vasyukov2013}
\bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Vasyukov}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Anahory}},
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Embon}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Halbertal}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Cuppens}},
  \bibinfo{author}{\bibfnamefont{L.}~\bibnamefont{Neeman}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Finkler}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Segev}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Myasoedov}},
  \bibinfo{author}{\bibfnamefont{M.~L.} \bibnamefont{Rappaport}},
  \bibnamefont{et~al.}, \bibinfo{journal}{Nat Nano}
  \textbf{\bibinfo{volume}{8}}, \bibinfo{pages}{639} (\bibinfo{year}{2013}),
  ISSN \bibinfo{issn}{1748-3387}.

\bibitem[{\citenamefont{Wang et~al.}(2015)\citenamefont{Wang, Kirtley, Katmis,
  Jarillo-Herrero, Moodera, and Moler}}]{Wang948}
\bibinfo{author}{\bibfnamefont{Y.~H.} \bibnamefont{Wang}},
  \bibinfo{author}{\bibfnamefont{J.~R.} \bibnamefont{Kirtley}},
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Katmis}},
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Jarillo-Herrero}},
  \bibinfo{author}{\bibfnamefont{J.~S.} \bibnamefont{Moodera}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{K.~A.} \bibnamefont{Moler}},
  \bibinfo{journal}{Science} \textbf{\bibinfo{volume}{349}},
  \bibinfo{pages}{948} (\bibinfo{year}{2015}), ISSN \bibinfo{issn}{0036-8075}.

\bibitem[{\citenamefont{Basinski and Dugdale}(1985)}]{dis3}
\bibinfo{author}{\bibfnamefont{Z.~S.} \bibnamefont{Basinski}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~S.} \bibnamefont{Dugdale}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{32}},
  \bibinfo{pages}{2149} (\bibinfo{year}{1985}).

\bibitem[{\citenamefont{McEuen et~al.}(1990)\citenamefont{McEuen, Szafer,
  Richter, Alphenaar, Jain, Stone, Wheeler, and Sacks}}]{McEuen1990}
\bibinfo{author}{\bibfnamefont{P.~L.} \bibnamefont{McEuen}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Szafer}},
  \bibinfo{author}{\bibfnamefont{C.~A.} \bibnamefont{Richter}},
  \bibinfo{author}{\bibfnamefont{B.~W.} \bibnamefont{Alphenaar}},
  \bibinfo{author}{\bibfnamefont{J.~K.} \bibnamefont{Jain}},
  \bibinfo{author}{\bibfnamefont{A.~D.} \bibnamefont{Stone}},
  \bibinfo{author}{\bibfnamefont{R.~G.} \bibnamefont{Wheeler}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{R.~N.} \bibnamefont{Sacks}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{64}},
  \bibinfo{pages}{2062} (\bibinfo{year}{1990}).

\bibitem[{\citenamefont{Wang and Goldman}(1991)}]{Wang1991}
\bibinfo{author}{\bibfnamefont{J.~K.} \bibnamefont{Wang}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{V.~J.} \bibnamefont{Goldman}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{67}},
  \bibinfo{pages}{749} (\bibinfo{year}{1991}).

\bibitem[{\citenamefont{Bardeen and Shockley}(1950)}]{dis0}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Bardeen}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{W.}~\bibnamefont{Shockley}},
  \bibinfo{journal}{Phys. Rev.} \textbf{\bibinfo{volume}{80}},
  \bibinfo{pages}{72} (\bibinfo{year}{1950}).

\bibitem[{\citenamefont{Hunter and Nabarro}(1953)}]{dis1}
\bibinfo{author}{\bibfnamefont{S.~C.} \bibnamefont{Hunter}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{F.~R.~N.} \bibnamefont{Nabarro}},
  \bibinfo{journal}{Proc. Roy. Soc. London A} \textbf{\bibinfo{volume}{220}},
  \bibinfo{pages}{542} (\bibinfo{year}{1953}).

\bibitem[{\citenamefont{Basinski et~al.}(1963)\citenamefont{Basinski, Dugdale,
  and Howie}}]{dis2}
\bibinfo{author}{\bibfnamefont{Z.~S.} \bibnamefont{Basinski}},
  \bibinfo{author}{\bibfnamefont{J.~S.} \bibnamefont{Dugdale}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Howie}},
  \bibinfo{journal}{Phil. Mag.} \textbf{\bibinfo{volume}{8}},
  \bibinfo{pages}{1989} (\bibinfo{year}{1963}).

\bibitem[{\citenamefont{Watts}(1989)}]{dis4}
\bibinfo{author}{\bibfnamefont{B.~R.} \bibnamefont{Watts}},
  \emph{\bibinfo{title}{{\rm in} Basic Problems and Applications, volume 8 of
  Dislocations in Solids, {\rm edited by F. R. N. Nabarro}}}
  (\bibinfo{publisher}{North-Holland}, \bibinfo{year}{1989}).

\bibitem[{\citenamefont{Watts}(1988)}]{dis5}
\bibinfo{author}{\bibfnamefont{B.~R.} \bibnamefont{Watts}},
  \bibinfo{journal}{J. Phys. F: Met. Phys.} \textbf{\bibinfo{volume}{18}},
  \bibinfo{pages}{1183} (\bibinfo{year}{1988}).

\bibitem[{\citenamefont{Bergmann et~al.}(1981)\citenamefont{Bergmann, Kaveh,
  and Wiser}}]{dis6}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Bergmann}},
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Kaveh}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Wiser}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{24}},
  \bibinfo{pages}{6807} (\bibinfo{year}{1981}).

\bibitem[{\citenamefont{Kasumov et~al.}(1981)\citenamefont{Kasumov, Kopetskii,
  Kokhanchik, and Matveev}}]{dis7}
\bibinfo{author}{\bibfnamefont{A.~U.} \bibnamefont{Kasumov}},
  \bibinfo{author}{\bibfnamefont{C.~H.~V.} \bibnamefont{Kopetskii}},
  \bibinfo{author}{\bibfnamefont{L.~S.} \bibnamefont{Kokhanchik}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{V.~N.}
  \bibnamefont{Matveev}}, \bibinfo{journal}{Sov. Phys. Solid State}
  \textbf{\bibinfo{volume}{23}}, \bibinfo{pages}{151} (\bibinfo{year}{1981}).

\bibitem[{\citenamefont{Karolik and Luhvich}(1994)}]{dis8}
\bibinfo{author}{\bibfnamefont{A.~S.} \bibnamefont{Karolik}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.~A.} \bibnamefont{Luhvich}},
  \bibinfo{journal}{J. Phys.: Condensed Matter} \textbf{\bibinfo{volume}{6}},
  \bibinfo{pages}{873} (\bibinfo{year}{1994}).

\bibitem[{\citenamefont{Viswanathan and Chandrasekar}(2014)}]{dis9}
\bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Viswanathan}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Chandrasekar}},
  \bibinfo{journal}{J. Appied Phys.} \textbf{\bibinfo{volume}{116}},
  \bibinfo{pages}{245103} (\bibinfo{year}{2014}).

\bibitem[{\citenamefont{Nagaosa et~al.}(2010)\citenamefont{Nagaosa, Sinova,
  Onoda, MacDonald, and Ong}}]{Nagaosa_AHE}
\bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Nagaosa}},
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Sinova}},
  \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Onoda}},
  \bibinfo{author}{\bibfnamefont{A.~H.} \bibnamefont{MacDonald}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{N.~P.} \bibnamefont{Ong}},
  \bibinfo{journal}{Rev. Mod. Phys.} \textbf{\bibinfo{volume}{82}},
  \bibinfo{pages}{1539} (\bibinfo{year}{2010}),
  \urlprefix\url{http://link.aps.org/doi/10.1103/RevModPhys.82.1539}.

\bibitem[{\citenamefont{Sakurai and Napolitano}(2010)}]{Sakurai}
\bibinfo{author}{\bibfnamefont{J.~J.} \bibnamefont{Sakurai}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.~J.} \bibnamefont{Napolitano}},
  \emph{\bibinfo{title}{Modern Quantum Mechanics}}
  (\bibinfo{publisher}{Addison-Wesley}, \bibinfo{year}{2010}),
  \bibinfo{edition}{2nd} ed.

\bibitem[{\citenamefont{Sundaram and Niu}(1999)}]{topo-dis1}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Sundaram}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Niu}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{59}},
  \bibinfo{pages}{14915} (\bibinfo{year}{1999}).

\bibitem[{\citenamefont{Lazar}(2002)}]{topo-dis2}
\bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Lazar}}, \bibinfo{journal}{J.
  of Phys. A: Math. and General} \textbf{\bibinfo{volume}{35}},
  \bibinfo{pages}{1983} (\bibinfo{year}{2002}).

\bibitem[{\citenamefont{Ran et~al.}(2009)\citenamefont{Ran, Zhang, and
  Vishwanath}}]{topo-dis3}
\bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Ran}},
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Zhang}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Vishwanath}},
  \bibinfo{journal}{Nature Phys.} \textbf{\bibinfo{volume}{5}},
  \bibinfo{pages}{298} (\bibinfo{year}{2009}).

\bibitem[{\citenamefont{Mesaros et~al.}(2010)\citenamefont{Mesaros, Sadri, and
  Zaanen}}]{topo-dis4}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Mesaros}},
  \bibinfo{author}{\bibfnamefont{D.}~\bibnamefont{Sadri}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Zaanen}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{82}},
  \bibinfo{pages}{073405} (\bibinfo{year}{2010}).

\bibitem[{\citenamefont{de~Juan et~al.}(2010)\citenamefont{de~Juan, Cortijo,
  and Vozmediano}}]{topo-dis5}
\bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{de~Juan}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Cortijo}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.~A.~H.} \bibnamefont{Vozmediano}},
  \bibinfo{journal}{Nucl. Phys. B} \textbf{\bibinfo{volume}{828}},
  \bibinfo{pages}{625} (\bibinfo{year}{2010}).

\bibitem[{\citenamefont{Wang and Zhang}(2013)}]{topo-dis6}
\bibinfo{author}{\bibfnamefont{Z.}~\bibnamefont{Wang}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{S.~C.} \bibnamefont{Zhang}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{87}},
  \bibinfo{pages}{161107(R)} (\bibinfo{year}{2013}).

\bibitem[{\citenamefont{Zhou et~al.}(2013)\citenamefont{Zhou, Jiang, Niu, and
  Shi}}]{topo-dis7}
\bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Zhou}},
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Jiang}},
  \bibinfo{author}{\bibfnamefont{Q.}~\bibnamefont{Niu}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Shi}},
  \bibinfo{journal}{Chin. Phys. Lett.} \textbf{\bibinfo{volume}{30}},
  \bibinfo{pages}{027101} (\bibinfo{year}{2013}).

\bibitem[{\citenamefont{Slager et~al.}(2014)\citenamefont{Slager, Mesaros,
  Juri\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode~\acute{c}\else \'{c}\fi{},
  and Zaanen}}]{PhysRevB.90.241403}
\bibinfo{author}{\bibfnamefont{R.-J.} \bibnamefont{Slager}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Mesaros}},
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Juri\ifmmode \check{c}\else
  \v{c}\fi{}i\ifmmode~\acute{c}\else \'{c}\fi{}}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Zaanen}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{90}},
  \bibinfo{pages}{241403} (\bibinfo{year}{2014}).

\bibitem[{\citenamefont{Juri\ifmmode \check{c}\else
  \v{c}\fi{}i\ifmmode~\acute{c}\else \'{c}\fi{}
  et~al.}(2012)\citenamefont{Juri\ifmmode \check{c}\else
  \v{c}\fi{}i\ifmmode~\acute{c}\else \'{c}\fi{}, Mesaros, Slager, and
  Zaanen}}]{PhysRevLett.108.106403}
\bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Juri\ifmmode \check{c}\else
  \v{c}\fi{}i\ifmmode~\acute{c}\else \'{c}\fi{}}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Mesaros}},
  \bibinfo{author}{\bibfnamefont{R.-J.} \bibnamefont{Slager}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Zaanen}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{108}},
  \bibinfo{pages}{106403} (\bibinfo{year}{2012}).

\bibitem[{\citenamefont{Slager et~al.}(2013)\citenamefont{Slager, Mesaros,
  Juricic, and Zaanen}}]{Slager2013}
\bibinfo{author}{\bibfnamefont{R.-J.} \bibnamefont{Slager}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Mesaros}},
  \bibinfo{author}{\bibfnamefont{V.}~\bibnamefont{Juricic}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.}~\bibnamefont{Zaanen}},
  \bibinfo{journal}{Nat Phys} \textbf{\bibinfo{volume}{9}}, \bibinfo{pages}{98}
  (\bibinfo{year}{2013}), ISSN \bibinfo{issn}{1745-2473}.

\bibitem[{\citenamefont{Gromov and Abanov}(2014)}]{Gromov2014}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Gromov}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Abanov}},
  \bibinfo{journal}{Physical Review Letters} \textbf{\bibinfo{volume}{114}},
  \bibinfo{pages}{016802} (\bibinfo{year}{2014}).

\bibitem[{\citenamefont{Bradlyn and Read}(2015)}]{Bradlyn2015low}
\bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Bradlyn}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Read}},
  \bibinfo{journal}{Physical Review B} \textbf{\bibinfo{volume}{91}},
  \bibinfo{pages}{125303} (\bibinfo{year}{2015}).

\bibitem[{\citenamefont{Shitade and Kimura}(2014)}]{Shitade-Kimura2014}
\bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Shitade}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Kimura}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{90}},
  \bibinfo{pages}{134510} (\bibinfo{year}{2014}).

\end{thebibliography}
