``Soft'' Anharmonic Vortex Glass in Ferromagnetic Superconductors
Leo Radzihovsky$^1$
A. M. Ettouhami$^1$
Karl Saunders$^2$
John Toner$^2$
Ferromagnetic order in superconductors can induce a {\em spontaneous} vortex (SV) state. For external field ${H}=0$, rotational symmetry guarantees a vanishing tilt modulus of the SV solid, leading to drastically different behavior than that of a conventional, external-field-induced vortex solid. We show that quenched disorder and anharmonic effects lead to elastic moduli that are wavevector-dependent out to arbitrarily long length scales, and non-Hookean elasticity. The latter implies that for weak external fields $H$, the magnetic induction scales {\em universally} like $B(H)\sim B(0)+ c H^{\alpha}$, with $\alpha\approx 0.72$. For weak disorder, we predict the SV solid is a topologically ordered vortex glass, in the ``columnar elastic glass'' universality class.
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\title{``Soft'' Anharmonic Vortex Glass in Ferromagnetic Superconductors}
\author{Leo Radzihovsky$^1$, A. M. Ettouhami$^1$, 
Karl Saunders$^2$, John Toner$^2$}

\address{$^1$ Department of Physics, University of Colorado,
Boulder, CO 80309}
\address{$^2$ Dept. of Physics,
Materials Science Inst., and Inst. of Theoretical Science, University
of Oregon, Eugene, OR 97403}

\date{\today}
\maketitle
\begin{abstract}
Ferromagnetic order in superconductors can induce a {\em spontaneous}
vortex (SV) state.  For external field ${\bf H}=0$, rotational
symmetry guarantees a vanishing tilt modulus of the SV solid, leading
to drastically different behavior than that of a conventional,
external-field-induced vortex solid.  We show that quenched disorder
and anharmonic effects lead to elastic moduli that are
wavevector-dependent out to arbitrarily long length scales, and
non-Hookean elasticity. The latter implies that for weak external
fields $H$, the magnetic induction scales {\em universally} like
$B(H)\sim B(0)+ c H^{\alpha}$, with $\alpha\approx 0.72$.  For weak
disorder, we predict the SV solid is a topologically ordered vortex
glass, in the ``columnar elastic glass'' universality class.
\end{abstract}
\pacs{64.60Fr,05.40,82.65Dp}

%\twocolumn
\begin{multicols}{2}
\narrowtext

Rare-earth borocarbide materials exhibit a rich phase diagram that
includes superconductivity, antiferromagnetism, ferromagnetism and
spiral magnetic order.\cite{PhysicsToday,experiments,Varma}
In particular, there is now ample experimental evidence that, at low
temperatures, superconductivity and ferromagnetism competitively
coexist in ErNi$_2$B$_2$C compounds. Other possible examples of such
ferromagnetic superconductors (FS) are the recently discovered high
temperature superconductor Sr$_2$Y Ru$_{1-x}$Cu$_x$O$_6$ and the
putative $p$-wave triplet strontium ruthenate superconductor, Sr$_2$Ru
O$_4$, which spontaneously breaks time reversal symmetry.
%While there are many interesting unanswered microscopic
%questions regarding the nature of such ferromagnetic superconductors
%(FS), much of their low-energy phenomenology is dictated by general
%symmetry principles.  
%As argued in the context of borocarbides\cite{Varma}, (but we believe
%applies more generally) such FS's are expected to exhibit a {\em
%spontaneous} vortex (SV) state driven by the {\em spontaneous}
%magnetization, rather than by an external magnetic field ${\bf
%H}$. 
%We believe that {\em all} such FS's exhibit a {\em spontaneous} vortex
%(SV) state driven by the spontaneous magnetization, rather than by an
%external magnetic field ${\bf H}$. Such states have previously been 
%postulated for borocarbides\cite{Varma}. 
For sufficiently strong ferromagnetism, such FS's have been
predicted\cite{Varma} to exhibit a {\em spontaneous} vortex (SV) state
driven by the spontaneous magnetization, rather than by an external
magnetic field ${\bf H}$.  The novel phenomenology of the associated
SV solid is the subject of this Letter.

Here we will show that for ${\bf H}=0$, the elastic properties of the
resulting SV solid differ dramatically and {\em qualitatively} from
those of a conventional Abrikosov lattice. The key underlying
difference is the {\em vanishing} of the tilt modulus, which is
guaranteed by the underlying rotational invariance (but see
below). Although this invariance is broken by the magnetization, ${\bf
M}$, the tilt modulus remains zero because this breaking is {\em
spontaneous}.  This contrasts strongly with a conventional vortex
solid, where the rotational symmetry is {\em explicitly} broken by the
{\em applied} field ${\bf H}$.  All of our conclusions, e.g., the
unusual $B(H)$ relation, illustrated in Fig.\ref{BvsH} are a direct
consequence of this important observation.

% In particular, we find that because of the ``softness'' of the SV
% crystal, the effects of thermal fluctuations and quenched disorder are
% strongly enhanced: (i) thermal fluctuations melt SV lattice at a
% melting temperature, $T_m$ that is significantly depressed below the
% melting transition of a conventional Abrikosov vortex lattice (induced
% by an external field) with an equivalent vortex density; (ii) quenched
% disorder destroys the translational order of the putative SV crystal
% beyond a correlation length that is analogous but qualitatively
% distinct from the usual Larkin length scale\cite{Larkin}; (iii) on
% scales longer than the nonlinear length scale $\xi_{NL}$, which we
% calculate below, nonlinear elastic effects become qualitatively
% important; consequently, the SV solid exhibits anomalous elasticity,
% characterized by effective elastic moduli that are
% wavevector-dependent out to {\em arbitrarily long length scales} and
% by non-Hookean elasticity, both universal properties of a nontrivial
% $T=0$ fixed point, perturbative around $d=7/2$; (iv) in $d=3$, for
% weak quenched disorder, we find that dislocation loops remain bound
% implying stability of the topologically ordered SV glass.

In particular, we find that this ``softness'' (i.e., vanishing tilt
modulus) of the SV lattice {\em drastically} enhances the effects of
quenched disorder. As in conventional vortex lattices\cite{Larkin},
{\em any} amount of disorder $\Delta_V$, however weak, is sufficient
to destroy translational order in SV lattices. Here the finite ordered
domains are {\em divergently anisotropic}, with dimensions
$\xi^L_\perp\propto 1/\Delta_V^{2/3}$ and $\xi^L_z\propto
1/\Delta_V^{1/3}$.  These lengths are measurable in scattering and
transport measurements.  Unlike conventional lattices, however, in SV
lattices the disorder also qualitatively alters the elastic behavior
at long distances, leading to ``anomalous elasticity'': a {\em
universal} scaling of elastic moduli with wavevector $\bf k$ out to
{\em arbitrarily} long length scales, with some elastic moduli
vanishing, and others diverging, as wavevector ${\bf k}\rightarrow
0$.\cite{Brandt}
%This phenomenon is completely distinct from the well known wavevector
%dependence of, e.g., the shear modulus $\mu (\bf k)$ in conventional
%vortex lattices \cite{Brandt}. The conventional behavior, which is due to the
%long-range (comparable to the London penetration length $\lambda_L$)
%interaction between vortex lines, only occurs for large ${\bf k}$
%(i.e., ${\bf k} $'s such that $k \lambda_L \gg 1$
%\cite{footnote1}), while the anomalous elasticity described here persists
%down to arbitrarily {\em small} ${\bf q}$'s (i.e., out to arbitrarily 
%{\em long} length scales)\cite{footnote2}. 
This behavior is characteristic of a new kind of topologically ordered
``columnar elastic glass''(CEG) phase of vortices, which is stable, for
weak disorder, against proliferation of dislocations.
% 
\begin{figure}[bth] 
\centering
\setlength{\unitlength}{1mm} 
\begin{picture}(25,60)(0,-12)
\put(-60,-50){\begin{picture}(20,20)(0,0) 
\special{psfile=B_H.ps vscale=55 hscale=55} 
\end{picture}} 
\end{picture} 
\caption{The nonlinear and universal power-law $B(H)$ scaling, 
which at weakest fields $H<H_{cr.}$ and strongest fields $H>H_{NL}$
is cutoff by the crystal symmetry breaking anisotropy and $\xi_{NL}$,
respectively.}
\label{BvsH} 
\end{figure} 
% 
%As is often the case with symmetries that are {\em spontaneously}
%broken, one of 

The best way to experimentally probe a {\em spontaneously} broken 
symmetry is to break it directly with an external field. 
%as is done for instance in measuring the magnetization $M$, in
%a normal ferromagnet, as a function of applied field $H$ near a
%critical point. Here, in the FS, we can also explicitly break the
%rotation invariance by applying 
We predict that as a consequence of the anomalous elasticity, the increase
in the magnetic induction,
$\delta B(H)
\equiv B(H)-B(0)$ 
%(i.e., the compressional strain $\epsilon_c$) 
over the {\em spontaneous} induction $B(0)$, due to a weak
applied field ${\bf H}$ along ${\bf M}$
%(i.e., the compressional stress $\sigma$) 
obeys a {\em universal} ``non-Hookean'' scaling law:
%
\begin{equation}
\delta B(H) \propto |H|^{\alpha}\;,
\label{nonhook}
\end{equation}
%
with the {\em universal} exponent $\alpha=0.72 \pm
0.04$,
%\cite{errorbar}
a prediction that should be experimentally testable.
%We call this behavior ``non-Hookean'' since ${\bf H}$ has the effect of
%uniformly compressing the 
%flux lattice, and hence acts as an external compressional {\em stress} 
%$\sigma$, while the compressional {\em strain} $\epsilon_c$ of the flux 
%lattice is proportional to $\delta B$. Hence, Eq.\ref{nonhook} is a nonlinear
%relation of the form: $\epsilon_c \propto \sigma^{\alpha}$, $\alpha \ne 1$.
%
%It should be straightforward to experimentally test the relation 
%Eq.\ref{nonhook}. Observation of such a nonlinear stress-strain relation 
%at {\em small} fields ${\bf H}$ would be a dramatic experimental confirmation 
%of the existence of anomalous elasticity in SV solids.
%
% , and, hence of the existence of the novel ``columnar 
% Bragg glass'' phase of disordered SV flux lattices.

Whatever the microscopic origin of the novel FS state, general symmetry
principles dictate that the long length scale phenomenology is described
by an effective Landau-Ginzburg free energy functional
%
\begin{eqnarray}
\hspace{0cm}F_{GL}&=&\int \! d^3 r\bigg[{\hbar^2\over 2m}|(\bbox{\nabla}- i
{2\pi\over\phi_0} {\bf A})\psi|^2+{a\over2}|\psi|^2+
{b\over4}|\psi|^4\nonumber\\
&+&{K\over2}|\bbox{\nabla}{\bf M}|^2+{t\over2}|{\bf M}|^2+
{u\over4}|{\bf M}|^4+{1\over8\pi}|{\bf B}-4\pi{\bf M}|^2\nonumber\\
&-&{1\over4\pi}{\bf B}\cdot{\bf H}\bigg]\;,
\label{F_GL}
\end{eqnarray}
%
where $\psi$ is the superconducting order parameter and ${\bf A}$ is the
electromagnetic vector potential with $\bf B=\bbox{\nabla}\times{\bf
  A}$.
%is the magnetic induction, and ${\bf M}$ is the spin magnetization. 
The constants $m,a,b,K,t$, and $u$ are experimentally measurable
phenomenological parameters, with $a$ and $t$ changing their signs at
the superconducting and ferromagnetic decoupled transition temperatures
respectively, and $\phi_0=hc/2e$ the elementary magnetic flux quantum.

The above model displays rich phenomenology\cite{Varma}. It has been
shown in detail by Greenside, et al.\cite{Varma}, that for a range of
physically realistic parameters (e.g., $\lambda_L/\xi$,
$\lambda_L/\sqrt{K/|t|}\approx O(10)$,\cite{Varma} with the large
Abrikosov ratio required for a robust mixed state, and large exchange
$K$ necessary to suppress the competing spiral phase, relative to the
SV state), systems described by $F_{GL}$, Eq.\ref{F_GL}, exhibit a
thermodynamically stable phase consisting of a {\em spontaneous}
(i.e. ${\bf H}=0$) vortex state, in which superconducting vortices are
generated by the spontaneous magnetization of the ferromagnetic
component.

Deep within this state, both $\psi({\bf r})$ and ${\bf M}({\bf r})$
are large, and therefore the London approximation applies. We take
${\bf M}({\bf r})=M_0\hat{\bf n}({\bf r})$ to be of approximately
uniform magnitude $M_0$
%(ignoring the small, irrelevant
%periodic component induced by the coupling to ${\bf B}$) 
and fluctuating direction $\hat{\bf n}({\bf
r})$. We also take $|\psi({\bf r})|$ to be a constant $\psi_0\neq0$,
everywhere except at the locations of the vortices (where $\psi=0$)
and allow its phase $\theta({\bf r})$ to vary subject to the
circulation condition that it increases by an amount $2\pi$ along a
closed path enclosing a vortex\cite{ReviewsSC}.  The magnetic
induction ${\bf B}({\bf r})$ is confined to thin flux filaments
defined by the zeros of $\psi$. Using the circulation condition and the
minimizing above free energy
$F_{GL}$ with respect to $\psi_0$, $\theta$ and ${\bf A}$, we
eliminate $\bbox{\nabla} \theta$ and ${\bf B}$ in favor of vortex line
conformational degrees of freedom and obtain:

% We take the magnetization to be approximately uniform ${\bf
% M}({\bf r})=M_0\hat{\bf n}$ (with a small periodic component induced
% by the coupling to ${\bf B}$) and $\psi({\bf r})=\psi_0\neq0$
% everywhere except at locations of vortices (where $\psi=0$) and
% magnetic induction $\bf B$ is confined to thin flux filaments defined
% by zero's of $\psi$.  Minimizing the above free energy, $F_{GL}$ with
% respect to $\psi_0$ and $\bf A$ and using the resulting equations to
% eliminate $\bf B$ in favor of vortex line conformational degrees of
% freedom we obtain an effective free energy
%
\begin{eqnarray}
F_L&=&{\phi_0^2\over8\pi}\int d^3r d^3r'{\bf t({\bf r})}\cdot{\bf
t({\bf r}')}V(|{\bf r}-{\bf r}'|)\nonumber\\
&+&\int d^3r\left[{1\over2}K M_0^2|\bbox{\nabla}\hat{\bf n}|^2-
\phi_0 M_0{\bf t}({\bf r})\cdot\hat{\bf n}\right]\;,
\label{F_L}
\end{eqnarray}
%
where we have dropped unimportant constant pieces. ${\bf t({\bf
r})}=\sum_n\int ds {d{\bf r}_n\over ds}\delta^3({\bf r}-{\bf r}_n(s))$
is the directed vortex line density, with magnitude given by the local
flux density (in units of $\phi_0$) and with direction tangent to the
local vorticity; ${\bf r}_n(s)$ is the locus of the $n$-th vortex
line, parameterized by an arc length $s$. The vortex interaction
$V(r)$ is approximately given by the standard expression
$V(r)=e^{-r/\lambda_L}/(4\pi\lambda_L^2 r)$, with $\lambda_L^2=
m\phi_0^2/(16\pi^3\hbar^2|\psi_0|^2)$\cite{Varma}, cut off at scales
smaller than the coherence length $\xi$.
% by the flux line core energy
% per unit of length
% $\epsilon_1=\phi_0^2\ln(\lambda_L/\xi)/(16\pi^2\lambda_L^2)$,

At low temperature $T$ and in the clean limit the vortex
lines freeze into a lattice (as has been observed\cite{PhysicsToday}),
directed on average along a {\em spontaneously} chosen magnetization 
direction $\hat{\bf n}_0$, which we choose to be the $\hat{\bf z}$ axis, 
and with the lattice constant determined by the average magnetization 
$M_0$. Vortex lattice fluctuations are described by the 2d phonon
displacement vector field ${\bf u}(z,{\bf R}_n)$, defined by ${\bf
r}_n(z)=(X_n+u_{x}(z,{\bf R}_n),Y_n+u_{y}(z,{\bf R}_n),z)$, where
${\bf R}_n=(X_n,Y_n)$ spans the vortex lattice, and we chose $s=z$,
%parameterize a vortex line by its projection along the z-axis, i.e.,
%the coordinate $z$. 
%${\bf M}({\bf r})$ fluctuations are characterized by 
and by a small 2d vector $\delta{\bf n}\perp \hat{\bf z}$
%(lying in the $xy$
%plane) describing $M_0{\bf\hat{n}}$the local magnetization deviation
%from the average magnetization $M_0{\bf\hat{z}}$ and 
defined by $\hat{\bf n}=(1-\delta{\bf n}\cdot\delta{\bf
n})^{1/2}\hat{\bf z}+\delta{\bf n}$.

Integrating out $\delta{\bf n}$ and going to the continuum, we
obtain\cite{Lubensky,unpublished} the effective elastic Hamiltonian:
%
\begin{equation}
H_{el}={1\over2}\int d^3r\left[\kappa|\partial^2_z{\bf u}|^2+2\mu
u_{ij}^2 +\lambda u_{ii}^2\right]\;,
\label{Helastic}
\end{equation}
%
where $\kappa$ is the vortex line curvature modulus and $\mu$ and
$\lambda$ are the Lam\'e coefficients.
%($\mu$ is the shear modulus and
%$\lambda$ should not be confused with the London penetration length
%$\lambda_L$)
$\mu$, $\lambda$ and $\kappa$ are
all expressible in terms of the parameters of the original model
Eq.\ref{F_GL}, and $u_{ij}={1\over2}(\partial_i u_j+\partial_j
u_i-\partial_z u_i\partial_z u_j)$ is the rotationally invariant {\em
nonlinear} strain tensor, with $i,j \in x,y$.  This form of $H_{el}$ is
strictly dictated by general symmetry considerations, specifically:
the underlying rotational symmetry of the SV solid about any arbitrary
axis 
%$\hat{\bf \omega}$ perpendicular to the magnetization axis
%$\hat{\bf n_0}=\hat{\bf z}$ (under which ${\bf u}\rightarrow{\bf
%u}+\theta z\hat{\bf n_0}\times\hat{\bf \omega}$), which 
guarantees that only the fully rotationally invariant
combination $u_{ij}$, given above, appears and that the vortex line tilt
modulus ($c_{44}$) vanishes identically (but see
below)\cite{unpublished}.

%As we will show below, the ``soft'' curvature, nonlinear elasticity
%and quenched disorder conspire to lead to effective elastic moduli of
%the SV solid that, unlike the conventional flux lattice, are actually
%wavevector dependent out to {\em arbitrarily long} length
%scales.\cite{Brandt,footnote2}
%
% We first focus on the clean limit. Extending mathematically related
% analysis of liquid crystals,\cite{GP} shows that the nonlinear
% elasticity of the SV lattice (last term in Eq.\ref{strain_tensor}) is
% only relevant for $d<d_{uc}=5/2$ and therefore is unimportant in clean
% bulk 3d superconductors. Nevertheless because of the soft
% ``curvature'' (rather than ``tension'') elasticity along the $z$, we
% expect the effect of thermal fluctuations to be substantially enhanced
% and to therefore lead to SV crystal phenomenology that is {\em
% qualitatively} different from that of a conventional vortex crystal.
%
% We first compute the phonon-phonon thermal correlation function
% $C^T_{ij}({\bf r})=\langle\left[u_i({\bf r})- u_i({\bf
% 0})\right]\left[u_j({\bf r})- u_j({\bf
% 0})\right]\rangle=P^\perp_{ij}C^T_\perp({\bf r})
% +P^{||}_{ij}C^T_{||}({\bf r})$, with the transverse and longitudinal
% parts given by
%
% \begin{eqnarray}
% C^T_s({\bf r})=2T\int{d^2{q_\perp}d q_z\over(2\pi)^3}
% {1-e^{i{\bf q}\cdot{\bf r}}\over\kappa q_z^4 + B_s q_\perp^2}\;,
% \label{CT}
% \end{eqnarray}
%
% where $s$ takes on $\perp$ and $||$ values and $B_\perp=\mu$ and
% $B_{||}=2\mu+\lambda$, and $P^s_{ij}$ are the transverse and
% longitudinal (to $\bf q_\perp$) projection operators. Lacking a
% detailed melting theory, we use the Lindemann criterion (temperature
% at which root-mean displacement is on the order of the vortex lattice
% spacing) to estimate the location of the melting line for the SV
% lattice. Performing above integrals we find upto factors of order
% $\pi$ an implicit expression for the melting temperature of the SV
% lattice
%
%  \begin{equation}
% T_m\approx\left({\phi_0\over M_0(T_m)}\right)^{5/4}
% \left(\kappa(T_m) B_s(T_m)^3\right)^{1/4}\;.
% \label{Tm}
% \end{equation}
%
% We observe that line nature of the SV crystal leads to ice-like
% (negative slope) melting curve, $T_m(M)\sim 1/M^{5/4}$, as previously
% noted in a different context in Ref.\onlinecite{IceMelting}.  As
% discussed in the introduction, because of the ``softness'' of the SV
% lattice, $T_m$ found here is strongly suppressed relative to that of
% an ordinary Abrikosov vortex lattice for an equivalent vortex
% density.\cite{ReviewsSC} However, because the Lindemann criterion
% integral, Eq.\ref{CT} is dominated by large wavevectors at the
% Brillouin zone boundary, our prediction for $T_m$, which neglects
% short-scale wavevector dependence of elastic moduli, is not
% quantitatively trustworthy. 
%
% In addition to qualitatively modified melting curve, we expect that
% the enhanced vortex fluctuations of the SV lattice will manifest
% themselves in neutron scattering, leading to a strongly reduced
% Debye-Waller factor. Despite of these enhanced thermal fluctuations,
% in 3d, $C_s^T({\bf r})$ clearly saturates to a constant at large $r$,
% and therefore, at low $T$, we expect clean SV lattice to display
% long-ranged translational order, accompanied by true Bragg peaks.
%
% We have first studied the melting of a clean SV lattice, by
% generalizing by now standard Lindermann criterion
% analysis\cite{ReviewsSC}. We found, that, because of the softness
% ($c_{44}=0$) of the SV lattice, the effects of thermal fluctuations
% are considerably enhanced and naively lead to upward shifts of the
% ice-like (negative slope in $B(T_m)$) melting curve.\cite{unpublished}
% However, because it is the phonon modes at Brillouin zone boundary
% that are responsible for melting the vortex lattice, our
% long-wavelength description is incapable of unambiguous and universal
% predictions (indeed such modes exhibit finite tilt modulus).

Although the softness of the SV lattice suggests, via the Lindemann
criterion, a lower melting temperature, our long wavelength
description cannot directly predict the melting temperature, because
thermal fluctuations are dominated by short distance modes (near the
Brillouin zone boundary).  Instead, here we focus on the much more
interesting, robust and universal effects of quenched disorder, all of
which (at long wavelengths) can be incorporated into our model by
adding to $H_{el}$
%
\begin{equation}
H_{dis}=\int d^3r\left[{\cal R}e \sum_{\bf G} V_{\bf G}({\bf r})e^{i {\bf G}
\cdot {\bf u}({\bf r})}+{\bf h}({\bf r})\cdot\partial_z{\bf
u}\right]\;.
\label{Hdisorder}
\end{equation}
%
The first term accounts for the random {\em positional} pinning (at
reciprocal lattice vectors ${\bf G}$) of the vortex lattice, with
$V_{\bf G}({\bf r})$ a complex random pinning potential, whose
statistics can be accurately represented by zero-mean, {\em
  short-ranged}, Gaussian correlations: $\overline{V_{\bf G}({\bf
    r})V_{\bf G'}^*({\bf r'})} = \Delta_V\delta_{{\bf G}{\bf
    G'}}^{\perp} \delta^d({\bf r}-{\bf r'})$\cite{RTaerogel}. The
second term reflects the presence of an additional {\em orientational,
  tilt} disorder.
%(unimportant in a conventional vortex lattice). 
The tilt field ${\bf h}({\bf r})$ is a random vector with {\em
short}-ranged {\em isotropic} spatial correlations, whose statistics
can also be taken zero-mean Gaussian, with $\overline{h_i({\bf r})
h_j({\bf r'})} =\Delta\delta_{ij}\delta^d({\bf r}-{\bf
r'})$\cite{RTaerogel}.  In the case of magnetic impurities, it is the
random local fields, acting on $\bf M$, i.e., ${\bf h}({\bf
r})\cdot{\hat{\bf n}}$, which via the Higg's mechanism, $\delta{\bf
n}\approx\partial_z{\bf u}$ lead to the tilt disorder piece of
$H_{dis}$.  In the case of nonmagnetic impurities, it is the
short-scale anisotropy in the spatial distribution of vortex line
pinning centers, together with lack of rotational symmetry in the SV
state, that leads to the tilt disorder in Eq.\ref{Hdisorder}, as can
be easily illustrated by short-scale modes thinning
procedure.\cite{unpublished} (As in conventional vortex lattices, in
$d>2$, the pinning of the {\em long-scale} vortex density distortions
is {\em irrelevant} at long scales).

Adapting Larkin's\cite{Larkin} calculation to the ``soft'' elasticity
of the SV lattice we find that in $d$ dimensions and at short length
scales, mean-squared lattice distortions (in domains of size
$L_\perp(\ll L_z)$) grow as $u_{rms}^2\sim L_\perp^{9/2-d}$. Hence,
translational order is destroyed for $d=3<d_c^{\Delta_V}=9/2$,
implying (in the weak disorder limit) divergently anisotropic Larkin
domains spanned by 3d Larkin lengths $L_\perp^L\approx\left(a^2(\kappa
  B_s^7)^{1/4}/\Delta_V\right)^{2/3}$ and $L_z^L\approx\left(a^2\kappa
  B_s/\Delta_V\right)^{1/3}$, with $a$ the vortex spacing, $s\in L,T$,
and $B_L=2\mu+\lambda$, $B_T=\mu$.  Beyond these Larkin length scales,
the Larkin approximation breaks down and the full nonlinear nature of
the random-field pinning potential must be taken into account, leading
to much more weakly (specifically, logarithmically) divergent lattice
fluctuations as in conventional vortex lattices\cite{FisherFRG}.  This
weak divergence is completely overwhelmed for $d<d_{uc}=7/2$ by the
far stronger fluctuations induced by the tilt disorder, which diverge
algebraically:
% As is well
% known in other contexts,\cite{FisherFRG} because of the short-scale
% periodicity of the random field potential, we find that if
% dislocations do {\em not} unbind, then beyond the Larkin scale, the SV
% lattice displacements grow logarithmically-slowly.
%
%In contrast, the tilt disorder alone, for $d<d_{uc}=7/2$, leads to
%{\em power-law} growth of SV lattice displacements,
$u_{rms}^2\sim
L_\perp^{7/2-d}$ (ignoring nonlinear elastic effects) out to
arbitrarily long length scales
% , and therefore in 3d strongly dominates
% over the logarithmic roughness of
%$\Delta_V$
\cite{unpublished,RTaerogel,JSRTstretched,SRTcolumnar}. 
%
%Hence,
%for the remainder of this Letter, we will confine our attention to the
%dominant tilt disorder, and study the model described by the
%Hamiltonian $H_{full}$ obtained by adding to $H_{el}$ only the {\em
%second} term in $H_{dis}$.

%Using $H_{full}$ and
Ignoring nonlinear elastic terms, we obtain
%(in d dimensions and up to factors of order $\pi$)
the disorder-averaged phonon correlation function $\overline{C_s({\bf
r})}\approx \mbox{Max}[\Delta/(\kappa^3 B_s^5)^{1/4}r_\perp^{7/2-d},
\Delta/(\kappa^{5-d} B_s^{d-1})^{1/2}z^{7-2d}]\times O(1)$.
The spatial (Larkin-like) correlation lengths, measurable in neutron
scattering experiments, are defined by
$C_s(\mbox{Min}[\xi_\perp,\xi_z])=a^2$ and are given by $\xi_{\perp}
\approx\big[a^2(\kappa^3 B_s^5)^{1/4}/\Delta\big]^{1/(7/2-d)}$, $\xi_z
\approx\xi_\perp^{1/2}(\kappa/B_s)^{1/4}$. Hence, in 3d, in the 
presence of arbitrarily weak disorder $\Delta$, on length scales
longer than $\xi_{\perp,z}$ the translational order of SV solid decays
rapidly, that is, it is {\em short-ranged}.
%dominated by the disorder part given in Fourier space by
%
%\begin{eqnarray}
%C^\Delta_s({\bf r})=2\Delta\int{d^2{q_\perp}d q_z\over(2\pi)^3}
%{q_z^2(1-e^{i{\bf q}\cdot{\bf r}})\over
%\left[\kappa q_z^4 + B_s q_\perp^2\right]^2}\;,
%\label{CDelta}
%\end{eqnarray}
%
This, however, does {\em not} imply that the system is equivalent to
the SV {\em liquid} state, or to a fully disordered vortex glass as it
would be if dislocation loops unbound. Using duality methods,
introduced recently by two of the authors\cite{RTaerogel}, we have
shown that a translationally disordered, but topologically ordered
``columnar elastic glass'' (CEG) phase is in fact possible in 3d.
This is analogous to our recent results in the context of randomly
pinned smectic liquid crystals\cite{RTaerogel,JSRTstretched} and
columnar phase of discotic liquid crystals\cite{SRTcolumnar} confined
in aerogel, as well as the Bragg glass conjecture for conventional
vortex lattices\cite{GL} and the random field XY model\cite{DSFisher}.
%, which also
%predict such an exotic possibility.

% Assuming that dislocations in fact remain bound (or, if they do
% unbind, considering the behavior on scales smaller than the
% dislocation unbinding length $\xi_{disl.}>>\xi_{NL}$ (see below)) we
% now turn our attention to the effects of full {\em anharmonic
% elasticity} defined by Eqs.\ref{Helastic}, \ref{strain_tensor}.
% Calculations similar to those of Ref.\onlinecite{RTaerogel} show that
% the perturbation theory in elastic nonlinearities diverges beyond the
% minimum of the nonlinear crossover length scales
% $\xi_{NL}^z=8\pi^2\kappa^2/\Delta$,
% $\xi_{NL}^\perp=(\xi_{NL}^z)^2(B_s/\kappa)^{1/2}$, which in the
% renormalization group (RG) language signals the instability of the
% Gaussian fixed point to 
In this topologically ordered phase 
%(or, considering the behavior on scales smaller than the dislocation 
%unbinding length $\xi_{disl.}>>\xi_{NL}$ (see below)) 
the anharmonic elastic terms in the Hamiltonian lead to drastically modified 
elastic scaling on length scales longer than the nonlinear crossover
length scales $\xi_{NL}^z=8\pi^2\kappa^2/\Delta$ and
$\xi_{NL}^\perp=(\xi_{NL}^z)^2(B_s/\kappa)^{1/2}$. In renormaliztion
group (RG) language, on scales longer than $\xi_{NL}^{z,\perp}$, our
system flows away from a zero-disorder Gaussian fixed point and
approaches a nontrivial $T=0$ anharmonic, disordered fixed point.
This new fixed point can
be studied using standard RG analysis\cite{RTaerogel}, and we predict
that at scales longer than $\xi_{NL}^{z,\perp}$, out to an arbitrary
length scale (but see below), the elastic moduli $\kappa$, $\mu$,
$\lambda$, and the disorder variance $\Delta$ become singular
functions of wavevector $\bf k$\cite{Brandt}:
%
\begin{mathletters}
\begin{eqnarray}
\kappa({\bf k})&=&\kappa_0\left(k_z\xi_{NL}^z\right)^{-\eta_\kappa}
f_\kappa(k_\perp\xi_{NL}^\perp/(k_z\xi_{NL}^{z})^\zeta)
\;,\label{kappa}\\
\mu({\bf k})&=&\mu_0\left(k_z\xi_{NL}^z\right)^{\eta_{\mu}}
f_\mu(k_{\perp}\xi_{NL}^{\perp}/(k_z\xi_{NL}^{z})^\zeta)
\;,\label{mu}\\
\lambda({\bf k})&=&g_*\mu_0\left(k_z\xi_{NL}^z\right)^{\eta_{\mu}}
f_\mu(k_\perp\xi_{NL}^\perp/(k_z\xi_{NL}^{z})^\zeta)
\;,\label{lambda}\\
\Delta({\bf k})&=&\Delta_0\left(k_z\xi_{NL}^z\right)^{1-2\eta_\kappa}
f_\Delta(k_\perp\xi_{NL}^\perp/(k_z\xi_{NL}^{z})^\zeta)
\;,\label{Delta}
\end{eqnarray}
\label{etas}
\end{mathletters}
%
where $f_{\kappa/\mu/\Delta}(x)$ is independent of $x$ if $x \ll 1$
and goes like $x$ to the power $-\eta_{\kappa}/\zeta$,
$\eta_{\mu}/\zeta$ and $(1-2\eta_{\kappa})/\zeta$, respectively, when
$x \gg 1$, so that, e.g., $\kappa({\bf k}) \propto
k_\perp^{-\eta_{\kappa}/\zeta}$ for $k_\perp\xi_{NL}^\perp \gg
(k_z\xi_{NL}^{z})^\zeta$.  $\kappa_0$, $\mu_0$ and $\Delta_0$ are the
bare values of the elastic constants and the anomalous exponents are
universal, and to leading order in $\epsilon={7\over2}-d$ are given by
$\eta_{\kappa}=1.478
\epsilon=0.74$ and $\eta_{\mu}=0.6919 \epsilon=0.346$, where the second
equalities are their values in 3d. The constant $g_*$ is also
universal, and to leading order in $\epsilon$ $g_*=-0.03272 +
O(\epsilon)$.  The anisotropy exponent $\zeta =
2-(\eta_{\kappa}+\eta_{\mu})/2$, and therefore the two independent
exponents $\eta_\kappa$ and $\eta_\mu$ completely characterize the
anomalous elasticity, with $\kappa({\bf k})$ and $\Delta({\bf k})$
diverging with linearly-related exponents, and $\mu({\bf k})$ and
$\lambda({\bf k})$ both vanishing in exactly the same way as ${\bf
k}\rightarrow\infty$. The negative universal amplitude ratio $g_*$
implies that the SV glass exhibits a {\em negative universal} Poisson
ratio $\sigma_p= -0.0338 + O(\epsilon)$.
%as in tethered membranes\cite{membranes}.
%Interestingly, as in all of the other examples that we know of, the
%universal Poisson ratio here is also negative, at least to leading
%order in $\epsilon$ (although is in this case so small that the higher
%order in $\epsilon$ corrections may very well change its sign).
%We stress that in contrast to the elastic moduli wavevector-dependence
%which arises from the finite range of the inter-vortex
%potential\cite{Brandt}, and only extends out to the London penetration
%length, in isotropic samples of randomly pinned {\em spontaneous}
%vortex solid, the singular behavior of the elastic moduli,
%Eqs.\ref{etas}, predicted here, persists all the way out to the system
%size. Physically, it describes the ``return to isotropy'', at long
%scales, of the highly anisotropic bare elastic energy,
%Eq.\ref{Helastic}, driven by the isotropic disorder roughness. As in
%other examples of anomalous
%elasticity\cite{RTaerogel,unpublished,JSRTstretched,SRTcolumnar,GP},
%the long scale divergence of $\kappa({\bf q})$ and vanishing of the
%Lam\'e coefficients, $\mu({\bf q})$ and $\lambda({\bf q})$, describes
%the fact that bending and shearing/compressing the disorder-roughened
%SV solid results in shearing/compressing and bending contributions,
%respectively. This therefore ``admixes'' stiff, $q_\perp^2$, shear and
%compression elasticity into the soft, $q_z^4$, bend elasticity, and
%visa versa.

For length scales longer than $\xi_{NL}$ this strong,
power-law anomalous elasticity will alter the behavior of
all physical observables of the SV solid, such as, e.g., the width of
the structure function peak $S({\bf k})$ measured in neutron
scattering and the behavior in its tails. It also leads to a
{\em non-Hookean}, i.e., nonlinear, stress-strain relation 
even for {\em arbitrarily small} applied stress $\sigma$.
%(Note here, that $\sigma$ and $\epsilon_c$ should not be confused with the 
%Poisson ratio, $\sigma_p$ and $\epsilon$-expansion parameter, respectively).

To see this, consider a purely compressive stress,
$\sigma_{ij}=\sigma\delta_{ij}$,
%(which, as we shall see shortly,
%arises from an externally applied magnetic field ${\bf H}$ along the
%flux lines).  
which adds
%\cite{footnote2}
a term $\sigma{\bbox\nabla}\cdot{\bf u}=\sigma u_{ii}+{1 \over
  2}|\partial_z{\bf u}|^2$ to the Hamiltonian.
%, $H_{full}$.
%The actual values of the dimensionless constants of
%proportionality are not important and for simplicity we set them equal
%to unity. 
In Fourier space, the second, symmetry breaking, term becomes $\sigma
k_z^2 |{\bf u}({\bf k})|^2$ and begins to dominate over the
$\kappa({\bf k}) k_z^4|{\bf u}({\bf k})|^2$ term once $\sigma k_z^2
\ge \kappa({\bf k})k_z^4$. This clearly happens for $k_z$'s less
than a critical $k_c$ given by $\sigma k_c^2=\kappa(k_z=k_c,{\bf
k}_{\perp}=0)k_c^4$. Using Eq.\ref{kappa} for $\kappa({\bf k})$ in
this expression and solving for $k_c$, we find
$k_c(\sigma)=(\sigma/\kappa_0)^{1/(2-\eta_{\kappa})}
(\xi_{NL}^z)^{\eta_{\kappa}/(2-\eta_{\kappa})}$.

Now, for sufficiently weak stress $\sigma$, $1/\xi^z_{NL}\gg
k_c(\sigma)$, the stress-induced rotational symmetry breaking term is
subdominant to the vortex curvature energy $\kappa({\bf k}) k_z^4|{\bf
u}({\bf k})|^2$ for wavevectors $k_z\gg k_c(\sigma)$. Consequently,
on intermediate length scales, $k_{c}(\sigma)<k_z<k_{NL}^z$, the SV glass
exhibits anomalous elasticity summarized by Eqs.\ref{etas}. However,
on longer length scales, $k_z\ll k_c(\sigma)$, the applied stress
energy dominates, suppressing fluctuations and cutting off
the anomalous elasticity at $k_c(\sigma)$. Therefore, the elastic
moduli {\em saturate}, for $k_z
\ll k_c(\sigma)$, at their values at $k_z=k_c(\sigma)$, $k_{\perp}=0$ given by
Eq.\ref{etas}, i.e., $\mu({\bf k}\rightarrow
0)\rightarrow\mu(k_z=k_c,{\bf k}_{\perp}= 0)\propto k_c^{\eta_{\mu}}$
and $\lambda({\bf k}\rightarrow 0)\rightarrow\lambda(k_z=k_c,{\bf
k}_{\perp}= 0)\propto k_c^{\eta_{\mu}}$. This implies that the Young's
modulus $Y(\sigma) = 4\mu(\mu +\lambda)/(2\mu +\lambda)=
4\mu_0(1+g_*)/(2+g_*)(k_c\xi_{NL}^z)^{\eta_\mu}\approx
2\mu_0(k_c\xi_{NL}^z)^{\eta_\mu}=2\mu_0(\sigma/\sigma_{NL})^\beta$,
with $\sigma\ll\sigma_{NL}$, $\beta=\eta_\mu/(2-\eta_\kappa)$, and
$\sigma_{NL}=\kappa_0\xi_{NL}^{-2}$.  Hence, we find for stress
$\sigma\le\sigma_{NL}$, a {\em nonlinear} (non-Hookean) strain-stress
relation: $\epsilon_c(\sigma)\equiv\bbox{\nabla}\cdot{\bf
u}=\sigma/Y(\sigma)\propto
\sigma^{\alpha}$, with $\alpha=1-\beta=1-\eta_{\mu}/(2-\eta_{\kappa})
\approx 0.72$.
%, where in the last approximate equality, the numerical values of
%$\eta_{\mu}$ and $\eta_{\kappa}$, obtained from the
%$\epsilon$-expansion have been used.

Now, it is easy to see that ${\bf H}=H\hat{z}$, applied {\em along}
the direction of the SV lattice, acts as a compressive stress,
$\sigma=H B(0)$, and the induced $\delta B(H)=B(H)-B(0)$ plays the
role of the compressive strain, $\epsilon_c=\delta B/B(0)$. Hence $H$
allows us to {\em directly} probe the anomalous elasticity,
with the non-Hookean elasticity leading to the nonlinear and universal
$B(H)$ relation, for
$H<H_{NL}=\sigma_{NL}/B(0)$, given in Eq.\ref{nonhook} and Fig.\ref{BvsH}.

%To see that the non-Hookean stress-strain relation,
%Eq.\ref{stressstrain}, leads to a nonlinear relation between $B$ and
%$H$, note that the presence of an external field, ${\bf H}$,
%necessitates the inclusion of an additional term $-{\bf B}\cdot{\bf
%H}$ in the Hamiltonian. A field ${\bf H}$ along the $\hat{\bf z}$
%direction (the direction of the initial magnetic induction ${\bf
%B}(H=0)$), will therefore induce an increase in the magnetic
%induction, $\delta B(H) \equiv B(H)-B(H=0)$. In the flux lattice
%state, however, since each flux line carries {\em exactly} one flux
%quantum, $\phi_0$, and no more, the only way to increase $B$ is to
%move the flux lines closer together; i.e., compress the lattice. This
%implies that the external field leads to a compressive strain term,
%proportional to $-H\bbox{\nabla}\cdot{\bf u}$. The external field
%${\bf H}$ also {\em explicitly} breaks rotational symmetry, leading to
%the addition of a term $H|\partial_z{\bf u}|^2$. Thus, the application
%of a field ${\bf H}$ leads to a compressional strain (proportional to
%$\delta B$) and to explicit rotational symmetry breaking, just as a
%uniform compressive stress does. The two cases are exactly analogous,
%with $\sigma$ replaced by $H$, and the non-Hookean stress-strain
%relation, Eq.\ref{stressstrain}, immediately translates into a
%nonlinear $\delta B(H)-H$ relation of exactly the same form; i.e.,
%into Eq.\ref{nonhook}.

Of course, as with all crystalline ferromagnets, crystalline
symmetry breaking fields will {\em explicitly} break rotational
invariance, even in the absence of ${\bf H}$. They will lead to a
non-zero tilt modulus, i.e., a term ${1\over2}V_{cr.}|\partial_z{\bf
u}|^2$ in the Hamiltonian, which will cut off the anomalous elasticity
on length scales longer than $\xi_{cr.}^z=({\kappa_0 \over
V_{cr.}})^{1/(2-\eta_\kappa)} (\xi_{NL}^z
)^{-\eta_{\kappa}/(2-\eta_\kappa)}$. For weak $V_{cr.}$, our
predictions will apply over a wide range of length scales
$L_{z,\perp}$ satisfying $\xi_{cr.}^{z,\perp} \gg L_{z,\perp} \gg
\xi_{NL}^{z,\perp}$.  On longer length scales the SV solid elasticity
will cross over to the conventional ``tension'' elasticity of vortex
lattices\cite{ReviewsSC}, exhibiting a {\em linear} relation $\delta B
\propto H$, for $H<H_{cr.}=V_{cr.}/B(0)$.

%Now, of course, in a real crystalline ferromagnet, there are always
%crystalline symmetry breaking fields, which pick out a preferred
%direction for the magnetization, thereby explicitly breaking the
%rotation invariance.  These lead to a non-zero tilt modulus, i.e., a
%term ${1\over2}V_x|\partial_z{\bf u}|^2$ in the Hamiltonian, which
%cuts off the anomalous elasticity even in the absence of an external
%field ${\bf H}$. Consequently, we expect the predictions of our
%rotationally invariant theory to break down beyond a crossover length
%scale, $\xi_a^z=({\kappa_0 \over V_x})^{1 \over \delta} (\xi_{NL}^z
%)^{-\eta_{\kappa} \over \delta}$.  If the crystal symmetry breaking
%fields are weak, $V_x$ will be small, and our anomalous elastic theory
%will be valid for a wide range of length scales $L_{z,\perp}$
%satisfying $\xi_a^{z,\perp} \gg L_{z,\perp} \gg \xi_{NL}^{z,\perp}$.
%For longer length scales the soft anisotropic elasticity of the SV
%lattice will cross over to the conventional ``tension'' elasticity of
%ordinary, random field, vortex lattices.  This crossover in length
%scales manifests itself in a crossover applied magnetic field $H_c$,
%below which the non-Hookean law Eq.\ref{nonhook} breaks down, to be
%replaced by a conventional, Hookean relation $\delta B \propto H$, as
%illustrated in Fig.\ref{BvsH}. The value of the crossover field $H_c$
%is proportional to the crystal symmetry breaking field $V_x$.

%The presence of the crystal symmetry breaking fields also make it
%experimentally possible to apply the external magnetic field {\em
%opposite} to the direction of the spontaneous magnetization. While the
%true ground state clearly has ``spontaneous '' magnetization {\em
%parallel} to the externally applied field, this anti-parallel
%configuration is naively metastable if $|{\bf H}| < V_{cr.}$. Writing
%${\bf H} = -H {\bf\hat{z}}$, we readily see that, when $H$ reaches the
%metastability limit $V_x$, the ${1\over2}V_x|\partial_z{\bf u}|^2$
%term is exactly canceled by the $-{1\over2}H|\partial_z{\bf u}|^2$
%term arising from the ${\bf H} \cdot {\bf M}$ interaction. Thus, at
%precisely this metastable limit, the flux lattice becomes ``soft'' in
%exactly the sense we have been discussing throughout the
%paper. Therefore, we expect anomalous elasticity out to {\em
%arbitrarily long} length scales at this metastable point, {\em even}
%in the presence of crystal symmetry breaking fields.

The crystal anisotropy also leads to metastability for
$|H|< H_{cr.}$, thereby allowing experimental studies of $\bf H$
applied in the direction {\em opposite} to that of $\bf M$. In the
limit $|H|\rightarrow H_{cr.}$, the vortex solid becomes ``soft''
again, with the crystal-field generated tilt modulus vanishing as
$H_{cr.}-|H|$, and the system exhibiting anomalous elasticity out to
arbitrary length scale at this finely field-tuned point. However,
because the rotational symmetry is nevertheless broken (with
restoration only at the quadratic $(\partial_z {\bf u})^2$ level), we expect
that the universality class and therefore the anomalous exponents will be
distinct from that of the CEG studied here.\cite{unpublished}

As argued in the case of vanishing crystal anisotropy, here too, the
anomalous elasticity leads to a non-trivial field dependence of the
Young's modulus.  We find $Y(H)\propto(H_{cr.}-|H|)^{\beta'}$, with
$\beta'>0$ a universal exponent analogous to, but distinct from the
$\beta$ of the CEG. However, in contrast to the rotationally invariant
case, here the flux density $\delta
B(H)=H/Y(H)=H(H_{cr}-|H|)^{-\beta'}$ diverges as $|H|$ approaches
$H_{cr.}$ from below. This signals a breakdown of the analysis, which
assumes $\delta B(H)\ll B(0)$, and suggests that the metastability
limit occurs at fields $H_{ms}$ determined by $\delta B(H_{ms})\approx
B(0)$.
%, and is therefore far
%below $H_{cr.}$ predicted by the {\em harmonic} theory.

%This exact anomalous elasticity is probably {\em not}, however,
%controlled by the (CEG) fixed point we have
%been discussing up to now. The reason for this is that the CEG fixed
%point described a model with {\em exact} rotation invariance, to {\em
%all} orders in the rotation angle $\theta$.  In contrast, at the
%metastable fixed point of a model with crystal symmetry breaking
%fields, we have simply {\em tuned} $H$ to cancel off the leading order
%in $\theta$ (i.e., the $|\partial_z{\bf u}|^2$) term in the
%Hamiltonian. The full model however, is {\em not} rotation invariant
%to all orders in $\theta$, since, for {\em large} $\theta$, the
%crystal symmetry breaking field has a different dependence on $\theta$
%than the ${\bf H} \cdot {\bf M}$ term (e.g., $\cos(\theta)$ vs.
%$\cos(4 \theta)$ for an underlying cubic crystal). This lower symmetry
%of the ``metastable limit'' model allows, e.g., the coefficients of
%the cubic $(\bbox{\nabla}\cdot{\bf u})|\partial_z{\bf u}|^2$ and
%quartic $|\partial_z{\bf u}|^4$ nonlinearities in the Hamiltonian to
%be {\em different} from one another and from those of the quadratic
%$|\bbox{\nabla}{\bf u}|^2$ terms. This should be contrasted with the
%CBG Hamiltonian, in which these cubic and quartic terms arise {\em
%only} from the square of the nonlinear strain tensor $u_{ij}$, and
%must therefore have the same coefficients (up to known factors of
%order unity) as each other and the quadratic terms.
%
%The departure of these cubic and quartic coefficients away from
%equality with the quadratic terms could be a relevant perturbation
%{\em away} from the rotationally invariant fixed point to a new
%non-rotationally invariant fixed point. Such a new fixed point (the
%existence of which we are currently investigating \cite{unpublished})
%would exhibit anomalous elasticity with {\em different} universal
%exponents than those given above for the rotationally invariant
%problem.

%As in the rotationally invariant problem, the anomalous elasticity at
%the metastable limit leads to non-Hookean elasticity, which, in turn,
%implies a nonlinear $\delta B(H)$ relation. Now, however, the
%situation is {\em qualitatively} quite different. In the rotationally
%{\em invariant} problem, the Young's modulus $Y(H)$ ($\propto
%H^{\beta}$, with $\beta < 1$) vanished as $H$ went to {\em zero},
%which was {\em precisely} the point at which the compressive stress
%($\propto H$) also vanished. As a result, $\delta B (H) \propto H/Y(H)
%\propto H^{1-\beta} \rightarrow 0$ as $H \rightarrow 0$, despite the
%vanishing of $Y(H)$ as $\rightarrow 0$.

%In the presence of the crystal symmetry breaking field $V_x$, however,
%the Young's modulus $Y(H)$ vanishes {\em not} as $H \rightarrow 0$,
%but as $H \rightarrow H_c= V_x$, (remembering that we have written
%${\bf H} = -H {\bf\hat{z}}$) because $H_c$ is the value of $H$ at
%which the Hamiltonian becomes soft and, hence, anomalously elastic.
%At this value of $H$, the compressive stress is {\em non}-zero (since
%$H$ is). The vanishing of the Young's modulus now leads to a {\em
%divergence} of $\delta B$ as $H \rightarrow H_c$ of the form
%
%\begin{equation}
%\delta B(H) \propto {-H \over Y(H)} \propto {-H \over (H_c-H)^{\beta'}} \propto {-H_c \over (H_c-H)^{\beta'}},
%\label{newstressstrain}
%\end{equation}
%
%where the last proportionality applies close to $H=H_c$, and the
%universal exponent $\beta' \neq \beta$, the analogous exponent for the
%rotationally invariant problem, since it is defined by a new (and as
%yet unanalyzed) fixed point.

%Clearly, the calculation just described will not remain valid all the
%way down to $H=H_c$, since it was predicated on an expansion for {\em
%small} departures from the $H=0$ ground state, where $B=B(H=0)$.
%Indeed, Eq.\ref{newstressstrain}, predicts that for some finite
%departure of $H$ from its critical value $H_c$, $|\delta B|$ actually
%becomes bigger than $B(H=0)$. Since $\delta B <0$, this would imply
%that the spins had flipped. Obviously, beyond this point (if not
%before), all metastability is lost, and the system spontaneously seeks
%out its true, flipped ground state. Nonetheless, if $H_c \ll B(H=0)$,
%it will be possible to get close enough to $H=H_c$ for
%Eq.\ref{newstressstrain} to hold {\em without} reaching the regime
%just described in which $|\delta B| \sim (B(H=0))$ and our theory
%breaks down. Hence, in a system with weak crystal symmetry breaking
%fields, it should be possible to test Eq.\ref{newstressstrain}, and,
%hence, determine the exponent $\beta'$.  We hope to have a theoretical
%calculation of this exponent soon\cite{unpublished}.

Finally, we note that ErNi$_2$B$_2$C has a strong uniaxial anisotropy,
with $\bf M$ preferentially lying in a nearly isotropic xz-plane. The
appropriate model in this case resembles that given by
Eq.\ref{Helastic}, except that in the strain tensor $u_{ij}$,
$u_i\rightarrow u_x$, and $u_y$ has a conventional quadratic, tension
elasticity. A similar model has recently been considered by us in the
context of the columnar phase of discotics confined in an anisotropic
random medium, such as, e.g., strained aerogel\cite{SRTcolumnar}. The
resulting ``$m=1$ elastic glass'' phase is also topologically stable
with anomalous elasticity, and has additional interesting features,
most notably, that it exhibits short-ranged translational order in one
direction ($x$) and {\em quasi}-long-ranged translational order in the
other ($y$), a prediction that should be testable via neutron
scattering and decoration experiments. We conclude by noting that the
``soft'' elasticity of the SV solid should also manifest itself in
many other physical observables, most notably in dynamics, exhibiting
novel current-voltage characteristics and enhanced critical currents,
sensitive to and with nontrivial dependence on $H$.

We thank Anton Andreev and Tom Lubensky for discussions and
acknowledge financial support by the NSF through DMR99-80123 (JT,KS),
DMR96-25111, MRSEC DMR-9809555, and the A.P. Sloan and The
David and Lucile Packard Foundations (LR,ME).
%
\vspace*{-0.5cm}
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\vspace*{-1.6cm}
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%\bibitem{footnote1} Such $q$'s need not be very large, of course,
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%\bibitem{footnote2} This assumes the absence of crystal symmetry breaking 
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%(non-crystalline) ferromagnetic superconductor (which is not forbidden
%by any fundamental principle known to us) exists, there would be no
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%\bibitem{membranes} J.A.Aronovitz and T.C.Lubensky, Phys. Rev. Lett.
%{\bf 60}, 2634 (1988); P. Le Doussal and L. Radzihovsky, Phys. Rev. Lett.
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%\bibitem{ClarkMeyer} N.C. Clark and R. B. Meyer,
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%\bibitem{GP} G. Grinstein, R. A. Pelcovits, Phys. Rev. Lett. {\bf 47}, 
%856 (1981); L. Golubovic, Z. G. Wang, ibid., {\bf 69}, 2535 (1992).

\end{references} 

% many open questions most interesting is dynamics; expect depinning 
% transition to be different universality class and enhancement of Et.
%
\end{multicols} 
\end{document}

cond-mat0103204.extracted_bib
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%
\bibitem{footnote1} Such $q$'s need not be very large, of course,
%since the magnetic penetration depth $\lambda_L$ itself is frequently 
%quite large. However, at sufficiently small $q$, {\em conventional}
%lattices always exhibit {\em conventional} elasticity (i.e., 
%wavevector independent elastic constants). 

%
\bibitem{footnote2} This assumes the absence of crystal symmetry breaking 
%fields, which we discuss later in the paper. If an {\em amorphous}
%(non-crystalline) ferromagnetic superconductor (which is not forbidden
%by any fundamental principle known to us) exists, there would be no
%such symmetry breaking fields and all of our predictions would persist
%all the way down to ${\bf q}= 0$ and ${\bf H}= 0$.

%
\bibitem{errorbar} The uncertainty in the precise numerical value of this
%exponent arises because we have only calculated it to first order in an 
%expansion in $\epsilon \equiv 7/2 - d$, where $d$ is the dimension of space. 

%
\bibitem{comment} Here we do not consider the putative 
%Larkin-Ovchinnikov-Fulde-Ferrel inhomogeneous superconducting states.


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%
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%
\bibitem{membranes} J.A.Aronovitz and T.C.Lubensky, Phys. Rev. Lett.
%{\bf 60}, 2634 (1988); P. Le Doussal and L. Radzihovsky, Phys. Rev. Lett.
%{\bf 69}, 1209 (1992).

%
\bibitem{ClarkMeyer} N.C. Clark and R. B. Meyer,
%Appl. Phys. Lett. {\bf 22}, 493 (1973).

%
\bibitem{GP} G. Grinstein, R. A. Pelcovits, Phys. Rev. Lett. {\bf 47}, 
%856 (1981); L. Golubovic, Z. G. Wang, ibid., {\bf 69}, 2535 (1992).

