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\begin{document}
\noindent
{\large \bf Comment on ``Competition between helimagnetism 
and commensurate quantum spin correlations in LiCu$_2$O$_2$''}

\vspace{0.1cm}


In a neutron scattering investigation of LiCu$_2$O$_2$ 
Masuda {\it et al.}\cite{masuda}  
reported the direct 
observation of an incommensurate (IC) magnetic structure
below 22 K. Though this study confirms 
similar indirect IC observations\cite{zvyagin,choi,gippius} 
pointing to the presence of frustrated magnetic interactions
they  deserve now more detailed work to elucidate 
the microscopic origin of that frustration. 
We will show that the adopted antiferromagnetic (afm) 
%$J_1$-$J_2$ 
double-chain 
(DC) Heisenberg model \cite{masuda,zvyagin,choi} 
(Fig.\ 1a) suggests
an unrealistic frustation scenario for LiCu$_2$O$_2$.
It should 
%to . 
be replaced by a ferromagnetic(fm)-afm
frustrated {\it single}-chain model (Fig.\ 1b).\cite{gippius,rempitch}
Based on electronic structure (LDA) and  cluster calculations as well as 
a phenomenolgical analysis of magnetic data, we arrive 
at opposite estimates 
compared with  Masuda {\it et al.}  
\cite{masuda} with respect to the magnitude/sign of the 
main couplings.
%$J_{DC}$, $J_1$ and $J_2$ 
The controversy concerns the following main points: 

(i) Most importantly, the signs of the n.n.\ inchain exchange $J_1$ 
are opposite: afm +1.68 meV in Ref.\ 1 vs.\ fm  
-11$\pm$ 3meV in our analysis.\cite{screening,remarkmicro}
For CuO$_2$ chains with   
Cu-O-Cu bond angles $\gamma$ near 94$^{\circ}$
as in Li$_2$CuO$_2$ (with fm inchain order), 
according to the Kanamori-Goodenough rule 
and to the fm direct Cu 3$d$-O $2p$ exchange,  
a total  {\it fm} $J_{1} <$0 can be expected.
However, its magnitude is sensitive to the
competition with a $\gamma$-dependent
afm contribution to $J_1$.\cite{gippius} Hence, to simplified
distance-only based suggestions \cite{masuda} that
$\mid$$J_1$$\mid$$ \gg J_2$, do note hold here.   

(ii) We found the  nnn inchain coupling $J_2$ afm (generic for
%edge-shared 
CuO$_2$ chains), i.e.\ frustrated with fm $J_1$ and any
$J_{DC}$. Moreover we estimated 
 $J_2 $$\sim$$\mid J_1\mid$. 
However, the important source of frustration 
$J_2$ is ignored in Ref.\ 1.


(iii) A dominant interchain coupling $J_{DC} \approx$ 5.8 meV
is claimed by Masuda {\it et al.}
%based on their afm DC model, 
whereas from our LDA analysis a tiny $J_{DC}\sim$ 
0.5 meV only follows. It can be neglected to first approximation.
The weak $J_{DC}$ is caused 
%among them.
by  
the tiny interchain (DC) overlap of the predominant
%standard
O 2$p_{x,y}$ orbitals of the
CuO$_4$ plaquettes forming the CuO$_2$ chains.
%Anyhow, even for larger $J_{DC}$ values, fm $J_1$ and afm $J_2$
%our scenario remains valid.
Note, that if $J_{1}$$<$0,  the DC 
is {\it  unfrustrated} for $J_{2}$=0.






 



 With $J_1$$\sim$-11meV,  we explain also
the measured magnetic susceptibility $\chi (T)$ (Fig.\ 1) and
the afm
Curie-Weiss constants $\Theta_{CW}$. \cite{zvyagin} 
Approximating the main couplings between two nn-chains
$J_{\perp}$ in the ab-plane  
in mean-field  theory and the  
inchain couplings
exactly in large clusters, we derive for the bulk value
%\begin{equation}
$\Theta_{CW}\approx \Theta_{CW,1D} -zJ_{\perp}/4k_B,$ where
%\end{equation}
$z$=2 is the number of interchain n.n.'s. 
From  
cluster studies we obtained  an afm 
$\Theta_{CW,1D}
\approx$ -42K for single chains with
frustrating afm $J_{2}>$0 and
$\alpha$$ \equiv$$J_2/J_1$=-1.1. With $J_{\perp}$$\approx $5.7meV  
\cite{gippius} one arrives at  $\Theta_{CW}$= -75K 
close to experimental values 
$\approx$ -80 to -90K.\cite{zvyagin,remsign}
%For the closely related Li$_2$CuO$_2$ 
%($\alpha=$--0.7, $z$=16,  
%and  a fm $\Theta_{CW,1D}$$\approx$+2K from   
%$\chi(T)$-fit), 
%our model yields with 
%$\Theta_{CW}$$\sim $--38 K 
%in accord with experiment.\cite{klingeler} 


%Irrespective on the sign of $J_{nn}$ (see below),
Finally, we note that Masuda et al.\ \cite{masuda}
 argue that their propagation vector $\zeta$ would 
contradict our $J$ ratio:
$\alpha$=-1/(4$\cos (2\pi\zeta))$.
However, this simple expression is
valid for  single-chains with classical spins $s \gg 1$. 
%with classical large $S$ spins,
In our case with $s$=1/2 quantum fluctuations\cite{bursill},  
 interchain coupling\cite{remarkli}, and spin-anisotropy 
do affect $\alpha$ strongly. 



To conclude, the application of  the afm DC-model of Ref.\ 1 
to LiCu$_2$O$_2$ 
is not justified whereas the proposed frustrated
single-chain model with fm $J_1$ and afm $J_2$ couplings
is consistent with  the experimental data
and the generally accepted CuO$_2$-chain physics.




We thank the  DFG (J.M.) Emmy-Noether-program (H.R.)) and 
RFBR(04-03-32876, A.G.)
%, and INTAS (01-0654, A.M) 
for support.\\


\noindent
S.-L. Drechsler$^1$, J.\ M\'alek$^2$, 
J.\ Richter$^3$, A.S.\ Moskvin, A.A.\ Gippius, and H.\ Rosner\\ 
\indent 
{\small $^1$ Inst.\  f.\ Festk\"orpertheorie im IFW Dresden}\\
\indent
\small{01171 Dresden, Germany}\\
\indent
\small{ $^2$ Institute of Physics, ASCR, Prague, Czech Republic}\\
\indent
\small{ $^3$ Inst.\ f.\ Theor.\ Physik, Universit\"at Magdeburg, Germany.}
\noindent
\\
\vspace{0.1cm}
\noindent
%{\small Received: \today; published:}\\
\noindent
%{\small DOI:}\\
\noindent
\pacs{75.10.Pq,75.25.+z,75.30.Hx}
\vspace{-1.3cm}
\begin{references}
\vspace{-0.3cm}
\bibitem[1]{masuda} T.\ Masuda {\it et al.}, Phys.\ Rev.\ Lett.\ {\bf 92},
177201 (2004).
\bibitem[2]{zvyagin}S.\ Zvyagin {\it et al.}, Phys.\
Rev.\ B {\bf 66}, 064424 (2000).
\bibitem[3]{choi}K.-Y.\ Choi {\it et al.}, Phys.\
Rev.\ B {\bf 69}, 104421 (2004).
\bibitem[4]{gippius}  A.A.\ Gippius  {\it et al.}, Phys.\ Rev.\ B {\bf 70}, 
020406 (2004).
%, cond-mat/0312576.
\bibitem[5]{rempitch}
The observed propagation vector
$q=(0.5, \zeta =0.174, 0)$ measured in standard units
2$\pi /b$ of the crystal structure  
yields a 
%{\it acute} 
pitch angle $\Phi$=$\pm 2\pi \zeta +2\pi n$, 
$n$=0,1,2,..., within the "fm"  
interval $-\pi /2$$ \leq $$\Phi$$ \leq $$\pi /2$
\cite{bursill}. 
However, within the 
system of coordinates adopted in Ref.\ 1 
with a 
lattice constant of $b/2$ there are two solutions
$\Phi_{DC}$=$\pm \pi ( n \pm \zeta )$ with  "afm"
pitch angles for odd (even) $n$, 
respectively, depending on the sign
of $J_{DC}$. 
%Anyhow, for weak $J_{DC}$ 
%the helix formed on a single chain is hardly affected.
\bibitem[6]{screening}The error bars stems from uncertainties 
in the screening of bare fm interactions.\cite{gippius}  
%\bibitem[7]{remarkge}An exception from this rule 
% occurs for CuGeO$_3$ with specifically 
%large $\gamma \approx$ 99$^{\circ}$ and  large crystal 
%field effects.
\bibitem[7]{remarkmicro}
From mapping low-lying magnetic excitations of the 5-band 
Cu 3$d$ O 2$p$ extended Hubbard 
model (fitted
to spectroscopic data) onto excitations of 
the Heisenberg model.
%\bibitem[8]{klingeler}R.\ Klingeler, PHD-Thesis, RWTH Aachen (2003).
\bibitem[8]{remsign}Notice the "+"-sign convention of 
$\Theta_{CW}$ used in Ref.\ 2.
 \bibitem[9]{bursill}R.\ Bursill {\it et al.}, J.\ Phys.\ C,  
{\bf 7}, 8605 (1995), Fig.\ 2.
\bibitem[10]{remarkli}Strong interchain exchange may even
prevent a helix formation, e.g.\ as in Li$_2$CuO$_2$
with fm inchain order below 8K.

\end{references}
\vspace{-1cm}
\begin{figure}[b]
\begin{center}
\indent
\psfig{width=5.5cm,angle=0,figure=chi.ps}
\end{center}
\vspace{-0.7cm}
\caption[]{Susceptibility of Heisenberg rings
%$\chi (T)$ for spin-1/2 
 with $-J_1=J_2$=8.2 meV, $J_{DC}$=0. $N$=16 sites,
and Lande-Factors $g_L$= 2.24 and 2.0, respectively
(full lines) 
compared with experiment (Ref.\ 1; $\Box$
magnetic field {\bf H} 
$\parallel$ c; 
%$g_L$=2.23; 
$\circ $ {\bf H} 
$\parallel$ (a,b).  
%$g_L$=2.04. 
In the inset the DC scenario ((a),\cite{masuda})
is compared with the
%The two scenarios under debate are sketched in the inset;
%DC\cite{masuda} (a) vs.\ our 
single-chain one (b).
Thickness of lines symbolizes the coupling strength.
The empirical $J$-values are in accord with LDA  and
microscopic estimates \cite{remarkmicro}. Naturally, the finite 
cluster approach cannot describe the low-$T$ behavior of $\chi(T)$.
}
\label{susci}
\end{figure}
\end{document}
(ii) 
%T.\ Masuda {\it et al.} 
 The diffraction data 
are interpreted
within a "DC"-helix with spins confined 
to the (a,b) crystallographic
plane. Such a helix is rather 
unexpected in view of the 
measured  easy-axis-type (c-axis)
magnetic anisotropy.\cite{masuda,zvyagin} 
The 
rotational plane (e.g.\ (a,c)) should include the c-axis.
Furthermore, an (ab)-helix (ABH) is 
\textcolor{black}{inconsistent with our NMR data (Ref.\ 4, Fig.3)}. In 
conflict with these data,
within the ABH-model \textcolor{black}{ in  H $\parallel$ c geometry one
would observe only a single $^7$Li NMR line near $H$=20k{\O} 
since local magnetic fields $H_{loc}$ are always perpendicular 
to $H_{ext}$.\cite{remcon}
}
\textcolor{black}{
Additionally, their fit
with 23 nonequivalent magnetic Bragg peaks
ignores the O contributions present by 
symmetry in an IC CuO$_2$ chain.\cite{chung}} 
%Their ABH is based only on 
%Cu-form factor calculations.

%closer to the {\it ferromagnetic} (fm, $\zeta$=0) than to a simple afm 
%inchain ($\zeta$=0.5) order. 
Thus, their 
%reported 
$q$-value
%propagation vector 
is 
inconsistent\cite{bursill} with their proposed afm helix. 
Due to the used inproper relation between $\phi$  
%the pitch angle
and $q$,
%the propagation vector, 
the afm $\chi(T)$
descriptions 
%\cite{chi} 
are not convincingly supported by the neutron data. 

\bibitem[7]{chung}Most important in case of
 fm inchain order. See e.g.\ the related Li$_2$CuO$_2$ (resolved with 409 
nonequivalent Bragg peaks): 
S.\ Chung {\it et al.}, Phys.\
Rev.\ B {\bf 68}, 144410 (2003). 
%They cancel only in the period-2 afm inchain order.

\begin{table}[b]
\begin{tabular}{|l|c|c|c|c|c|}
\hline
[meV]&$y$&2$y$&$x$&$\tilde{x}$&$xy$\\ \hline
$|t|$&64&109&73&18&25\\
$J^{afm}$& \quad 4.1 \ (5.5)&11.9 (15.8)&7.1(5.3)&0.3(0.4)&0.6(0.8)\\
$J^{fm}$&-17.9 (-13.6)&-1.8(1.4)&-1.4(1.8)&-&-\\
$J^{eff}$&\ -13.9 (-8.1)&10.1(14.4)&3.5(5.7)&0.3(0.4)&0.6(0.8)\\ \hline
\end{tabular}
\caption{\label{tab} LDA hopping $t$ and exchange
integrals $J$ for Hubbard $U$ =4(3) eV. For notation
see Ref.\ 4 (y(x)- n.n.\ in(inter)chain, $\tilde{x}$-DC).
 The total 
%exchange integrals 
$J^{eff}$ 
consist of 
%differently calculated 
afm and fm
contributions: $J^{afm}$=$4t^2/U$ (superexchange) 
and $J^{fm}$ (direct exchange)
from Wannier-functions, respectively.
%The value of 
The bare $J^{fm}$'s are screened as $U$
by factors of 2.46(3.3), respectively.\cite{gippius}}
\end{table}

Thus,  we are left with a frustrated $J_1$-$J_2$-model but now with a
fm-afm {\it single-chain} 
%$J$-
interpretation.
% of the $'s.
cond-mat0411418.extracted_bib
\bibitem[1]{masuda} T.\ Masuda {\it et al.}, Phys.\ Rev.\ Lett.\ {\bf 92},
177201 (2004).

\bibitem[2]{zvyagin}S.\ Zvyagin {\it et al.}, Phys.\
Rev.\ B {\bf 66}, 064424 (2000).

\bibitem[3]{choi}K.-Y.\ Choi {\it et al.}, Phys.\
Rev.\ B {\bf 69}, 104421 (2004).

\bibitem[4]{gippius}  A.A.\ Gippius  {\it et al.}, Phys.\ Rev.\ B {\bf 70}, 
020406 (2004).
%, cond-mat/0312576.

\bibitem[5]{rempitch}
The observed propagation vector
$q=(0.5, \zeta =0.174, 0)$ measured in standard units
2$\pi /b$ of the crystal structure  
yields a 
%{\it acute} 
pitch angle $\Phi$=$\pm 2\pi \zeta +2\pi n$, 
$n$=0,1,2,..., within the "fm"  
interval $-\pi /2$$ \leq $$\Phi$$ \leq $$\pi /2$
\cite{bursill}. 
However, within the 
system of coordinates adopted in Ref.\ 1 
with a 
lattice constant of $b/2$ there are two solutions
$\Phi_{DC}$=$\pm \pi ( n \pm \zeta )$ with  "afm"
pitch angles for odd (even) $n$, 
respectively, depending on the sign
of $J_{DC}$. 
%Anyhow, for weak $J_{DC}$ 
%the helix formed on a single chain is hardly affected.

\bibitem[6]{screening}The error bars stems from uncertainties 
in the screening of bare fm interactions.\cite{gippius}  
%
\bibitem[7]{remarkge}An exception from this rule 
% occurs for CuGeO$_3$ with specifically 
%large $\gamma \approx$ 99$^{\circ}$ and  large crystal 
%field effects.

\bibitem[7]{remarkmicro}
From mapping low-lying magnetic excitations of the 5-band 
Cu 3$d$ O 2$p$ extended Hubbard 
model (fitted
to spectroscopic data) onto excitations of 
the Heisenberg model.
%
\bibitem[8]{klingeler}R.\ Klingeler, PHD-Thesis, RWTH Aachen (2003).

\bibitem[8]{remsign}Notice the "+"-sign convention of 
$\Theta_{CW}$ used in Ref.\ 2.
 
\bibitem[9]{bursill}R.\ Bursill {\it et al.}, J.\ Phys.\ C,  
{\bf 7}, 8605 (1995), Fig.\ 2.

\bibitem[10]{remarkli}Strong interchain exchange may even
prevent a helix formation, e.g.\ as in Li$_2$CuO$_2$
with fm inchain order below 8K.


\bibitem[7]{chung}Most important in case of
 fm inchain order. See e.g.\ the related Li$_2$CuO$_2$ (resolved with 409 
nonequivalent Bragg peaks): 
S.\ Chung {\it et al.}, Phys.\
Rev.\ B {\bf 68}, 144410 (2003). 
%They cancel only in the period-2 afm inchain order.

\begin{table}[b]
\begin{tabular}{|l|c|c|c|c|c|}
\hline
[meV]&$y$&2$y$&$x$&$\tilde{x}$&$xy$\\ \hline
$|t|$&64&109&73&18&25\\
$J^{afm}$& \quad 4.1 \ (5.5)&11.9 (15.8)&7.1(5.3)&0.3(0.4)&0.6(0.8)\\
$J^{fm}$&-17.9 (-13.6)&-1.8(1.4)&-1.4(1.8)&-&-\\
$J^{eff}$&\ -13.9 (-8.1)&10.1(14.4)&3.5(5.7)&0.3(0.4)&0.6(0.8)\\ \hline
