Fermionic description of spin-gap states of antiferromagnetic Heisenberg ladders\\ in a magnetic field
Hiromitsu Hori and Shoji Yamamoto
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\begin{document}
\draft
\preprint{28 February 2002}
\title{Fermionic description of spin-gap states of antiferromagnetic
       Heisenberg ladders\\
       in a magnetic field}
\author{Hiromitsu Hori and Shoji Yamamoto}
\address
{Department of Physics, Okayama University,
 Tsushima, Okayama 700-8530, Japan}

%\date{Received \hspace{6cm}}
\date{Received 28 February 2002}
\maketitle
%\begin{abstract}
%\end{abstract}
\pacs{PACS numbers: 75.10.Jm, 75.60.Ej}
\narrowtext

   Spin gaps$-$the energy gaps in magnetic excitation spectra$-$of
low-dimensional Heisenberg antiferromagnets have been attracting
considerable interest in recent years.
This fascinating subject was initiated by Haldane \cite{H1153},
renewed at the opportunity of high-temperature superconductivity
being discovered \cite{B188}, and further developed via the
synthesis of ladder materials \cite{J219,H230}.
Dagotto, Riera and Scalapino \cite{D5744} pioneeringly pointed out a
possible electronic mechanism for the formation of spin gaps in
two-leg ladders.
Following investigations \cite{D618} revealed that assembling chains
into ladders, the crossover between one and two dimensions is far
from smooth$-$Heisenberg ladders with an even number of legs have a
spin gap, while those with an odd number of legs have no gaps.

   Spin ladders in a magnetic field provide further interesting
topics.
Their ground-state magnetization curves have extensively been
studied \cite{C5126,C6241,C1768,T396} in an attempt to find {\it
quantized plateaux}.
Spin-$S$ $L$-leg Heisenberg ladders
\begin{equation}
   {\cal H}
   =\sum_{n=1}^N
    \left(
     J\sum_{l=1}^{L}
      \mbox{\boldmath$S$}_{n,l}\cdot\mbox{\boldmath$S$}_{n+1,l}
    +J'\sum_{l=1}^{L-1}
      \mbox{\boldmath$S$}_{n,l}\cdot\mbox{\boldmath$S$}_{n,l+1}
    \right)\,,
   \label{E:H}
\end{equation}
may exhibit magnetization plateaux at $M/M_{\rm sat}=m/LS$ provided
$LS-m\in\mbox{\boldmath Z}$ \cite{C5126,O1984,T103}, where $M$ and
$M_{\rm sat}$ are the magnetization and its saturated value,
respectively.
Since coupled-chain materials are likely to be systematically
obtained in the case of spin $\frac{1}{2}$ \cite{A3463}, most of
numerical efforts are being devoted to $S=\frac{1}{2}$ ladders.
Then, without any bond polymerization, the most tractable system of
our interest is the three-leg ladder.
Lanczos diagonalization of finite systems \cite{C5126}, a
series-expansion technique \cite{C6241} and density-matrix
renormalization-group calculations \cite{T396} are all in agreement
to support the existence of the plateau at $M/M_{\rm sat}=1/3$ for
strong interchain coupling $J'\agt J$.
As for more-than-three-leg ladders, the existence of plateaux as well
as their surviving region is still left to be verified.

   Spin-$\frac{1}{2}$ two- and three-leg ladders are indeed realized
in layer materials Sr$_{n-1}$Cu$_{n+1}$O$_{2n}$ ($n=3,5,\cdots$)
\cite{H230,A3463}, while four- and five-leg ones in
La$_{4+4n}$Cu$_{8+2n}$O$_{14+8n}$ ($n=2,3,\cdots$) \cite{B}.
It is unfortunate that nevertheless large degrees of freedom prevent
us from making numerical access to multi-leg ladders.
In such circumstances, we propose a systematic approach to Heisenberg
ladders in order to describe their spin-gap states in a field.
Employing the Jordan-Wigner transformation on a unique path and then
making a mean-field treatment of the fermionic Hamiltonian, we
semiquantitatively visualize the appearance of plateaux and estimate
the corresponding critical interactions $J'_{\rm c}$ at an
arbitrary number of legs, $L$.

   It is along a snake-like path,
$(n,l)
=(1,1)\rightarrow(1,2)\rightarrow\cdots\rightarrow(1,L)\rightarrow
 (2,L)\rightarrow(2,L-1)\rightarrow\cdots\rightarrow(2,1)\rightarrow
 (3,1)\rightarrow\cdots$,
that we define spinless fermions.
This elaborately ordered path was first proposed by Dai and Su
\cite{D964} and turned out to describe the spin gap as a function of
$L$ much better than the naively ordered path \cite{A6233}, which is
usually employed.
When we introduce renumbered spin operators
$\widetilde{\mbox{\boldmath$S$}}_{n,l}=\mbox{\boldmath$S$}_{n,l}$
($\mbox{\boldmath$S$}_{n,L-l+1}$) for an odd (even) $n$, the spinless
fermions are created as
$c_{n,l}^\dagger=\widetilde{S}_{n,l}^+
 {\rm exp}[-{\rm i}\pi
 (\sum_{i=1}^{n-1}\sum_{j=1}^L
   \widetilde{S}_{i,j}^+\widetilde{S}_{i,j}^-
 +\sum_{j=1}^{l-1}
   \widetilde{S}_{n,j}^+\widetilde{S}_{n,j}^-)]$.
Making a standard mean-field treatment \cite{D964} of the fermionic
Hamiltonian and assuming the spatial homogeneity
$\langle\widetilde{S}_{n,l}^z\rangle
 =\langle c_{n,l}^\dagger c_{n,l}\rangle-1/2=M/LN$, we obtain
\begin{eqnarray}
   &&
   {\cal H}
   =\frac{M}{LN}\sum_k\sum_{l=1}^L
    \Bigl[
     2J+(2-\delta_{l,1}-\delta_{l,L})J'
    \Bigr]
    c_{k,l}^\dagger c_{k,l}
   \nonumber \\
   &&\qquad
   +\frac{J'}{2}\sum_k\sum_{l=1}^{L-1}
    \left(
     c_{k,l}^\dagger c_{k,l+1}+c_{k,l+1}^\dagger c_{k,l}
    \right)
   \nonumber \\
   &&\qquad
   +\frac{J}{2}\sum_k\sum_{l=1}^L
    \Bigl[
     {\rm e}^{-{\rm i}\pi(L-l)(2M/LN+1)}
     {\rm e}^{-{\rm i}k}
   \nonumber \\
   &&\qquad\quad
    +{\rm e}^{ {\rm i}\pi(l-1)(2M/LN+1)}
     {\rm e}^{ {\rm i}k}
    \Bigr]
    c_{k,l}^\dagger c_{k,L-l+1}
   \nonumber \\
   &&\qquad
   -\Bigl(J+\frac{L-1}{L}J'\Bigr)M\Bigl(1+\frac{M}{LN}\Bigr)\,,
   \label{E:MFH}
\end{eqnarray}
where
$c_{k,l}^\dagger=N^{-1/2}\sum_n{\rm e}^{-{\rm i}kn}c_{n,l}^\dagger$.
The effective Hamiltonian (\ref{E:MFH}), together with the Zeeman
term $-H\sum_k\sum_{l=1}^L(c_{k,l}^\dagger c_{k,l}-1/2)$, is
numerically diagonalized, adopting the open boundary condition along
rungs, while taking the thermodynamic limit under the periodic
boundary condition along legs.

   In order to verify the reliability of the present approach, we
plot in Fig. \ref{F:Hc} the thus-obtained spin gaps ($H_{{\rm c}1}$)
and saturation fields ($H_{{\rm c}2}$), the former of which are
compared with highly precise numerical estimates \cite{W886}, whereas
the latter of which with the exact solutions
$H_{{\rm c}2}=2J+J'[1+\cos(\pi/L)]$.
Although the present scheme somewhat overestimates $H_{{\rm c}1}$
with increasing $L$, it correctly tells whether the gap survives or
not.
As for $H_{{\rm c}2}$, the present calculation can be regarded as
exact.
We further show in Fig. \ref{F:M} typical calculations of the
ground-state magnetization.
All the plateaux satisfying the criterion
$L/2-M/N\in\mbox{\boldmath Z}$ indeed appear with
increasing $J'$.
Mean-field approaches generally underestimate quantum fluctuations
and therefore necessarily overestimate the magnetization.
Allow us, however, to stress that the length of a plateau is still
well describable in our treatment, which is essential to estimate
the lower boundaries of the plateau-surviving region.
\vspace*{-40mm}
\begin{figure}
\centerline
{\mbox{\psfig{figure=Fig1.eps,width=99mm,angle=0}}}
\vspace*{2mm}
\caption{Spin gaps $H_{\rm c1}$ (a) and saturation fields
         $H_{\rm c2}$ (b) for the $L$-leg ladders with $J'=J$, where
         the present culculations ($\circ$) are compared with
         density-matrix renormalization-group estimates [17]
         ($\times$) in (a) and with the exact values ($\times$) in
         (b).}
\label{F:Hc}
\vspace*{-36mm}
\centerline
{\mbox{\psfig{figure=Fig2.eps,width=99mm,angle=0}}}
\vspace*{2mm}
\caption{Ground-state magnetization curves of the three- and five-leg
         ladders with $J'=3J$ (a) and $J'=5J$ (b), respectively.
         Numerical-diagonalization results [7] are also shown by
         a dotted line in (a).}
\label{F:M}
\end{figure}

   Now we explore the main issue in Fig. \ref{F:J'c}.
The critical value $J'_{\rm c}=1.04\pm 0.01$ for $L=3$ is in good
agreement with the previous estimate $J'_{\rm c}\simeq J$
\cite{C6241} obtained through the series expansion from the
strong-rung-coupling limit $J/J'\rightarrow 0$.
$J'_{\rm c}$ appears to increase with $L$, where even- and odd-$L$
ladders may form distinguishable series of their own.
Here is a conclusive remark:
Among possible nontrivial quantized magnetizations,
$M/M_{\rm sat}=1-2/L,1-4/L,\cdots,2/L\,(1/L)$,
{\it the inner plateaux are easier to induce}.
The plateaux at the end-value magnetizations
$M/M_{\rm sat}=1-2/L,2/L\,(1/L)$ are generally induced with larger
$J'$ than those at the inner-value magnetizations are.
Practical observation of multi-plateau magnetization curves may
less be feasible with increasing $L$.
At the least, however, an available five-leg ladder material
(La$_2$CuO$_4$)$_3$La$_2$Cu$_4$O$_7$ \cite{B} encourages us
to make theoretical explorations into the unique way from one- to
two-dimensional antiferromagnets in a field.
It is also important, further numerical verification of the plateau
at $M/M_{\rm sat}=3/5$ more surviving than that at
$M/M_{\rm sat}=1/5$ for $L=5$.
\vspace*{-46mm}
\begin{figure}
\centerline
{\mbox{\psfig{figure=Fig3.eps,width=99mm,angle=0}}}
\vspace*{2mm}
\caption{Critical interchain interaction to induce magnetization
         plateaux, $J'_{\rm c}$, for the $L$-leg ladders, where the
         corresponding magnetization values $M/M_{\rm sat}$ are
         indicated beside symbols.}
\label{F:J'c}
\end{figure}

   We are grateful to Prof. M. Takahasi for useful comments.
This work was supported by the Japanese Ministry of Education,
Science, Sports and Culture, and by the Sumitomo Foundation.

\begin{references}

\bibitem{H1153}
   F. D. M. Haldane:
      Phys. Rev. Lett. {\bf 50} (1983) 1153.

\bibitem{B188}
   J. G. Bednorz and K. A. M\"uller:
      Z. Phys. B {\bf 64} (1986) 188.

\bibitem{J219}
   D. C. Johnston, J. W. Johnson, D. P. Goshorn and A. J. Jacobson:
      Phys. Rev. B {\bf 35} (1987) 219.

\bibitem{H230}
   Z. Hiroi, M. Azuma, M. Takano and Y. Bando:
      J. Solid State Chem. {\bf 95} (1991) 230.

\bibitem{D5744}
   E. Dagotto, J. Riera and D. Scalapino:
      Phys. Rev. B {\bf 45} (1992) R5744.

\bibitem{D618}
   E. Dagotto and T. M. Rice:
      Science {\bf 271} (1996) 618.

\bibitem{C5126}
   D. C. Cabra, A. Honecker and P. Pujol:
      Phys. Rev. Lett. {\bf 79} (1997) 5126.

\bibitem{C6241}
   D. C. Cabra, A. Honecker and P. Pujol:
      Phys. Rev. B {\bf 58} (1998) 6241.

\bibitem{C1768}
   D. C. Cabra and M. D. Grynberg:
      Phys. Rev. Lett. {\bf 82} (1999) 1768.

\bibitem{T396}
   K. Tandon, S. Lal, S. K. Pati, S. Ramasesha and D. Sen:
      Phys. Rev. B {\bf 59} (1999) 396.

\bibitem{O1984}
   M. Oshikawa, M. Yamanaka and I. Affleck:
      Phys. Rev. Lett. {\bf 78} (1997) 1984.

\bibitem{T103}
   K. Totsuka:
      Phys. Lett. A {\bf 228} (1997) 103.

\bibitem{A3463}
   M. Azuma, Z. Hiroi, M. Takano, K. Ishida and Y. Kitaoka:
      Phys. Rev. Lett. {\bf 73} (1994) 3463.

\bibitem{B}
   B. Batlog and R. Cava,
      unpublished.

\bibitem{D964}
   X. Dai and Z. Su:
      Phys. Rev. B {\bf 57} (1998) 964.

\bibitem{A6233}
   M. Azzouz, L. Chen and S. Moukouri:
      Phys. Rev. B {\bf 50} (1994) 6233.

\bibitem{W886}
   S. R. White, R. M. Noack and D. J. Scalapino:
      Phys. Rev. Lett. {\bf 73} (1994) 886.

\end{references}

\widetext
\end{document}

cond-mat0206554.extracted_bib
\bibitem{H1153}
   F. D. M. Haldane:
      Phys. Rev. Lett. {\bf 50} (1983) 1153.


\bibitem{B188}
   J. G. Bednorz and K. A. M\"uller:
      Z. Phys. B {\bf 64} (1986) 188.


\bibitem{J219}
   D. C. Johnston, J. W. Johnson, D. P. Goshorn and A. J. Jacobson:
      Phys. Rev. B {\bf 35} (1987) 219.


\bibitem{H230}
   Z. Hiroi, M. Azuma, M. Takano and Y. Bando:
      J. Solid State Chem. {\bf 95} (1991) 230.


\bibitem{D5744}
   E. Dagotto, J. Riera and D. Scalapino:
      Phys. Rev. B {\bf 45} (1992) R5744.


\bibitem{D618}
   E. Dagotto and T. M. Rice:
      Science {\bf 271} (1996) 618.


\bibitem{C5126}
   D. C. Cabra, A. Honecker and P. Pujol:
      Phys. Rev. Lett. {\bf 79} (1997) 5126.


\bibitem{C6241}
   D. C. Cabra, A. Honecker and P. Pujol:
      Phys. Rev. B {\bf 58} (1998) 6241.


\bibitem{C1768}
   D. C. Cabra and M. D. Grynberg:
      Phys. Rev. Lett. {\bf 82} (1999) 1768.


\bibitem{T396}
   K. Tandon, S. Lal, S. K. Pati, S. Ramasesha and D. Sen:
      Phys. Rev. B {\bf 59} (1999) 396.


\bibitem{O1984}
   M. Oshikawa, M. Yamanaka and I. Affleck:
      Phys. Rev. Lett. {\bf 78} (1997) 1984.


\bibitem{T103}
   K. Totsuka:
      Phys. Lett. A {\bf 228} (1997) 103.


\bibitem{A3463}
   M. Azuma, Z. Hiroi, M. Takano, K. Ishida and Y. Kitaoka:
      Phys. Rev. Lett. {\bf 73} (1994) 3463.


\bibitem{B}
   B. Batlog and R. Cava,
      unpublished.


\bibitem{D964}
   X. Dai and Z. Su:
      Phys. Rev. B {\bf 57} (1998) 964.


\bibitem{A6233}
   M. Azzouz, L. Chen and S. Moukouri:
      Phys. Rev. B {\bf 50} (1994) 6233.


\bibitem{W886}
   S. R. White, R. M. Noack and D. J. Scalapino:
      Phys. Rev. Lett. {\bf 73} (1994) 886.

