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arXiv:cond-mat/9908224 · Physical review. B, Condensed matter

Spin-1 ladder: A bosonization study

D. Allen, David Sénéchal

Published May 1, 2000• 31 citations
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Abstract

We construct a field-theoretic description of two coupled spin-1 Heisenberg chains, starting with the known representation of a single spin-1 chain in terms of Majorana fermions (or Ising models). After reexamining the bosonization rules for two Ising models, taking particular care of order and disorder operators, we obtain a bosonic description of the spin-1 ladder. From renormalization-group and mean-field arguments, we conclude that, for a small interchain coupling, the spin-1 ladder is approximately described by three decoupled, two-frequency sine-Gordon models. We then predict that, starting with decoupled chains, the spin gap decreases linearly with interchain coupling, in both the ferromagnetic and antiferromagnetic directions. Finally, we discuss the possibility of an incommensurate phase in the spin-1 zigzag chain.

DOI
10.1103/physrevb.61.12134

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We construct a field-theoretic description of two coupled spin-1 Heisenberg chains, starting with the known representation of a single spin-1 chain in terms of Majorana fermions (or Ising models). After reexamining the bosonization rules for two Ising models, taking particular care of order and disorder operators, we obtain a bosonic description of the spin-1 ladder. From renormalization-group and mean-field arguments, we conclude that, for a small interchain coupling, the spin-1 ladder is approximately described by three decoupled, two-frequency sine-Gordon models. We then predict that, starting with decoupled chains, the spin gap decreases linearly with interchain coupling, in both the ferromagnetic and antiferromagnetic directions. Finally, we discuss the possibility of an incommensurate phase in the spin-1 zigzag chain.
The spin-1/2 Heisenberg spin ladder model is studied by via a perturbation calculation and via bosonization. We mainly focus on the case of ferromagnetic interchain coupling in relation to the Haldane problem. Using the GKO coset construction of the conformal field theory, we bosonize the system in terms of the SU(2) (spin) sector and a residual Ising sector. The interchain interaction generically drives the system to strong-coupling massive phases (the Haldane phase or the dimerized phase). However, it is suggested that the criticality appears for particular choices of the couplings.
We investigate the phase diagram of two coupled quantum spin-1/2 chains, a square ladder, allowing for anisotropies both in the interchain and intrachain coupling, using the recently introduced density-matrix renormalization-group (DMRG) algorithm. We consider open chains up to a length of 2\ifmmode\times\else\texttimes\fi{}200, while retaining up to 120 states in the truncated basis of the Hilbert space. The phase diagram is discussed in connection with exact results and bosonization analysis. Numerically, we show that the crossover from the critical behavior of the uncoupled disordered spin-1/2 chain to a gapped quantum liquid sets in for infinitesimal interchain coupling, both for ferromagnetic and antiferromagnetic interchain coupling. The Haldane phase of the coupled chains is, even for small interchain couplings, generically described by the Affleck-Kennedy-Lieb-Tasaki model. We also discuss typical shortcomings of the DMRG for some choices of the interaction parameters. \textcopyright{} 1996 The American Physical Society.
We study a model of two weakly coupled isotropic spin-1/2 Heisenberg chains with an antiferromagnetic coupling along the chains (spin ladder). It is shown that the system always has a spectral gap and the lower lying excitations are triplets. For the case of identical chains the model in the continuous limit is shown to be equivalent to four decoupled noncritical Ising models with the Z2{\mathit{Z}}_{2}Z2​\ifmmode\times\else\texttimes\fi{}SU(2) symmetry. For this case we obtain the exact expressions for asymptotics of spin-spin correlation functions. It is shown that when the chains have different exchange integrals J1{\mathit{J}}_{1}J1​\ensuremath{\gg}J2{\mathit{J}}_{2}J2​ the spectrum at low energies is described by the O(3)-nonlinear \ensuremath{\sigma} model. We discuss the topological order parameter related to the gap formation and give a detailed description of the dynamical magnetic susceptibility. \textcopyright{} 1996 The American Physical Society.
The Heisenberg spin ladder is studied in the semiclassical limit, via a mapping to the nonlinear \ensuremath{\sigma} model. Different treatments are needed if the interchain coupling K is small, intermediate, or large. For intermediate coupling a single nonlinear \ensuremath{\sigma} model is used for the ladder. It predicts a spin gap for all nonzero values of K if the sum s+s\ifmmode \tilde{}\else \~{}\fi{} of the spins of the two chains is an integer, and no gap otherwise. For small K, a better treatment proceeds by coupling two nonlinear \ensuremath{\sigma} models, one for each chain. For integer s=s\ifmmode \tilde{}\else \~{}\fi{}, the saddle point approximation predicts a sharp drop in the gap as K increases from zero. A Monte Carlo simulation of a spin-1 ladder is presented which supports the analytical results.
We analyze the phase diagram of a system of spin-1/2 Heisenberg antiferromagnetic chains interacting through a zig-zag coupling, also called zig-zag ladders. Using bosonization techniques we study how a spin-gap or more generally plateaux in magnetization curves arise in different situations. While for coupled XXZ chains, one has to deal with a recently discovered chiral perturbation, the coupling term which is present for normal ladders is restored by an external magnetic field, dimerization or the presence of charge carriers. We then proceed with a numerical investigation of the phase diagram of two coupled Heisenberg chains in the presence of a magnetic field. Unusual behaviour is found for ferromagnetic coupled antiferromagnetic chains. Finally, for three (and more) legs one can choose different inequivalent types of coupling between the chains. We find that the three-leg ladder can exhibit a spin-gap and/or non-trivial plateaux in the magnetization curve whose appearance strongly depends on the choice of coupling.
The effect of antiferromagnetic interchain coupling in alternating spin-(1,1/2) chains is studied by means of spin-wave theory and density-matrix renormalization group (DMRG). In particular, two limiting cases are investigated, the two-leg ladder and its two-dimensional (2D) generalization. Results of the ground-state properties like energy, spin gap, magnetizations, and correlation functions are reported for the whole range of the interchain coupling ${J}_{\ensuremath{\perp}}.$ For the 2D case the spin-wave results predict a smooth dimensional crossover from 1D to 2D keeping the ground state always ordered. For the ladder system, the DMRG results show that any ${J}_{\ensuremath{\perp}}>0$ drives the system to a gapped ground state. Furthermore the behavior of the correlation functions closely resemble the uniform spin-1/2 ladder. For ${J}_{\ensuremath{\perp}}$ lower than 0.3, however, the gap behaves quadratically as $\ensuremath{\Delta}\ensuremath{\sim}{0.6J}_{\ensuremath{\perp}}^{2}.$ Finally, it is argued that the behavior of the spin gap for an arbitrary number of mixed coupled spin chains is analogous to that of the uniform spin-1/2 chains.