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International Journal of Modern Physics B

σ-MODEL ANALYSIS OF QUANTUM SPIN LADDERS

Olav F. Syljuåsen

Published Feb 20, 2000• 0 citations
Publisher

Abstract

The σ-model approach to quantum spin ladders, described here, allows one to analytically calculate correlation lengths and gaps for both even- and odd-legged ladders, at any temperature. Comparison with high precision quantum Monte Carlo simulations show that these analytic results are surprisingly accurate. A self-contained account of these analytic methods is presented.

DOI
10.1142/s0217979200000431

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The σ-model approach to quantum spin ladders, described here, allows one to analytically calculate correlation lengths and gaps for both even- and odd-legged ladders, at any temperature. Comparison with high precision quantum Monte Carlo simulations show that these analytic results are surprisingly accurate. A self-contained account of these analytic methods is presented.
Analytic expressions for the temperature dependences of the correlation length are given for antiferromagnetic spin- 12\frac{1}{2}21​ Heisenberg ladders using a finite-size nonlinear \ensuremath{\sigma} model approach. These calculations rely on identifying three successive crossover regimes as a function of temperature. In each of these regimes, precise and controlled approximations are formulated. The analytical results are found to be in excellent agreement with Monte Carlo simulations for the Heisenberg Hamiltonian.
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.
In order to shed light whether the `even-odd conjecture' (even numbers of legs will superconduct accompanied by a spin gap while odd ones do not) for correlated electrons in ladder systems, the pairing correlation is studied for the Hubbard model on a two- and three-leg ladders. We have employed both the weak-coupling renormalization group and the quantum Monte Carlo (QMC) method for strong interactions. For the two-leg Hubbard ladder, a systematic QMC (with a controlled level spacings) has detected an enhanced pairing correlation, which is consistent with the weak-coupling prediction. We also calculate the correlation functions in the three-leg Hubbard ladder and show that the weak-coupling study predicts the dominant superconductivity, which refutes the naive even-odd conjecture. A crucial point is a spin gap for only some of the multiple spin modes is enough to make the ladder superconduct with a pairing symmetry (d-like here) compatible with the gapped mode. A QMC study for the three-leg ladder endorses the enhanced pairing correlation.
Quantum Monte Carlo simulations provide one of the more powerful and versatile numerical approaches to condensed matter systems. However, their application to frustrated quantum spin models, in all relevant temperature regimes, is hamstrung by the infamous “sign problem.” Here we exploit the fact that the sign problem is basis-dependent. Recent studies have shown that passing to a dimer (two-site) basis eliminates the sign problem completely for a fully frustrated spin model on the two-leg ladder. We generalize this result to all partially frustrated two-leg spin-1/2 ladders, meaning those where the diagonal and leg couplings take any antiferromagnetic values. We find that, although the sign problem does reappear, it remains remarkably mild throughout the entire phase diagram. We explain this result and apply it to perform efficient quantum Monte Carlo simulations of frustrated ladders, obtaining accurate results for thermodynamic quantities such as the magnetic specific heat and susceptibility of ladders up to L = 200 L = 200 rungs (400 spins 1/2) and down to very low temperatures.
The temperature dependence of the uniform susceptibility and the ground-state energy of antiferromagnetic Heisenberg ladders with up to six legs has been calculated, using the Monte Carlo loop algorithm. The susceptibilities of even-leg ladders show spin gaps while those of odd-leg ladders remain finite in the zero-temperature limit. For small ratios of intra- to interleg couplings, odd-leg ladders can be mapped at low temperatures to single chains. For equal couplings, the logarithmic corrections at low temperatures increase markedly with the number of legs.
We study antiferromagnetic spin-\textonehalf{} Heisenberg ladders, comprised of nc{n}_{c}nc​ chains ($2\ensuremath{\le}{n}_{c}\ensuremath{\le}6$) with ratio $\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}$ of inter-chain to intra-chain couplings. The correlation length $\ensuremath{\xi}(T)$ is deduced from measurements of the correlation function. For even nc{n}_{c}nc​, the static structure factor exhibits a peak at a temperature below the corresponding spin gap. Results for isotropically coupled ladders ($\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}=1$) are compared to those for the single chain and the square lattice. For $\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}\ensuremath{\le}0.5$, the correlation function of the two-chain ladder is in excellent agreement with analytic results from conformal field theory, and $\ensuremath{\xi}(T)$ exhibits simple scaling behavior.