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arXiv:cond-mat/9605068 · Physical Review Letters

Monte Carlo Study of Correlations in Quantum Spin Ladders

M. Greven, R. J. Birgeneau, U.‐J. Wiese

Published Aug 26, 1996• 160 citations
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Abstract

We study antiferromagnetic spin-\textonehalf{} Heisenberg ladders, comprised of ${n}_{c}$ 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 ${n}_{c}$, 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.

DOI
10.1103/physrevlett.77.1865

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We study antiferromagnetic spin-\textonehalf{} Heisenberg ladders, comprised of nc{n}_{c}nc​ chains (2\ensuremath≤nc\ensuremath≤62\ensuremath{\le}{n}_{c}\ensuremath{\le}62\ensuremath≤nc​\ensuremath≤6) with ratio J\ensuremath⊥J\ensuremath∥\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}J\ensuremath∥​J\ensuremath⊥​​ of inter-chain to intra-chain couplings. The correlation length \ensuremathξ(T)\ensuremath{\xi}(T)\ensuremathξ(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 (J\ensuremath⊥J\ensuremath∥=1\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}=1J\ensuremath∥​J\ensuremath⊥​​=1) are compared to those for the single chain and the square lattice. For J\ensuremath⊥J\ensuremath∥\ensuremath≤0.5\frac{{J}_{\ensuremath{\perp}}}{{J}_{\ensuremath{\parallel}}}\ensuremath{\le}0.5J\ensuremath∥​J\ensuremath⊥​​\ensuremath≤0.5, the correlation function of the two-chain ladder is in excellent agreement with analytic results from conformal field theory, and \ensuremathξ(T)\ensuremath{\xi}(T)\ensuremathξ(T) exhibits simple scaling behavior.
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.
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In this thesis the quantum Monte Carlo loop algorithm is applied to study the properties of correlation functions in low-dimensional quantum antiferromagnets. Two different types of lattice models are investigated. The first system is the quantum critical XXZ chain. The second class of models emerges when coupling the square lattice Heisenberg model to phonons. In case of the quantum critical XXZ chain, the loop algorithm is used to compute the two-point spin and dimer correlation functions at low temperatures. Starting from the well-known asymptotic expressions from conformal field theory for the decay of the correlations, the crossover from the finite-temperature to the ground state behavior is investigated. In addtion, the Monte Carlo data are used to extract numerical estimates for the correlation amplitudes. The deviations from asymptotic scaling at low finite temperatures and intermediate distances are examined. The results are also used to study the algebraic and logarithmic corrections in the ground state. The method is further applied to compute the low-temperature spin correlations of the square lattice Heisenberg model coupled to dispersionless Einstein phonons. Also, statically dimerized Heisenberg models in two dimensions are investigated. Here the quantum Monte Carlo technique is used to determine the specific pattern of dimerized bonds which has the largest gain in the magnetic energy. A direct comparison of ground state energies is only possible for a small number of dimerized systems. For this reason an alternative approach is developed, which allows to study a significantly larger class of models. Starting from a Landau expansion of the free energy, this amounts to a computation of specific structure factors of the dimer correlations of the uniform Heisenberg model. As a consequence, a Landau theory for the two-dimensional counterpart of the spin-Peierls transition is established. This adiabatic approach also applies to the Heisenberg model coupled to dispersive phonons. In this case the data for the dimer correlations are used to study the impact of the spin-phonon coupling on the phonon dispersion relations. In particular, the softening of phonon modes is investigated, which is relevant for experimental studies on substances with a quasi two-dimensional magnetic structure.
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