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arXiv:cond-mat/9709110 · Physical Review Letters

Square-Lattice Heisenberg Antiferromagnet at Very Large Correlation Lengths

Bernard B. Beard, R. J. Birgeneau, M. Greven, U.‐J. Wiese

Published Feb 23, 1998• 122 citations
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Abstract

The correlation length of the square-lattice spin-1/2 Heisenberg antiferromagnet is studied in the low-temperature (asymptotic-scaling) regime. Our novel approach combines a very efficient loop cluster algorithm---operating directly in the Euclidean time continuum---with finite-size scaling. This enables us to probe correlation lengths up to $\ensuremath{\xi}\ensuremath{\approx}350000$ lattice spacings, more than 3 orders of magnitude larger than in any previous study. We resolve a conundrum concerning the applicability of asymptotic-scaling formulas to experimentally and numerically determined correlation lengths. Our results have direct implications for the zero-temperature behavior of spin-1/2 ladders.

DOI
10.1103/physrevlett.80.1742

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The correlation length of the square-lattice spin-1/2 Heisenberg antiferromagnet is studied in the low-temperature (asymptotic-scaling) regime. Our novel approach combines a very efficient loop cluster algorithm---operating directly in the Euclidean time continuum---with finite-size scaling. This enables us to probe correlation lengths up to \ensuremathξ\ensuremath≈350000\ensuremath{\xi}\ensuremath{\approx}350000\ensuremathξ\ensuremath≈350000 lattice spacings, more than 3 orders of magnitude larger than in any previous study. We resolve a conundrum concerning the applicability of asymptotic-scaling formulas to experimentally and numerically determined correlation lengths. Our results have direct implications for the zero-temperature behavior of spin-1/2 ladders.
Critical behavior of the quantum phase transition of a site-diluted Heisenberg antiferromagnet on a square lattice is investigated by means of the quantum Monte Carlo simulation with the continuous-imaginary-time loop algorithm. Although the staggered spin-correlation function decays in a power law with the exponent definitely depending on the spin size S, the correlation-length exponent is classical, i.e., $\ensuremath{\nu}=4/3.$ This implies that the length scale characterizing the nonuniversal quantum phase transition is nothing but the mean size of connected spin clusters.
We present an analysis of high precision Monte Carlo data for the two dimensional S=1/2 quantum Heisenberg antiferromagnet up to ξ=95.7(3)ξ= 95.7(3)ξ=95.7(3) obtained by the continuous time version of the loop algorithm. Our data are in good agreement with a prediction of the recent theory (Phys. Rev. Lett. vol. 77, 3439 (1996)) in very high temperature regime, but they strongly disagree with it in the low temperature regime.
Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).
The S=1/2S=1/2S=1/2 and S=1S=1S=1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for S=1,S=1,S=1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended-parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchain-coupling, case.
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.
The dimensional crossover in a spin-S nearest-neighbor Heisenberg antiferromagnet is discussed as it is tuned from a two-dimensional square lattice, of lattice spacing a, towards a spin chain by varying the width Ly{L}_{y}Ly​ of a semi-infinite strip ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$. For integer spins and arbitrary Ly{L}_{y}Ly​, and for half integer spins with Ly/a{L}_{y}/aLy​/a an arbitrary even integer, explicit analytical expressions for the zero temperature correlation length and the spin gap are given. For half integer spins and Ly/a{L}_{y}/aLy​/a an odd integer, it is argued that the c=1c\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1c=1 behavior of the SU(2)1{)}_{1})1​ Wess-Zumino-Witten fixed point is squeezed out as the width ${L}_{y}\ensuremath{\rightarrow}\ensuremath{\infty}$; here c is the conformal charge. The results specialized to S=12S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}S=21​ are applied to spin-ladder systems.