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arXiv:2008.11919 · Physical review. B./Physical review. B

Ground-state energy density, susceptibility, and Wilson ratio of a two-dimensional disordered quantum spin system

J.-H. Peng, D.-R. Tan, Fu-Jiun Jiang

Published Dec 21, 2020• 0 citations
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Abstract

A two-dimensional (2D) spin-1/2 antiferromagnetic Heisenberg model with a specific kind of quenched disorder is investigated, using the first-principles nonperturbative quantum Monte Carlo calculations (QMCs). The employed disorder distribution has a tunable parameter $p$ which can be considered as a measure of randomness ($p=0$ correponds to the clean model). Through large-scale QMCs, the dynamic critical exponents $z$, the ground-state energy densities ${E}_{0}$, and the Wilson ratios $W$ of various $p$ are determined with high precision. Interestingly, we find that the $p$ dependencies of $z$ and $W$ are likely to be complementary to each other. For instance, while the $z$ values of $0.4\ensuremath{\le}p\ensuremath{\le}0.9$ match well among themselves and are statistically different from that of $p=0$, the $W$ values for $p<0.7$ are in reasonably good agreement with $W\ensuremath{\sim}0.1243$ of the clean case. Surprisingly, our study indicates that a threshold of randomness, ${p}_{W}$, associated with $W$ exists. In particular, beyond this threshold the magnitude of $W$ grows with $p$. This is somehow counterintuitive since one expects the spin correlations should diminish accordingly. Similarly, there is a threshold ${p}_{z}$ related to $z$ after which a constant value is obtained for $z$. The results presented here are not only interesting from a theoretical perspective but also can serve as benchmarks for future related studies.

DOI
10.1103/physrevb.102.214206

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A two-dimensional (2D) spin-1/2 antiferromagnetic Heisenberg model with a specific kind of quenched disorder is investigated, using the first-principles nonperturbative quantum Monte Carlo calculations (QMCs). The employed disorder distribution has a tunable parameter ppp which can be considered as a measure of randomness (p=0p=0p=0 correponds to the clean model). Through large-scale QMCs, the dynamic critical exponents zzz, the ground-state energy densities E0{E}_{0}E0​, and the Wilson ratios WWW of various ppp are determined with high precision. Interestingly, we find that the ppp dependencies of zzz and WWW are likely to be complementary to each other. For instance, while the zzz values of 0.4\ensuremath≤p\ensuremath≤0.90.4\ensuremath{\le}p\ensuremath{\le}0.90.4\ensuremath≤p\ensuremath≤0.9 match well among themselves and are statistically different from that of p=0p=0p=0, the WWW values for p<0.7p<0.7p<0.7 are in reasonably good agreement with W\ensuremath∼0.1243W\ensuremath{\sim}0.1243W\ensuremath∼0.1243 of the clean case. Surprisingly, our study indicates that a threshold of randomness, pW{p}_{W}pW​, associated with WWW exists. In particular, beyond this threshold the magnitude of WWW grows with ppp. This is somehow counterintuitive since one expects the spin correlations should diminish accordingly. Similarly, there is a threshold pz{p}_{z}pz​ related to zzz after which a constant value is obtained for zzz. The results presented here are not only interesting from a theoretical perspective but also can serve as benchmarks for future related studies.
Inspired by the recent results regarding whether the Harris criterion is valid for quantum spin systems, we have simulated a two-dimensional spin-1/2 Heisenberg model on the square lattice with a specific kind of quenched disorder using the quantum Monte Carlo calculations. In particular, the considered quenched disorder has a tunable parameter $0\ensuremath{\le}p\ensuremath{\le}1$ which can be considered as a measure of randomness. Interestingly, when the magnitude of ppp increases from 0 to 0.95, at the associated quantum phase transitions the numerical value of the correlation length exponent $\ensuremath{\nu}$ grows from a number compatible with the O(3)O(3)O(3) result 0.7112(5) to a number slightly greater than 1. In other words, by varying ppp, $\ensuremath{\nu}$ can reach an outcome between 0.7112(5) and 1 (or greater). Furthermore, among the studied values of ppp, all the associated $\ensuremath{\nu}$ violate the Harris criterion except the ones corresponding to $p\ensuremath{\ge}0.9$. Considering the form of the employed disorder here, the above described scenario should remain true for other randomness if it is based on an idea similar to the one used in this study. This is indeed the case according to our preliminary results stemming from investigating another disorder distribution.
Using quantum Monte Carlo simulations, we study effects of disorder on the S=1/2S=1/2S=1/2 Heisenberg model with exchange constant JJJ on the square lattice supplemented by multispin interactions QQQ. It was found recently [L. Liu et al., Phys. Rev. X 8, 041040 (2018)] that the ground state of this $J\ensuremath{-}Q$ model with random couplings undergoes a quantum phase transition from the N\'eel antiferromagnetic state into a randomness-induced spin-liquid-like state that is a close analog to the well known random-singlet (RS) state of the Heisenberg chain with random couplings. This 2D RS state arises from a spontaneously symmetry-broken fourfold degenerate columnar valence-bond solid that is broken up by the disorder into finite domains, with spinons localized at topological defects. The interacting spinons form a critical collective many-body state without magnetic long range order but with the mean spin-spin correlations decaying with distance rrr as ${r}^{\ensuremath{-}2}$, as in the one-dimensional RS state. The dynamic exponent $z\ensuremath{\ge}2$, varying continuously with the model parameters. In this work, we further investigate the properties of the RS state in the $J\ensuremath{-}Q$ model with random QQQ couplings. We study the temperature dependence of the specific heat and various susceptibilities for large enough systems to reach the thermodynamic limit. We also analyze the size dependence of the critical magnetic order parameter and its susceptibility in the ground state. For all these quantities, we find consistency with the conventional quantum-critical scaling laws when the condition implied by the ${r}^{\ensuremath{-}2}$ form of the spin correlations is imposed. In particular, all the different quantities can be explained by the same value of the dynamic exponent zzz at fixed model parameters. We argue that the RS state identified in the $J\ensuremath{-}Q$ model corresponds to a generic renormalization group fixed point that can be reached in many quantum magnets with random couplings and that it has already been observed experimentally.
The quantum critical regimes (QCR) of both a two-dimensional (2D) disordered spin-12\frac{1}{2}21​ antiferromagnet and a 2D clean dimerized quantum Heisenberg model are studied using the first-principles nonperturbative quantum Monte Carlo simulations. In particular, the three well-known universal coefficients associated with QCR are investigated in detail. While in our investigation we find the obtained results are consistent with the related analytic predictions, non-negligible finite-temperature (TTT) effects are observed. Moreover, a striking finding in our study is that the numerical value for one of the universal coefficients we determine is likely to be different significantly from the corresponding (theoretical) result(s) established in the literature. To better understand the sources for the discrepancy observed here, apart from carrying out the associated analytic calculations not considered previously, it will be desirable as well to conduct a comprehensive examination of the exotic features of QCR for other disordered and clean spin systems than those investigated in this study.
Abstract Motivated by the need for high-precision benchmark results for basic quantum many-body models, several ground state properties of the square lattice S = 1 / 2 Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures T sufficiently low to realize the T → 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L × L lattices with L ∈ [ 6 , 96 ] are tabulated versus L and extrapolations to infinite system size are carried out. The relative statistical errors of finite-size energies are below 10 −7 . The extrapolated ground state energy density is e 0 = − 0.669 441 857 ( 7 ) , where the number within parentheses is the statistical error (one standard deviation) of the preceding digit. This value represents an improvement in precision of three orders of magnitude over the previous best result. The leading and subleading finite-size corrections to e 0 are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is m s = 0.307 447 ( 2 ) , which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in L −1 correction to m s 2 agrees with the value from chiral perturbation theory and the presence of a factor ln γ ⁡ ( L ) in the second-order correction is also confirmed, with the previously unknown value of the exponent being γ = 0.82 ( 4 ) . The finite-size corrections to the staggered susceptibility point to logarithmic corrections in this quantity as well. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.
Using an adaptive strategy which enables the study of quenched disordered system via the density-matrix renormalization-group method, we compute the various ground-state spin-spin correlation measures of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain with random coupling constants, namely, the mean values of the bulk and of the end-to-end correlations, the typical value of the bulk correlations, and the distribution of the bulk correlations. Our results are in agreement with the predictions of the strong-disorder renormalization-group method. We do not find any hint of logarithmic corrections either in the bulk average correlations, which were recently reported by Shu et al. [Phys. Rev. B 94, 174442 (2016)], or in the end-to-end average correlations. We report the existence of a logarithmic correction on the end-to-end correlations of the clean chain. Finally, we have determined that the distribution of the bulk correlations, when properly rescaled by an associated Lyapunov exponent, is a narrow and universal (disorder-independent) probability function.
A two-leg quenched random bond disordered antiferromagnetic spin-1/2 Heisenberg ladder system is investigated by means of stochastic series expansion quantum Monte Carlo (QMC) method. Thermal properties of the uniform and staggered susceptibilities, the structure factor, the specific heat, and the spin gap are calculated over a large number of random realizations in a wide range of disorder strength. According to our QMC simulation results, the considered system has a special temperature point at which the specific heat takes the same value regardless of the strength of the disorder. Moreover, the uniform susceptibility is shown to display the same character except for a small difference in the location of the special point. Finally, the spin gap values are found to decrease with increasing disorder parameter and the smallest gap value found in this study is well above the weak coupling limit of the clean case.