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arXiv:1910.12705 · Physical review. B./Physical review. B

Validity of the Harris criterion for two-dimensional quantum spin systems with quenched disorder

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

Published May 4, 2020• 4 citations
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Abstract

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 $p$ 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 $O(3)$ result 0.7112(5) to a number slightly greater than 1. In other words, by varying $p$, $\ensuremath{\nu}$ can reach an outcome between 0.7112(5) and 1 (or greater). Furthermore, among the studied values of $p$, 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.

DOI
10.1103/physrevb.101.174404

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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≤p\ensuremath≤10\ensuremath{\le}p\ensuremath{\le}10\ensuremath≤p\ensuremath≤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 \ensuremathν\ensuremath{\nu}\ensuremathν 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, \ensuremathν\ensuremath{\nu}\ensuremathν can reach an outcome between 0.7112(5) and 1 (or greater). Furthermore, among the studied values of ppp, all the associated \ensuremathν\ensuremath{\nu}\ensuremathν violate the Harris criterion except the ones corresponding to p\ensuremath≥0.9p\ensuremath{\ge}0.9p\ensuremath≥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.
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 $0.4\ensuremath{\le}p\ensuremath{\le}0.9$ 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 $W\ensuremath{\sim}0.1243$ 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.
We use quantum Monte Carlo simulations to study effects of disorder on the quantum phase transition occurring versus the ratio $g=J/{J}^{\ensuremath{'}}$ in square-lattice dimerized S=1/2S=1/2S=1/2 Heisenberg antiferromagnets with intradimer and interdimer couplings JJJ and ${J}^{\ensuremath{'}}$. The dimers are either randomly distributed (as in the classical dimer model), or come in parallel pairs with horizontal or vertical orientation. In both cases the transition violates the Harris criterion, according to which the correlation-length exponent should satisfy $\ensuremath{\nu}\ensuremath{\ge}1$. We do not detect any deviations from the three-dimensional O(3) universality class obtaining in the absence of disorder (where $\ensuremath{\nu}\ensuremath{\approx}0.71$). We discuss special circumstances which allow $\ensuremath{\nu}<1$ for the type of disorder considered here.
We use quantum Monte Carlo simulations to study a disordered S=1/2S=1/2S=1/2 Heisenberg quantum spin model with three different nearest-neighbor interactions, ${J}_{1}\ensuremath{\le}{J}_{2}\ensuremath{\le}{J}_{3}$, on the square lattice. We consider the regime in which J1{J}_{1}J1​ represents weak bonds, and J2{J}_{2}J2​ and J3{J}_{3}J3​ correspond to two kinds of stronger bonds (dimers) which are randomly distributed on columns forming coupled two-leg ladders. When increasing the average intradimer coupling (J2+J3)/2({J}_{2}+{J}_{3})/2(J2​+J3​)/2, the system undergoes a N\'eel to quantum glass transition of the ground state and later a second transition into a quantum paramagnet. The quantum glass phase is of the gapless Mott glass type (i.e., in boson language it is incompressible at temperature T=0T=0T=0), and we find that the temperature dependence of the uniform magnetic susceptibility follows the stretched exponential form $\ensuremath{\chi}\ensuremath{\sim}exp(\ensuremath{-}b/{T}^{\ensuremath{\alpha}})$, with $0<\ensuremath{\alpha}<1$. At the N\'eel-glass transition, we observe the standard O(3) critical exponents, which implies that the Harris criterion for the relevance of the disorder is violated in this system.
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.
We consider the Ising model on the square lattice with biaxially correlated random ferromagnetic couplings, the critical point of which is fixed by self-duality. The disorder, which has a correlator, $G(r)\ensuremath{\sim}{r}^{\ensuremath{-}1}$, represents a relevant perturbation according to the extended Harris criterion. Critical properties of the system are studied by large scale Monte Carlo simulations. The correlation length critical exponent $\ensuremath{\nu}=2.005(5)$ corresponds to that expected in a system with isotropic correlated long-range disorder, whereas the scaling dimension of the magnetization density ${x}_{m}=\ensuremath{\beta}∕\ensuremath{\nu}=0.1294(7)$ is somewhat larger than in the pure system. Conformal properties of the magnetization and energy density profiles are also examined numerically.
The influence of uncorrelated, quenched disorder on the phase transition of two dimensional Potts models will be reviewed. After an introduction where the conditions of relevance of quenched randomness on phase transitions are exemplified by experimental measurements, the results of perturbative and numerical investigations in the case of the Potts model will be presented. The Potts model is of particular interest, since it can have in the pure case a second-order or a first-order transition, depending on the number of states per spin. In 2D, transfer matrix calculations and Monte Carlo simulations were used in order to check the validity of conformal invariance methods in disordered systems. These techniques were then used to investigate the universality class of the disordered Potts model, in both regimes of the pure model phase transitions. A test of replica symmetry became possible through a study of multiscaling properties and a detailed analysis of the probability distribution of the correlation functions was also made possible.