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arXiv:1407.1113 · Physical Review B

Criticality and Mott glass phase in a disordered two-dimensional quantum spin system

Nvsen Ma, Anders W. Sandvik, Dao‐Xin Yao

Published Sep 26, 2014• 13 citations
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Abstract

We use quantum Monte Carlo simulations to study a disordered $S=1/2$ 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 ${J}_{1}$ represents weak bonds, and ${J}_{2}$ and ${J}_{3}$ 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 $({J}_{2}+{J}_{3})/2$, 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 $T=0$), 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.

DOI
10.1103/physrevb.90.104425

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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, J1\ensuremath≤J2\ensuremath≤J3{J}_{1}\ensuremath{\le}{J}_{2}\ensuremath{\le}{J}_{3}J1​\ensuremath≤J2​\ensuremath≤J3​, 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 \ensuremathχ\ensuremath∼exp(\ensuremath−b/T\ensuremathα)\ensuremath{\chi}\ensuremath{\sim}exp(\ensuremath{-}b/{T}^{\ensuremath{\alpha}})\ensuremathχ\ensuremath∼exp(\ensuremath−b/T\ensuremathα), with 0<\ensuremathα<10<\ensuremath{\alpha}<10<\ensuremathα<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.
We use quantum Monte Carlo simulations to study a dimer-diluted S = 1/2 Heisenberg model on a bilayer square lattice with intralayer interaction J 1 and interlayer interaction J 2 . Below the classical percolation threshold p c , the system has three phases reachable by tuning the interaction ratio g = J 2 / J 1 : a Néel ordered phase, a gapless quantum glass phase, and a gapped quantum paramagnetic phase. We present the ground-state phase diagram in the plane of dilution p and interaction ratio g . The quantum glass phase is certified to be of the gapless Mott glass type, having a uniform susceptibility vanishing at zero temperature T and following a stretched exponential form at T > 0; χ u exp(- b / T α ) with α < 1. At the phase transition point from Neel ordered to Mott glass, we find that the critical exponents are different from those of the clean system described by the standard O (3) universality class in 2+1 dimensions.
We employ the stochastic series expansion quantum Monte Carlo method to investigate the S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg model on a bilayer square lattice with diluted interlayer couplings. Both regular and random dilution patterns are considered. In systems with regular dilution, tuning the interlayer interaction drives a quantum phase transition from a N\'eel-ordered phase to a quantum disordered phase, consistent with the O(3)O(3)O(3) universality class. In contrast, random dilution gives rise to a two-step transition: from the N\'eel phase to an intermediate Mott glass (MG) phase, followed by a transition to the quantum disordered phase. Within the MG phase, the uniform magnetic susceptibility exhibits a stretched-exponential temperature dependence ${\ensuremath{\chi}}_{u}\ensuremath{\sim}exp(\ensuremath{-}b/{T}^{\ensuremath{\alpha}}), 0<\ensuremath{\alpha}<1$. At the N\'eel-to-glass transition, the quenched disorder modifies the critical exponents in a manner consistent with the Harris criterion. These findings provide new insights into disorder-driven quantum phase transitions and the emergence of glassy phases in diluted bilayer quantum magnets.
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.
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.
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.
We study the quenched disordered magnetic system, which is obtained from the 2D SO(3) quantum Heisenberg model, on a square lattice, with nearest neighbors interaction, by taking a Gaussian random distribution of couplings centered in an antiferromagnetic coupling, J¯>0 and with a width ΔJ. Using coherent spin states we can integrate over the random variables and map the system onto a field theory, which is a generalization of the SO(3) nonlinear sigma model with different flavors corresponding to the replicas, coupling parameter proportional to J¯ and having a quartic spin interaction proportional to the disorder (ΔJ). After deriving the CP1 version of the system, we perform a calculation of the free energy density in the limit of zero replicas, which fully includes the quantum fluctuations of the CP1 fields zi. We, thereby obtain the phase diagram of the system in terms of (T,J¯,ΔJ). This presents an ordered antiferromagnetic (AF) phase, a paramagnetic (PM) phase and a spin-glass (SG) phase. A critical curve separating the PM and SG phases ends at a quantum critical point located between the AF and SG phases, at T=0. The Edwards–Anderson order parameter, as well as the magnetic susceptibilities are explicitly obtained in each of the three phases as a function of the three control parameters. The magnetic susceptibilities show a Curie-type behavior at high temperatures and exhibit a clear cusp, characteristic of the SG transition, at the transition line. The thermodynamic stability of the phases is investigated by a careful analysis of the Hessian matrix of the free energy. We show that all principal minors of the Hessian are positive in the limit of zero replicas, implying in particular that the SG phase is stable.