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arXiv:1006.1598 · Physical Review B

Quantum phase transitions in disordered dimerized quantum spin models and the Harris criterion

Dao‐Xin Yao, Jonas Gustafsson, E. W. Carlson, Anders W. Sandvik

Published Nov 19, 2010• 22 citations
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Abstract

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 $S=1/2$ Heisenberg antiferromagnets with intradimer and interdimer couplings $J$ 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.

DOI
10.1103/physrevb.82.172409

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We use quantum Monte Carlo simulations to study effects of disorder on the quantum phase transition occurring versus the ratio g=J/J\ensuremath′g=J/{J}^{\ensuremath{'}}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′{J}^{\ensuremath{'}}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 \ensuremathν\ensuremath≥1\ensuremath{\nu}\ensuremath{\ge}1\ensuremathν\ensuremath≥1. We do not detect any deviations from the three-dimensional O(3) universality class obtaining in the absence of disorder (where \ensuremathν\ensuremath≈0.71\ensuremath{\nu}\ensuremath{\approx}0.71\ensuremathν\ensuremath≈0.71). We discuss special circumstances which allow \ensuremathν<1\ensuremath{\nu}<1\ensuremathν<1 for the type of disorder considered here.
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 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.
The two-dimensional $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ dimerized quantum Heisenberg model is studied on the square lattice by means of (stochastic series expansion) quantum Monte Carlo simulations as a function of the coupling ratio $\ensuremath{\alpha}={J}^{\ensuremath{'}}/J$. The critical point of the order-disorder quantum phase transition in the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ model is determined as ${\ensuremath{\alpha}}_{c}=2.5196(2)$ by finite-size scaling for up to approximately 10 000 quantum spins. By comparing six dimerized models we show, contrary to the current belief, that the critical exponents of the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ model are not in agreement with the three-dimensional classical Heisenberg universality class. This lends support to the notion of nontrivial critical excitations at the quantum critical point.
Using a combination of unbiased quantum Monte Carlo simulations and a decoupled dimer mean-field theory, we investigate the thermal and quantum phase transitions of the spin-1/2 Heisenberg model on the dimerized diamond lattice. We find that at sufficiently strong dimerization the system exhibits a quantum disordered ground state, in contrast to the antiferromagnetic phase stabilized at weak dimerization. We determine the quantum critical point and examine the thermodynamic responses in both regimes. The ratio for the critical interdimer (JcJ_cJc​) to intradimer (JDJ_DJD​) coupling is obtained as Jc/JD=0.3615(5)J_c/J_D = 0.3615(5)Jc​/JD​=0.3615(5). Our results show that the decoupled dimer mean-field theory well captures the competition between the local singlet formation and the antiferromagnetic ordering tendency, and thus provides an appropriate qualitative description of this three-dimensional quantum magnet, in contrast to the conventional mean-field decoupling. We also examine the differences in the ordering temperatures between the antiferromagnetic and the ferromagnetic spin-1/2 Heisenberg model on the diamond lattice based on quantum Monte Carlo simulations.
Using a combination of unbiased quantum Monte Carlo simulations and a decoupled dimer mean-field theory, we investigate the thermal and quantum phase transitions of the spin- 1 2 Heisenberg model on the dimerized diamond lattice. We find that at sufficiently strong dimerization the system exhibits a quantum disordered ground state, in contrast to the antiferromagnetic phase stabilized at weak dimerization. We determine the quantum critical point and examine the thermodynamic responses in both regimes. The ratio for the critical interdimer ( J c ) to intradimer ( J D ) coupling is obtained as J c / J D = 0.3615 ( 5 ) . Our results show that the decoupled dimer mean-field theory well captures the competition between the local singlet formation and the antiferromagnetic ordering tendency, and thus provides an appropriate qualitative description of this three-dimensional quantum magnet, in contrast to the conventional mean-field decoupling. We also examine the differences in the ordering temperatures between the antiferromagnetic and the ferromagnetic spin- 1 2 Heisenberg model on the diamond lattice based on quantum Monte Carlo simulations.
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.