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arXiv:1111.0420 · Physical Review B

Quantum Monte Carlo study of the transverse-field quantum Ising model on infinite-dimensional structures

Seung Ki Baek, Jaegon Um, Sudo Yi, Beom Jun Kim

Published Nov 18, 2011• 6 citations
arXivHTMLPDF

Abstract

In a number of classical statistical-physical models, there exists a characteristic dimensionality called the upper critical dimension above which one observes the mean-field critical behavior. Instead of constructing high-dimensional lattices, however, one can also consider infinite-dimensional structures, and the question is whether this mean-field character extends to quantum-mechanical cases as well. We therefore investigate the transverse-field quantum Ising model on the globally coupled network and on the Watts-Strogatz small-world network by means of quantum Monte Carlo simulations and the finite-size scaling analysis. We confirm that both of the structures exhibit critical behavior consistent with the mean-field description. In particular, we show that the existing cumulant method has difficulty in estimating the correct dynamic critical exponent and suggest that an order parameter based on the quantum-mechanical expectation value can be a practically useful numerical observable to determine critical behavior when there is no well-defined dimensionality.

DOI
10.1103/physrevb.84.174419

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In a number of classical statistical-physical models, there exists a characteristic dimensionality called the upper critical dimension above which one observes the mean-field critical behavior. Instead of constructing high-dimensional lattices, however, one can also consider infinite-dimensional structures, and the question is whether this mean-field character extends to quantum-mechanical cases as well. We therefore investigate the transverse-field quantum Ising model on the globally coupled network and on the Watts-Strogatz small-world network by means of quantum Monte Carlo simulations and the finite-size scaling analysis. We confirm that both of the structures exhibit critical behavior consistent with the mean-field description. In particular, we show that the existing cumulant method has difficulty in estimating the correct dynamic critical exponent and suggest that an order parameter based on the quantum-mechanical expectation value can be a practically useful numerical observable to determine critical behavior when there is no well-defined dimensionality.
I study the properties of the quantum critical point of the transverse-field quantum Ising model on various fractal lattices such as the Sierpiński carpet, Sierpiński gasket, and Sierpiński tetrahedron. Using a continuous-time quantum Monte Carlo simulation method and finite-size scaling analysis, I identify the quantum critical point and investigate its scaling properties. Among others, I calculate the dynamic critical exponent and find that it is greater than one for all three structures. The fact that it deviates from one is a direct consequence of the fractal structures not being integer-dimensional regular lattices. Other critical exponents are also calculated. The exponents are different from those of the classical critical point and satisfy the quantum scaling relation, thus confirming that I have indeed found the quantum critical point. I find that the Sierpiński tetrahedron, of which the dimension is exactly 2, belongs to a different universality class than that of the two-dimensional square lattice. I conclude that the critical exponents depend on more details of the structure than just the dimension and the symmetry.
Quantum tricriticality of a ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnetic Ising model on a square lattice is studied using the mean-field (MF) theory, scaling theory, and the unbiased worldline quantum Monte-Carlo (QMC) method based on the Feynman path integral formula. The critical exponents of the quantum tricritical point (QTCP) and the qualitative phase diagram are obtained from the MF analysis. By performing the unbiased QMC calculations, we provide the numerical evidence for the existence of the QTCP and numerically determine the location of the QTCP in the case of J1=J2{J}_{1}={J}_{2}J1​=J2​. From the systematic finite-size scaling analysis, we conclude that the QTCP is located at HQTCP/J1=3.260(2){H}_{\mathrm{QTCP}}/{J}_{1}=3.260(2)HQTCP​/J1​=3.260(2) and ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{QTCP}}/{J}_{1}=4.10(5)$. We also show that the critical exponents of the QTCP are identical to those of the MF theory, because the QTCP in this model is in the upper critical dimension. The QMC simulations reveal that unconventional proximity effects of the ferromagnetic susceptibility appear close to the antiferromagnetic QTCP, and the proximity effects survive for the conventional quantum critical point. We suggest that the momentum dependence of the dynamical and static spin structure factors is useful for identifying the QTCP in experiments.
I investigate the quantum phase transition of the transverse-field quantum Ising model in which nearest neighbors are defined according to the connectivity of scale-free networks. Using a continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, I identify the quantum critical point and study its scaling characteristics. For the degree exponent λ=6, I obtain results that are consistent with the mean-field theory. For λ=4.5 and 4, however, the results suggest that the quantum critical point belongs to a non-mean-field universality class. Further simulations indicate that the quantum critical point remains mean-field-like if λ>5, but it continuously deviates from the mean-field theory as λ becomes smaller.
The random transverse-field Ising ferromagnet (RTFIF) is a highly disordered quantum system which contains randomness in the coupling strengths as well as in the transverse-field strengths. In one dimension, the critical properties are governed by an infinite-randomness fixed point (IRFP), and renormalization-group studies argue that the two-dimensional (2D) model is also governed by an IRFP. However, even the location of the critical point remains unsettled among quantum Monte Carlo (QMC) studies. In this work, we perform extensive QMC simulations to locate the quantum critical point and attempt a finite-size scaling analysis to observe the critical behavior. We estimate the critical field strength of the 2D RTFIF as ${\mathrm{\ensuremath{\Gamma}}}_{c}=7.52(2)$, together with critical exponents such as $\ensuremath{\beta}=1.5(3), \ensuremath{\nu}=1.6(3)$, and z=3.3(3)z=3.3(3)z=3.3(3) or $\ensuremath{\psi}=0.50(3)$. We have also considered the McCoy-Wu model, which has randomness in the ferromagnetic coupling strengths but not in the transverse-field strength. Our QMC calculation shows that the critical behavior of the 2D McCoy-Wu model is closer to that of the 2D transverse-field Ising spin glass than to that of the 2D RTFIF. These numerical findings enhance our understanding of disordered 2D quantum systems.
We study the ferromagnetic transverse-field Ising model with quenched disorder at T = 0 T = 0 in one and two dimensions by means of stochastic series expansion quantum Monte Carlo simulations using a rigorous zero-temperature scheme. Using a sample-replication method and averaged Binder ratios, we determine the critical shift and width exponents \nu_\mathrm{s} ν s and \nu_\mathrm{w} ν w as well as unbiased critical points by finite-size scaling. Further, scaling of the disorder-averaged magnetisation at the critical point is used to determine the order-parameter critical exponent \beta β and the critical exponent \nu_{\mathrm{av}} ν a v of the average correlation length. The dynamic scaling in the Griffiths phase is investigated by measuring the local susceptibility in the disordered phase and the dynamic exponent z' z ′ is extracted. By applying various finite-size scaling protocols, we provide an extensive and comprehensive comparison between the different approaches on equal footing. The emphasis on effective zero-temperature simulations resolves several inconsistencies in existing literature.
We study the critical behavior of the transverse-field quantum Ising model on a fractal structure, namely the Sierpinski carpet. When a magnetic field Δ is applied perpendicular to the Ising spin direction, quantum fluctuations affect the transition between the ferromagnetic and the paramagnetic phases. Employing the continuous-time quantum Monte Carlo simulation method and the finite-size scaling analysis, we investigate the interplay between the quantum fluctuations and the exotic dimensionality of the fractal structure and its effect on the critical behavior. As the transverse magnetic field increases, the critical temperature monotonically decreases until it apparently vanishes at a critical field Δ(c), beyond which the system becomes paramagnetic at all temperatures. However, the critical exponents are independent of Δ and remain the same as in the purely classical(Δ=0) case.