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arXiv:2505.07627 · Physical review. B./Physical review. B

Quantum Monte Carlo study of the bond- and site-diluted transverse-field Ising model

Calvin Krämer, Max Hörmann, Kai Phillip Schmidt

Published Jul 8, 2025• 1 citation
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Abstract

We study the transverse-field Ising model on a square lattice with bond- and site-dilution at zero temperature by stochastic series expansion quantum Monte Carlo simulations. Tuning the transverse field $h$ and the dilution $p$, the quantum phase diagram of both models is explored. Both quantum phase diagrams show long-range order for small $h$ and small $p$. The ordered phase of each is separated from the disordered (quantum) Griffiths phase by second-order phase transitions on two critical lines touching at a multi-critical point. Using Binder ratios we locate quantum critical points with high accuracy. The order-parameter critical exponent $β$ and the average correlation-length exponent $ν_{\mathrm{av}}$ are determined along the critical lines and at the multi-critical points via finite-size scaling. We find three internally consistent sets of critical exponents and compare them with potentially connected universality classes. The quantum Griffiths phase in the vicinity of the phase transition lines is analyzed through the local susceptibility. Our results indicate that activated scaling occurs not only at the percolation transition, but also at the phase transition line for $p$ smaller than the percolation threshold $p_c$.

DOI
10.1103/wsv6-t3rl

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We study the transverse-field Ising model on a square lattice with bond- and site-dilution at zero temperature by stochastic series expansion quantum Monte Carlo simulations. Tuning the transverse field hhh and the dilution ppp, the quantum phase diagram of both models is explored. Both quantum phase diagrams show long-range order for small hhh and small ppp. The ordered phase of each is separated from the disordered (quantum) Griffiths phase by second-order phase transitions on two critical lines touching at a multi-critical point. Using Binder ratios we locate quantum critical points with high accuracy. The order-parameter critical exponent βββ and the average correlation-length exponent νavν_{\mathrm{av}}νav​ are determined along the critical lines and at the multi-critical points via finite-size scaling. We find three internally consistent sets of critical exponents and compare them with potentially connected universality classes. The quantum Griffiths phase in the vicinity of the phase transition lines is analyzed through the local susceptibility. Our results indicate that activated scaling occurs not only at the percolation transition, but also at the phase transition line for ppp smaller than the percolation threshold pcp_cpc​.
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.
The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing autocorrelation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
We use numerical linked cluster expansions (NLCEs) to study the site-diluted transverse-field Ising model on the square lattice at T=0. NLCE with a self-consistent mean field on the boundary of the clusters is used to obtain the ground-state magnetization, susceptibility, and structure factor as a function of transverse field h and exchange constant J. Adding site dilution to the model turns NLCE into a series expansion in the dilution parameter p. Studying the divergence of the structure factor allows us to establish the phase diagram in the h/J and p plane. By studying the magnetization of the system in a longitudinal field, we investigate the Griffiths-McCoy singularities. We find that the magnetization develops nonlinearities in the Griffiths phase with exponents that vary continuously with h. Additionally, the probability distribution of the local susceptibility develops long tails in the Griffiths phase, which is studied in terms of its moments.
The quantum phase transition of the one-dimensional long-range transverse-field Ising model is explored by combining the quantum Monte Carlo method and stochastic parameter optimization, specifically achieved by tuning correlation ratios so that space and imaginary time are isotropic. In our simulations, the simulator automatically determines the parameters to sample from, even without prior knowledge of the critical point and universality class. The leading-order finite-size corrections are eliminated by comparing two systems with different sizes; this procedure is also performed automatically. Varying the decay exponent of the long-range interaction, σ, we investigate σ dependence of the dynamical exponent and the other critical exponents precisely in the mean-field, nonuniversal, and two-dimensional classical Ising universality regimes. We successfully obtained numerical evidence supporting σ=7/4 as the universality boundary between the latter two.
We study the effects of dilution to the critical properties of site-diluted Ising model in two dimensions using Monte Carlo simulations. Quenched disorder from the dilution is incorporated into the Ising model via random empty sites on the square lattice of Ising spins. Thermodynamic quantities such as the magnetization MMM per spin, energy EEE per spin, magnetic susceptibility χχχ per spin, and specific heat CCC per spin are then calculated after the system has equilibrated. At small dilution concentrations d<0.1d<0.1d<0.1, we find that the value of the critical exponent βββ does not deviate from its pure Ising value. At higher dilution concentrations d>0.1d>0.1d>0.1, however, we find βββ to strongly depend on the value of ddd. We are able to locate a critical temperature TcT_cTc​ and a critical dilution concentration dcd_cdc​ where the phase transition occurs. We find TcT_cTc​ to depend linearly on ddd. In the phase diagrams of MMM, EEE, χχχ, and CCC, we find that the phase transition line eventually disappears at high dilutions. Our results suggest that there is a transition from Strong Universality at low dilution to Weak Universality at high dilution. Lastly, we find a wide and nonsingular increase in the magnetic susceptibility χχχ at the low temperature and high dilution region.