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arXiv:1811.05504 · Physical review. E

Griffiths-McCoy singularities in the dilute transverse-field Ising model: A numerical linked cluster expansion study

Foster Thompson, Rajiv R. P. Singh

Published Mar 25, 2019• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreve.99.032129

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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.
We study the ±J transverse-field Ising spin-glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong-field limit. In the SK model and in high dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension d=6, which is below the upper critical dimension of d=8. In contrast, at lower dimensions we find a rich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence of the equal-time structure factor allows us to locate the critical coupling where the correlation length diverges, implying the onset of a thermodynamic phase transition. We find that the spin-glass susceptibility as well as various power moments of the local susceptibility become singular in the paramagnetic phase before the critical point. Griffiths-McCoy singularities are very strong in two dimensions but decrease rapidly as the dimension increases. We present evidence that high enough powers of the local susceptibility may become singular at the pure-system critical point.
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 quantum quenches in the transverse-field Ising model defined on different lattice geometries such as chains, two- and three-leg ladders, and two-dimensional square lattices. Starting from fully polarized initial states, we consider the dynamics of the transverse and the longitudinal magnetization for quenches to weak, strong, and critical values of the transverse field. To this end, we rely on an efficient combination of numerical linked cluster expansions (NLCEs) and a forward propagation of pure states in real time. As a main result, we demonstrate that NLCEs comprising solely rectangular clusters provide a promising approach to study the real-time dynamics of two-dimensional quantum many-body systems directly in the thermodynamic limit. By comparing to existing data from the literature, we unveil that NLCEs yield converged results on time scales which are competitive to other state-of-the-art numerical methods.
We study Yang-Lee zeros as they approach the thermodynamic limit of the two-dimensional (2D) nearest-neighbor antiferromagnetic Ising model on square and triangular lattices. We employ the numerical linked cluster expansion (NLCE) equipped with exact enumeration (EE) of the partition function to estimate the Laplacian of the free energy, which is proportional to the zeros density. Using a modified NLCE, where the expansion can be carried directly on the Yang-Lee zeros of the involved clusters, we estimate the density of Yang-Lee zeros in the thermodynamic limit. NLCE gives significantly more zeros than EE in the complex field plane providing more insights on how the root curves look in the thermodynamic limit. For the square lattice at $T\ensuremath{\ll}{T}_{c}$, the results suggest that two vertical lines at $\ifmmode\pm\else\textpm\fi{}{h}_{c}(T)$ in the complex field plane (i.e., two concentric circles in the complex fugacity plane) are the thermodynamic root curves. A similar picture is found for the triangular lattice for phase transitions at large values of magnetic field while further study is needed for phase transitions at smaller values of magnetic field. The convergence of the NLCE and (EE) calculations of the partition function to the thermodynamic limit is studied in both lattices and the temperature-field phase diagram is obtained from Yang-Lee zeros using both methods. This NLCE-based approach will facilitate the study of different types of phase transitions using Yang-Lee zeros in future research.
In many-body systems with quenched disorder, dynamical observables can be singular not only at the critical point, but in an extended region of the paramagnetic phase as well. These Griffiths singularities are due to rare regions, which are locally in the ordered phase and contribute to a large susceptibility. Here, we study the geometrical properties of rare regions in the transverse Ising model with dilution or with random couplings and transverse fields. In diluted models, the rare regions are percolation clusters, while in random models the ground state consists of a set of spin clusters, which are calculated by the strong disorder renormalization method. We consider the so called energy cluster, which has the smallest excitation energy and calculate its mass and linear extension in one-, two- and three-dimensions. Both average quantities are found to grow logarithmically with the linear size of the sample. Consequently, the energy clusters are not compact: for the diluted model they are isotropic and tree-like, while for the random model they are quasi-one-dimensional.
The Griffiths-McCoy singularity is a phenomenon characteristic of low-dimensional disordered quantum spin systems, in which the magnetic susceptibility shows singular behavior as a function of the external field even within the paramagnetic phase. We study whether this phenomenon is observed in the transverse-field Ising model with disordered ferromagnetic interactions on the quasi-two-dimensional diluted Chimera graph both by quantum Monte Carlo simulations and by extensive experiments on the D-Wave quantum annealer used as a quantum simulator. From quantum Monte Carlo simulations, evidence is found for the existence of the Griffiths-McCoy singularity in the paramagnetic phase. The experimental approach on the quantum hardware produces results that are less clear cut due to the intrinsic noise and errors in the analog quantum device but can nonetheless be interpreted to be consistent with the existence of the Griffiths-McCoy singularity as in the Monte Carlo case. This is the first experimental approach based on an analog quantum simulator to study the subtle phenomenon of Griffiths-McCoy singularities in a disordered quantum spin system, through which we have clarified the capabilities and limitations of the D-Wave quantum annealer as a quantum simulator.