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Journal of Computational Physics

The method of minimal representations in 2D Ising model calculations

Beresford Ν. Parlett, Wee-Liang Heng

Published Oct 1, 1994• 2 citations
Publisher
DOI
10.1006/jcph.1994.1164

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The minimally entangled typical thermal states (METTS) are an ensemble of pure states, equivalent to the Gibbs thermal state, designed with an efficient tensor network representation in mind. In this article, we use the projected entangled pair states (PEPS) as their representation on a two-dimensional (2D) lattice. Unlike matrix product states (MPS), which for 2D systems are limited by an exponential computational barrier in the lattice size, PEPS provides a more tractable approach. To substantiate the prowess of PEPS in modeling METTS (dubbed PEPS-METTS), we benchmark it against the purification method in the context of the 2D quantum Ising model at its critical temperature. Our analysis reveals that PEPS-METTS achieves accurate results with significantly lower bond dimensions. We further corroborate this finding in the 2D Fermi-Hubbard model. At a technical level, we introduce an efficient zipper method to obtain PEPS boundary MPS needed to compute expectation values and perform sampling. The imaginary time evolution is done with the neighborhood tensor update.
Four-loop expansions of the coefficients of the Callan-Symanzik equation for 3D and 2D dilute Ising models are calculated. Fixed-point coordinates and critical exponents are estimated by two methods of summation of double series: a generalisation of the Pade-Borel approximation and the first confluent form of the in algorithm of Wynn. Summation of the double series for the 2D dilute Ising model gives exponents very close to exponents of the pure Ising model, in accordance with the exact solution. The two methods are also applied to summation of single-variable series of pure Ising and polymer models, both in 3D and 2D cases. Both methods are shown to provide close agreement between present estimates and results obtained earlier either exactly in conformal invariant theories (2D) or numerically with high accuracy (3D). An additional test is provided by estimation of critical exponents of the 2D O(n) model for n=-1. The methods tested this way are used to calculate critical exponents of the 3D dilute Ising model. The values obtained are consistent with recent experimental results.
The critical behavior of 2D anisotropic systems with weak quenched disorder described by the so{called generalized Ashkin{Teller model (GATM), including the Ising model with random bonds, the dilute Baxter model, the impure N{color Ashkin{Teller model, and minimal conformal eld theory models (MCFTM) with c < 1 (c is the central charge) perturbed by randomness is discussed.All these models except MCFTM were found to belong to the Ising model universality class.Critical exponents of disordered MCFTM are calculated within perturbative expansions in the powers of = c 1=2. RG ows exhibit the rounding of uctuation{driven rst{order phase transitions by disorder.