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Physical review. B, Condensed matter

Hamiltonian studies of the d = 2 Ashkin-Teller model

Mahito Kohmoto, Marcel den Nijs, Leo P. Kadanoff

Published Nov 1, 1981• 226 citations
Publisher

Abstract

We study a one-dimensional quantum Hamiltonian problem which is equivalent to a highly anisotropic version of the two-dimensional Ashkin-Teller model. This problem is studied by using its duality and other symmetry properties, by a consideration of its limiting cases, and by mapping it into an $\mathrm{XXZ}$ linear chain which is equivalent to a highly anisotropic six-vertex model. In addition eleventh-order strong-coupling series are used to derive numerical data about the Hamiltonian problem. By these means a phase diagram is obtained. The essentially new feature of this diagram is a "critical fan," i.e., a region where a line of continuously varying criticality "fans out" and becomes an area of critical behavior.

DOI
10.1103/physrevb.24.5229

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We study a one-dimensional quantum Hamiltonian problem which is equivalent to a highly anisotropic version of the two-dimensional Ashkin-Teller model. This problem is studied by using its duality and other symmetry properties, by a consideration of its limiting cases, and by mapping it into an XXZ\mathrm{XXZ}XXZ linear chain which is equivalent to a highly anisotropic six-vertex model. In addition eleventh-order strong-coupling series are used to derive numerical data about the Hamiltonian problem. By these means a phase diagram is obtained. The essentially new feature of this diagram is a "critical fan," i.e., a region where a line of continuously varying criticality "fans out" and becomes an area of critical behavior.
The phase diagram and the critical properties of the Hamiltonian version of the two-dimensional n-component cubic model are investigated. Judging from the results of simple limits, mean-field calculation and RG transformations, the phase diagram of the one-dimensional quantum system is similar to that of the two-dimensional classical model. Several RG transformations were used to investigate the critical properties using different cell sizes in the transformation. The coincidence of critical and tricritical fixed points and the presence of a marginal operation showed the formation of the Ashkin-Teller fixed line and the breaking of the universality. The cubic transition is found to be first order for n>2.
High- and low-temperature series, Monte Carlo simulations, mean-field calculations, and renormalization-group ideas are used to analyze the behavior of the Ashkin-Teller model in three dimensions. The data generated by series and Monte Carlo methods in the neighborhood of the decoupling point are consistent with the renormalization-group conclusion that this system cannot show a line of continuously varying exponents. Thus the phase diagram for the d=3d=3d=3 system is quite different from that in d=2d=2d=2. Indeed a new phase (not seen in d=2d=2d=2) is observed, one in which the system spontaneously breaks its natural symmetry between its two kinds of Ising spins. A variety of continuous transitions are observed and explained in terms of an analysis which is fully consonant with renormalization-group concepts.
The exact solution of an Ashkin-Teller model, with layer and spatial anisotropy and Z 2 *Z 2 symmetry, is presented. The model is related to the eight-vertex model by a sublattice or orbifold duality and it has the same free energy and magnetisation. The exact solution manifold is a three-dimensional submanifold of the full six-dimensional thermodynamic space. It admits a Z 3 symmetry relating three regimes each exhibiting different partial order. In the spatially isotropic case, these regimes are separated by three lines of continuously varying critical behaviour which coalesce at the four-state Potts critical point. The magnetisations and polarisation are derived using corner transfer matrices giving rise to the appearance of chiral and Virasoro characters. The associated critical exponents confirm the known values. In particular, one obtains the value x sigma =1/8 for the magnetic scaling dimension and identify the next-to-leading thermal field.
The N -color quantum Ashkin–Teller spin chain is a prototypical model for the study of strong-randomness phenomena at first-order and continuous quantum phase transitions. In this paper, we first review the existing strong-disorder renormalization group approaches to the random quantum Ashkin–Teller chain in the weak-coupling as well as the strong-coupling regimes. We then introduce a novel general variable transformation that unifies the treatment of the strong-coupling regime. This allows us to determine the phase diagram for all color numbers N , and the critical behavior for all . In the case of two colors, N = 2, a partially ordered product phase separates the paramagnetic and ferromagnetic phases in the strong-coupling regime. This phase is absent for all , i.e., there is a direct phase boundary between the paramagnetic and ferromagnetic phases. In agreement with the quantum version of the Aizenman–Wehr theorem, all phase transitions are continuous, even if their clean counterparts are of first order. We also discuss the various critical and multicritical points. They are all of infinite-randomness type, but depending on the coupling strength, they belong to different universality classes.
The critical behavior of a bond-disordered Ashkin-Teller model on a square lattice is investigated by intensive Monte Carlo simulations. A duality transformation is used to locate a critical plane of the disordered model. This critical plane corresponds to the line of critical points of the pure model, along which critical exponents vary continuously. Along this line the scaling exponent corresponding to randomness \ensuremath{\varphi}=(\ensuremath{\alpha}/\ensuremath{\nu}) varies continuously and is positive so that the randomness is relevant, and different critical behavior is expected for the disordered model. We use a cluster algorithm for the Monte Carlo simulations based on the Wolff embedding idea, and perform a finite size scaling study of several critical models, extrapolating between the critical bond-disordered Ising and bond-disordered four-state Potts models. The critical behavior of the disordered model is compared with the critical behavior of an anisotropic Ashkin-Teller model, which is used as a reference pure model. We find no essential change in the order parameters' critical exponents with respect to those of the pure model. The divergence of the specific heat C is changed dramatically. Our results favor a logarithmic type divergence at Tc{\mathit{T}}_{\mathit{c}}Tc​, C\ensuremath{\sim}lnL for the random-bond Ashkin-Teller and four-state Potts models and C\ensuremath{\sim}ln lnL for the random-bond Ising model.