Journal of Physics A Mathematical and General

Hamiltonian studies of the two-dimensional n-component cubic model. I

Ferenc Iglói

Published Mar 11, 1986• 3 citations

Abstract

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.

DOI: 10.1088/0305-4470/19/4/017

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Journal of Physics A Mathematical and General

Hamiltonian studies of the two-dimensional n-component cubic model. I

Ferenc Iglói

Published Mar 11, 1986• 3 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/19/4/017

Related papers

Critical behavior of the n -component cubic model and the Ashkin-Teller fixed line

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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 $\mathrm{XXZ}$ 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.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.

Physical review. B, Condensed matterNov 1, 1981226 citations

The cubic lattice gas, a generalization of the

n

-component face-cubic model, is studied in two dimensions by variational renormalization-group techniques for general

n

, and by an exact mapping onto a solid-on-solid model for

n=2

. This approach reveals new interconnections among the class of cubic, Potts, and Ashkin-Teller phase-transition phenomena. In particular, it is shown that the fixed line of the Ashkin-Teller lattice gas serves as the backbone for a hierarchy of pairs of fixed points which branch off from it, as

n

is decreased from 2, at special points where two operators are marginal. The phase diagrams of the pure and dilute

n

-component cubic models are obtained and the orders of all transitions are determined. The cubic phase transition is found to be first order for all

n>{n}_{c}=2

. The evolution of the results as a function of dimensionality is discussed briefly.

N=2

(XY

model with cubic anisotropy) are compatible with the existence of a line of fixed points joining the Ising and the

O(2)

fixed points. Along this line the exponent $\ensuremath{\eta}$ has the constant value

1/4,

while the exponent $\ensuremath{\nu}$ runs in a continuous and monotonic way from 1 to $\ensuremath{\infty}$ [from Ising to

O(2)].

In the four-loop approximation, for

N>~3

we find a cubic fixed point in the region

u,v>~0.

The phase transition of the classical two-dimensional Potts model, in particular the order of the transition as the number of components

q

increases, is studied by constructing renormalization-group transformations on the equivalent one-dimensional quantum problem. It is shown that the block transformation with two sites per cell indicates the existence of a critical

{q}_{c}

separating the small-

q

and large-

q

regions with different critical behaviors. The physically accessible fixed point for

q>{q}_{c}

is a discontinuity fixed point where the specific-heat exponent $\ensuremath{\alpha}=1$, and therefore the transition is of first order.

\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.