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arXiv:hep-th/0406139 · Journal of Physics A Mathematical and General

A nonperturbative study of phase transitions in the multi-frequency sine-Gordon model

Gábor Zsolt Tóth

Published Sep 30, 2004• 20 citations
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Abstract

The phase spaces of the two- and three-frequency sine-Gordon models are examined in the framework of the truncated conformal space approach. The focus is mainly on a tricritical point in the phase space of the three-frequency model. We give substantial evidence that this point exists. We also find the critical line in the phase space and present TCSA data showing the change of the spectrum on the critical line as the tricritical endpoint is approached. We find a few points of the line of first-order transition as well.

DOI
10.1088/0305-4470/37/41/003

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The phase spaces of the two- and three-frequency sine-Gordon models are examined in the framework of the truncated conformal space approach. The focus is mainly on a tricritical point in the phase space of the three-frequency model. We give substantial evidence that this point exists. We also find the critical line in the phase space and present TCSA data showing the change of the spectrum on the critical line as the tricritical endpoint is approached. We find a few points of the line of first-order transition as well.
The tricritical Ising universality class emerges at the end point of a line of continuous Ising phase transitions, above which the transition is first order. In 1+1 dimensions, it is described by the second minimal model of conformal field theory (CFT) with central charge c=7/10c=7/10c=7/10. Recently, there has been a revival of interest in the tricritical Ising model due to its emergent space-time supersymmetry. Combining numerical (infinite density-matrix renormalization group) and field theoretical (bosonization and CFT) techniques, there authors demonstrate here that Ising tricriticality occurs in a paradigmatic extended one dimensional Hubbard model with explicit bond dimerization. The low-energy sector can be described by a triple sine-Gordon model, which, in the transition region, reduces to the tricritical Ising CFT. The theoretically predicted decay of various two-point correlation functions is found to be in excellent agreement with the density-matrix renormalization group data.
We discuss the nonequilibrium time evolution of the phase field in the sine-Gordon model using two very different approaches: the truncated Wigner approximation and the truncated conformal space approach. We demonstrate that the two approaches agree for a period covering the first few oscillations, thereby giving a solid theoretical prediction in the framework of the sine-Gordon model, which is thought to describe the dynamics of two bosonic condensates in quasi-one-dimensional traps coupled via a Josephson tunneling term. We conclude, however, that the recently observed phase-locking behavior cannot be explained in terms of homogeneous sine-Gordon dynamics, which hints at the role of other degrees of freedom or inhomogeneity in the experimental system.
We analyze a version of the sine-Gordon model in which the strength of the cosine potential has a periodic dependence on time. This model can be considered as the continuum limit of the many-body generalization of the Kapitza pendulum. Based on the perturbed CFT point of view, we develop a truncated conformal space approach (TCSA) to investigate the Floquet quasienergy spectrum. We focus on the effective behaviour for large driving frequencies, which we also derive exactly. Depending on the driving protocol, we can recover the original sine-Gordon model or its two-frequency version. The rich structure of the two-frequency model implies that the time-periodic drive can break integrability, can lead to new states in the spectrum, or can result in a phase transition. Our method is applicable for any periodically driven perturbed conformal field theories.
The critical behavior of a two-dimensional scalar Euclidean field theory with a potential term that allows for three minima is analyzed using an approximate position-space renormalization-group transformation on the equivalent quantum spin Hamiltonian. The global phase diagram shows a tricritical point separating a critical line from a line of first-order transitions. Other critical properties are examined, and good agreement is found with results on classical spin models belonging to the same universality class.
A three-dimensional anisotropic sine-Gordon model, derived as the spin-wave approximation to the biaxial (m=2m=2m=2) Lifshitz point problem in a uniform magnetic field, is shown to possess [in close analogy to the isotropic two-dimensional (2D) sine-Gordon theory which is well known to describe the critical behavior of the 2D XY\mathrm{XY}XY model], a surface of infinite-order phase transitions. This critical surface separates a phase characterized by infinite correlation length $\ensuremath{\xi}$ and power-law decay of correlations, and controlled by a stable fixed line, from one with finite $\ensuremath{\xi}$ and exponential decay. As the critical surface is approached from the latter phase, $\ensuremath{\xi}$ diverges as exp ($\ensuremath{\sigma}{t}^{\ensuremath{-}\ensuremath{\nu}}$) where $\ensuremath{\nu}=1$ is a universal number, ttt measures the distance from the critical surface, and $\ensuremath{\sigma}$ is nonuniversal. On the critical surface correlations decay like ${r}^{\ensuremath{-}\ensuremath{\eta}}{(\mathrm{ln}r)}^{\ensuremath{-}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}}$, where $\ensuremath{\eta}=4$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}=0.88\ensuremath{\cdots}$. Speculations on the occurrence of an infinite-order transition in liquid-crystal mixtures exhibiting nematic, smectic-AAA, and smectic-CCC phases are advanced.