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Physical review. B, Condensed matter

Phase diagrams of surface structures from Bethe-ansatz solutions of the quantum sine-Gordon model

F. D. M. Haldane, Per Bak, Tomas Bohr

Published Sep 1, 1983• 93 citations
Publisher

Abstract

Phase diagrams of uniaxial two-dimensional systems with commensurate, incommensurate, and liquid phases are derived by combining exact results for the quantum sine-Gordon model with the Kosterlitz-Thouless theory of melting. The phase diagram depends on the order of commensurability, $p$. In particular, for $p=3$ (the "chiral Potts" case), we conjecture that the phase diagram contains no Lifshitz point, in contrast to previous authors; for $p=1$, dislocations remove the original $\mathrm{CI}$ transition completely.

DOI
10.1103/physrevb.28.2743

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Phase diagrams of uniaxial two-dimensional systems with commensurate, incommensurate, and liquid phases are derived by combining exact results for the quantum sine-Gordon model with the Kosterlitz-Thouless theory of melting. The phase diagram depends on the order of commensurability, ppp. In particular, for p=3p=3p=3 (the "chiral Potts" case), we conjecture that the phase diagram contains no Lifshitz point, in contrast to previous authors; for p=1p=1p=1, dislocations remove the original CI\mathrm{CI}CI transition completely.
Recently Haldane has derived the Bethe ansatz equations describing the ground state properties of the massive Thirring model with finite `hole' density (the quantum sine-Gordon model with finite soliton density). His derivation is explicitly confirmed here using a particular cutoff scheme, and his solutions of the equations in various limits are reviewed with complementary results. The commensurate-incommensurate transition in the two-dimensional sine-Gordon model with a uniaxial misfit is examined at finite temperatures using the results obtained in the massive Thirring model.
Using the recently obtained exact results for the expectation values of operators in the sine- and sinh-Gordon models [A. B. Zamolodchikov and S. Lukyanov, Nucl. Phys. B 493, 571 (1997); V. Fateev, S. Lukyanov, A. B. Zamolodchikov, and Al. B. Zamolodchikov, Phys. Lett. B 406, 83 (1997)] we calculate the specific heat of the corresponding two-dimensional Euclidean (classical) models. We show that the temperature dependence of the specific heat of the sine-Gordon model, in the commensurate phase, has a maximum well below the Kosterlitz-Thouless transition and that the sinh-Gordon model is thermodynamically unstable in the strong-coupling regime. We give also the temperature dependence of the specific heat in the incommensurate phase of the sine-Gordon model.
We investigate the classical chiral Ashkin-Teller model on a square lattice with the corner transfer matrix renormalization group algorithm. We show that the melting of the period-4 phase in the presence of a chiral perturbation takes different forms depending on the coefficient of the four-spin term in the Ashkin-Teller model. Close to the clock limit of two decoupled Ising models, the system undergoes a two-step commensurate-incommensurate transition as soon as the chirality is introduced, with an intermediate critical floating phase bounded by a Kosterlitz-Thouless transition at high temperature and a Pokrovsky-Talapov transition at low temperature. By contrast, close to the four-states Potts model, we argue for the existence of a unique commensurate-incommensurate transition that belongs to the chiral universality class, and for the presence of a Lifshitz point where the ordered, disordered, and floating phases meet. Finally, we map the whole phase diagram, which turns out to be in qualitative agreement with the 40-year-old prediction by Huse and Fisher.
In this paper we study the critical exponents of the quantum sine-Gordon model and U(1) Thirring models in the incommensurate phase. This phase appears when the chemical potential h exceeds a critical value and is characterized by a finite density of solitons. The low-energy sector of this phase is critical and is described by the Gaussian model (Tomonaga-Luttinger liquid) with the compactification radius dependent on the soliton density and the sine-Gordon model coupling constant $\ensuremath{\beta}.$ For a fixed value of $\ensuremath{\beta},$ we find that the Luttinger parameter K is equal to 1/2 at the commensurate-incommensurate transition point and approaches the asymptotic value ${\ensuremath{\beta}}^{2}/8\ensuremath{\pi}$ away from it. We describe a possible phase diagram of the model consisting of an array of weakly coupled chains. The possible phases are Fermi liquid, spin density wave, spin-Peierls, and Wigner crystal.
There are two different results concerning the functional relationship between the chemical potential of the 1D massive Thirring model and the misfit parameter of the 1D quantum sine-Gordon model which have been reported [4-7]. In order to investigate this problem, the Jordan boson representation of fermion field operators is used to obtain the bosonized expression of the 1D massive Thirring model. The equivalence of this model with the 1D quantum sine-Gordon model is proved, all zero-mode contributions taken into account. The problem of controversial results previously reported is solved. Thereafter, by means of Bethe-Ansatz method, a new expression for the critical curve of the commensurate-incommensurate transition in the two-dimensional sine-Gordon model is found.