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Journal of the Physical Society of Japan

Commensurate-Incommensurate Transition in the Two-Dimensional Sine-Gordon System: the Bethe Ansatz Calculation

Yukio Okwamoto

Published Mar 1, 1983• 6 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.52.942

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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.
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.
The commensurate-incommensurate (C-IC) phase transition in the one dimensional quantum sineGordon model at zero temperature is exactly solved with the use of the Bethe ansatz technique for the lattice massive Thirring model. The energy difference between C and IC phases is derived based on the same ground state which is valid in the whole parameter region. It is due to the fact that there is no change in the ground state of the lattice massive Thirring model even in the strongly repulsive region in contrast to the continuum massive Thirring model even in the strongly repulsive region in contrast to the continuum massive Thirring model. It is proved in the whole parameter region that the IC phase can be realized with the soliton density proportional to x f E s (Es: formation energy of soliton), when Es becomes negative.
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.