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Progress of Theoretical Physics

Commensurate-Incommensurate Cross-Over in the One-Dimensional Quantum Sine-Gordon System (g2=4 ). II: Dynamical Properties

H. Yamamoto, A. D. S. Nagi

Published Sep 1, 1984• 0 citations
Publisher

Abstract

The dynamical behavior of the one-dimensional quantum sine-Gordon system with a misfit parameter is investigated at a particular value of the quantum parameter g2=4π. For this purpose the dynamical correlation functions are calculated by using the Fermi-Bose relation. From the behavior of the displacement correlation function we find that the thermal velocity is given by ∼√T/m0 in the commensurate state (m0-| µ 8730|≫T) or in the cross-over region (m0≃| µ \tilde |≫T) and approaches value 1 as T or | µ \tilde | is increased. Here m0 and µ \tilde, respectively, are the gap and the chemical potential in the corresponding fermion system. For the case of denity-density correlation function, we give in the commensurate state expressions which are similar to those derived on the basis of the soliton phenomenology in the classical system. A particular case of the density-density correlation function is discussed in detail by following another route.

DOI
10.1143/ptp.72.456

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The dynamical behavior of the one-dimensional quantum sine-Gordon system with a misfit parameter is investigated at a particular value of the quantum parameter g2=4π. For this purpose the dynamical correlation functions are calculated by using the Fermi-Bose relation. From the behavior of the displacement correlation function we find that the thermal velocity is given by ∼√T/m0 in the commensurate state (m0-| µ 8730|≫T) or in the cross-over region (m0≃| µ \tilde |≫T) and approaches value 1 as T or | µ \tilde | is increased. Here m0 and µ \tilde, respectively, are the gap and the chemical potential in the corresponding fermion system. For the case of denity-density correlation function, we give in the commensurate state expressions which are similar to those derived on the basis of the soliton phenomenology in the classical system. A particular case of the density-density correlation function is discussed in detail by following another route.
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.
The commensurate-incommensurate transition of a two-dimensional classical sine-Gordon system is studied by the equivalent one-dimensional quantum system. The latter is expanded around its classical limit, corresponding to a low-temperature expansion of the two-dimensional system. It is found that both mass and wavefunction renormalisation are required. The critical exponent for incommensurability is beta =0, but if the momentum cut-off is kept finite one obtains beta =1. The classical limit is a singular point and the results are reliable when they are not too close to the transition.