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Journal of Physics A Mathematical and General

Quantum field ground state of the sine-Gordon model with finite soliton density: exact results

F. D. M. Haldane

Published Feb 1, 1982• 55 citations
Publisher

Abstract

The Bethe ansatz solutions of the massive Thirring model are renormalised and used to obtain exactly the ground-state properties of the sine-Gordon model with a finite density of topological solitons. Using a theory of 1D quantum fluids, the correlation function exponents and low-energy excitation spectrum are obtained. Expansions in various limits are given, and various crossovers described.

DOI
10.1088/0305-4470/15/2/021

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The Bethe ansatz solutions of the massive Thirring model are renormalised and used to obtain exactly the ground-state properties of the sine-Gordon model with a finite density of topological solitons. Using a theory of 1D quantum fluids, the correlation function exponents and low-energy excitation spectrum are obtained. Expansions in various limits are given, and various crossovers described.
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.
For the quantum sine-Gordon--massive Thirring model at finite temperature, the relationship between the Bethe ansatz ``elementary excitations'' and physical excitations of the system is examined. The thermodynamic equations are written in terms of a set of dressed particles in the general case. We find that the recently predicted soliton mass discontinuity (on varying coupling strength at finite temperature) is not a real effect--the function exhibiting the discontinuity does not correspond to the mass of the soliton.
The thermodynamics of the massive Thirring model is discussed with the help of the Bethe Ansatz approach in a way different from Fowler and Zotos (1982). In particular a finite Debye cut-off is introduced to investigate effects of the lattice discreteness. The specific heat seems to have a structure as a sum of the Debye-type specific heat and a 'soliton' peak, when the authors take into account the relation of the massive Thirring model to the quantum sine-Gordon model. The finite temperature excitations are calculated. The excitation energy proposed by Fowler and Zotos is shown to have significance which is insensitive to the way of formulation.
We study the sine-Gordon (SG) model at finite densities of the topological charge and small SG interaction constant, related to the one-dimensional Hubbard model near half filling. Using the modified Wentzel-Kramers-Brillouin approach, we find that the spectrum of the Gaussian fluctuations around the classical solution reproduces the results of the Bethe ansatz studies. The modification of the collective coordinate method allows us to write down the action, free from infrared divergencies. The behavior of the density-type correlation functions is nontrivial and we demonstrate the existence of leading and subleading asymptotes. A consistent definition of the charge-raising operator is discussed. The superconducting-type correlations are shown to decrease slowly at small soliton densities, while the spectral weight of right (left) moving fermions is spread over neighboring ‘‘4kF′′``{4k}_{F}''‘‘4kF′′​ harmonics.
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.