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Physical review. B, Condensed matter

Post-Gaussian approximation to the soliton of the (1+1)-dimensional sine-Gordon model

Hermann G. Kümmel

Published Jun 6, 2001• 7 citations
Publisher

Abstract

Starting from the Gaussian-variational approximation the state of the single sine-Gordon soliton is constructed using the coupled cluster method. The quantum contributions to the soliton mass and those to the space dependence of the soliton are computed and found to be non-negligible. If the coupling constant ${\ensuremath{\beta}}^{2}$ approaches the critical value $4\ensuremath{\pi}$ the mass becomes classical, but in other respects the soliton remains a quantum object.

DOI
10.1103/physrevb.64.014301

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Starting from the Gaussian-variational approximation the state of the single sine-Gordon soliton is constructed using the coupled cluster method. The quantum contributions to the soliton mass and those to the space dependence of the soliton are computed and found to be non-negligible. If the coupling constant \ensuremathβ2{\ensuremath{\beta}}^{2}\ensuremathβ2 approaches the critical value 4\ensuremathπ4\ensuremath{\pi}4\ensuremathπ the mass becomes classical, but in other respects the soliton remains a quantum object.
Making use of thermal-Green's-function technique, we study the quantum-statistical mechanics of a sine-Gordon system in 1 + 1 dimensions. In the weak-coupling limit, the temperature dependences of the soliton energy, Es{E}_{s}Es​, the soliton inertial mass, and the soliton density are determined. At high temperatures (T>mT>mT>m, where mmm is the mass of the fundamental field), Es{E}_{s}Es​ decreases monotonically as the temperature increases, and Es{E}_{s}Es​ jumps to zero around T=TcrT={T}_{\mathrm{cr}}T=Tcr​ ($\ensuremath{\equiv}{e}^{\ensuremath{-}1}{{E}_{s}}^{0}$), where Es0{{E}_{s}}^{0}Es​0 is the soliton energy at T=0T=0T=0 K. The soliton density agrees with the classical statistical-mechanics results for ${T}_{\mathrm{cr}}>T\ensuremath{\gg}m$, if Es{E}_{s}Es​ in the classical theory is replaced by the temperature-dependent one of the present theory.
We present an alternative way to determine the unknown parameter associated to a gaussian approximation in a generic two-dimensional model. Instead of the standard variational approach, we propose a procedure based on a quantitative prediction of conformal invariance, valid for systems in the scaling regime, away from criticality. We illustrate our idea by considering, as an example, the sine-Gordon model. Our method gives a good approximation for the soliton mass as function of βββ.
Despite the fact that it is not integrable, the (1 + 2)-dimensional Sine-Gordon equation has N-soliton solutions, whose velocities are lower than the speed of light (c = 1), for all N ≥ 1. Based on these solutions, a quantum-mechanical system is constructed over a Fock space of particles. The coordinate of each particle is an angle around the unit circle. U, a nonlinear functional of the particle number-operators, which obeys the Sine-Gordon equation in (1 + 2) dimensions, is constructed. Its eigenvalues on N-particle states in the Fock space are the slower-than-light, N-soliton solutions of the equation. A projection operator (a nonlinear functional of U), which vanishes on the single-particle subspace, is a mass-density generator. Its eigenvalues on multi-particle states play the role of the mass density of structures that emulate free, spatially extended, relativistic particles. The simplicity of the quantum-mechanical system allows for the incorporation of perturbations with particle interactions, which have the capacity to “annihilate” and “create” solitons – an effect that does not have an analog in perturbed classical nonlinear evolution equations.
The (1+1)-dimensional sine-Gordon model is studied in the framework of the variational Gaussian wave functionals. The renormalized mass depends on two dimensionless parameters. In the phase diagram a first-order phase transition will replace the second-order phase transition for large bare mass. The spectrum of the two-particle states consists of a continuum of scattering states and the isolated values connected to the excitonlike bound states. The energy of the bound states and the phase shifts of the scattering states are explicitly calculated.
The Bethe-ansatz equations for the thermodynamic properties of the quantum sine-Gordon systems are derived in the zero-charge-sector attractive case. For rational values of the coupling parameter $\frac{\ensuremath{\mu}}{\ensuremath{\pi}}$ these reduce to a finite set, solved here numerically for $\ensuremath{\mu}=[\frac{(n\ensuremath{-}1)}{n}]\ensuremath{\pi}$, for several values of nnn, to give the specific heat as a function of temperature. The "soliton" contribution peaks at $\ensuremath{\simeq}0.4$ soliton masses for $\ensuremath{\mu}=\frac{4}{5}\ensuremath{\pi}$, shifting downward for higher $\ensuremath{\mu}$. A detailed analysis of the sine-Gordon limit of the XYZ\mathrm{XYZ}XYZ spin chain is presented, and a non-Lorentz-invariant feature of that limit is noted.
The integrability of a quantum sine-Gordon model allows finite densities of zero binding energy 'bound states' at finite temperatures for certain values of the coupling constant. At these same coupling strengths, the usual Yang and Yang formulation (1969) of Bethe ansatz thermodynamics appears to predict discontinuities in some elementary excitation masses, for example solitons. This is examined in two limits-free fermions and free bosons. The apparent discontinuity turns out to be an artefact of the formalism, common to all cases where zero binding energy bound states exist.