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Physical review. B, Condensed matter

Quantum-statistical mechanics of extended objects. I. Kinks in the one-dimensional sine-Gordon system

Kazumi Maki, Hajime Takayama

Published Oct 15, 1979• 128 citations
Publisher

Abstract

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, ${E}_{s}$, the soliton inertial mass, and the soliton density are determined. At high temperatures ($T>m$, where $m$ is the mass of the fundamental field), ${E}_{s}$ decreases monotonically as the temperature increases, and ${E}_{s}$ jumps to zero around $T={T}_{\mathrm{cr}}$ ($\ensuremath{\equiv}{e}^{\ensuremath{-}1}{{E}_{s}}^{0}$), where ${{E}_{s}}^{0}$ is the soliton energy at $T=0$ K. The soliton density agrees with the classical statistical-mechanics results for ${T}_{\mathrm{cr}}>T\ensuremath{\gg}m$, if ${E}_{s}$ in the classical theory is replaced by the temperature-dependent one of the present theory.

DOI
10.1103/physrevb.20.3223

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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≡e\ensuremath−1Es0\ensuremath{\equiv}{e}^{\ensuremath{-}1}{{E}_{s}}^{0}\ensuremath≡e\ensuremath−1Es​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 Tcr>T\ensuremath≫m{T}_{\mathrm{cr}}>T\ensuremath{\gg}mTcr​>T\ensuremath≫m, if Es{E}_{s}Es​ in the classical theory is replaced by the temperature-dependent one of the present theory.
We study the quantum-statistical mechanics, at low temperatures, of a one-dimensional antiferromagnetic Heisenberg model with two anisotropies. In the weak-coupling limit we determine the temperature dependences of the soliton energy and the soliton density. We have found that the leading correction to the sine-Gordon (SG) expression for the soliton density and the quantum soliton energy comes from the out-of-plane magnon mode, not present in the pure SG model. We also show that when an external magnetic field is applied, the chain supports a new type of kink, where the sublattices rotate in opposite directions.
The thermal renormalization of the Bethe-ansatz excitation spectrum for the sine-Gordon system is presented. For the soliton mass, the results agree closely with those of Maki and Takayama over a significant temperature range, but the sudden decrease in mass they predict near a particular temperature is not found. For the Bethe-ansatz excitation corresponding to the phonon at zero temperature, the rest mass changes rapidly as the temperature increases, even near the classical limit. This suggests that at finite temperature this excitation (the one-string) no longer corresponds to the physical phonon.
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.
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.
We give two fundamental methods for evaluation of classical free energies of all the integrable models admitting soliton solutions; the sine-Gordon equation is one example. Periodic boundary conditions impose integral equations for allowed phonon and soliton momenta. From these, generalized Bethe-Ansatz and functional-integration methods using action-angle variables follow. Results for free energies coincide, and coincide with those that we find by transfer-integral methods. Extension to the quantum case, and quantum Bethe Ansatz, on the lines to be reported elsewhere for the sinh-Gordon equation, is indicated.
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.