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arXiv:cond-mat/9908048 · Physical Review Letters

Flow Equation Solution for the Weak- to Strong-Coupling Crossover in the Sine-Gordon Model

Stefan K. Kehrein

Published Dec 13, 1999• 53 citations
arXivHTMLPDF

Abstract

A continuous sequence of infinitesimal unitary transformations, combined with an operator product expansion for vertex operators, is used to diagonalize the quantum sine-Gordon model for ${\ensuremath{\beta}}^{2}\ensuremath{\in}(2\ensuremath{\pi},\ensuremath{\infty})$. The leading order of this approximation already gives very accurate results for the single-particle gap in the strong-coupling phase. This approach can be understood as an extension of perturbative scaling theory since it links weak- to strong-coupling behavior in a systematic expansion. The method should also be useful for other strong-coupling problems that can be formulated in terms of vertex operators.

DOI
10.1103/physrevlett.83.4914

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A continuous sequence of infinitesimal unitary transformations, combined with an operator product expansion for vertex operators, is used to diagonalize the quantum sine-Gordon model for \ensuremathβ2\ensuremath∈(2\ensuremathπ,\ensuremath∞){\ensuremath{\beta}}^{2}\ensuremath{\in}(2\ensuremath{\pi},\ensuremath{\infty})\ensuremathβ2\ensuremath∈(2\ensuremathπ,\ensuremath∞). The leading order of this approximation already gives very accurate results for the single-particle gap in the strong-coupling phase. This approach can be understood as an extension of perturbative scaling theory since it links weak- to strong-coupling behavior in a systematic expansion. The method should also be useful for other strong-coupling problems that can be formulated in terms of vertex operators.
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I study the Sine-Gordon (SG) and Sinh-Gordon (ShG) quantum field theories with a recently introduced variational method, the relativistic continuous matrix product states (RCMPS). The main advantage is to work directly in the thermodynamic limit, and without any UV regulator. The SG model is well understood and integrable, which provides a convenient benchmark for the variational method and serves as a warm-up. RCMPS approximate the ground state of the SG model arbitrary well up to the free Fermion point [coupling β=4πβ=\sqrt{4π}β=4π​ in equal-time quantization convention, or b=1/2b=1/\sqrt{2}b=1/2​ in CFT convention], where the ground energy collapses to −∞-\infty−∞, and some renormalized ansatz would be needed. The ShG model, while integrable, is less understood and its strong coupling regime β≈1β\approx 1β≈1 is subject to some controversy. RCMPS also fit the ground state of the ShG model up to approximately b=1/2b=1/\sqrt{2}b=1/2​, after which their predictions start to deviate substantially from the "exact" results. This is more puzzling as nothing is expected to happen physically for the ShG model at that point (eg, the ground energy density does not diverge). Either the "exact" ShG results are not exact (the analytic continuation of the SG Bethe Ansatz solution is unwarranted), or, more likely, the physical structure of the ShG ground state changes in such a way that it becomes out of reach of the RCMPS manifold for reasonable bond dimensions.
The linear problem associated with the Lax operator of the classical sine-Gordon theory can be recast into the monodromy matrix form that can be extended to quantum theory as well. Product of the quantum monodromy matrices will have contributions from the singularities arising out of the operator product expansions of sine-Gordon field. This enables one to find the star-triangle relations. This is a generalization of the method used by Thacker for the non-relativistic nonlinear Schrödinger field theory. In the infinite volume limit, it leads to an unambiguous description of the algebra involving the scattering data operators. Starting from a vacuum the module of physical states are constructed by the application of chains of the scattering operators and they turn out to have definite eigenvalues of energy and momentum. The mass-spectra and two-particle S matrix elements can be calculated straightforwardly and they coincide with the well-known results of Zamolodchikov and Zamolodchikov. It is also shown that the particle states can be created by a generalized Jordon-Wigner transformation of the free fermionic creation operators — thus establishing a link between the S matrix element and certain co-cycles.
A systematic renormalisation group technique for studying the 2D sine-Gordon theory (Coulomb gas, XY model) near its phase transition is presented. The new results are (a) higher order terms in the flow equations beyond those of Kosterlitz (1974) give rise to a new universal quantity; (b) this in turn gives the universal form as well as the relative coefficient of the next-to-leading term in the correlation function of the XY model; (c) the free energy (1PI vacuum sum) is calculated after the singularity at beta 2 =4 pi is treated; (d) vortices with multiple charges are shown to be irrelevant; (e) symmetry breaking fields are analysed systematically. The main ideas that the sine-Gordon theory can be defined as a double expansion in alpha (fugacity) and delta = beta 2 /8 pi -1 (distance from the critical temperature at alpha =0). Wave-function and coupling constant ( alpha ) renormalisations are necessary and sufficient, around beta 2 =8 pi where cos phi acquires dimension 2, for functions with elementary SG fields. This gives rise to renormalisation of beta . The renormalisability is proved to the order calculated in the context of the SG theory, and in general, by using the equivalence to the Thirring-Schwinger model. The renormalised beta 2 plays a role analogous to the dimension in a phi 4 theory-8 pi being the critical dimension. beta 2 >8 pi gives an infrared asymptotically free theory which leads to the well-known fixed line. The infrared properties are understood by analogy with the non-linear sigma model.
In this paper we derive Klümper-Batchelor-Pearce-Destri-de Vega type nonlinear integral equations for describing the thermodynamics in the sine-Gordon model, when a chemical potential coupled to the topological charge is also present in the theory. The equations are valid at any value of the coupling constant and are particularly useful for computing the expectation values of local operators and some currents as functions of the temperature and the chemical potential. The equations are also extended to the case, when an extra term corresponding to the momentum conservation, is added to the thermodynamic potential. The benefits of this description are twofold. On the one hand, it can serve as an appropriate testing ground for other promising theoretical methods, like the method of random surfaces. On the other hand, the efficient computation of vertex operator expectation values, allows one to provide useful theoretical data, for the experimentally measurable coherence factors of a tunnel-coupled cold atomic condensate system.
We have developed a quantum-mechanical approach to study the critical properties of the quantum sine-Gordon equation describing a massless scalar field in 1+1 space-time dimensions with interaction density proportional to cos(${\mathrm{\ensuremath{\beta}}}_{0}$\ensuremath{\varphi}). The main feature of the present theory is to find an appropriate starting point for perturbation expansion, which can directly treat the infrared divergences encountered in the conventional perturbation theory. Exact critical conditions for the phase transition of the model are derived; and the quantum sine-Gordon Hamiltonian is found to correspond exactly to a free-field model. Near the critical regime, a gap will open near zero momentum in the excitation spectrum, and the corresponding ground-state wave function is a pairing quasiparticle state, analogous to the BCS superconducting state. It is proved that this ground state will possess quasi-long-range and quasi-off-diagonal long-range correlations. Comparisons with the Kosterlitz-Thouless classical theory of the two-dimensional Coulomb gas are also made.