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Free massive fermions inside the quantum discrete sine-Gordon model

N. Kutz, Technische Univ. Berlin (Germany). Sonderforschungsbereich 288 -Differentialgeometrie und Quantenphysik

Published Jan 1, 1996• 1 citation

Abstract

We extend the notion of space shifts introduced in [FV3] for certain quantum light cone lattice equations of sine-Gordon type at root of unity (e.g. [FV1][FV2][BKP][BBR]). As a result we obtain a compatibility equation for the roots of central elements within the algebra of observables (also called current algebra). The equation which is obtained by exponentiating these roots is exactly the evolution equation for the "classical background" as described in [BBR]. As an application for the introduced constructions, a one to one correspondence between a special case of the quantum light cone lattice equations of sine-Gordon type and free massive fermions on a lattice as constructed in [DV] is derived. Supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 288 e-mail: nadja@math.tu-berlin.de 1 Introduction Let us consider a class of integrable lattice systems defined by an evolution equation of the following type: g u = V 0 (g l \\Gamma g r ) + g d = 0; (1) where...

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We extend the notion of space shifts introduced in [FV3] for certain quantum light cone lattice equations of sine-Gordon type at root of unity (e.g. [FV1][FV2][BKP][BBR]). As a result we obtain a compatibility equation for the roots of central elements within the algebra of observables (also called current algebra). The equation which is obtained by exponentiating these roots is exactly the evolution equation for the "classical background" as described in [BBR]. As an application for the introduced constructions, a one to one correspondence between a special case of the quantum light cone lattice equations of sine-Gordon type and free massive fermions on a lattice as constructed in [DV] is derived. Supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 288 e-mail: nadja@math.tu-berlin.de 1 Introduction Let us consider a class of integrable lattice systems defined by an evolution equation of the following type: g u = V 0 (g l \Gamma g r ) + g d = 0; (1) where...

The discrete quantum Sine-Gordon model at roots of unity remarkably combines a classical integrable system with an integrable quantum spin system, whose parameters obey classical equations of motion. We show that the fundamental

R

-matrix of the model (which satisfies a difference property Yang–Baxter equation) naturally splits into a product of a singular "classical" part and a finite dimensional quantum part. The classical part of the

R

-matrix itself satisfies the quantum Yang–Baxter equation, and therefore can be factored out producing, however, a certain "twist" of the quantum part. We show that the resulting equation exactly coincides with the star-triangle relation of the

N

-state chiral Potts model. The associated spin model on the whole lattice is, in fact, more general than the chiral Potts and reduces to the latter only for the simplest (constant) classical background. In a general case the model is inhomogeneous: its Boltzmann weights are determined by non-trivial background solutions of the equations of motion of the classical discrete sine-Gordon model.

The light-cone approach is reviewed. This method allows to find the underlying quantum field theory for any integrable lattice model in its gapless regime. The relativistic spectrum and S-matrix follows straightforwardly in this way through the Bethe Ansatz. We show here how to derive the infinite number of local commuting and non-local and non-commuting conserved charges in integrable QFT, taking the massive Thirring model (sine-Gordon) as an example. They are generated by quantum monodromy operators and provide a representation of

q-

deformed affine Lie algebras $U_q({\hat\G})$. Based on lectures delivered at the

XXX_q

Karpacz Winter School, Poland, February 14-26, 1994.

We propose TBA integral equations for multiparticle soliton (fermion) states in the Sine-Gordon (massive Thirring) model. This is based on T-system and Y-system equations, which follow from the Bethe Ansatz solution in the light-cone lattice formulation of the model. Even and odd charge sectors are treated on an equal footing, corresponding to periodic and twisted boundary conditions, respectively. The analytic properties of the Y-system functions are conjectured on the basis of the large volume solution of the system, which we find explicitly. A simple relation between the TBA Y-functions and the counting function variable of the alternative non-linear integral equation (Destri-deVega equation) description of the model is given.

In this paper we demonstrate, that the light-cone lattice approach for the Massive-Thirring (sine-Gordon) model, through the quantum inverse scattering method, admits an appropriate framework for computing the finite volume form-factors of local operators of the model. In this work we compute the finite volume diagonal matrix elements of the U(1) conserved current in the pure soliton sector of the theory. Based on the systematic large volume expansion of our results, we conjecture an exact expression for the finite volume expectation values of local operators in pure soliton states. At large volume in leading order these expectation values have the same form as in purely elastic scattering theories, but exponentially small corrections differ from previous Thermodynamic Bethe Ansatz conjectures of purely elastic scattering theories.

In this paper we demonstrate, that the light-cone lattice approach for the Massive-Thirring (sine-Gordon) model, through the quantum inverse scattering method, admits an appropriate framework for computing the finite volume form-factors of local operators of the model. In this work we compute the finite volume diagonal matrix elements of the U(1) conserved current in the pure soliton sector of the theory. Based on the systematic large volume expansion of our results, we conjecture an exact expression for the finite volume expectation values of local operators in pure soliton states. At large volume in leading order these expectation values have the same form as in purely elastic scattering theories, but exponentially small corrections differ from previous Thermodynamic Bethe Ansatz conjectures of purely elastic scattering theories.