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Journal of the Physical Society of Japan

Massless Thirring Model and Bethe Ansatz Wavefunction

Yasushi Komori, Miki Wadati

Published Mar 15, 1996• 6 citations
Publisher

Abstract

The quantum one-dimensional massless Thirring model is studied. A transformation which casts the massless Thirring model into a quadratic form is found. The transformed field operators satisfy interesting commutation relations, fermion anti-commutation relations among the same kinds of fields and anyon-like commutation relations between different kinds. Bethe Ansatz wavefunctions are constructed algebraically using the transformation and the commutation relations in coordinate space.

DOI
10.1143/jpsj.65.722

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The quantum one-dimensional massless Thirring model is studied. A transformation which casts the massless Thirring model into a quadratic form is found. The transformed field operators satisfy interesting commutation relations, fermion anti-commutation relations among the same kinds of fields and anyon-like commutation relations between different kinds. Bethe Ansatz wavefunctions are constructed algebraically using the transformation and the commutation relations in coordinate space.
Introduction The algebraic Bethe Ansatz is presented in this chapter. This is an important generalization of the coordinate Bethe Ansatz presented in Part I, and is one of the essential achievements of QISM. The algebraic Bethe Ansatz is based on the idea of constructing eigenfunctions of the Hamiltonian via creation and annihilation operators acting on a pseudovacuum. The matrix elements of the monodromy matrix play the role of these operators. The transfer matrix (the sum of the diagonal elements of the monodromy matrix) commutes with the Hamiltonian; thus constructing eigenfunctions of τ(µ) determines the eigenfunctions of the Hamiltonian. The basis of the algebraic Bethe Ansatz is stated in section 1. The commutation relations between matrix elements of the monodromy matrix are specified by the R-matrix. The explicit form of the commutation relations allows the construction of eigenfunctions of the transfer matrix (the trace of the monodromy matrix). (Recall that the Hamiltonian may also be obtained from the transfer matrix via the trace identities.) Further developments of the algebraic Bethe Ansatz necessary for the computation of correlation functions are given in section 2. The general scheme is illustrated with some examples in section 3. The NS model, the sine-Gordon model and spin models are considered in detail. The Pauli principle for interacting one-dimensional bosons plays an important role in constructing the ground state of the system and is discussed in section 4. The eigenvalues of the shift operator acting on the monodromy matrix are calculated in section 5.
We develop a generalized coordinate Bethe-ansatz framework for the massive Thirring model with an explicitly time-dependent fermion mass. Unlike the generalized Bethe ansatz for chiral quantum field theories, where particle transport acts in internal spin spaces and leads to quantum Knizhnik-Zamolodchikov equations, the present construction acts directly on particle coordinates and physical rapidities. Its central new feature is a rapidity lattice. Because the time-dependent mass couples right- and left-moving components already in the one-particle sector, a fixed-rapidity block is not closed under spatial winding. Periodicity therefore requires all sectors generated by the finite shift β→β−pβ\toβ-pβ→β−p. Particle exchange relates amplitudes on this lattice through scalar analytic difference equations of Ruijsenaars type, whose solutions determine the complete time-dependent many-body wavefunction. Consistency of the coordinate construction also selects an exponential evolution of the bare mass, providing a new realization of the time dependent integrability-renormalization group correspondence. In the static limit, the rapidity-lattice construction reduces to the conventional massive-Thirring Bethe equations. Through the massive-Thirring/sine Gordon equivalence, the result also connects this exact quantum construction to the generalized space-time dependent Lax formalism.