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arXiv:math-ph/0401003 · Journal of Physics A Mathematical and General

Exact solution of a 1D quantum many-body system with momentum-dependent interactions

Harald Grosse, Edwin Langmann A, Cornelius Paufler

Published Apr 5, 2004• 33 citations
arXivHTMLPDF

Abstract

We discuss a 1D many-body model of distinguishable particles with local, momentum dependent two-body interactions. We show that the restriction of this model to fermions corresponds to the non-relativistic limit of the massive Thirring model. This fermion model can be solved exactly by a mapping to the 1D boson gas with inverse coupling constant. We provide evidence that this mapping is the non-relativistic limit of the duality between the massive Thirring model and the quantum sine-Gordon model. We also investigate the question if the generalization of this model to distinguishable particles is exactly solvable by the coordinate Bethe ansatz and find that this is not the case.

DOI
10.1088/0305-4470/37/16/008

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We discuss a 1D many-body model of distinguishable particles with local, momentum dependent two-body interactions. We show that the restriction of this model to fermions corresponds to the non-relativistic limit of the massive Thirring model. This fermion model can be solved exactly by a mapping to the 1D boson gas with inverse coupling constant. We provide evidence that this mapping is the non-relativistic limit of the duality between the massive Thirring model and the quantum sine-Gordon model. We also investigate the question if the generalization of this model to distinguishable particles is exactly solvable by the coordinate Bethe ansatz and find that this is not the case.
As is well known, there exists a four-parameter family of local interactions in 1D. We interpret these parameters as coupling constants of delta-type interactions which include different kinds of momentum-dependent terms, and determine all cases leading to many-body systems of distinguishable particles which are exactly solvable by the coordinate Bethe ansatz. We find two such families of systems, one with two independent coupling constants deforming the well-known delta-interaction model to non-identical particles, and the other with a particular one-parameter combination of the delta and (the so-called) delta-prime interaction. We also find that the model of non-identical particles gives rise to a somewhat unusual solution of the Yang–Baxter relations. For the other model we write down explicit formulae for all eigenfunctions.
As is well known, there exists a four-parameter family of local interactions in 1D. We interpret these parameters as coupling constants of delta-type interactions which include different kinds of momentum-dependent terms, and determine all cases leading to many-body systems of distinguishable particles which are exactly solvable by the coordinate Bethe ansatz. We find two such families of systems, one with two independent coupling constants deforming the well-known delta-interaction model to non-identical particles, and the other with a particular one-parameter combination of the delta and (the so-called) delta-prime interaction. We also find that the model of non-identical particles gives rise to a somewhat unusual solution of the Yang–Baxter relations. For the other model we write down explicit formulae for all eigenfunctions.
We consider two particular one-dimensional quantum many-body systems with local interactions related to the root system CN. Both models describe identical particles moving on the half-line with nontrivial boundary conditions at the origin, but in the first model the particles interact with the delta interaction while in the second via a particular momentum dependent interaction commonly known as delta-prime interaction. We show that the Bethe ansatz solution of the delta-interaction model is consistent even for the general case where the particles are distinguishable, whereas for the delta-prime interaction it only is consistent and nontrivial in the fermion case. We also establish a duality between the bosonic delta- and the fermionic delta-prime model, and we elaborate on the physical interpretations of these models.
This thesis is devoted to the study of various examples of exactly solved quantum many-body systems in one-dimension. It is divided into two parts: the first provides background and complementary results to the second, which consists of three scientific papers. The first paper concerns a particu- lar extension, corresponding to the root system CN, of the delta-interaction model. We prove by construction that its exact solution, even in the gen- eral case of distinguishable particles, can be obtained by the coordinate Bethe ansatz. We also elaborate on the physical interpretation of this model. It is well known that the delta-interaction is included in a four parameter family of local interactions. In the second paper we interpret these parameters as cou- pling constants of certain momentum dependent interactions and determine all cases leading to a many-body system of distinguishable particles which can be solved by the coordinate Bethe ansatz. In the third paper we consider the so-called rational Calogero-Sutherland model, describing an arbitrary number of particles on the real line, confined by a harmonic oscillator potential and interacting via a two-body interaction proportional to the inverse square of the inter-particle distance. We construct a novel solution algorithm for this model which enables us to obtain explicit formulas for its eigenfunctions. We also show that our algorithm applies, with minor changes, to all extensions of the rational Calogero-Sutherland model which correspond to a classical root system.
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.
We study a massive Thirring-like model in 2-dimensional space-time, which contains two different species of fermions. This model is a field theoretical version of the quantum mechanical model originally proposed by Glöckle, Nogami and Fukui, where different fermions interact with each other through δ-function potentials. We derive a corresponding boson model by the bosonization technique in the path integral formulation. This is a simple but non-trivial extension of the freedom of the bosonization technique. Operator correspondences between fermion and boson fields are given. One of these could not be realistically expected from the naive correspondence of the original single-species models. It is essential for this point that in our model fermions of the same kind do not interact with each other directly. We find that for a specific value of the coupling constant, one boson field becomes free while the other is a Sine-Gordon field. For this case, therefore, our two-species model is equivalent to the ordinary Sine-Gordon model of a single boson field.