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arXiv:hep-th/9608066 · Journal of Physics A Mathematical and General

The Maxwell - Bloch equation and correlation functions for the penetrable Bose gas

T Kojima, V. E. Korepin

Published Jul 21, 1997• 1 citation
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Abstract

We consider the quantum nonlinear Schrödinger equation in one space and one time dimension. We are interested in the non-free-fermionic case. We consider static temperature-dependent correlation functions. The determinant representation for the correlation functions simplifies in the small-mass limit of the Bose particle. In this limit we describe the correlation functions by the vacuum expectation value of a boson-valued solution for Maxwell - Bloch differential equation. We evaluate long-distance asymptotics of the correlation functions in the small-mass limit.

DOI
10.1088/0305-4470/30/14/020

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We consider the quantum nonlinear Schrödinger equation in one space and one time dimension. We are interested in the non-free-fermionic case. We consider static temperature-dependent correlation functions. The determinant representation for the correlation functions simplifies in the small-mass limit of the Bose particle. In this limit we describe the correlation functions by the vacuum expectation value of a boson-valued solution for Maxwell - Bloch differential equation. We evaluate long-distance asymptotics of the correlation functions in the small-mass limit.
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The quantum nonlinear Schrödinger equation (one dimensional Bose gas) is considered. Classification of representations of Yangians with highest weight vector permits us to represent correlation function as a determinant of a Fredholm integral operator. This integral operator can be treated as the Gelfand-Levitan operator for some new differential equation. These differential equations are written down in the paper. They generalize the fifth Painlève transcendent, which describe equal time, zero temperature correlation function of an impenetrable Bose gas. These differential equations drive the quantum correlation functions of the Bose gas. The Riemann problem, associated with these differential equations permits us to calculate asymp-totics of quantum correlation functions. Quantum correlation function (Fredholm determinant) plays the role of τ functions of these new differential equations. For the impenetrable Bose gas space and time dependent correlation function is equal to τ function of the nonlinear Schrödinger equation itself, For a penetrable Bose gas (finite coupling constant c) the correlator is τ-function of an integro-differentiation equation.
Introduction In the previous chapters correlation functions were represented as determinants of a special form of Fredholm integral operator. First, a differential equation was obtained for the time-independent zero temperature field correlation function for the impenetrable Bose gas (it depends only on one variable—distance). By means of an isomonodromic deformation technique A. Jimbo, T. Miwa, Y. Mori and M. Sato wrote down the ordinary differential equation (of Painlevè type). In the previous chapter differential equations for correlation functions in more general situations (with time, temperature and for finite coupling) were constructed. A.R. Its came up with the idea of applying the matrix Riemann problem for the description of these correlation functions (Fredholm determinants). This permits us to write down partial differential equations (completely integrable) for correlation functions and evaluate the asymptotics of correlation functions. Below we follow this line. Integrable nonlinear partial differential equations for correlation functions of the one-dimensional nonrelativistic Bose gas with point-like repulsion between gas particles were given in section XIV. So quantum correlation functions can be expressed in terms of the solutions of classical nonlinear integrable evolution systems. These classical systems have been investigated already by means of the classical inverse scattering method. Our approach allows one, in particular, to solve the particularly interesting problem of calculating the asymptotics of correlators at large time and distance; this is done in the next section.
Introduction Impenetrable bosons in one space dimension are a special case (at coupling constant c → ∞) of the one-dimensional Bose gas (NS model) considered in detail in Chapter I. The results obtained there for the NS model can be specialized to the case of impenetrable bosons. It is to be said that all the formulæe are considerably simplified in this case. From the point of view of physics this is due to the fact that the δ-function potentials in the N -particle Hamiltonian (I.1.11) are now infinitely strong and the bosons cannot penetrate one another, so that the wave function should be equal to zero if the bosons' coordinates coincide. In this respect impenetrable bosons are rather similar to free fermions (see Appendix I.I). In this chapter correlation functions of impenetrable bosons are considered and their representations as Fredholm determinants of linear integral operators are given. The (very essential) simplification with respect to the general case is due to the fact that now all the auxiliary quantum fields can be set equal to zero (see IX.6), so that the kernels of these operators do not contain quantum fields. The determinant representation for the time-dependent correlator is easy to obtain in this case. Temperature correlation functions are considered as well as zero temperature ones.
We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions ψ(x 1 , 0)ψ † (x 2 , t) ±,T .We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz.For the special case x 1 = 0, we express correlation functions with Neumann boundary conditions ψ(0, 0)ψ † (x 2 , t) +,T , in terms of solutions of nonlinear partial differential equations which were introduced in [1] as a generalization of the nonlinear Schrödinger equations.We generalize the Fredholm minor determinant formulae of ground state correlation functions ψ(x 1 )ψ † (x 2 ) ±,0 in [2], to the Fredholm determinant formulae for the time and temperature dependent correlation functions ψ(x 1 , 0)ψ † (x 2 , t) ±,T , t ∈ R, T ≥ 0.