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Cambridge University Press eBooks

The Algebraic Bethe Ansatz and Asymptotics of Correlation Functions

V. E. Korepin, N. M. Bogoliubov, Anatoli G. Izergin

Published Aug 19, 1993• 94 citations
Publisher

Abstract

Introduction In the previous chapter we evaluated the asymptotics of the field correlation function using differential equations. In this chapter we present a completely different approach. Instead of determinant representations we shall write some special series (which emphasize the role of the R-matrix). This series is especially efficient for the current ( j ( x ) = Ψ † ( x )Ψ( x )) correlation function. It helps us to evaluate the asymptotics at zero temperature. The asymptotics of temperature correlation functions also can be obtained at any value of coupling constant for the Bose gas. The series for the correlation function is based on the classification of all exactly solvable models (section VII.6) related to the fixed R-matrix. The series explicitly separates the contribution of the R-matrix and of the arbitrary functions a (λ) and d (λ). The Fourier coefficients of the irreducible part depend only on the R-matrix. Let us emphasize once more that in this chapter we shall consider the penetrable Bose gas (0 < c < ∞). In section 1 the algebraic foundation of the new approach to correlation functions is given. In section 2 the series representation for the current correlator 〈 j ( x ) j (0)〉 at zero temperature (in the thermodynamic limit) is given. In section 3 temperature correlations (for the penetrable Bose gas, 0 < c < ∞) are constructed. In section 4 explicit formulæe for asymptotics are presented. In section 5 the emptiness formation probability (the probability of absence of particles in some space interval due to thermal fluctuations) is evaluated.

DOI
10.1017/cbo9780511628832.022

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We study models of quantum-statistical mechanics which can be solved by the algebraic Bethe ansatz. The general method of calculating correlation functions is based on the method of determinant representations. The auxiliary Fock space and auxiliary Bose fields are introduced in order to remove the two-body scattering and represent correlation functions as a mean value of a determinant of a Fredholm integral operator; the representation has a simple form for large space and time separations. In this paper we explain how to calculate the mean value in the auxiliary Fock space of an asymptotic expression of the Fredholm determinant. It is necessary for the evaluation of the asymptotic form of the physical correlation functions.

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Introduction In the previous chapter we evaluated the asymptotics of the field correlation function using differential equations. In this chapter we present a completely different approach. Instead of determinant representations we shall write some special series (which emphasize the role of the R-matrix). This series is especially efficient for the current ( j ( x ) = Ψ † ( x )Ψ( x )) correlation function. It helps us to evaluate the asymptotics at zero temperature. The asymptotics of temperature correlation functions also can be obtained at any value of coupling constant for the Bose gas. The series for the correlation function is based on the classification of all exactly solvable models (section VII.6) related to the fixed R-matrix. The series explicitly separates the contribution of the R-matrix and of the arbitrary functions a (λ) and d (λ). The Fourier coefficients of the irreducible part depend only on the R-matrix. Let us emphasize once more that in this chapter we shall consider the penetrable Bose gas (0 < c < ∞). In section 1 the algebraic foundation of the new approach to correlation functions is given. In section 2 the series representation for the current correlator 〈 j ( x ) j (0)〉 at zero temperature (in the thermodynamic limit) is given. In section 3 temperature correlations (for the penetrable Bose gas, 0 < c < ∞) are constructed. In section 4 explicit formulæe for asymptotics are presented. In section 5 the emptiness formation probability (the probability of absence of particles in some space interval due to thermal fluctuations) is evaluated.
The algebraic Bethe Ansatz is a powerful method for the calculation of quantum correlation functions. In Part III we shall start this calculation. We shall arrive at an extremely important conclusion: quantum correlation functions can be represented as determinants of certain matrices. The dimension of these matrices is equal to the number of particles in the ground state. In order to better understand the nature of this matrix, one should look once more through section 6 of Chapter VI and section 10 of Chapter VII, where the determinant representation of the partition function for the six-vertex model with a special type of boundary conditions (domain wall) was obtained. Starting from determinant formulæe (VII.10.1) and (VII.10.2) and using also the formula for the determinant of the sum of two matrices (in the Appendix to Chapter IX), one can reproduce all the determinant formulæe in Part III. In the thermodynamic limit, the correlation functions will be represented as determinants of a Predholm integral operator. In Part IV, we shall explain how to use this determinant representation to write down the differential equation for a quantum correlation function. In Part IV we shall also discover that the differential equation for a quantum correlation function is closely related to the initial classical nonlinear differential equation, which was quantized (in our example this is the nonlinear Schrödinger equation). The quantum correlation functions play the role of the τ functions of a classical differential 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 The aim of this chapter is to obtain the determinant representation for the equal-time correlation function 〈Ψ † ( x )Ψ(0)〉 in the nonrelativistic one-dimensional Bose gas (NS model). To apply the approach of the previous chapters, one has to introduce local fields Ψ † ( x ), Ψ(0) into the two-site generalized model. This is done in section 1. In section 2, the formula for the matrix elements of the operator Ψ † ( x )Ψ(0) in the generalized model are derived. In terms of the auxiliary quantum fields introduced in section 3, this formula is transformed to the vacuum mean value of the determinant with respect to the vacuum in the auxiliary Fock space. The determinant representation for the mean value of the operator Ψ † ( x )Ψ(0) with respect to the N -particle Bethe eigenstate is derived in section 4. The derivation is similar to that of the previous chapter. It is also similar to that of section 6 of Chapter IX. The derivation is based on the representation of the determinant of the sum of two matrices in terms of the minors of the individual matrices. The explicit formula is given in the Appendix to Chapter IX. In section 5 expressions for the correlation function in the ground state of the one-dimensional Bose gas (zero and finite temperature) are obtained and the thermodynamic limit is also considered. Local fields in the generalized model To consider local quantum fields Ψ † ( x )Ψ(0) in the frame of the two-site generalized model, one has to clarify the structure of the monodromy matrices T(1|λ), T(2|λ) (see section XI.1, formulæe (XI.1.6)–(XI.1.12)).
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.
Introduction The natural language for the description of correlation functions in exactly solvable models is the language of differential equations. The determinant representation of correlation functions in the previous chapter can be used to obtain these differential equations. An ordinary differential equation (of the Painlevètype) for the equal-time, zero temperature correlation function of the impenetrable Bose gas was obtained in. In that paper (starting from Lenard's formula for the two-point zero temperature equal-time correlator of impenetrable bosons) it was shown that the correlator is described by an ordinary differential equation (reducible to the Painlevè trascendent), and multipoint zero temperature equal-time correlators were also described as solutions of differential equations. The considerations in that paper were based on the spectral isomonodromic deformation method. Our approach here is different. We begin by considering the logarithm of the determinant of the linear integral operator entering the representation for the correlator as a τ-function for some integrable nonlinear evolutionary system. Then the linear integral equations playing the role of the Gel'fand-Levitan-Marchenko equations for this integrable system are constructed. The differential equations for correlators are derived, in fact, from these integral equations (it is worth mentioning that the idea of obtaining the differential equations from the integral ones can be found in. This approach permits one to include time- and temperature-dependent correlators for impenetrable bosons in the same framework. It is remarkable that space- and time-dependent correlation functions are derived by partial differential equations which almost coincide with the original classical nonlinear Schrödinger equation, the only difference being that complex involution is dropped.
We study models of quantum-statistical mechanics which can be solved by the algebraic Bethe ansatz. The general method of calculating correlation functions is based on the method of determinant representations. The auxiliary Fock space and auxiliary Bose fields are introduced in order to remove the two-body scattering and represent correlation functions as a mean value of a determinant of a Fredholm integral operator; the representation has a simple form for large space and time separations. In this paper we explain how to calculate the mean value in the auxiliary Fock space of an asymptotic expression of the Fredholm determinant. It is necessary for the evaluation of the asymptotic form of the physical correlation functions.