Quantum Inverse Scattering Method and Correlation Functions

Abstract

The quantum inverse scattering method is a means of finding exact solutions of two-dimensional models in quantum field theory and statistical physics (such as the sine-Gordon equation or the quantum non-linear Schrödinger equation). These models are the subject of much attention amongst physicists and mathematicians. The present work is an introduction to this important and exciting area. It consists of four parts. The first deals with the Bethe ansatz and calculation of physical quantities. The authors then tackle the theory of the quantum inverse scattering method before applying it in the second half of the book to the calculation of correlation functions. This is one of the most important applications of the method and the authors have made significant contributions to the area. Here they describe some of the most recent and general approaches and include some new results. The book will be essential reading for all mathematical physicists working in field theory and statistical physics.

Parametric spectral statistics in unitary random matrix ensembles: from distribution functions to intra-level correlations

“…However, as we will now show for the case q = 0, it can be rewritten in the form which is (i) more amenable to numerical analysis, and (ii) makes the integrability of the kernel (in the sense discussed in Ref. [25]) manifest. Without loss of generality, we can set ǫ = 0, and shift the variables so as to define the determinant on the quadrant (−∞, 0] ⊗ (−∞, 0].…”

Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions

“…Suppose Ψ| is an eigenstate of Let us define the state |0 = ⊗| ↑ j which now has nothing to do with the eigenstate of the transfer matrix τ (u) as it does in the conventional algebraic Bethe ansatz [13]. On the other hand, the components of the double-row monodromy matrix T(u) act on the states, giving rise to:…”

“…Since Yang and Baxter's pioneering works [4,5,1], the quantum Yang-Baxter equation (QYBE), which define the underlying algebraic structure, has become a cornerstone for constructing and solving the integrable models. There are several well-known methods for deriving the Bethe ansatz (BA) solution of integrable models: the coordinate BA [6,1,7,8,9], the T-Q approach [1,10], the algebraic BA [11,12,13], the analytic BA [14], the functional BA [15] and others [16,17,18,19,20,21,22,23,24,25,26,27,28,29].…”

Integrable inhomogeneous spin chains in generalized Lunin-Maldacena backgrounds

“…The study of the planar dilatation operator's integrability is very important because it not only enable us to test the Maldacena's correspondence as it is an generator of nontrivial integrable models. Exactly solvable models are of interest in high energy physics, condensed matter physics, statistical mechanics and mathematics since the pioneering work of Hans Bethe [6] (see, e.g., [7][8][9][10] for reviews). According to this ansatz the amplitudes of the eigenfunction are expressed by a nonlinear combination of properly defined plane waves.…”