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Form Factors and Correlation Functions

Giuseppe Mussardo

Published Mar 26, 2020• 0 citations
Publisher

Abstract

Abstract At the heart of a quantum field theory are the correlation functions of the various fields. In the case of integrable models, the correlators can be expressed in terms of the spectral series based on the matrix elements on the asymptotic states. These matrix elements, also known as form factors, satisfy a set of functional and recursive equations that can exactly solved in many cases of physical interest. Chapter 19 covers general properties of form factors, Faddeev–Zamolodchikov algebra, symmetric polynomials, kinematical and bound state poles, the operator space and kernel functions, the stress-energy tensor and vacuum expectation values and the Ising model in a magnetic field.

DOI
10.1093/oso/9780198788102.003.0019

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Abstract At the heart of a quantum field theory are the correlation functions of the various fields. In the case of integrable models, the correlators can be expressed in terms of the spectral series based on the matrix elements on the asymptotic states. These matrix elements, also known as form factors, satisfy a set of functional and recursive equations that can exactly solved in many cases of physical interest. Chapter 19 covers general properties of form factors, Faddeev–Zamolodchikov algebra, symmetric polynomials, kinematical and bound state poles, the operator space and kernel functions, the stress-energy tensor and vacuum expectation values and the Ising model in a magnetic field.
Abstract Quantum field theory with boundaries is a very rich subject both from purely theoretical point of view or applicative aims. Chapter 21 presents the basis results relative to critical systems and massive integrable field theories. It covers stress-energy tensor in boundary conformal field theories and integrable deformations of these theories (including the single Virasoro algebra), reflection amplitudes, conformal boundary operators and states (including the Hilbert space and Ishibashi state), non-renormalizable states of the Hilbert space, operator product expansion involving a boundary operator, massive integrable boundary field theory, the massive boundary Ising model, boundary magnetic fields and correlation functions.
Using the "Liouville space'' (the space of operators) of the massive Ising model of quantum field theory, there is a natural definition of form factors in any mixed state. These generalize the usual form factors, and are building blocks for mixed-state correlation functions. We study the cases of mixed states that are diagonal in the asymptotic particle basis, and obtain exact expressions for all mixed-state form factors of order and disorder fields. We use novel techniques based on deriving and solving a system of nonlinear functional differential equations. We then write down the full form factor expansion for mixed-state correlation functions of these fields. Under weak analytic conditions on the eigenvalues of the density matrix, this is an exact large-distance expansion. The form factors agree with the known finite-temperature form factors when the mixed state is specialized to a thermal Gibbs ensemble. Our results can be used to analyze correlation functions in generalized Gibbs ensembles (which occur after quantum quenches). Applying this to the density matrix for non-equilibrium steady states with energy flows, we observe that non-equilibrium form factors have branch cuts in rapidity space. We verify that this is in agreement with a non-equilibrium generalization of the KMS relations, and we conjecture that the leading large-distance behavior of order and disorder non-equilibrium correlation functions contains oscillations in the log of the distance between the fields.
In this thesis we investigate finite size effects in 1+1 dimensional integrable QFT. In particular we consider matrix elements of local operators (finite volume form factors) and vacuum expectation values and correlation functions at finite temperature. In the first part of the thesis we give a complete description of the finite volume form factors in terms of the infinite volume form factors (solutions of the bootstrap program) and the S-matrix of the theory. The calculations are correct to all orders in the inverse of the volume, only exponentially decaying (residual) finite size effects are neglected. We also consider matrix elements with disconnected pieces and determine the general rule for evaluating such contributions in a finite volume. The analytic results are tested against numerical data obtained by the truncated conformal space approach in the Lee-Yang model and the Ising model in a magnetic field. In a separate section we also evaluate the leading exponential correction (the μμμ-term) associated to multi-particle energies and matrix elements. In the second part of the thesis we show that finite volume factors can be used to derive a systematic low-temperature expansion for correlation functions at finite temperature. In the case of vacuum expectation values the series is worked out up to the third non-trivial order and a complete agreement with the LeClair-Mussardo formula is observed. A preliminary treatment of the two-point function is also given by considering the first nontrivial contributions.
For the integrable spin- s XXZ chain we express explicitly any given spin- s form factor in terms of a sum over the scalar products of the spin-1/2 fundamental operators of the algebraic Bethe ansatz. Here, they are given by the operator-valued matrix elements of the monodromy matrix of the spin-1/2 XXZ spin chain. We derive the reduction formula of higher-spin form factors by the fusion method in detail. We call an arbitrary matrix element of a local operator between two Bethe eigenstates a form factor of the operator in this paper. We thus revise the derivation of the higher-spin XXZ form factors given in a previous paper. The revised method has several interesting applications in mathematical physics. For instance, we express the spin- s XXZ correlation function of an arbitrary entry at zero temperature in terms of a sum of multiple integrals.
We review the concept of finite-temperature form factor that was introduced recently by the author in the context of the Majorana theory.Finite-temperature form factors can be used to obtain spectral decompositions of finite-temperature correlation functions in a way that mimics the form-factor expansion of the zero temperature case.We develop the concept in the general factorised scattering set-up of integrable quantum field theory, list certain expected properties and present the full construction in the case of the massive Majorana theory, including how it can be applied to the calculation of correlation functions in the quantum Ising model.In particular, we include the "twisted construction", which was not developed before and which is essential for the application to the quantum Ising model.
ntegrable models in statistical mechanics and quantum field theory constitute a rich research field at the crossroads of modern mathematics and theoretical physics. An important issue to understand is the space of local operators in the system and, ultimately, their correlation functions and form factors. This book is the first published monograph on this subject. It treats integrable lattice models, notably the six-vertex model and the XXZ Heisenberg spin chain. A pair of fermions is introduced and used to create a basis of the space of local operators, leading to the result that all correlation functions at finite distances are expressible in terms of two transcendental functions with rational coefficients. Step-by-step explanations are given for all materials necessary for this construction, ranging from algebraic Bethe ansatz, representations of quantum groups, and the Bazhanov-Lukyanov-Zamolodchikov construction in conformal field theory to Riemann surfaces and their Jacobians. Several examples and applications are given along with numerical results.