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.
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.
Abstract Using the ‘Liouville space’ (the space of operators) of the massive Dirac theory, we define mixed-state form factors of U (1) twist fields. We consider mixed states with density matrices diagonal in the asymptotic particle basis. This includes the thermal Gibbs state as well as all generalized Gibbs ensembles of the Dirac theory. When the mixed state is specialized to a thermal Gibbs state, using a Riemann–Hilbert problem and low-temperature expansion, we obtain finite-temperature form factors of U (1) twist fields. We then propose the expression for form factors of U (1) twist fields in general diagonal mixed states. We verify that these form factors satisfy a system of nonlinear functional differential equations, which is derived from the trace definition of mixed-state form factors. At last, under weak analytic conditions on the eigenvalues of the density matrix, we write down the large distance form factor expansions of two-point correlation functions of these twist fields. Using the relation between the Dirac and Ising models, this provides the large-distance expansion of the Rényi entropy (for integer Rényi parameter) in the Ising model in diagonal mixed states.
We study the large distance expansion of correlation functions in the free massive Majorana theory at finite temperature, alias the Ising field theory at zero magnetic field on a cylinder. We develop a method that mimics the spectral decomposition, or form factor expansion, of zero-temperature correlation functions, introducing the concept of ‘finite-temperature form factors’. Our techniques are different from those of previous attempts in this subject. We show that an appropriate analytical continuation of finite-temperature form factors gives form factors in the quantization scheme on the circle. We show that finite-temperature form factor expansions are able to reproduce expansions in form factors on the circle. We calculate finite-temperature form factors of non-interacting fields (fields that are local with respect to the fundamental fermion field). We observe that they are given by a mixing of their zero-temperature form factors and of those of other fields of lower scaling dimension. We then calculate finite-temperature form factors of order and disorder fields. For this purpose, we derive the Riemann–Hilbert problem that completely specifies the set of finite-temperature form factors of general twist fields (order and disorder fields and their descendants). This Riemann–Hilbert problem is different from the zero-temperature one, and so are its solutions. Our results agree with the known form factors on the circle of order and disorder fields.
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 We study the form factors of local operators of integrable QFT’s between states with finite energy density. These states arise, for example, at finite temperature, or from a generalized Gibbs ensemble. We generalize Smirnov’s form factor axioms, formulating them for a set of particle/hole excitations on top of the thermodynamic background, instead of the vacuum. We show that exact form factors can be found as minimal solutions of these new axioms. The thermodynamic form factors can be used to construct correlation functions on thermodynamic states. The expression found for the two-point function is similar to the conjectured LeClair-Mussardo formula, but using the new form factors dressed by the thermodynamic background, and with all singularities properly regularized. We study the different infrared asymptotics of the thermal two-point function, and show there generally exist two different regimes, manifesting massive exponential decay, or effectively gapless behavior at long distances, respectively. As an example, we compute the few-excitations form factors of vertex operators for the sinh-Gordon model.
We study the form factors of local operators of integrable QFT's between states with finite energy density. These states arise, for example, at finite temperature, or from a generalized Gibbs ensemble. We generalize Smirnov's form factor axioms, formulating them for a set of particle/hole excitations on top of the thermodynamic background, instead of the vacuum. We show that exact form factors can be found as minimal solutions of these new axioms. The thermodynamic form factors can be used to construct correlation functions on thermodynamic states. The expression found for the two-point function is similar to the conjectured LeClair-Mussardo formula, but using the new form factors dressed by the thermodynamic background, and with all singularities properly regularized. We study the different infrared asymptotics of the thermal two-point function, and show there generally exist two different regimes, manifesting massive exponential decay, or effectively gapless behavior at long distances, respectively. As an example, we compute the few-excitations form factors of vertex operators for the sinh-Gordon model.
We propose a form factor approach for the computation of the large distance asymptotic behavior of correlation functions in quantum critical (integrable) models. In the large distance regime we reduce the summation over all excited states to one over the particle/hole excitations lying on the Fermi surface in the thermodynamic limit. We compute these sums, over the so-called critical form factors, exactly. Thus we obtain the leading large distance behavior of each oscillating harmonic of the correlation function asymptotic expansion, including the corresponding amplitudes. Our method is applicable to a wide variety of integrable models and yields precisely the results stemming from the Luttinger liquid approach, the conformal field theory predictions and our previous analysis of the correlation functions from their multiple-integral representations. We argue that our scheme applies to a general class of non-integrable quantum critical models as well.