Integrable Quantum Field Theories Research Articles

Completing the bootstrap program for TT― -deformed massive integrable quantum field theories

In recent years a considerable amount of attention has been devoted to the investigation of 2D quantum field theories perturbed by certain types of irrelevant operators. These are the composite field TT― —constructed out of the components of the stress-energy tensor—and its generalisations—built from higher-spin conserved currents. The effect of such perturbations on the infrared and ultraviolet properties of the theory has been extensively investigated. In the context of integrable quantum field theories, a fruitful perspective is that of factorised scattering theory. In fact, the above perturbations were shown to preserve integrability. The resulting deformed scattering matrices—extensively analysed with the thermodynamic Bethe ansatz—provide the first step in the development of a bootstrap program. In this paper we present a systematic approach to computing matrix elements of operators in generalised TT― -perturbed models, based on employing the standard form factor program. We show that for theories with diagonal scattering and certain types of fields the deformed form factors, factorise into the product of the undeformed ones and of a perturbation- and theory-dependent term. From these solutions, correlation functions can be obtained and their asymptotic properties studied.

Symmetry-resolved measures in quantum field theory: A short review

In this short review, we present the key definitions, ideas and techniques involved in the study of symmetry resolved entanglement measures, with a focus on the symmetry resolved entanglement entropy. In order to be able to define such entanglement measures, it is essential that the theory under study possess an internal symmetry. Then, symmetry-resolved entanglement measures quantify the contribution to a particular entanglement measure that can be associated to a chosen symmetry sector. Our review focuses on conformal (gapless/massless/critical) and integrable (gapped/massive) quantum field theories, where the leading computational technique employs symmetry fields known as (composite) branch point twist fields.

TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document}-deformed entanglement entropy for IQFT

We calculate the TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document}-deformed entanglement entropy for integrable quantum field theories (IQFTs) using the form factor bootstrap approach. We solve the form factor bootstrap axioms for the branch-point twist fields and obtain the deformed form factors. Using these form factors, we compute the deformed von Neuman entropy up to two particle contributions. The solution of the form factor axioms is not unique. We find that for the simplest solution of the bootstrap axioms, the UV limit of the entanglement entropy takes the same form as the undeformed one, but the effective central charge is deformed. For solutions with additional CDD-like factors, we can have different behaviors. The IR corrections, which only depends on the particle spectrum is untouched.

Finite volume form factors in integrable theories

We develop a new method to calculate finite size corrections for form factors in two-dimensional integrable quantum field theories. We extract these corrections from the excited state expectation value of bilocal operators in the limit when the operators are far apart. We elaborate the finite size effects explicitly up to the 3rd Lüscher order and conjecture the structure of the general form. We also fully recover the explicitly known massive fermion finite volume form factors.

TT¯ Deformation: A Lattice Approach

Integrable quantum field theories can be regularized on the lattice while preserving integrability. The resulting theories on the lattice are integrable lattice models. A prototype of such a regularization is the correspondence between a sine-Gordon model and a six-vertex model on a light-cone lattice. We propose an integrable deformation of the light-cone lattice model such that in the continuum limit we obtain the TT¯-deformed sine-Gordon model. Under this deformation, the cut-off momentum becomes energy dependent and the underlying Yang–Baxter integrability is preserved. Therefore, this deformation is integrable but non-local: similar to the TT¯ deformation of quantum field theory.

Entanglement entropy from form factors in textrm{T}overline{textrm{T}} -deformed integrable quantum field theories

In two recent papers [1, 2] we have proposed a program of study which allows us to compute the correlation functions of local and semi-local fields in generalised textrm{T}overline{textrm{T}} -deformed integrable quantum field theories. This new program, based on the construction of form factors, opens many avenues for future study, one of which we address in this paper: computing entanglement measures employing branch point twist fields. Indeed, over the past 15 years, this has become one the leading methods for the computation of entanglement measures, both in conformal field theory and integrable quantum field theory. Thus the generalisation of this program to textrm{T}overline{textrm{T}} -perturbed theories offers a promising new tool for the study of entanglement measures in the presence of irrelevant perturbations. In this paper, we show that the natural two-particle form factor solution for branch point twist fields in replica theories with diagonal scattering admits a simple generalisation to a solution for textrm{T}overline{textrm{T}} -perturbed theories. Starting with this solution, some of the known properties of entanglement measures in massive integrable quantum field theories can be generalised to the perturbed models. We show this by focusing on the Ising field theory. During the completion of this paper, we became aware of the recent publication [3] where the same problem has been addressed.

Extending the thermodynamic form factor bootstrap program: multiple particle-hole excitations, crossing symmetry, and reparameterization invariance

In this study, we further the thermodynamic bootstrap program which involves a set of recently developed ideas used to determine thermodynamic form factors of local operators in integrable quantum field theories. These form factors are essential building blocks for dynamic correlation functions at finite temperatures or non-equilibrium stationary states. In this work we extend this program in three ways. Firstly, we demonstrate that the conjectured annihilation pole axiom is valid in the low energy particle-hole excitations. Secondly, we introduce a crossing relation, which establishes a connection between form factors with different excitation content. Typically, the crossing relation is a consequence of Lorentz invariance, but due to the finite energy density of the considered states, Lorentz invariance is broken. Nonetheless a crossing relation involving excitations with both particles and holes can established using the finite volume representation of the thermodynamic form factors. Finally, we demonstrate that the thermodynamic form factors satisfy a reparameterization invariance, an invariance which encompasses crossing. Reparameterization invariance exploits the fact that the details of the representation of the thermodynamic state are unimportant. In the course of developing these results, we demonstrate the internal consistency of the thermodynamic form factor bootstrap program in a number of ways. Finally, we provide explicit computations of form factors of conserved charges and densities with crossed excitations and show our results can be used to infer information about thermodynamic form factors in the Lieb-Liniger model.

Integrable crosscap states: from spin chains to 1D Bose gas

The notion of a crosscap state, a special conformal boundary state first defined in 2d CFT, was recently generalized to 2d massive integrable quantum field theories and integrable spin chains. It has been shown that the crosscap states preserve integrability. In this work, we first generalize this notion to the Lieb-Liniger model, which is a prototype of integrable non-relativistic many-body systems. We then show that the defined crosscap state preserves integrability. We derive the exact overlap formula of the crosscap state and the on-shell Bethe states. As a byproduct, we prove the conjectured overlap formula for integrable spin chains rigorously by coordinate Bethe ansatz. It turns out that the overlap formula for both models take the same form as a ratio of Gaudin-like determinants with a trivial prefactor. Finally we study quench dynamics of the crosscap state, which turns out to be surprisingly simple. The stationary density distribution is simply a constant. We also derive the analytic formula for dynamical correlation functions in the Tonks-Girardeau limit.

The full analytic trans-series in integrable field theories

We analyze a family of generalized energy densities in integrable quantum field theories in the presence of an external field coupled to a conserved charge. By using the Wiener-Hopf technique to solve the linear thermodynamic Bethe ansatz equations we derive the full analytic trans-series for these observables in terms of a perturbatively defined basis. We show how to calculate these basis elements to high orders analytically and reveal their complete resurgence structure. We demonstrate that the physical value of the energy density is obtained by the median resummation of the perturbative series.

Comment on the Vacuum Energy Density for λϕ4 Theory in d Spacetime Dimensions

In a recent article we showed that the vacuum energy density in two spacetime dimensions for a wide variety of integrable quantum field theories has the form ρvac=−m2/2g where m is a physical mass and g is a generalized coupling, where in the free field limit g→0, ρvac diverges. This vacuum energy density has the form ⟨Tμν⟩=−ρvacgμν, and has previously been considered as a contribution to the stress energy tensor in Einstein’s gravity as a “cosmological constant”. We speculated that in four spacetime dimensions ρvac takes a similar form ρvac=−m4/2g, but did not support this idea in any specific model. In this article, we study this problem for λϕ4 theory in d spacetime dimensions. We show how to obtain the exactρvac for the sinh–Gordon theory in the weak coupling limit by using a saddle point approximation. This calculation indicates that the vacuum energy can be well-defined, positive or negative, without spontaneous symmetry breaking. We also show that ρvac satisfies a Callan–Symanzik type of renormalization group equation. For the most interesting case physically, ρvac is positive and can arise from a marginally relevant negative coupling g and the vacuum energy flows to zero at low energies.

Mingling of the infrared and ultraviolet and the “cosmological constant” for interacting QFT in 2d

We propose a proper definition of the vacuum expectation value of the stress energy tensor 〈0| Tμν| 0〉 for integrable quantum field theories in two spacetime dimensions, which is the analog of the cosmological constant in 4d. For a wide variety of models, massive or massless, we show {rho}_{textrm{vac}}=-{m}^2/2mathfrak{g} exactly, where mathfrak{g} is a generalized coupling which we compute and m is a basic mass scale. The kinds of models we consider are the massive sinh-Gordon and sine-Gordon theories and perturbations of the Yang-Lee and 3-state Potts models, pure Toverline{T} perturbations of infra-red QFT’s, and UV completions of the latter which are massless flows between UV and IR fixed points. In the massive case m is the physical mass of the lightest particle and mathfrak{g} is related to parameters in the 2-body S-matrix. In some examples ρvac = 0 due to a fractional supersymmetry. For massless cases, m can be a scale of spontaneous symmetry breaking. The “cosmological constant problem” generically arises in the free field limit mathfrak{g} → 0, thus interactions can potentially resolve the problem at least for most cases considered in this paper. We speculate on extensions of these results to 4 spacetime dimensions and propose {rho}_{textrm{vac}}={m}^4/2mathfrak{g} , however without integrability we cannot yet propose a precise manner in which to calculate mathfrak{g} . Nevertheless, based on cosmological data on ρvac, if mathfrak{g} ~ 1 then it is worth pointing out that the lightest mass particle is on the order of experimental values of proposed neutrino masses.

On convergence of form factor expansions in the infinite volume quantum Sinh-Gordon model in 1+1 dimensions

This paper develops a technique allowing one to prove the convergence of a class of series of multiple integrals which corresponds to the form factor expansion of space-like separated two-point functions in the 1+1 dimensional massive integrable Sinh-Gordon quantum field theory.

Asymptotics in an asymptotic CFT

In this work we illustrate the resurgent structure of the λ-deformation; a two-dimensional integrable quantum field theory that has an RG flow with an SU(N)k Wess-Zumino-Witten conformal fixed point in the UV. To do so we use modern matched asymptotic techniques applied to the thermodynamic Bethe ansatz formulation to compute the free energy to 38 perturbative orders in an expansion of large applied chemical potential. We find numerical evidence for factorial asymptotic behaviour with both alternating and non-alternating character which we match to an analytic expression. A curiosity of the system is that the leading non-alternating factorial growth vanishing when k divides N. The ambiguities associated to Borel resummation of this series are suggestive of non-perturbative contributions. This is verified with an analytic study of the TBA system demonstrating a cancellation between perturbative and non-perturbative ambiguities.

Deforming the ODE/IM correspondence with textrm{T}overline{textrm{T}}

The ODE/IM correspondence is an exact link between classical and quantum integrable models. The primary purpose of this work is to show that it remains valid after textrm{T}overline{textrm{T}} perturbation on both sides of the correspondence. In particular, we prove that the deformed Lax pair of the sinh-Gordon model, obtained from the unperturbed one through a dynamical change of coordinates, leads to the same Burgers-type equation governing the quantum spectral flow induced by textrm{T}overline{textrm{T}} . Our main conclusions have general validity, as the analysis may be easily adapted to all the known ODE/IM examples involving integrable quantum field theories.

Symmetry resolved entanglement of excited states in quantum field theory. Part I. Free theories, twist fields and qubits

The excess entanglement resulting from exciting a finite number of quasiparticles above the ground state of a free integrable quantum field theory has been investigated quite extensively in the literature. It has been found that it takes a very simple form, depending only on the number of excitations and their statistics. There is now mounting evidence that such formulae also apply to interacting and even higher-dimensional quantum theories. In this paper we study the entanglement content of such zero-density excited states focusing on the symmetry resolved entanglement, that is on 1+1D quantum field theories that possess an internal symmetry. The ratio of charged moments between the excited and grounds states, from which the symmetry resolved entanglement entropy can be obtained, takes a very simple and universal form, which in addition to the number and statistics of the excitations, now depends also on the symmetry charge. Using form factor techniques, we obtain both the ratio of moments and the symmetry resolved entanglement entropies in complex free theories which possess U(1) symmetry. The same formulae are found for simple qubit states.

Y-systems for generalised Gibbs ensembles in integrable quantum field theory

The thermodynamic Bethe ansatz approach to the study of integrable quantum field theories was introduced in the early 90s. Since then it has been known that the thermodynamic Bethe ansatz equations can be recast in the form of Y-systems. These Y-systems have a number of interesting properties, notably in the high-temperature limit their solutions are constants from which the central charge of the ultraviolet fixed point can be obtained and they are typically periodic functions, with period proportional to the dimension of the perturbing field. In this letter we discuss the derivation of Y-systems when the standard thermodynamic Bethe ansatz equations are replaced by generalised versions, describing generalised Gibbs ensembles. We shown that for many integrable quantum field theories, there is a large class of distinct generalised Gibbs ensembles which share the same Y-system.

Renormalization in TT¯ -deformed nonintegrable theories

That the exact quantum S-matrix of $\text{T}\bar{\text{T}}$-deformed field theories is known has interesting consequences for their perturbative renormalisation. Recent investigations into the interplay between renormalisation and integrable deformations have focused on those cases where the (undeformed) seed theories are integrable. In this paper, we study the perturbative renormalisation of non-integrable $\text{T}\bar{\text{T}}$-deformed field theories, focusing on a theory of scalars with a quartic interaction. We show that in this theory, the 4-point 1-loop S-matrix exhibits features similar to its integrable cousins. We also carry out a similar analysis for the case of an integrable quantum field theory comprising a pair of chiral bosons. Our discussion of this latter theory supplies the first explicit example of a perturbative field theory computation in the formalism due to Sen.

Entanglement dynamics of thermofield double states in integrable models

We study the entanglement dynamics of thermofield double (TFD) states in integrable spin chains and quantum field theories. We show that, for a natural choice of the Hamiltonian eigenbasis, the TFD evolution may be interpreted as a quantum quench from an initial state which is low-entangled in the real-space representation and displays a simple quasiparticle structure. Based on a semiclassical picture analogous to the one developed for standard quantum quenches, we conjecture a formula for the entanglement dynamics, which is valid for both discrete and continuous integrable field theories, and expected to be exact in the scaling limit of large space and time scales. We test our conjecture in two prototypical examples of integrable spin chains, where numerical tests are possible. First, in the XY-model, we compare our predictions with exact results obtained by mapping the system to free fermions, finding excellent agreement. Second, we test our conjecture in the interacting XXZ Heisenberg model, against numerical iTEBD calculations. For the latter, we generally find good agreement, although, for some range of the system parameters and within the accessible simulation times, some small discrepancies are visible, which we attribute to finite-time effects.

Toverline{T} deformations and the width of fundamental particles

We provide a simple geometric meaning for deformations of so-called Toverline{T} type in relativistic and non-relativistic systems. Deformations by the cross products of energy and momentum currents in integrable quantum field theories are known to modify the thermodynamic Bethe ansatz equations by a “CDD factor”. In turn, CDD factors may be interpreted as additional, fixed shifts incurred in scattering processes: a finite width added to the fundamental particles (or, if negative, to the free space between them). We suggest that this physical effect is a universal way of understanding Toverline{T} deformations, both in classical and quantum systems. We first show this in non-relativistic systems, with particle conservation and translation invariance, using the deformation formed out of the densities and currents of particles and momentum. This holds at the level of the equations of motion, and for any interaction potential, integrable or not. We then argue, and show by similar techniques in free relativistic particle systems, that Toverline{T} deformations of relativistic systems produce the equivalent phenomenon, accounting for length contractions. We also show that, in both the relativistic and non-relativistic cases, the width of particles is equivalent to a state-dependent change of metric, where the distance function discounts the particles’ widths, or counts the additional free space. This generalises and explains the known field-dependent coordinate change describing Toverline{T} deformations. The results connect such deformations with generalised hydrodynamics, where the relations between scattering shifts, widths of particles and state-dependent changes of metric have been established.

Thermodynamics of perturbations of some single particle field theories

We study the thermodynamic Bethe ansatz (TBA) equations for pure perturbations of some simple integrable quantum field theories with a single bosonic or fermionic particle, in particular the massive sinh-Gordon model and its ultraviolet (UV) limit which is a deformation of the conformally invariant free massless boson. Whereas the TBA equations for deformations of massive theories are in principle known, the TBA equations we propose for the deformations of conformal field theories are relatively new and require a special factorization in rapidity variables of the CDD factor for the scattering of the massless particles. The latter TBA equations can be solved exactly and reproduce the known results for the ground state energy on a cylinder of circumference R which were previously obtained using different methods based for instance on the Burgers differential equation. Special attention is paid to the c-theorem in this context which is discussed in some detail. For positive infra-red central charge c IR, for flows consistent with the c-theorem the ground state energy develops a (previously known) square-root singularity toward the UV, which strongly suggests the theories are UV incomplete in these physically important cases. We suggest that the singularity indicates a tachyonic vacuum instability. Other cases with c IR < 0 do not have this singularity are interpreted as being UV complete with c UV = 0. We extend our results to a continuously variable c IR by introducing a chemical potential and suggest this as a possible toy model for the perturbed Liouville theory.