Thermodynamic Bethe Ansatz Research Articles

The folded spin-1/2 XXZ model: II. Thermodynamics and hydrodynamics with a minimal set of charges

We study the (dual) folded spin-1/2 XXZ model in the thermodynamic limit. We focus, in particular, on a class of ``local'' macrostates that includes Gibbs ensembles. We develop a thermodynamic Bethe Ansatz description and work out generalised hydrodynamics at the leading order. Remarkably, in the ballistic scaling limit the junction of two local macrostates results in a discontinuity in the profile of essentially any local observable.

Generalized hydrodynamics in complete box-ball system for $U_q(\widehat{sl}_n)$

We introduce the complete box-ball system (cBBS), which is an integrable cellular automaton on 1D lattice associated with the quantum group U_q(\widehat{sl}_n)Uq(sl̂n). Compared with the conventional (n-1)(n−1)-color BBS, it enjoys a remarkable simplification that scattering of solitons is totally diagonal. We also submit the cBBS to randomized initial conditions and study its non-equilibrium behavior by thermodynamic Bethe ansatz and generalized hydrodynamics. Excellent agreement is demonstrated between theoretical predictions and numerical simulation on the density plateaux generated from domain wall initial conditions including their diffusive broadening.

A general definition of $JT_a$ -- deformed QFTs

We propose a general path-integral definition of two-dimensional quantum field theories deformed by an integrable, irrelevant vector operator constructed from the components of the stress tensor and those of a U(1) current. The deformed theory is obtained by coupling the original QFT to a flat dynamical gauge field and ``half'' a flat dynamical vielbein. The resulting partition function is shown to satisfy a geometric flow equation, which perfectly reproduces the flow equations for the deformed energy levels that were previously derived in the literature. The S-matrix of the deformed QFT differs from the original S-matrix only by an overall phase factor that depends on the charges and momenta of the external particles, thus supporting the conjecture that such QFTs are UV complete, although intrinsically non-local. For the special case of an integrable QFT, we check that this phase factor precisely reproduces the change in the finite-size spectrum via the Thermodynamic Bethe Ansatz equations.

Periodicity and integrability for the cube recurrence

Zamolodchikov periodicity is a property of T- and Y-systems, arising in the thermodynamic Bethe ansatz. Zamolodchikov integrability was recently considered as its affine analog in our joint work with P. Pylyavskyy. Here we prove periodicity and integrability for similar discrete dynamical systems based on the cube recurrence, also known as the discrete BKP equation. The periodicity part was conjectured by Henriques in 2007.

Free energy fluxes and the Kubo–Martin–Schwinger relation

A general, multi-component Eulerian fluid theory is a set of nonlinear, hyperbolic partial differential equations. However, if the fluid is to be the large-scale description of a short-range many-body system, further constraints arise on the structure of these equations. Here we derive one such constraint, pertaining to the free energy fluxes. The free energy fluxes generate expectation values of currents, akin to the specific free energy generating conserved densities. They fix the equations of state and the Euler-scale hydrodynamics, and are simply related to the entropy currents. Using the Kubo–Martin–Schwinger relations associated to many conserved quantities, in quantum and classical systems, we show that the associated free energy fluxes are perpendicular to the vector of inverse temperatures characterising the state. This implies that all entropy currents can be expressed as averages of local observables. In few-component fluids, it implies that the averages of currents follow from the specific free energy alone, without the use of Galilean or relativistic invariance. In integrable models, in implies that the thermodynamic Bethe ansatz must satisfy a unitarity condition. The relation also guarantees physical consistency of the Euler hydrodynamics in spatially-inhomogeneous, macroscopic external fields, as it implies conservation of entropy, and the local-density approximated Gibbs form of stationarity states. The main result on free energy fluxes is based on general properties such as clustering, and we show that it is mathematically rigorous in quantum spin chains.

Quantum periods and TBA equations for [formula omitted]SU(2) Nf = 2 SQCD with flavor symmetry

We apply the exact WKB analysis to the quantum Seiberg-Witten curve for 4-dimensional N=2SU(2)Nf=2 SQCD with the flavor symmetry. The discontinuity and the asymptotic behavior of the quantum periods define a Riemann-Hilbert problem. We derive the thermodynamic Bethe ansatz (TBA) equations as a solution to this problem. We also compute the effective central charge of the underlying CFT, which is shown to be proportional to the one-loop beta function of the SQCD.

Groundstate finite-size corrections and dilogarithm identities for the twisted , and models

We consider the Y-systems satisfied by the , , vertex and loop models at roots of unity with twisted boundary conditions on the cylinder. The vertex models are the 6-, 15- and Izergin–Korepin 19-vertex models respectively. The corresponding loop models are the dense, fully packed and dilute Temperley–Lieb loop models respectively. For all three models, our focus is on roots of unity values of e iλ with the crossing parameter λ corresponding to the principal and dual series of these models. Converting the known functional equations to nonlinear integral equations in the form of thermodynamic Bethe ansatz equations, we solve the Y-systems for the finite-size corrections to the groundstate eigenvalue following the methods of Klümper and Pearce. The resulting expressions for c − 24Δ, where c is the central charge and Δ is the conformal weight associated with the groundstate, are simplified using various dilogarithm identities. Our analytic results are in agreement with previous results obtained by different methods.

Note on quantum periods and a TBA-like system

There is an interesting relation between the quantum periods on a certain limit of local P1×P1 Calabi-Yau space and a TBA (Thermodynamic Bethe Ansatz) system appeared in the studies of ABJM (Aharony-Bergman-Jafferis-Maldacena) theory. We propose a one-parameter generalization of the relation. Furthermore, we derive the differential operators for quantum periods and the TBA-like equation in various limits of the generalized relation.

Quantum periods and TBA-like equations for a class of Calabi-Yau geometries

We continue the study of a novel relation between quantum periods and TBA(Thermodynamic Bethe Ansatz)-like difference equations, generalize previous works to a large class of Calabi-Yau geometries described by three-term quantum operators. We give two methods to derive the TBA-like equations. One method uses only elementary functions while the other method uses Faddeev’s quantum dilogarithm function. The two approaches provide different realizations of TBA-like equations which are nevertheless related to the same quantum period.

Long strings and symmetric product orbifold from the AdS3Bethe equations

A particularly rich class of integrable systems arises from the AdS/CFT duality. There, the two-dimensional quantum field theory living on the string worldsheet may be understood in terms of a non-relativistic factorized S matrix, and the energy spectrum may be derived by techniques such as the mirror thermodynamic Bethe ansatz or the quantum spectral curve. In the case of AdS3/CFT2 without Ramond-Ramond fluxes, the worldhseet theory is a Wess-Zumino-Witten model with continous and discrete representations which, for the lowest allowed level, is dual to the symmetric product orbifold of a free theory. I will show how continuous representations may arise from integrability, and that at the lowest level the Bethe equations yield the symmetric product orbifold partition function on the nose.

Bound state scattering simplified

In the description of the AdS5/CFT4 duality by an integrable system the scattering matrix for bound states plays a crucial role: it was initially constructed for the evaluation of finite size corrections to the planar spectrum of energy levels/anomalous dimensions by the thermodynamic Bethe ansatz, and more recently it re-appeared in the context of the glueing prescription of the hexagon approach to higher-point functions. In this work we present a simplified form of this scattering matrix and we make its pole structure manifest. We find some new relations between its matrix elements and also present an explicit form for its inverse. We finally discuss some of its properties including crossing symmetry. Our results will hopefully be useful for computing finite-size effects such as the ones given by the complicated sum-integrals arising from the glueing of hexagons, as well as help towards understanding universal features of the AdS5/CFT4 scattering matrix.

QQ-system and non-linear integral equations for scattering amplitudes at strong coupling

We provide the two fundamental sets of functional relations which describe the strong coupling limit in AdS3 of scattering amplitudes in mathcal{N} = 4 SYM dual to Wilson loops (possibly extended by a non-zero twist l): the basic QQ-system and the derived TQ-system. We use the TQ relations and the knowledge of the main properties of the Q-function (eigenvalue of some Q-operator) to write the Bethe Ansatz equations, viz. a set of (‘complex’) non-linear-integral equations, whose solutions give exact values to the strong coupling amplitudes/Wilson loops. Moreover, they have some advantages with respect to the (‘real’) non-linear-integral equations of Thermodynamic Bethe Ansatz and still reproduce, both analytically and numerically, the findings coming from the latter. In any case, these new functional and integral equations give a larger perspective on the topic also applicable to the realm of mathcal{N} = 2 SYM BPS spectra.

Emergence and Disruption of Spin-Charge Separation in One-Dimensional Repulsive Fermions.

At low temperature, collective excitations of one-dimensional (1D) interacting fermions exhibit spin-charge separation, a unique feature predicted by the Tomonaga-Luttinger liquid (TLL) theory, but a rigorous understanding remains challenging. Using the thermodynamic Bethe ansatz (TBA) formalism, we analytically derive universal properties of a 1D repulsive spin-1/2 Fermi gas with arbitrary interaction strength. We show how spin-charge separation emerges from the exact TBA formalism, and how it is disrupted by the interplay between the two degrees of freedom that brings us beyond the TLL paradigm. Based on the exact low-lying excitation spectra, we further evaluate the spin and charge dynamical structure factors (DSFs). The peaks of the DSFs exhibit distinguishable propagating velocities of spin and charge as functions of interaction strength, which can be observed by Bragg spectroscopy with ultracold atoms.

Generalized hydrodynamics in box-ball system

Box-ball system (BBS) is a prominent example of integrable cellular automata in one dimension connected to quantum groups, Bethe ansatz, ultradiscretization, tropical geometry and so forth. In this paper we study the generalized Gibbs ensemble of BBS soliton gas by thermodynamic Bethe ansatz and generalized hydrodynamics. The results include the solution to the speed equation for solitons, an intriguing connection of the effective speed with the period matrix of the tropical Riemann theta function, an explicit description of the density plateaux that emerge from domain wall initial conditions including their diffusive corrections.

Functional equations and separation of variables for exact g-function

The g-function is a measure of degrees of freedom associated to a boundary of two-dimensional quantum field theories. In integrable theories, it can be computed exactly in a form of the Fredholm determinant, but it is often hard to evaluate numerically. In this paper, we derive functional equations — or equivalently integral equations of the thermodynamic Bethe ansatz (TBA) type — which directly compute the g-function in the simplest integrable theory; the sinh-Gordon theory at the self-dual point. The derivation is based on the classic result by Tracy and Widom on the relation between Fredholm determinants and TBA, which was used also in the context of topological string. We demonstrate the efficiency of our formulation through the numerical computation and compare the results in the UV limit with the Liouville CFT. As a side result, we present multiple integrals of Q-functions which we conjecture to describe a universal part of the g-function, and discuss its implication to integrable spin chains.

TBA equations for the Schrödinger equation with a regular singularity

We derive the thermodynamic Bethe ansatz (TBA) equations for the Schrödinger equation with an arbitrary polynomial potential and a regular singular (simple and double pole) term. The TBA equations provide a non-trivial generalization of the ODE/IM correspondence and also give a solution for the Riemann–Hilbert problem in the exact WKB method. We study the TBA equations in detail for the linear and the harmonic oscillator potentials together with inverse and centrifugal terms. As an application, we also compute numerically the Voros spectrum for these potentials using the Bohr–Sommerfeld quantization condition.

Structure constants in mathcal{N} = 4 SYM at finite coupling as worldsheet g-function

We develop a novel nonperturbative approach to a class of three-point functions in planar mathcal{N} = 4 SYM based on Thermodynamic Bethe Ansatz (TBA). More specifically, we study three-point functions of a non-BPS single-trace operator and two determinant operators dual to maximal Giant Graviton D-branes in AdS5×S5. They correspond to disk one-point functions on the worldsheet and admit a simpler and more powerful integrability description than the standard single-trace three-point functions. We first introduce two new methods to efficiently compute such correlators at weak coupling; one based on large N collective fields, which provides an example of open-closed-open duality discussed by Gopakumar, and the other based on combinatorics. The results so obtained exhibit a simple determinant structure and indicate that the correlator can be interpreted as a generalization of g-functions in 2d QFT; an overlap between an integrable boundary state and a state corresponding to the single-trace operator. We then determine the boundary state at finite coupling using the symmetry, the crossing equation and the boundary Yang-Baxter equation. With the resulting boundary state, we derive the ground-state g-function based on TBA and conjecture its generalization to other states. This is the first fully nonperturbative proposal for the structure constants of operators of finite length. The results are tested extensively at weak and strong couplings. Finally, we point out that determinant operators can provide better probes of sub-AdS locality than single-trace operators and discuss possible applications.

Exact WKB analysis and TBA equations for the Mathieu equation

We derive the Thermodynamic Bethe Ansatz (TBA) equations for the exact WKB periods of the Mathieu equation in the weak coupling region. We will use the TBA equations to calculate the quantum corrections to the WKB periods, which are regarded as the quantum periods of N=2SU(2) super Yang-Mills theory at strong coupling. We calculate the effective central charge of the TBA equations, which is found to be proportional to the coefficient of the one-loop beta function of the 4d theory. We also study the spectral problem for the Mathieu equation based on the TBA equations numerically.

Quantum Generalized Hydrodynamics.

Physical systems made of many interacting quantum particles can often be described by Euler hydrodynamic equations in the limit of long wavelengths and low frequencies. Recently such a classical hydrodynamic framework, now dubbed generalized hydrodynamics (GHD), was found for quantum integrable models in one spatial dimension. Despite its great predictive power, GHD, like any Euler hydrodynamic equation, misses important quantum effects, such as quantum fluctuations leading to nonzero equal-time correlations between fluid cells at different positions. Focusing on the one-dimensional gas of bosons with delta repulsion, and on states of zero entropy, for which quantum fluctuations are larger, we reconstruct such quantum effects by quantizing GHD. The resulting theory of quantum GHD can be viewed as a multicomponent Luttinger liquid theory, with a small set of effective parameters that are fixed by the thermodynamic Bethe ansatz. It describes quantum fluctuations of truly nonequilibrium systems where conventional Luttinger liquid theory fails.

Integrability and cycles of deformed [formula omitted] gauge theory

To analyse pure N=2SU(2) gauge theory in the Nekrasov-Shatashvili (NS) limit (or deformed Seiberg-Witten (SW)), we use the Ordinary Differential Equation/Integrable Model (ODE/IM) correspondence, and in particular its (broken) discrete symmetry in its extended version with two singular irregular points. Actually, this symmetry appears to be ‘manifestation’ of the spontaneously broken Z2 R-symmetry of the original gauge problem and the two deformed SW one-cycle periods are simply connected to the Baxter's T and Q functions, respectively, of the Liouville conformal field theory at the self-dual point. The liaison is realised via a second order differential operator which is essentially the ‘quantum’ version of the square of the SW differential. Moreover, the constraints imposed by the broken Z2 R-symmetry acting on the moduli space (Bilal-Ferrari equations) seem to have their quantum counterpart in the TQ and the T periodicity relations, and integrability yields also a useful Thermodynamic Bethe Ansatz (TBA) for the periods (Y(θ,±u) or their square roots, Q(θ,±u)). A latere, two efficient asymptotic expansion techniques are presented. Clearly, the whole construction is extendable to gauge theories with matter and/or higher rank groups.