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arXiv:2212.09416 · Physics Letters B

The full analytic trans-series in integrable field theories

Zoltán Bajnok, János Balog, István Vona

Published Jul 13, 2023• 18 citations
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Abstract

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.

DOI
10.1016/j.physletb.2023.138075

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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.
We determine the complete trans-series solution for the (non-relativistic) moments of the rapidity density in the Lieb-Liniger model. The trans-series is written explicitly in terms of a perturbative basis, which can be obtained from the already known perturbative expansion of the density by solving several ordinary differential equations. Unknown integration constants are fixed from Volin’s method. We have checked that our solution satisfies the analytical consistency requirements including the newly derived resurgence relations and agrees with the high precision numerical solution. Our results also provides the full analytic trans-series for the capacitance of the coaxial circular plate capacitor.
In this thesis we explore the physics of renormalons in integrable models under the framework of resurgence. In the first part, we review some background on resurgence, integrability and renormalons, including a discussion of large N renormalons and ring diagrams. In the second part, we start from the Bethe ansatz integral equations and obtain exact trans-series for the free energy in multiple models. These trans-series include non-perturbative effects which correspond to renormalons at unexpected positions in the Borel plane. We test the trans-series numerically and at large N. We also study what happens to these trans-series under a topological angle. In the third part, we use apply the techniques of resurgence in non-relativistic theories. We find a relation between the energy gap of the system and the positions of singularities in the Borel plane. By studying the asymptotic behaviour of ring diagrams, we identify this relation with renormalons.
We propose an exact summation method to compute thermodynamic observables in integrable quantum field theories. The key idea is to use the matrix-tree theorem to write the Gaudin determinants that appear in the cluster expansion as a sum over graphs. For theories with a diagonal S-matrix, this method is more powerful than the standard Thermodynamic Bethe Ansatz (TBA) technique as it is exact to all orders of powers in inverse volume. We have obtained using this method the TBA equation, the excited state energies in finite volume, the Leclair-Mussardo formula for one point functions, the finite-temperature boundary entropy and cumulants of conserved charges in Generalized Gibbs Ensembles. Moreover, the graph expansion can also be regarded as an alternative to algebraic manipulations involving Gaudin determinants. We have applied this idea to derive the equations of state and other transport properties in Generalized Hydrodynamics. For theories with a non-diagonal S-matrix, the description of a complete set of states is more involved and it is not known how a cluster expansion can be implemented. It is nevertheless possible to apply the direct summation method in reverse and interpret known TBA equations with complex strings in terms of diagrams.
We present a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisemberg chain we derive, for arbitrary values of the anysotropy, a {\bf single} non--linear integral equation from which the free energy can be exactly calculated. The high--temperature expansion follows in a sistematic and relatively simple way. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around T=0T=0T=0. Furthermore, we derive a single non-linear integral equation describing the finite--size ground--state energy of the Sine--Gordon quantum field theory. PACS: 05.30, 03.70. 75.10.5
We consider expectation values of local operators in (continuum) integrable models in a situation when the mean value is calculated in a single Bethe state with a large number of particles. We develop a form factor expansion for the thermodynamic limit of the mean value, which applies whenever the distribution of Bethe roots is given by smooth density functions. We present three applications of our general result: (i) in the framework of integrable quantum field theory (IQFT) we present a derivation of the LeClair–Mussardo formula for finite temperature one-point functions. We also extend the results to boundary operators in boundary field theories. (ii) We establish the LeClair–Mussardo formula for the non-relativistic 1D Bose gas in the framework of the algebraic Bethe ansatz (ABA). This way we obtain an alternative derivation of the results of Kormos et al for the (temperature dependent) local correlations using only the concepts of the ABA. (iii) In IQFT we consider the long-time limit of one-point functions after a certain type of global quench. It is shown that our general results imply the integral series found by Fioretti and Mussardo. We also discuss the generalized eigenstate thermalization hypothesis in the context of quantum quenches in integrable models. It is shown that a single mean value always takes the form of a thermodynamic average in a generalized Gibbs ensemble, although the relation to the conserved charges is rather indirect.
We report on a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisenberg chain we derive, for arbitrary values of the anysotropy, a single non-linear integral equation from which the free energy can be exactly calculated.These equations are shown to be equivalent to an infinite set of algebraic equations of Bethe type which provide alternatively the thermodinamics. The high-temperature expansion follows in a sistematic and relatively simple way from our non-linear integral equations. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around T=0. Furthermore, we derive a single non-linear integral equation describing the finite-size ground-state energy of the Sine-Gordon quantum field theory.