In this note, inspired partly by the works on the analytic trans-series of free-energies in 2D integrable QFTs in the UV limit, we study the corresponding asymptotic expansions in the IR limit of the integrable XXX spin chains, at the anti-ferromagnetic critical point. Starting from the linear TBA integral equation, we generate the perturbative series in the minimal-scheme coupling constants, up to the first order where the
ζ3 appears. To all perturbative orders, at spin one-half, the perturbative series relates to that of the
β2=8π− massive sine-Gordon by a sign flip of the coupling constant, and this relation generalizes for an arbitrary spin, to certain massive deformation of
su(2)k WZW. This relation partially supports the conjectured field-theoretical interpretations of such critical points, at the leading-power
&& all-logarithmic accuracy. We also compute the perturbative coefficients attached to the first exponentially-small corrections up to the order where
ζ3 appears, in case
s>1. An interesting feature is that the largest exponentially-small corrections at the locations
h2+s2k are non-vanishing, but the corresponding Borel singularities in the perturbative series are canceled for generic
s>1, not necessarily integers or half-integers. This becomes most dramatic in the a 't Hooft type large
s limit, in which the perturbative series has a finite convergence radius, while these exponentially small corrections still survive.
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.
A bstract We develop a novel approach to the Wilsonian renormalisation of Hamiltonians for 2-dimensional quantum field theories on the cylinder described in the UV by marginally relevant deformations of conformal field theories. To introduce a Wilsonian short-distance cutoff we make essential use of free field realisations of the full vertex operator algebra in the UV. Our method is intrinsically non-perturbative; we derive a Hamiltonian analogue of Polchinski’s equation describing the flows of all couplings. As a primary example of our general method, we apply it to the marginal anisotropic deformation of the
su2 su 2 Wess-Zumino-Witten model at level 1, which is equivalent to the sine-Gordon model on the cylinder. In particular, we reproduce the standard renormalisation group flow of the sine-Gordon model near the Kosterlitz-Thouless point to second order in the couplings, a result usually derived using Lagrangian/path-integral methods.
Abstract We study the thermodynamic Bethe ansatz (TBA) equations for pure T T ¯ 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 T T ¯ 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 T T ¯ perturbed Liouville theory.
This thesis is concerned with perturbed conformal field theory, the thermodynamic Bethe ansatz technique and applications to statistical mechanics. In particular, the phase space of two dimensional Z(_N)-symmetric statistical models is examined using these techniques. The aim of the first two chapters is to review some general material concerning statistical mechanics, perturbed conformal field theory, integrable two-dimensional quantum field theory and the thermodynamic Bethe ansatz (TBA) technique. In the third chapter Z(_N)-symmetric statistical theories are discussed and the known features of the phase space of such models are surveyed. The field content of the conformal models in this space (called parafermionic models) is investigated in order to determine which perturbations can be used to investigate the phase space. In the fourth and fifth chapters TBA equations are proposed to describe massless and massive renormalisation flows from the Z(_N)-symmetric conformal theories under self-dual Z(_N)-symmetric perturbations. According to the sign of the perturbation parameter the infrared limits are shown to be either conformal c = 1 or massive theories. The ground state energies of these models can be discovered in all perturbative regimes via the TBA method and the results agree with perturbation theory in ultraviolet and infrared limits. Results from detailed studies of the N = 5, 6..10 models are presented throughout. It is also deduced that the parafermionic models lie exactly at the bifurcation point of the first-order transition region into the Kosterlitz-Thouless region of the Z(_N)-symmetric phase space. The sixth and seventh chapters deal solely with massive perturbations. In chapter six, results from the TBA equations are used to deduce the mass spectrum and the vacuum structure of the underlying scattering theory. In chapter seven, proposals for the massive S-matrices are made. For N odd the mass spectra proposed by the TBA method and that predicted by the S-matrix approach (using the minimality principle) differ. It is suggested therefore, that the N odd S-matrices contain zeroes in the physical strip, violating the minimality principle.
We develop a novel approach to the Wilsonian renormalisation of Hamiltonians for 2-dimensional quantum field theories on the cylinder described in the UV by marginally relevant deformations of conformal field theories. To introduce a Wilsonian short-distance cutoff we make essential use of free field realisations of the full vertex operator algebra in the UV. Our method is intrinsically non-perturbative; we derive a Hamiltonian analogue of Polchinski's equation describing the flows of all couplings. As a primary example of our general method, we apply it to the marginal anisotropic deformation of the
su2 Wess--Zumino--Witten model at level 1, which is equivalent to the sine-Gordon model on the cylinder. In particular, we reproduce the standard renormalisation group flow of the sine-Gordon model near the Kosterlitz--Thouless point to second order in the couplings, a result usually derived using Lagrangian/path-integral methods.