The finite-volume spectrum of an integrable massive perturbation of a rational conformal field theory interpolates between massive multiparticle states in infinite-volume (IR limit) and conformal states, which are approached at zero volume (UV limit). Each state is labeled in the IR by a set of “Bethe-ansatz quantum numbers,” while in the UV limit it is characterized primarily by the conformal dimensions of the conformal field creating it. We present explicit conjectures for the UV conformal dimensions corresponding to any IR state in the ϕ1, 3-perturbed minimal models ℳ(2, 5) and ℳ(3, 5). The conjectures, which are based on a combinatorial interpretation of the Rogers-Ramanujan-Schur identities, are consistent with numerical results obtained previously for low-lying energy levels.
We study the energy level spacing of perturbed conformal minimal models in finite volume, considering perturbations of such models that are massive but not necessarily integrable.We compute their spectrum using a renormalization group improved truncated conformal spectrum approach.With this method we are able to study systems where more than 40 000 states are kept and where we determine the energies of the lowest several thousand eigenstates with high accuracy.We find, as expected, that the level spacing statistics of integrable perturbed minimal models are Poissonian while the statistics of nonintegrable perturbations are GOE-like.However, by varying the system size (and so controlling the positioning of the theory between its IR and UV limits) one can induce crossovers between the two statistical distributions.
Starting from the lattice A 3 realization of the Ising model defined on a strip with integrable boundary conditions, the exact spectrum (including excited states) of all the local integrals of motion is derived in the continuum limit by means of thermodynamic Bethe ansatz techniques. It is also possible to follow the massive flow of this spectrum between the UV c = 1/2 conformal fixed point and the massive IR theory. The UV expression of the eigenstates of such integrals of motion in terms of Virasoro modes is found to have only rational coefficients and their fermionic representation turns out to be simply related to the quantum numbers describing the spectrum.
A (1+1)-dimensional quantum field theory with a degenerate vacuum (in finite volume) can contain particles, known as kinks, which interpolate between different vacua and have non-trivial restrictions on their multi-particle Hilbert space. Assuming such a theory to be integrable, we show how to calculate the multi-kink energy levels in finite volume given its factorizable S-matrix. In massive theories this can be done exactly up to contributions due to off-shell and tunneling effects that fall off exponentially with volume. As a first application we compare our analytical predictions for the kink scattering theories conjectured to describe the sub-leading thermal and magnetic perturbations of the tricritical Ising model with numerical results from the truncated conformal space approach. In particular, for the sub-leading magnetic perturbation our results allow us to decide between the two different S-matrices proposed by Smirnov and Zamolodchikov.
We develop a method of computing the excited state energies in Integrable Quantum Field Theories (IQFT) in finite geometry, with the spatial coordinate compactified on a circle of circumference R. The IQFT “commuting transfer matrices” introduced earlier [Commun. Math. Phys. 177 (1996) 381] for Conformal Field Theories (CFT) are generalized to non-conformal IQFT obtained by perturbing CFT with the operator Φ1,3. We study the models in which the fusion relations for these “transfer matrices” truncate and provide closed integral equations which generalize the equations of the thermodynamic Bethe ansatz to excited states. The explicit calculations are done for the first excited state in the “scaling Lee-Yang model”.
Abstract It is well understood that 2d conformal field theory (CFT) deformed by an irrelevant
TT T T ¯ perturbation of dimension 4 has universal properties. In particular, for the most interesting cases, the theory develops a singularity in the ultra-violet (UV), signifying a shortest possible distance, with a Hagedorn transition in applications to string theory. We show that by adding an infinite number of higher [
TT T T ¯ ]s>1 irrelevant operators of positive integer scaling dimension 2(s+1) with tuned couplings, this singularity can be resolved and the theory becomes UV complete with a Virasoro central charge cUV> cIR consistent with the c-theorem. We propose an approach to classifying the possible UV completions of a given CFT perturbed by [
TT T T ¯ ]s that are integrable. The main tool utilized is the thermodynamic Bethe ansatz. We study this classification for theories with scalar (diagonal) factorizable S-matrices. For the Ising model with cIR =
21 1 2 we find 3 UV completions based on a single massless Majorana fermion description with cUV =
107 7 10 and
23 3 2 , which both have
N N = 1 SUSY and were previously known, and we argue that these are the only solutions to our classification problem based on this spectrum of particles. We find 3 additional ones with a spectrum of 8 massless particles related to the Lie group E8 appropriate to a magnetic perturbation with cUV =
2221,1215 21 22 , 15 12 , and
231 31 2 . We argue that it is likely there are more cases for this E8 spectrum. We also study simpler cases based on su(3) and su(4) where we can propose complete classifications. For su(3) the infrared (IR) theory is the 3-state Potts model with cIR =
54 4 5 and we find 3 completions with
54 4 5 < cUV≤
516 16 5 . For the su(4) case, which has 3 particles and cIR = 1, and we find 11 UV completions with 1 < cUV ≤ 5, most of which were previously unknown.
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.