This thesis contains three main parts, which are largely independent. In the first part we deal with the boundary bootstrap in supersymmetric factorized scattering theory. We give a description of supersymmetry in the case when the space is a half-line and present rules for the determination of the representations in which higher level boundary bound states transform, and for the determination of the supersymmetric one-particle reflection matrix factors for the higher level boundary bound states. These rules apply under the condition that the bulk particles transform in the kink or in the boson-fermion representation. Examples for the application of these rules to specific models are also given. In the second part we investigate the problem whether the TCSA spectrum can be approximated by the spectrum of the original Hamiltonian operator in which the coefficients of the terms are suitably changed. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. Another truncation method that preserves the solvability of the model is also considered. The results of perturbative and numerical calculations show that the above approximation is possible and that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method. In the third part we investigate the phase structure of the two- and three-frequency sine-Gordon models using the TCSA. In the case of the three-frequency model the tricritical point, several points of the critical line and a few points of the line of first order transition are found.
This thesis contains three main parts, which are largely independent. In the first part we deal with the boundary bootstrap in supersymmetric factorized scattering theory. We give a description of supersymmetry in the case when the space is a half-line and present rules for the determination of the representations in which higher level boundary bound states transform, and for the determination of the supersymmetric one-particle reflection matrix factors for the higher level boundary bound states. These rules apply under the condition that the bulk particles transform in the kink or in the boson-fermion representation. Examples for the application of these rules to specific models are also given. In the second part we investigate the problem whether the TCSA spectrum can be approximated by the spectrum of the original Hamiltonian operator in which the coefficients of the terms are suitably changed. The investigation is done in the case of the critical Ising model on a strip with an external magnetic field on one of the boundaries. Another truncation method that preserves the solvability of the model is also considered. The results of perturbative and numerical calculations show that the above approximation is possible and that the qualitative behaviour of the truncated spectrum as a function of the coupling constant depends on the truncation method. In the third part we investigate the phase structure of the two- and three-frequency sine-Gordon models using the TCSA. In the case of the three-frequency model the tricritical point, several points of the critical line and a few points of the line of first order transition are found.
This thesis is concerned with several aspects of the Ising and tri-critical Ising models in two-dimensions. These are much-studied models relevant in both condensed matter physics as descriptions of the critical phenomena of two- dimensional systems and in String theory as building blocks of the string world sheet theory. The first part of the thesis is concerned with the derivation of differential equations for the critical four-point function in the Ising model. We present a method which provides the well-known standard solutions by a new and efficient route. The second part of the thesis is concerned with off-critical behaviour, and in particular the numerical study of perturbations of conformal field theory through the truncated conformal space approach. We show that the coupling constant undergoes significant renormalization in this scheme, and in particular the Ising model can be found as a fixed point for a finite value of the bare coupling constant. The renormalization group equations we find are of general use in the TCSA approach. The final part of the thesis considers off-critical boundary conditions in the tri-critical Ising model. We study them using a variant of the mean-field method and find a qualitative description of the space of boundary conditions that is in accord with the exact conformal field theory description. This is both a test of the method and its applicability in new domains, and also shows that previously published results are in error.
A relevant perturbation of a conformal field theory (CFT) on the half-plane, by both a bulk and boundary operator, often leads to a massive theory with a particle description in terms of the bulk S-matrix and boundary reflection factor R. The link between the particle basis and the CFT in the bulk is usually made with the thermodynamic Bethe ansatz effective central charge C(_eff). This allows a conjectured S-matrix to be identified with a specific perturbed CFT. Less is known about the links between the reflection factors and conformal boundary conditions, but it has been proposed that an exact, off-critical version of Affleck and Ludwig's g-function could be used, analogously to C(_eff), to identify the physically realised reflection factors and to match them with the corresponding boundary conditions. In the first part of this thesis, this exact g-function is tested for the purely elastic scattering theories related to the ADET Lie algebras. Minimal reflection factors are given, and a method to incorporate a boundary parameter is proposed. This enables the prediction of several new flows between conformal boundary conditions to be made. The second part of this thesis concerns the three-parameter family of PT-symmetric Hamiltonians H(M,o,1) = p(^2) – (ix) (^2M) – α(ix) The positions where the eigenvalues merge and become complex correspond to quadratic and cubic exceptional points. The quasi-exact solvability of the models for M = 3 is exploited to exploreaway from M = 3 is investigated using both numerical and perturbative approaches.
This thesis investigates two aspects of Conformal Field Theories (CFTs) in D dimensions. Its first part is devoted to conformal blocks, special functions that arise in the partial wave expansion of CFT four-point functions. We prove that these conformal blocks admit an expansion in terms of polar coordinates and show that the expansion coefficients are determined by recursion relations. Conformal blocks are naturally defined on the complex plane: we study their restriction to the real line, and show that they obey a fourth-order differential equation there. This ODE can be used to efficiently compute conformal blocks and their derivatives in general D. Several applications to the conformal bootstrap program are mentioned.The second half of this thesis investigates RG flows that are defined by perturbing a CFT by a number of relevant operators. We study such flows using the Truncated Conformal Space Approach (TCSA) of Yurov and Zamolodchikov, a numerical method that allows for controlled computations in strongly coupled QFTs. Two different RG flows are considered: the free scalar field deformed by a mass term, and phi^4 theory. The former is used as a benchmark, in order to compare numerical TCSA results to exact predictions. TCSA results for phi^4 theory display spontaneous Z_2 symmetry breaking at strong coupling: we study the spectrum of this theory both in the Z_2-broken and preserved phase, and we compare the critical exponents governing the phase transition to known values. In a separate chapter, we show how truncation errors can be reduced by adding suitable counterterms to the bare TCSA action, following earlier work in two dimensions.
We study different aspects of integrable boundary quantum field theories, focusing mostly on the ``boundary sine-Gordon model'' and its applications to condensed matter physics. The first part of the review deals with formal problems. We analyze the classical limit and perform semi-classical quantization. We show that the non-relativistic limit corresponds to the Calogero-Moser model with a boundary potential. We construct a lattice regularization of the problem via the XXZ chain. We classify boundary bound states. We generalize the Destri de Vega method to compute the ground state energy of the theory on a finite interval. The second part deals with some applications to condensed matter physics. We show how to compute analytically time and space dependent correlations in one-dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasiparticles. Correlators follow then from matrix elements of local operators between multiparticle states -- the massless form-factors. Although, in general an infinite sum of these form-factors has to be considered, we find that for the current, spin and energy operators only a few (two or three) are necessary to obtain an accuracy of more than 1\%. Our results hold for arbitrary impurity strength, in contrary to the perturbative expansions in the coupling constants. As an example, we compute the frequency dependent condunctance, at zero temperature, in a Luttinger liquid with an impurity, and also discuss the susceptibility in the Kondo model and the time-dependent properties of the two-state problem with dissipation.
In this thesis two rather different topics are addressed: • the application of the Truncated Conformal Spectrum Approach (TCSA) to pertur- bations of SU(2)k Wess-Zumino-Witten (WZW) models, and to Landau-Ginzburg (LG) theories; • the analysis of the role of quantum statistics in the steady flow patterns of 2d driven ideal Fermi gases in a non-integrable geometry. A common methodological question unifies the two topics: how to simulate numerically quantum systems with infinite spaces of states. This problem is typically faced either introducing a lattice in the real space or directly in the continuum, and the two addressed topics can be seen as examples of these two cases. The TCSA is aimed to tackle perturbed conformal field theories directly in the continuum. Instead, to study the ideal Fermi gas we introduce a lattice which naturally captures the features of the chosen non-integrable geometry. Chapter 1: This part of the thesis is mainly methodological and it is focused on the application of the TCSA to c = 1 conformal field theories. In particular, our interest is concerned with perturbations of WZW models with SU(2)k symmetry, which capture low energy properties of k/2-Heisenberg spin-chains. The new problem faced is how to apply the method in the presence of marginal perturbations, where UV divergences result in non-universal contributions in the simulations. A generic framework to cure this pathology is proposed, and tested in the case of the current-current deformation of SU(2)1. Among c = 1 theories, Landau-Ginzburg theories are of great interest, since their classical structure of the vacua captures the massive phases of deformation of minimal models. The most important methodological problem we faced here is that LG theories have an uncountable space of states. We show that the TCSA applies also in this case, once the target space of the LG field is properly compactified. The TCSA is thus used to very an existing conjecture on the number of stable neutral bound states. Chapter 2: Interacting quantum gases, such as the electron liquid, are known to admit a hydrodynamic description and, as for classical interacting fluids, they develop turbulence at sufficiently strong driving. On the other hand, ideal classical fluids cannot develop any turbulent flow. We ask whether this property extends to ideal quantum gases. Indeed, Fermi gases satisfy the exclusion principle that could play the role of an effective interaction, resulting in non-trivial flows. We are going to describe an ideal Fermi gas which flows form a narrow channel into a wider region. A lattice which approximates the geometry is introduced, and the system is tackled within two approaches: the micro-canonical formalism and the Lindblad equation. In the first case the system is closed, while in the second one it is coupled to external baths. Despite this difference, the observed physics turns out to be the same far from the boundaries. We show that, in specific regimes, quantum statistics induces non-trivial patterns in the vorticity, which generates local magnetic moments of a measurable intensity.
This thesis is devoted to the study of ghost theories out of the critical point, in two dimensions. The first chapter offers a bird's eye view of the most important applications of ghosts to condensed matter physics. After a brief exposition of the basic (and less basic) facts concerning ghosts at the critical point, an outline of the non-perturbative methods, used in Part I and Part II, is furnished. Essentially, they rely on the so-called integrable approach, which is based on the possibility of describing all the states of an integrable quantum field theory in terms of pseudo-particles in a Hilbert space. The scattering properties of such excitations are encoded into a matrix (the S-matrix), which can be exactly determined by imposing a set of stringent constraints, and which allows to specify completely the particle content of the theory (masses, multiplicities, bound states) [38, 39]. The knowledge of the scattering amplitude, then, permits to extract all the thermodynamic quantities of the system (free energy) by means of the Thermodynamic Bethe ansatz (TBA) technique [40] and, at least in principle, to determine the off-critical correlation functions of the local operators of the theory, thanks to the so-called Form Factor bootstrap approach [41-43]. Part I contains the simplest examples of off-critical ghost theories, namely the massive versions of the conformal free bosonic and fermionic ones [44]. Despite their non-interacting nature, still there are non-local sectors of the models, which exhibit a highly interacting behavior. Correlation functions of operators belonging to these sectors are computed exactly and a comparison with the massive ordinary counterparts is performed. Afterwards, the effects;produced by the introduction of impurities are considered [45]. At the moment, such models lack a physical realization, but they are important as 'prototype' systems, shedding light on some crucial basic aspects (e.g. the choice of the most convenient basis for the space of states). Part II deals with a deceptively simple representative of the aforementioned nonlinear sigma models defined on supersymmetric manifolds, where the vector field, with one commuting component and two anticommuting ones, transforms under the global symmetry OSP(ll2). This system has a simple physical realization in terms of a dense loop model, where crossings of loops are allowed [28, 46]. At long wavelength, the theory is gapless and the Goldstone excitations are nothing but free fermionic ghosts [25, 28]. We propose the exact S-matrix for this system and present TBA calculations, supporting such conjecture. The bootstrap form factor approach is outlined, including a detailed discussion about the symmetry properties of the model and the explicit derivation of some basic objects, such as the minimal form factors. Moreover, we compute explicitly the two-point correlation function of a suitably chosen operator of the theory, comparing its large distance limit with the result expected on the basis of conformal field theory considerations. Since the work is still in progress [47], we conclude sketching the main goals and the route we intend to take, in order to pursue them.