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.
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.
Introduction to Integrable Many-Body systems III This is the third part of a three-volume introductory course about integrable systems of interacting bodies. The emphasis is put onto the method of Thermodynamic Bethe ansatz. Two kinds of integrable models are studied. Systems of itinerant electrons, forming a part of Condensed Matter Physics, involve the Hubbard lattice model of electrons with short-ranged one-site interactions (Sect. 20) and the s-d exchange Kondo model (Sect. 21), describing the scattering of conduction electrons on a spin-s impurity. Methods and basic concepts used in Quantum Field Theory are explained on the integrable (1 + 1)-dimensional sine-Gordon model. We start with the classical description of the model in Sect. 22, analyze its finite energy field configurations (soliton, anti-soliton and breathers) and show its classical integrability. The model is quantized by using two schemes: the conformal (Sect. 23) and Lagrangian (Sect. 24) quantizations. The scattering matrix of the sine-Gordon theory is derived at the full quantum level in the bootstrap scheme and is compared to its classical limit in Sect. 25. The parameters of the scattering matrix are related to those of the Lagrangian by calculating the ground-state energy in an applied magnetic field in two ways: Conformal perturbation theory and Thermodynamic Bethe ansatz (Sect. 26). The relation of the sine-Gordon theory to the XXZ Heisenberg model, which provides a complete solution of the sine-Gordon model in a finite volume, is pointed out in Sect. 27. The obtained results are applied in Sect. 28. to the derivation of the exact thermodynamics for the (symmetric) two-component Coulomb gas; this is the first classical two-dimensional fluid with exactly solvable thermodynamics.
Impurities are ubiquitous in condensed matter. Boundary Conformal Field Theory (BCFT) provides a powerful method to study a localized quantum impurity interacting with a gapless continuum of excitations. The results can also be implied to nanoscopic devices like quantum dots. In these lecture notes, I review this field, including the following topics: I. General Renormalization Group (RG) framework for quantum impurity problems: example of simplest Kondo model II. Multi-channel Kondo model III. Quantum Dots: experimental realizations of one and two channel Kondo models IV. Impurities in Luttinger liquids: point contact in a quantum wire V. Quantum impurity entanglement entropy VI. Y-junctions of Luttinger liquids VII. Boundary condition changing operators and the X-ray edge problem.
We investigate various topics in theoretical quantum condensed matter physics.First, we show that a one-dimensional quantum wire with as few as 2 channels of interacting fermions can host metallic Luttinger liquid states of matter that are stable against all perturbations up to $q^\\text{th}$-order in fermion creation/annihilation operators for any fixed finite q.The leading relevant perturbations are thus complicated operators that are expected to modify the physics only at very low energies, below accessible temperatures. The stability of these Luttinger liquid fixed points is due to strong interactions between the channels, which can (but need not) be chosen to be purely repulsive. Our results might enable elementary physical realizations of these phases, and may also serve as a useful paradigm for thinking about higher-dimensional non-Fermi liquids. Separately, we present an elementary but general description of relaxation to gaussian and equilibrium generalized Gibbs states in lattice models of fermions or bosons with quadratic hamiltonians. Our analysis applies to arbitrary initial states that satisfy a mild condition on clustering of correlations.We obtain quantitive, model-independent predictions for how quickly local quantities relax in such systems.These predictions can be tested in near-term quantum gas experiments.Finally, we study chaotic many-body quantum systems that obey the eigenstate thermalization hypothesis (ETH).We show that a known bound on the growth rate of the out-of-time-order four-point correlator in such systems follows directly from the general ETH structure of operator matrix elements.This ties together two key paradigms of thermal behavior in isolated many-body quantum systems.We also consider a bipartition of the system, and study the entanglement properties of an energy eigenstate with nonzero energy density.When the two subsystems have nearly equal volumes, we find a universal correction to the entanglement entropy that is proportional to the square root of the system's heat capacity (or a sum of capacities, if there are conserved quantities in addition to energy).
The properties of exactly integrable two-dimensional quantum systems are reviewed and discussed. The nature of exact integrability as a physical phenomenon and various aspects of the mathematical formalism are explored by discussing several examples, including detailed treatments of the nonlinear Schr\"odinger (delta-function gas) model, the massive Thirring model, and the six-vertex (ice) model. The diagonalization of a Hamiltonian by Bethe's Ansatz is illustrated for the nonlinear Schr\"od\'{\i}nger model, and the integral equation method of Lieb for obtaining the spectrum of the many-body system from periodic boundary conditions is reviewed. Similar methods are applied to the massive Thirring model, where the fermion-antifermion and bound-state spectrum are obtained explicitly by the integral equation method. After a brief review of the classical inverse scattering method, the quantum inverse method for the nonlinear Schr\"odinger model is introduced and shown to be an algebraization of the Bethe Ansatz technique. In the quantum inverse method, an auxiliary linear problem is used to define nonlocal operators which are functionals of the original local field on a fixed-time string of arbitrary length. The particular operators for which the string is infinitely long (free boundary conditions) or forms a closed loop around a cylinder (periodic boundary conditions) correspond to the quantized scattering data and have a special significance. One of them creates the Bethe eigenstates, while the other is the generating function for an infinite number of conservation laws. The analogous operators on a lattice are constructed for the symmetric six-vertex model, where the object which corresponds to a solution of the auxiliary linear problem is a string of vertices contracted over horizontal links (arrows). The relationship between the quantum inverse method and the transfer matrix formalism is exhibited. The inverse Gel'fand-Levitan transform which expresses the local field operator as a functional of the quantized scattering data is formulated for the nonlinear Schr\"odinger equation, and some interesting properties of this transformation are noted, including its reduction to a Jordan-Wigner transformation in the limit of infinitely repulsive coupling.
This thesis studies exact S-matrices and integrability aspects for several two-dimensional quantum systems. The emphasis is on the algebraic structures that underlie exact solvability in two dimensions. The first project details the construction of an optics model for n-level atoms interacting with an electromagnetic field. In the slowly varying envelope and rotating wave approximations, the equations of motion are shown to have a zero-curvature representation, implying an infinite set of quantum conservation laws. The second system investigated is a disordered field theory of Dirac fermions, which after averaging over the potential can be expressed as an osp(2|2) current- current model. The current-current model in turn can be viewed as the fully renormalized version of an osp(2]2) (1) (marginal) Toda system. Non-local charges satisfying the Uq[osp(2|2)(1)] quantum superalgebra are constructed. The corresponding quantum affine symmetry is used to build the Toda S-matrix for the vector representation. The exact S-matrix for the fundamental “solitons” of the osp(2|2) current-current model is then obtained by taking the marginal limit of the Toda S-matrix. The remaining two projects shift the focus to boundary theories. In the first boundary study, two massive versions of the anisotropic spin 1/2 Kondo model are introduced. The two models have the same bulk sine-Gordon interactions but differ in their boundary terms. At the free fermion point, each of the models can be understood as two decoupled Ising models in boundary magnetic fields. Integrability of the models is discussed and boundary S-matrices at and away from the free fermion point are constructed. The final system studied describes an Anderson impurity in a BCS superconductor. This model can be mapped onto an Ising boundary field theory. The boundary S - matrix is shown to exhibit Andreev scattering and the existence of bound states lying inside the superconducting gap.
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.