The thesis, consisting of three parts, is focusing on different physical problems that are related to two dimensional Conformal Field Theory (CFT).The first part deals with nonequilibrium transport properties across a junction of quantum wires. Three models are studied. The first one describes the wires by a free compactified bosonic field, seen as the bosonization of the Luttinger liquid of electrons. The junction of the wires is modeled by a boundary condition that ensures nontrivial scattering of the charges between the wires. Combining canonical quantization and functional integral, we compute exactly the current correlation functions in equilibrium, but also in a nonequilibrium stationary state, as well as the full counting statistics of charge and energy between the wires set at different temperatures and potentials. The two other models of quantum wire junction are based on Wess-Zumino-Witten theory (WZW). In the first one, the junction is described by a “cyclic brane” and in the second, by a “coset brane”. The results in the first case are as complete as for the free field, but the charges are fully transmitted from one wire to the next one. In the second case, the scattering is nontrivial, but the model turns out to be difficult to solve.The second part of the thesis studies the global gauge anomalies in “coset” models of CFT, realized as gauged WZW theories. The (almost) complete classification of such anomalies, that lead to some inconsistent coset models, is presented. It is based on Dynkin classification of subalgebras of simple Lie algebras.Finally, the third part of the thesis describes the geometric construction of index from unitary operator families obtained from valence band projectors of a two-dimensional time-reversal invariant topological insulator. The index is related on one hand to the square root of the Wess-Zumino amplitude of such a family, and, on the other hand, it reproduces the Kane-Mele invariant of the insulator. The last identification requires a nontrivial argument that uses a new gauge anomaly of WZW models with boundary.The three parts of the thesis use similar geometrical tool of CFT, that permits to obtain several original results. The unity in the method, as well as the topic of anomalies, builds a bridge between the different components of the manuscript.
The thesis, consisting of three parts, is focusing on different physical problems that are related to two dimensional Conformal Field Theory (CFT).The first part deals with nonequilibrium transport properties across a junction of quantum wires. Three models are studied. The first one describes the wires by a free compactified bosonic field, seen as the bosonization of the Luttinger liquid of electrons. The junction of the wires is modeled by a boundary condition that ensures nontrivial scattering of the charges between the wires. Combining canonical quantization and functional integral, we compute exactly the current correlation functions in equilibrium, but also in a nonequilibrium stationary state, as well as the full counting statistics of charge and energy between the wires set at different temperatures and potentials. The two other models of quantum wire junction are based on Wess-Zumino-Witten theory (WZW). In the first one, the junction is described by a “cyclic brane” and in the second, by a “coset brane”. The results in the first case are as complete as for the free field, but the charges are fully transmitted from one wire to the next one. In the second case, the scattering is nontrivial, but the model turns out to be difficult to solve.The second part of the thesis studies the global gauge anomalies in “coset” models of CFT, realized as gauged WZW theories. The (almost) complete classification of such anomalies, that lead to some inconsistent coset models, is presented. It is based on Dynkin classification of subalgebras of simple Lie algebras.Finally, the third part of the thesis describes the geometric construction of index from unitary operator families obtained from valence band projectors of a two-dimensional time-reversal invariant topological insulator. The index is related on one hand to the square root of the Wess-Zumino amplitude of such a family, and, on the other hand, it reproduces the Kane-Mele invariant of the insulator. The last identification requires a nontrivial argument that uses a new gauge anomaly of WZW models with boundary.The three parts of the thesis use similar geometrical tool of CFT, that permits to obtain several original results. The unity in the method, as well as the topic of anomalies, builds a bridge between the different components of the manuscript.
We consider a model of quantum-wire junctions where the latter are described by conformal-invariant boundary conditions of the simplest type in the multicomponent compactified massless scalar free field theory representing the bosonized Luttinger liquids in the bulk of wires. The boundary conditions result in the scattering of charges across the junction with nontrivial reflection and transmission amplitudes. The equilibrium state of such a system, corresponding to inverse temperature β and electric potential V, is explicitly constructed both for finite and for semi-infinite wires. In the latter case, a stationary nonequilibrium state describing the wires kept at different temperatures and potentials may be also constructed. The main result of the present paper is the calculation of the full counting statistics (FCS) of the charge and energy transfers through the junction in a nonequilibrium situation. Explicit expressions are worked out for the generating function of FCS and its large-deviations asymptotics. For the purely transmitting case they coincide with those obtained in the literature, but numerous cases of junctions with transmission and reflection are also covered. The large deviations rate function of FCS for charge and energy transfers is shown to satisfy the fluctuation relations and the expressions for FCS obtained here are compared with the Levitov–Lesovik formulae.
This dissertation discusses some applications of conformal field theories (CFTs) in topological phases of matter. The first part is devoted to a discussion of coupled wire constructions of some novel quantum Hall systems. Through a theoretical coupled wire model, we construct strongly correlated electronic integer quantum Hall states with filling factor 16. The edge state is a bosonic chiral (E8)1 CFT, which is closely related to topological paramagnets in (3+1)d. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law. We further construct two descendant non-Abelian quantum Hall states at filling 8, each carrying bosonic chiral (G2)1 or (F4)1 edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the E8 state that relates the G2 and F4 Fibonacci states, which is reminiscent of similar physics in half-filled Landau level. The second part is focused on the surface topological orders of 3D bulk topological systems. Symmetry protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this part, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the ADE classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models. The third part is to discuss the relation between the conformal boundary state and (2+1)d SPT phases. We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group G in 2 + 1 dimensions. Our method is based on directly studying the 1 + 1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the non-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability and gappability in the presence of G. We study several examples including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 × Z2 symmetric topological superconductors.
Abstract Motivated by questions about quantum information and classification of quantum field theories, we consider Conformal Field Theories (CFTs) in spacetime dimension d ≥ 5 with a conformally-invariant spatial boundary (BCFTs) or 4-dimensional conformal defect (DCFTs). We determine the boundary or defect contribution to the Weyl anomaly using the standard algorithm, which includes imposing Wess-Zumino consistency and fixing finite counterterms. These boundary/defect contributions are built from the intrinsic and extrinsic curvatures, as well as the pullback of the ambient CFT’s Weyl tensor. For a co-dimension one boundary or defect (i.e. d = 5), we reproduce the 9 parity-even terms found by Astaneh and Solodukhin, and we discover 3 parity-odd terms. For larger co-dimension, we find 23 parity-even terms and 6 parity-odd terms. The coefficient of each term defines a “central charge” that characterizes the BCFT or DCFT. We show how several of the parity-even central charges enter physical observables, namely the displacement operator two-point function, the stress-tensor one-point function, and the universal part of the entanglement entropy. We compute several parity-even central charges in tractable examples: monodromy and conical defects of free, massless scalars and Dirac fermions in d = 6; probe branes in Anti-de Sitter (AdS) space dual to defects in CFTs with d ≥ 6; and Takayanagi’s AdS/BCFT with d = 5. We demonstrate that several of our examples obey the boundary/defect a-theorem, as expected.
The work presented in this thesis meanly addresses two topics in theoretical physics which are quantum thermodynamics and topological order. In the first case, physicists are trying to build up a theory able to describe quite in general phenomena involving heat and energy exchanges in quantum systems. The second topic, instead, is related to exotic phenomena and states of matter like the quantum Hall effect (QHE) or topological insulators and topological superconductors. In the first part od the thesis we define the quantum dynamics for closed and open systems. This is a key ingredient to address the field of quantum thermodynamics. Then, after an introductory part about the quantum thermodynamic transformations, we move toward the field of nonequilibrium fluctuation relations. We address the problem of irreversibility in classical as well as quantum mechanics. Here we present one of our main result. We characterize the "thermodynamic" irreversible adiabatic evolution of a quantum system starting such branch in a thermal equilibrium state at inverse temperature ßi. We give the amount of thermodynamic entropy growth for the process. As direct application of the preceding result we then address a quantum Otto cycle (QOC) working at finite power. We saw that the increasing of irreversible character of the evolution affects the main figures of merit of the cycle. The second part of the thesis addresses the field of topological order. At first we introduce the concept of topological orders, classes and invariants. Then we introduce the well known Kitaev model for 1 D superconductors. This model predicts Majorana zero mode at the ends of the wire (the 1 D system). MZM are topological states showing great resistance against disorder, local perturbations and any dissipative element. Then we consider a generalized Kitaev model where long range interactions are accounted. We get rich topological phase diagrams showing the presence of several MZM per edge. We study the appearing/disappearing dynamics of the modes according to the time reversal symmetry, that is fundamental in the study of topological phase. The phase diagrams we obtained also show the presence of massive edge modes. In this last case the topological invariants do not well describe any transition. At last we focused on a very limit cases where MZM are obtained at finite length of the wire. Such cases are really interesting since the great advance we can get from the finiteness of the wire in an experimental setup. The last part is about single electron tunneling devices. Here we got a different ability to work as "heat-to-current harvester" for a device using quantum dots respect to an analogue one using metallic dots. These different arguments find their unity by considering recent scientific works in which heat transport is addressed in single electron transistor devices where some element of the circuit shows a topological behaviour. This is a perfect system from hich we can get new transport phenomena.
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.
Quantum systems display counterintuitive features, escaping our everyday-life intuition, whose origin is related to entanglement. This phenomenon arises when different parts of the same systems display correlations, which cannot be understood via classical mechanics, and they are dubbed as quantum. To characterize the correlations above, it is necessary to find good entanglement measures which are able to capture quantitatively the quantum features of a given system. This motivates the investigation and the computation of quantities such as the von Neumann entropy of regions, which provides information about how a system is entangled. Our focus is quantum one-dimensional systems close to their critical point, where quantum correlations are enhanced, and a characterization via Quantum Field Theory (QFT) can be provided. In particular, we aim to understand the role of symmetries and localized defects on the entanglement properties. Those aspects display an intimate connection. First, via replica trick, the entropy is expressed via a replica permutation symmetry. Then, single defects break symmetries that are present in the corresponding homogeneous systems. Due to these reasons, the characterization of symmetry actions on subsystems can be viewed as a unifying framework for quantifying entanglement. We split this thesis into two parts. The first part is devoted to the symmetry-resolved entanglement. We first compute it in massive integrable field theories, giving analytical predictions for free fermions/bosons and the 3-state Potts model. For instance, we characterize the universal corrections to area law via form factor bootstrap. We also study excited states of field theories, and we characterize both low-energy states of Conformal Field Theory (CFT) and quasi-particle states. The second part deals with quantum correlations across defects. We provide a CFT framework to compute Rényi entropy and negativity in the ground state of conformal junctions, giving analytical predictions for free fermions and bosons. We find logarithmic growth of the entanglement measures above, and we relate the universal prefactor to the boundary conditions, encoded in a boundary scattering matrix. Then, we focus on the entanglement dynamics, and we study the time evolution of correlations for two paradigmatic classes of quench protocols. For a global quench, we find a linear growth of entropy across the defect whose slope is computed both in CFT, for thermal states, and for free fermions, via a modification of the quasi-particle picture. For a domain wall melting protocol, also known as inhomogeneous quench, we find that, while local observables are easily understood via hydrodynamics and local generalized Gibbs ensembles, the same is not true for the entropy. In contrast, by employing the quasi-particle picture, we provide precise predictions for the dynamics of entanglement, revealing the influence of long-range correlations between points that are specular with respect to the defect.