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.
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.
The growing field of topological orders has been extensively studied both form the communities of condensed matter and quantum simulation. However, very little is known about the fate of topological order in the presence of disturbing effects such as external noise or dissipation. In the first part of this thesis, we start by studying how the edge states of a topological insulator become unstable when interacting with thermal baths. Motivated by these results, we generalise the notion of Chern insulators from the well-known Hamiltonian case to Liouvillian dynamics. We achieve this goal by defining a new topological witness that is still related to the quantum Hall conductivity at finite temperature. The mixed character of edge states is also well captured by our formalism, and explicit models for topological insulators and dissipative channels are considered. Additionally, we find new topological phases that remain quantised at finite temperature. The construction is based on the Uhlmann phase, a geometric quantum phase defined for general density matrices. Using this new tool, we are able to characterise topological insulators and superconductors at finite temperature both in one and two spatial dimensions. From the experimental side, we propose a state-independent protocol to measure the topological Uhlmann phase in the context of quantum simulation. Symmetry-protected topological orders have traditionally emerged from shortrange interactions. It remains very much unknown what the role played by longrange interactions is, within the physics of these topological systems. In the second part of this thesis, we analyse how topological superconducting phases are affected by the inclusion of long-range couplings. Remarkably, we unveil new topological quasi-particles due to long-range interactions, that were absent in short-range models. We also study how topological invariants are modified by the presence of long-range effects. In the appendix section of the thesis, we explore new numerical methods for driven-dissipative phase transitions. We consider quantum systems with a dissipative term driving the system into a non-equilibrium steady state. The inclusion of short-range fluctuations out-of-equilibrium deeply modifies the shape of the phase-diagram, something never observed in equilibrium thermodynamics.
This thesis addresses different aspects of the dynamics of isolated quantum many-body systems far from equilibrium, ranging from transient phenomena at short times to questions about relaxation and equilibration at late times. The first part deals with nonequilibrium critical phenomena. Dynamical quantum phase transitions are characterized in a two-dimensional spin system by studying the distribution of Fisher zeros in the complex time plane. Moreover, a new topologically driven transition in the nonequilibrium steady state of quenched Chern insulators is investigated, resulting in the identification of universal nonanalytic behavior of the Hall conductance as a function of the quench parameter. The second part of the thesis comprises a study of transient dynamics of the transverse-field Ising model in two and three dimensions based on a perturbative derivation of classical networks to encode the time-evolved wave function; this construction is also applicable to represent the state as an artificial neural network. In the last part of the thesis a possible definition of the irreversibility of quantum many-body dynamics is introduced, which is based on the echo dynamics of observables under imperfect effective time reversal. Echo dynamics is studied in generic and integrable quantum systems as well as in the semiclassical limit. The relation to out-of-time-order correlators is explored, which were recently introduced to probe the quantum butterfly effect.
Gauge theories and topological phases play a fundamental role in different areas of physics. The first ones are at the basis of the Standard Model in the field of particle physics, describing the electroweak and strong interactions through a non-Abelian gauge theory. In condensed matter and statistical physics, gauge theories arise as low-energy effective descriptions of strongly correlated phenomena, such as quantum spin liquids and quantum Hall effect. In this realm there is a strong connection with topological phases and order, as emergent gauge fermions and bosons often describe collective excitations of new exotic states of matter of spin models. The discretization on a lattice is a possible way of dealing with the strongly coupled nature of these theories, due to the fact that this formulation at finite volume provides natural regularizing cut-offs, i.e. the lattice size and spacing. This allows for the investigation of different non-perturbative properties both numerically and analytically. Despite the success of these methods, there are various aspects which remain intractable due to the sign or complex action problems, like the out of equilibrium real time evolution or the analysis of quantum chromodynamics with finite chemical potential. In this respect, quantum simulators of many-body systems to simulate high energy physics arise as promising alternatives to face these problems in the near future. They are quantum systems that can be controlled and used to simulate more complicated systems, whose properties could not be analysed with classical computational, experimental or theoretical tools. In the last decades, there has been a huge development in the fields of quantum optics and atomic physics, allowing for the realization highly precise and controllable platforms by means of trapped ions, superconducting circuits, Rydberg atoms and ultracold atoms in optical lattices. This work of Thesis is part of these fields and has different purposes, all of them connected with the study of gauge theories and topological phases. Firstly, we want to develop a reformulation of lattice gauge theories in terms of gauge invariant fields, in a way to deal solely with physical variables directly in the action. Among several possible advantages, a crucial point is that this can be particularly helpful for the construction of consistent approximation schemes, such as mean-field theories, in order to understand and capture some of the physical features of the theory and make the mean-field approximation consistent with the Elitzur theorem, stating that a local gauge symmetry can not be spontaneously broken. As a second point, we face the problem of simulating higher dimensional gauge theories with ultracold atoms. One of the challenges in more than one dimension is indeed the realization of plaquette interaction terms in the Hamiltonian of lattice gauge theories: these can be engineered through four correlated hoppings in perturbation theory, or by means of constrained hoppings in the dual formulation. In this respect, our main target is to set up an ultracold atomic platform generating the plaquette term in two dimensions using only two correlated hoppings, and protecting gauge invariance through angular momentum conservation. The last point we address in the Thesis is related to the analysis of particles moving in static background gauge potentials, i.e. when the considered gauge field has no dynamical term in the action of the system. Indeed, the dynamics of quantum particles in the presence of static gauge fields gives rise to intriguing physical phenomena. In particular, using ultracold atom setups, the realization of artificial gauge potentials can be used to investigate the physics of topological semimetals, such as Weyl or Dirac type. In the last ten years, a lot of attention has been paid to their characterization, due to the appearence of clear theoretical predictions and very well-controlled experimental techniques. We investigate here the relation between topological phase transitions and van Hove singularities, i.e. the discontinuities in the energy derivative of the density of states, in three-dimensional gapless systems. In such materials, topological phase transitions can be defined by changes of the topological invariants of the Fermi sheets, happening at specific singular points. Moreover, Fermi surface singularities result in the presence of the so-called van Hove points. We then present a general argument to relate topological phase transitions and van Hove singularities, and show observable consequences that are related to the transport properties of the system. We exemplify our argument in Weyl systems by analyzing the three-dimensional Hofstadter model for various commensurate fluxes, which offers the opportunity to consider different kinds of Weyl metals and to understand the features of their density of states.
This dissertation deals with two broad topics - Majorana modes in Kitaev chain and parallels of black hole phenomena in the quantum Hall effect. Majorana modes in topological superconductors are of fundamental importance as realizations of real solutions to the Dirac equation and for their anyonic exchange statistics. They are realised as zero energy edge modes in one-dimensional topological superconductors, modeled by the Kitaev chain Hamiltonian. Here an extensive study is made on the wavefunction features of these Majorana modes. It is shown that the Majorana wavefunction has two distinct features- a decaying envelope and underlying oscillations. The latter becomes important when one considers the coupling between the Majorana modes in a finite-sized chain. The coupled Majorana modes form a non-local Dirac fermionic state which determines the ground state fermion parity. The dependance of the fermion parity on the parameters of the system is purely determined by the oscillatory part of the Majorana wavefunctions. Using transfer matrix method, one can uncover a new boundary in the phase diagram, termed as `circle of oscillations', across which the oscillations in the wavefunction and the ground-state fermionic parity cease to exist. This is closely related to the circle that appears in the context of transverse field XY spin chain, within which the spin-spin correlations have oscillations. For a finite sized system, the circle is further split into mutliple ellipses called `parity sectors'. The parity oscillations have a scaling behaviour i.e oscillations for different superconducting gaps can be scaled to collapse to a single plot. Making use of results from random matrix theory for class D systems, one can also predict the robustness of certain features of fermion parity switches in the presence of disorder and comment on the critical properties of the MBS wavefunctions and level crossings near zero energy. These results could provide directions for making measurements on zero-bias conductance oscillations and the parameter range of operations for robust parity switches in realistic disordered system. On the front of non-equilibrium dynamics, the effect of Majorana modes on the dynamical evolution of the ground state under time variation of a Hamiltonian parameter is studied. The key result is the failure of the ground state to evolve into opposite parity sectors under the dynamical tuning of the system within the topological phase. This dramatic lack of adiabaticity is termed as ‘parity blocking’. A real-space time-dependent formalism is also developed using Pfaffian correlations, where simple momentum space methods fail. This formalism can be used for calculating the non-equilibrium quantities, such as adiabatic fidelity and the residual energy in a system with open boundaries. The consideration of Majorana modes in non-equilibrium dynamics lead to deviation from Kibble-Zurek physics and non-analyticities in adiabatic fidelity even within the topological phase. The second part of the thesis deals with uncovering structural parallels of black hole phenomena such as the Hawking-Unruh effect and quasinormal modes in quantum Hall systems. The Hawking-Unruh effect is the emergence of a thermal state when a vacuum of a quantum field theory on a given spacetime is restricted to a submanifold bounded by an event horizon. The thermal state manifests as Hawking radiation in the context of a black hole spacetime with an event horizon. The Unruh effect is a simpler example where a family of accelerating observers in Minkowski spacetime are confined by the lightcone structure and the Minkowski vacuum looks as a thermal state to them. The key element in understanding the Hawking-Unruh effect is the Rindler Hamiltonian or the boost. The boost acts as the generator of time translation for the quantum states in the Rindler wedge giving rise to thermality. In this thesis it is shown that due to an exact isomorphism between the Lorentz algebra in Minkowksi spacetime and the algebra of area preserving transformations in the lowest Landau level of quantum hall effect, an applied saddle potential acts as an equivalent to the Rindler Hamiltonian giving rise to a parallel of Hawking-Unurh effect. In the lowest Landau level, the saddle potential is reduced to the problem of scattering off an inverted harmonic oscillator(IHO) and the tunneling probability assumes the form of a thermal distribution. The IHO also has scattering resonances which are poles of the scattering matrix in the complex energy plane. The scattering resonances are states with time-decaying behavior and have purely incoming/outgoing probability current. These states are identified as quasinormal modes analogous to those occurring the scattering off an effective potential in black hole spacetimes. The quasinormal decay is an unexplored effect in quantum Hall systems and provides a new class of time-dependent probe of quantum Hall physics. The parallels between the relativistic symmetry generators and the potentials applied in the lowest Landau level also open up an avenue for studying Lorentz Kinematics and symplectic phase space dynamics in the lowest Landau level. These parallels open up new avenues of exploration in the quantum Hall effect.
Topological systems are an exciting frontier of condensed matter physics, as well as having deep relationships to quantum computing. Progress continues to be made both towards building practical quantum computers and experimentally identifying topological features in real materials. However, these attempts have to contend with the messiness of the real world including the effects of finite temperatures. In quantum computing thermal errors are studied through the framework of quantum error correction. This allows a characterisation of how damaging these errors are, and has provided many general results on the limits of how well given quantum codes can endure thermal errors. In the first part of this thesis we study the typical response of one of the best known quantum codes (the two-dimensional toric code) when subjected to thermal noise. This typical behaviour is compared to the limiting bounds and we note differences that arise due to entropic effects. Topological features are typically properties of the ground state of a system. In some cases this means finite temperatures can be damaging, washing out the topological properties with excitations above the ground state. But in other cases it has been shown that topological systems can have their own unique finite temperature behaviours, which can be used to identify them. In the second part of the thesis we study the topological model Kitaev's honeycomb. Fixing certain conserved quantities restricts the model to special sectors, some of which carry topological phases. We begin by investigating an intrinsically finite temperature property of these topological phases, thermal edge currents. At too high temperatures, close to the bulk gap, the topological behaviour of these currents breaks down. However, we show that at lower temperatures they can be identified and shown to have the expected properties such as robustness to perturbations. Following this we investigate the thermal states of the full model. Our approach to this allows us to probe the topological order of the model in a way that has not been done before and we identify a number of interesting new features of the finite temperature physics.
For a long time, it was believed that the Ginzburg-Landau formalism was able to classify all different types of phase transitions. This view changed with the discovery of the quantum Hall effect and topological insulators. Topological insulators (superconductors) are materials that host symmetry-protected metallic edge states in an insulating (superconducting) bulk. Although topological insulators are well understood by now, a thermodynamic description of their behavior remained elusive, firstly because the edges are lost in the thermodynamic limit, when the system size goes to infinity, and secondly because topological quantum field theories involve non-local order parameters, and hence cannot be captured by the conventional Ginzburg-Landau formalism. Recently, this challenge has been overcome: by using Hill thermodynamics to describe the paradigmatic Bernevig-Hughes-Zhang model in two dimensions, it has been shown that at the topological phase change the thermodynamic potential does not scale extensively due to the boundary effects.\n\nHere, we extend this thermodynamic approach firstly to the Kitaev chain in one-dimension and then to different topological models in various dimensions (the Su-Schrieffer-Heeger model in one dimension, the Kane-Mele model in two dimensions and the Bernevig-Hughes-Zhang model in three dimensions) at zero temperature. Surprisingly, we find that all these models exhibit the same universal behavior in the order of the topological-phase transition, which depends on the dimensionality of the system. Next, we use this thermodynamic approach to describe topological phases at finite-temperatures. We calculate the entropy and heat capacity for the Kitaev chain and verify its behavior in comparison with conformal field theory. Furthermore, we extend the topological phase transition to finite-temperatures, by numerically deriving the topological phase diagram for the bulk of the system, and show that it displays a good agreement with the one calculated analytically from the Uhlmann phase. Finally, we calculate measurable quantities such as the density of states and the heat capacity for different topological insulators. Our work reveals unexpected universalities and opens the path to a thermodynamic description of systems with a non-local order parameter, provided that nonlinear terms are appropriately taken into account for the thermodynamic extensive variables.