In this thesis, I study dissipation in a number of one-dimensional models from condensed matter physics, by coupling them to a quantum bath. The main purpose is to see how the dissipation interacts with other properties, such as topological order and different types of symmetries. The first system studied is the Kitaev chain, a simple non-interacting model with topological order. The presence of a bath drives the system out of equilibrium and into a mixed quantum state, where conventional measures of topological order do not apply. I propose looking for certain dynamical signatures, that can tell us what happens to the topological order of the original system, once the bath is included. In addition, we study the dissipative Kitaev chain with periodic driving, which introduces a host of exotic new topological properties. Finally, I investigate the effect of dissipation on interacting models, such as the XXZ Heisenberg spin chain, and how symmetries might help us analyze this effect. While the dynamics of such a model are very difficult to compute, certain observables like the total magnetization of the spin chain behave in a peculiar way. By studying the symmetry properties of both the system and the bath, I show that the dissipation causes the expectation value of the magnetization to relax in a simple and coherent manner, only adding an overall damping factor to the dynamics of the isolated system.
In this thesis, I study dissipation in a number of one-dimensional models from condensed matter physics, by coupling them to a quantum bath. The main purpose is to see how the dissipation interacts with other properties, such as topological order and different types of symmetries. The first system studied is the Kitaev chain, a simple non-interacting model with topological order. The presence of a bath drives the system out of equilibrium and into a mixed quantum state, where conventional measures of topological order do not apply. I propose looking for certain dynamical signatures, that can tell us what happens to the topological order of the original system, once the bath is included. In addition, we study the dissipative Kitaev chain with periodic driving, which introduces a host of exotic new topological properties. Finally, I investigate the effect of dissipation on interacting models, such as the XXZ Heisenberg spin chain, and how symmetries might help us analyze this effect. While the dynamics of such a model are very difficult to compute, certain observables like the total magnetization of the spin chain behave in a peculiar way. By studying the symmetry properties of both the system and the bath, I show that the dissipation causes the expectation value of the magnetization to relax in a simple and coherent manner, only adding an overall damping factor to the dynamics of the isolated system.
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.
Topological phases of matter are understood to be characterized by particular configurations of entanglement encoded within the ground states of many-body quantum systems. In this thesis, we discuss how novel phenomena associated with topological phases can arise in systems that are driven out of equilibrium or coupled to their surroundings, which are not described by ground state physics. Firstly, we consider systems undergoing unitary time evolution due to some external driving. We show how the entanglement features of the time-dependent wavefunction can be topologically characterized, and demonstrate that these topological properties are reflected in the dynamics of various quantities usually associated with topological phases in equilibrium. We introduce a new non-equilibrium topological classification scheme which can be used to predict whether or not the topological features of a given system will be preserved under time evolution. In brief, this classification captures the fact that certain symmetries (namely antiunitary symmetries) are inevitably broken once the system is driven out of equilibrium; thus any topological phenomena protected by such symmetries cannot be expected to persist in non-equilibrium scenarios. Secondly, we investigate how topologically non-trivial systems are affected by dissipative effects, i.e.~coupling to an external environment. We demonstrate that system-environment interactions facilitate processes in the system that effectively break any antiunitary symmetries, regardless of the symmetries of the microscopic Hamiltonian. Accordingly, those phases that were shown to be unstable against time-dependent driving are also fragile against coupling to an environment. To illustrate the consequences of this fragility, we consider the effects of dissipation on the coherence properties of topological bound states, as well as the conductance properties of chiral and helical topological edge modes. We find that the decoherence rate of the former and the deviation from quantized conductivity of the latter are (not) exponentially suppressed in the inverse temperature when the phase in question is protected by unitary (antiunitary) symmetries. These results regarding open systems can be connected to the same non-equilibrium classification developed in the context of isolated systems undergoing unitary dynamics. Our findings highlight a distinction between topological phases that are either robust or fragile against non-equilibrium effects and/or system-environment coupling. The ramifications for the use of such systems in quantum technologies are discussed.
The discovery of topological matter has revolutionized the field of condensed matter physics giving rise to many interesting phenomena, and fostering the development of new quantum technologies. In this thesis we study the quantum dynamics that take place in low dimensional topological systems, specifically 1D and 2D lattices that are instances of topological insulators. First, we study the dynamics of doublons, bound states of two fermions that appear in systems with strong Hubbard-like interactions. We also include the effect of periodic drivings and investigate how the interplay between interaction and driving produces novel phenomena. Prominent among these are the disappearance of topological edge states in the SSH-Hubbard model, the sublattice confinement of doublons in certain 2D lattices, and the long-range transfer of doublons between the edges of any finite lattice. Then, we apply our insights about topological insulators to a rather different setup: quantum emitters coupled to the photonic analogue of the SSH model. In this setup we compute the dynamics of the emitters, regarding the photonic SSH model as a collective structured bath. We find that the topological nature of the bath reflects itself in the photon bound states and the effective dipolar interactions between the emitters. Also, the topology of the bath affects the single-photon scattering properties. Finally, we peek into the possibility of using these kind of setups for the simulation of spin Hamiltonians and discuss the different ground states that the system supports.
With the discovery of the integer quantum Hall effect by von Klitzing and collaborators in 1980, the mathematical field of topology entered the world of condensed matter physics. Almost three decades later, this eventually led to the theoretical prediction and the experimental realization of many intriguing topological materials and topology-based devices. In this Ph.D. thesis, we will study the interplay between topology and another key topic in condensed matter physics, namely the study of inter-particle interactions in many-body systems. This interplay is analyzed from two different perspectives. Firstly, we studied how the presence of electron-electron interactions affects single-electron injection into a couple of counter-propagating one-dimensional edge channels. The latter appear at the edges of topologically non-trivial systems in the quantum spin Hall regime and they can also be engineered by exploiting the integer quantum Hall effect. Because of inter-channel interactions, the injected electron splits up into a couple of counter-propagating fractional excitations. Here, we carefully study and discuss their properties by means of an analytical approach based on the Luttinger liquid theory and the bosonization method. Our results are quite relevant in the context of the so-called electron quantum optics, a fast developing field which deeply exploits the topological protection of one-dimensional edge states to study the coherent propagation of electrons in solid-state devices. As an aside, we also showed that similar analytical techniques can also be used to study the time-resolved dynamics of a Luttinger liquid subject to a sudden change of the interaction strength, a protocol known as quantum quench which is gaining more and more attention, especially within the cold-atoms community. Secondly, we study how inter-particle interactions can enhance the topological properties of strictly one-dimensional fermionic systems. More precisely, the starting point is the seminal Kitaev chain, a free-fermionic lattice model which hosts exotic Majorana zero-energy modes at its ends. The latter are extremely relevant in the context of topological quantum computation because of their non-Abelian anyonic exchange statistics. Here we show that, by properly adding electron-electron interactions to the Kitaev chain, it is possible to obtain lattice models which feature zero-energy parafermionic modes, an even more intriguing generalization of Majoranas. To this end, we develop at first an exact mapping between Z4 parafermions and ordinary fermions on a lattice. We subsequently exploit this mapping to analytically obtain an exactly solvable fermionic model hosting zero-energy parafermions. We study their properties and numerically investigate their signatures and robustness even when parameters are tuned away from the exactly solvable point.
The discovery of topological matter has revolutionized the field of condensed matter physics giving rise to many interesting phenomena, and fostering the development of new quantum technologies. In this thesis we study the quantum dynamics that take place in low dimensional topological systems, specifically 1D and 2D lattices that are instances of topological insulators. First, we study the dynamics of doublons, bound states of two fermions that appear in systems with strong Hubbard-like interactions. We also include the effect of periodic drivings and investigate how the interplay between interaction and driving produces novel phenomena. Prominent among these are the disappearance of topological edge states in the SSH-Hubbard model, the sublattice confinement of doublons in certain 2D lattices, and the long-range transfer of doublons between the edges of any finite lattice. Then, we apply our insights about topological insulators to a rather different setup: quantum emitters coupled to the photonic analogue of the SSH model. In this setup we compute the dynamics of the emitters, regarding the photonic SSH model as a collective structured bath. We find that the topological nature of the bath reflects itself in the photon bound states and the effective dipolar interactions between the emitters. Also, the topology of the bath affects the single-photon scattering properties. Finally, we peek into the possibility of using these kind of setups for the simulation of spin Hamiltonians and discuss the different ground states that the system supports.
A central goal in the study of modern condensed matter physics is the characterization of the dynamical properties of quantum systems. Many decades of effort towards this goal, studying a diverse range of (near-equilibrium) quantum matter, from Fermi liquids, to quantum two-level systems, to interacting spin models, and more, has revealed a remarkable pervasiveness of the simple dynamical description of these complex systems in terms of quasi-particles that carry spin, charge, and heat, and that are generally able to equilibrate systems. \n\nThis thesis is an examination of some exceptions to this rule. Specifically, we study a number of instances of quantum matter where equilibration phenomena happens at rather long time scales, or does not occur at all. Particular emphasis is laid on the role of dimensionality, disorder, and dissipation in engendering such novel dynamical behavior.\n\nFirst, we consider non-equilibrium dynamics in one-dimensional quasi-condensates. Low dimensionality inhibits scattering in these systems, and low-energy excitations are long-lived phase fluctuations that exhibit an enriched conformal symmetry. Utilizing this symmetry, we generalize sudden quenches typically used to study non-equilibrium dynamics to quenches along general relativistic and conformal trajectories. Gases never truly equilibrate after such a quench; instead, they evolve into a `prethermal' state with thermal-looking correlations and a chiral asymmetry. \n\nWe then study the problem of the dynamical transition driven by disorder, from an ergodic to a non-ergodic phase, in one-dimensional quantum spin chains. In particular, in XXZ chains with on-site disorder, we find a unique intermediate phase straddling the boundary of the dynamical phase transition, wherein rare-region effects lead to long-time tails in equilibration and vanishing DC conduction before the onset of non-ergodicity. We propose generalizations of such `Griffiths' behavior to arbitrary dimensions. We also study the dynamics of random-bond Heisenberg chains by developing a strong-disorder renormalization group protocol for these systems. We discuss how magnetic noise from such disordered systems contains signatures of their anomalous dynamical properties. \n\nNext, we re-examine the phenomenological theory of two-level systems in amorphous materials in the light of new experimental evidence that these states have large electric/magnetic dipole moments. We propose and justify an interpretation of the model as one of tunneling electrons slowed down by a large phonon drag and discuss the dynamical consequences of such polaronic effects. \n\nFinally, we discuss how magnetic noise measurements can be used to non-invasively access the anomalous properties of systems such as those discussed above. In particular, we examine how scattering properties of isolated magnetic impurities and non-local transport in a variety of two-dimensional materials can be probed experimentally using NV centers as noise magnetometers.