Fases topológicas de la materia y sistemas cuánticos abiertos

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Físicas, Departamento de Física Teórica I Física Teórica II (Métodos Matemáticos de la física), leída el 02/11/2016. Tesis formato europeo (compendio de artículos)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.Una transición de fase es una transformación entre dos estados de la materia con propiedades físicas diferentes, por ejemplo cuando el agua líquida se convierte en hielo. Tradicionalmente, la física de las transiciones de fase ha sido perfectamente descrita por la teoría de Landau. Esta teoría propone la existencia de un parámetro de orden local que es capaz de distinguir entre dos fases distintas. Además, al atravesar la transición de fase se rompe espontáneamente una simetría del sistema. A partir de los años 80 se empezaron a encontrar un tipo de transiciones de fase que no estaban bien descritas por la teoría de Landau. Estas fases de la materia se denominan órdenes topológicos y constituyen el principal objeto de esta tesis doctoral. Para estas transiciones no existe un parámetro de orden local que pueda distinguir entre fases con propiedades físicas distintas. Por el contrario, vienen caracterizadas por un parámetro de orden global que es capaz de retener la información topológica del sistema. La otra principal diferencia con respecto a las transiciones de orden, descritas por la teoría de Landau, es el papel que juegan las simetrías. En las transiciones de fase topológicas, cuando se cambia de una fase a otra, no se rompe ninguna simetría. De manera adicional, las fases topológicas de la materia vienen caracterizadas por un conjunto de propiedades distintivas: (1) el estado fundamental está separado por un gap del resto de excitaciones y está degenerado, (2) el sistema presenta estados gapless localizados en el borde, (3) las excitaciones son anyones con estadística exótica, etc...Depto. de Física TeóricaFac. de Ciencias FísicasTRUEunpu

Density-matrix Chern insulators: finite-temperature generalization of topological insulators

Thermal noise can destroy topological insulators (TI). However, we demonstrate how TIs can be made stable in dissipative systems. To that aim, we introduce the notion of band Liouvillian as the dissipative counterpart of band Hamiltonian, and show a method to evaluate the topological order of its steady state. This is based on a generalization of the Chern number valid for general mixed states (referred to as density-matrix Chern value), which witnesses topological order in a system coupled to external noise. Additionally, we study its relation with the electrical conductivity at finite temperature, which is not a topological property. Nonetheless, the density-matrix Chern value represents the part of the conductivity which is topological due to the presence of quantum mixed edge states at finite temperature. To make our formalism concrete, we apply these concepts to the two-dimensional Haldane model in the presence of thermal dissipation, but our results hold for arbitrary dimensions and density matrices

Two-dimensional density-matrix topological fermionic phases: topological Uhlmann numbers

We construct a topological invariant that classifies density matrices of symmetry-protected topological orders in two-dimensional fermionic systems. As it is constructed out of the previously introduced Uhlmann phase, we refer to it as the topological Uhlmann number n_(U). With it, we study thermal topological phases in several two-dimensional models of topological insulators and superconductors, computing phase diagrams where the temperature T is on an equal footing with the coupling constants in the Hamiltonian. Moreover, we find novel thermal-topological transitions between two nontrivial phases in a model with high Chern numbers. At small temperatures we recover the standard topological phases as the Uhlmann number approaches to the Chern number

Topological massive Dirac edge modes and long-range superconducting Hamiltonians

We discover novel topological effects in the one-dimensional Kitaev chain modified by long-range Hamiltonian deformations in the hopping and pairing terms. This class of models display symmetry-protected topological order measured by the Berry/Zak phase of the lower-band eigenvector and the winding number of the Hamiltonians. For exponentially decaying hopping amplitudes, the topological sector can be significantly augmented as the penetration length increases, something experimentally achievable. For power-law decaying superconducting pairings, the massless Majorana modes at the edges get paired together into a massive nonlocal Dirac fermion localized at both edges of the chain: a new topological quasiparticle that we call topological massive Dirac fermion. This topological phase has fractional topological numbers as a consequence of the long-range couplings. Possible applications to current experimental setups and topological quantum computation are also discussed

Thermal instability of protected end states in a one-dimensional topological insulator

We have studied the dynamical thermal effects on the protected end states of a topological insulator (TI) when it is considered as an open quantum system in interaction with a noisy environment at a certain temperature T . As a result, we find that protected end states in a TI become unstable and decay with time. Very remarkably, the interaction with the thermal environment (fermion-boson) respects chiral symmetry, which is the symmetry responsible for the protection (robustness) of the end states in this TI when it is isolated from the environment. Therefore, this mechanism makes end states unstable while preserving their protecting symmetry. Our results have immediate practical implications in recently proposed simulations of TIs using cold atoms in optical lattices. Accordingly, we have computed lifetimes of topological end states for these physical implementations that are useful to make those experiments realistic