Investigations of topological phases for quasi-1D systems

For a long time, quantum states of matter have been successfully characterized by the Ginzburg-Landau formalism that was able to classify all different types of phase transitions. This view changed with the discovery of the quantum Hall effect and topological insulators. The latter are materials that host metallic edge states in an insulating bulk, some of which are protected by the existing symmetries. Complementary to the search of topological phases in condensed matter, great efforts have been made in quantum simulations based on cold atomic gases. Sophisticated laser schemes provide optical lattices with different geometries and allow to tune interactions and the realization of artificial gauge fields. At the same time, new concepts coming from quantum information, based on entanglement, are pushing the frontier of our understanding of quantum phases as a whole. The concept of entanglement has revolutionized the description of quantum many-body states by describing wave functions with tensor networks (TN) that are exploited for numerical simulations based on the variational principle. This thesis falls within the framework of the studies in condensed matter physics: it focuses indeed on the so-called synthetic realization of quantum states of matter, more specifically, of topological ones, which may have on the long-run outfalls towards robust quantum computers. We propose a theoretical investigation of cold atoms in optical lattice pierced by effective (magnetic) gauge fields and subjected to experimentally relevant interactions, by adding a modern numerical approach based on TN algorithms. More specifically, this work will focus on (i) interacting topological phases in quasi-1D systems and, in particular, the Creutz-Hubbard model, (ii) the connection between condensed matter and high energy physics studying the Gross-Neveu model and the discretization of Wilson-Hubbard model, (iii) implementing tensor network-based algorithms.Durante mucho tiempo, los estados cuánticos de la materia se han caracterizado con éxito por el formalismo de Ginzburg-Landau que permitió de clasificar todos los diferentes tipos de transiciones de fase. Esta visión cambió con el descubrimiento del efecto Hall cuántico y los aislantes topológicos. Estos últimos son materiales que albergan estados de borde metálicos en una masa aislante, algunos de los cuales están protegidos por las simetrías existentes. Conjuntamente a la búsqueda de fases topológicas en materia condensada, se han hecho grandes esfuerzos en simulaciones cuánticas basadas en gases atómicos fríos. Los sofisticados esquemas láser proporcionan redes ópticas con diferentes geometrías y permiten ajustar las interacciones y la realización de campos de gauge artificial. Al mismo tiempo, los nuevos conceptos que provienen de la información cuántica, basados en el entanglement, están empujando la frontera de nuestra comprensión de las fases cuánticas en su conjunto. El concepto de entanglement ha revolucionado la descripción de los estados cuánticos de muchos cuerpos al describir las funciones de onda con redes tensoras (TN) que se explotan para simulaciones numéricas basadas en el principio de variación. Esta tesis se enmarca en los estudios de física de la materia condensada: en particular, se centra en la llamada realización sintética de los estados cuánticos de la materia, más específicamente, de los topológicos, que pueden tener en las salidas a largo plazo hacia computadoras cuánticas robustas. Se propone una investigación teórica de los átomos fríos en la red óptica con campos de gauge efectivos y sometidos a interacciones relevantes experimentalmente, agregando un enfoque numérico moderno basado en algoritmos TN. Más específicamente, este trabajo se centrará en (i) fases topológicas en los sistemas cuasi-1D y, en particular, el modelo Creutz-Hubbard, (ii) la conexión entre la materia condensada y la física de alta energía estudiando el modelo Gross-Neveu y el discretización del modelo Wilson-Hubbard, (iii) implementación de algoritmos basados en redes tensoras

Characterizing the quantum field theory vacuum using temporal Matrix Product states

In this paper we construct the continuous Matrix Product State (MPS) representation of the vacuum of the field theory corresponding to the continuous limit of an Ising model. We do this by exploiting the observation made by Hastings and Mahajan in [Phys. Rev. A \textbf{91}, 032306 (2015)] that the Euclidean time evolution generates a continuous MPS along the time direction. We exploit this fact, together with the emerging Lorentz invariance at the critical point in order to identify the matrix product representation of the quantum field theory (QFT) vacuum with the continuous MPS in the time direction (tMPS). We explicitly construct the tMPS and check these statements by comparing the physical properties of the tMPS with those of the standard ground MPS. We furthermore identify the QFT that the tMPS encodes with the field theory emerging from taking the continuous limit of a weakly perturbed Ising model by a parallel field first analyzed by Zamolodchikov.Comment: The results presented in this paper are a significant expansion of arXiv:1608.0654

Measurement-induced transitions beyond Gaussianity: a single particle description

Repeated measurements can induce entanglement phase transitions in the dynamics of quantum systems. Interacting models, both chaotic and integrable, generically show a stable volume-law entangled phase at low measurement rates which disappears for free, Gaussian fermions. Interactions break the Gaussianity of a dynamical map in its unitary part, but non-Gaussianity can be introduced through measurements as well. By comparing the entanglement and non-Gaussianity structure of different protocols, we propose a new single-particle indicator of the measurement-induced phase transition and we use it to argue in favour of the stability of the transition when non-Gaussianity is purely provided by measurementsComment: 11 pages, 4 figure

An efficient perturbation theory of density matrix renormalization group

Density matrix renormalization group (DMRG) is one of the most powerful numerical methods available for many-body systems. In this work, we develop a perturbation theory of DMRG (PT-DMRG) to largely increase its accuracy in an extremely simple and efficient way. By using the canonical matrix product state (MPS) representation for the ground state of the considered system, a set of orthogonal basis functions $\left\lbrace | \psi_i \rangle \right\rbrace$ is introduced to describe the perturbations to the ground state obtained by the conventional DMRG. The Schmidt numbers of the MPS that are beyond the bond dimension cut-off are used to define such perturbation terms. The perturbed Hamiltonian is then defined as $\tilde{H}_{ij}= \langle \psi_i | \hat{H} | \psi_j \rangle$; its ground state permits to calculate physical observables with a considerably improved accuracy as compared to the original DMRG results. We benchmark the second-order perturbation theory with the help of one-dimensional Ising chain in a transverse field and the Heisenberg chain, where the precision of DMRG is shown to be improved $\rm O(10)$ times. Furthermore, for moderate length $L$ the errors of DMRG and PT-DMRG both scale linearly with $L^{-1}$. The linear relation between the dimension cut-off of DMRG and that of PT-DMRG with the same precision shows a considerable improvement of efficiency, especially for large dimension cut-off's. In thermodynamic limit we show that the errors of PT-DMRG scale with $\sqrt{L^{-1}}$. Our work suggests an effective way to define the tangent space of the ground state MPS, which may shed lights on the properties beyond the ground state. Such second-order PT-DMRG can be readily generalized to higher orders, as well as applied to the models in higher dimensions

Many-body magic via Pauli-Markov chains -- from criticality to gauge theories

We introduce a method to measure many-body magic in quantum systems based on a statistical exploration of Pauli strings via Markov chains. We demonstrate that sampling such Pauli-Markov chains gives ample flexibility in terms of partitions where to sample from: in particular, it enables to efficiently extract the magic contained in the correlations between widely-separated subsystems, which characterizes the nonlocality of magic. Our method can be implemented in a variety of situations. We describe an efficient sampling procedure using Tree Tensor Networks, that exploits their hierarchical structure leading to a modest $O(\log N)$ computational scaling with system size. To showcase the applicability and efficiency of our method, we demonstrate the importance of magic in many-body systems via the following discoveries: (a) for one dimensional systems, we show that long-range magic displays strong signatures of conformal quantum criticality (Ising, Potts, and Gaussian), overcoming the limitations of full state magic; (b) in two-dimensional $\mathbb{Z}_2$ lattice gauge theories, we provide conclusive evidence that magic is able to identify the confinement-deconfinement transition, and displays critical scaling behavior even at relatively modest volumes. Finally, we discuss an experimental implementation of the method, which only relies on measurements of Pauli observables.Comment: 16 pages, 12 figure

Lecture Notes of Tensor Network Contractions

Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.Comment: 134 pages, 68 figures. In this version, the manuscript has been changed into the format of book; new sections about tensor network and quantum circuits have been adde

Large-S and Tensor-Network Methods for Strongly-Interacting Topological Insulators

The study of correlation effects in topological phases of matter can benefit from a multidisciplinary approach that combines techniques drawn from condensed matter, high-energy physics and quantum information science. In this work, we exploit these connections to study the strongly-interacting limit of certain lattice Hubbard models of topological insulators, which map onto four-Fermi quantum field theories with a Wilson-type discretisation and have been recently shown to be at reach of cold-atom quantum simulators based on synthetic spin-orbit coupling. We combine large-S and tensor-network techniques to explore the possible spontaneous symmetry-breaking phases that appear when the interactions of the topological insulators are sufficiently large. In particular, we show that varying the Wilson parameter r of the lattice discretisations leads to a novel Heisenberg–Ising compass model with critical lines that flow with the value of r.</jats:p

Quantifying non-stabilizerness through entanglement spectrum flatness

Non-stabilizerness - also colloquially referred to as magic - is the a resource for advantage in quantum computing and lies in the access to non-Clifford operations. Developing a comprehensive understanding of how non-stabilizerness can be quantified and how it relates other quantum resources is crucial for studying and characterizing the origin of quantum complexity. In this work, we establish a direct connection between non-stabilizerness and entanglement spectrum flatness for a pure quantum state. We show that this connection can be exploited to efficiently probe non-stabilizerness even in presence of noise. Our results reveal a direct connection between non-stabilizerness and entanglement response, and define a clear experimental protocol to probe non-stabilizerness in cold atom and solid-state platforms

Cold atoms meet lattice gauge theory

The central idea of this review is to consider quantum field theory models relevant for particle physics and replace the fermionic matter in these models by a bosonic one. This is mostly motivated by the fact that bosons are more ‘accessible’ and easier to manipulate for experimentalists, but this ‘substitution’ also leads to new physics and novel phenomena. It allows us to gain new information about among other things confinement and the dynamics of the deconfinement transition. We will thus consider bosons in dynamical lattices corresponding to the bosonic Schwinger or Z2 Bose–Hubbard models. Another central idea of this review concerns atomic simulators of paradigmatic models of particle physics theory such as the Creutz–Hubbard ladder, or Gross–Neveu–Wilson and Wilson–Hubbard models. This article is not a general review of the rapidly growing field—it reviews activities related to quantum simulations for lattice field theories performed by the Quantum Optics Theory group at ICFO and their collaborators from 19 institutions all over the world. Finally, we will briefly describe our efforts to design experimentally friendly simulators of these and other models relevant for particle physics. This article is part of the theme issue ‘Quantum technologies in particle physics’

Relaxation and (pre)thermalization in the one dimensional Bose-Hubbard model

Il presente lavoro di tesi si colloca nell’ambito della Meccanica Statistica Quantistica e si propone di studiare la dinamica di sistemi a molti corpi. In particolare, questo lavoro si focalizza sui sistemi fuori equilibrio che rappresentano ancora una frontiera dell’attuale ricerca fisica sia teorica sia sperimentale. Per portare un sistema fuori equilibrio il protocollo più utilizzato è il "Quantum Quench" che consiste nel preparare il sistema in un autostato di una certa Hamiltoniana iniziale e portarlo fuori equilibrio cambiando improvvisamente i parametri dell’ Hamiltoniana. Dallo studio dell’evoluzione temporale del sistema, descritta da un Hamiltoniana post-quench, è possibile dunque indagare l’ ergodicità e la termalizzazione, principi introdotti dapprima in meccanica statistica classica ed estesi, successivamente, da Von Neumann alla meccanica statistica quantistica. Sia teoricamente che sperimentalmente si è mostrato come tali principi sono ben definiti per sistemi quantistici non integrabili e si è osservato come gli stati stazionari sono descritti dall’ ensemble di Gibbs. Per quanto riguarda, invece, i sistemi integrabili, di fondamentale importanza sono stati gli esperimenti di Kinoshita et al. e Trotzky et al. che hanno mostrato come vengono meno sia l’ergodicità che la termalizzazione. Tali sistemi, inoltre, raggiungono uno stato di equilibrio non descritto dall’ensemble di Gibbs. Pertanto, è stato necessario definire l’ ensemble di Gibbs generalizzato (GGE) che descrive tali stati stazionari. La matrice densità assume dunque la seguente forma: ρ=(1/Z)e^{−λα Iα} dove Iα sono gli integrali del moto mentre λα sono i moltiplicatori di Lagrange. Nel contesto di studi descritto finora si situa il fulcro di questo lavoro riguardante l’analisi numerica di un particolare quench per una catena quantistica in cui sono presenti delle interazioni che rompono debolmente l’integrabilità. Nello specifico,sono state effettuate delle simulazioni mediante l’algoritmo TEBD (Time-evolving block decimation) su uno stato di partenza |ψ0> che descrive una catena infinita molto diluita, la cui evoluzione temporale è descritta dall’Hamiltoniana del modello Bose-Hubbard, che rappresenta uno dei più semplici ma anche tra i più potenti strumenti per descrivere le caratteristiche di sistemi di bosoni interagenti su reticoli ottici. Nella tesi sono riportati i risultati di diverse simulazioni numeriche effettuate per diversi valori dell’interazione del modello BH ed al variare dello stato iniziale. Nel limite di interazione nulla e di interazione infinita il sistema è integrabile e risolvibile analiticamente. In tali casi le osservabili locali raggiungono uno stato stazionario descritto dall’esemble di Gibbs generalizzato. Anche un’interazione debolissima comporta la rottura dell’integrabilità e in questo caso ci aspettiamo che lo stato stazionario del sistema sia termico. Le simulazioni mostrano che per tempi molto lunghi le osservabili locali rilassano a uno stato di pretermalizzazione descritto da una matrice densità che è una deformazione di quella trovata per l’ensemble di Gibbs generalizzato con delle cariche quasi conservate che caratterizzano il sistema, di cui si presenta una esplicita espressione. Solo per tempi molto lunghi e non accessibili sia alle simulazioni che agli esperimenti il sistema dovrebbe rilassare al vero stato stazionario. Infine, si è mostrato che al crescere dell’interazione il sistema ha un comportamento compatibile con un’eventuale termalizzazione