In this thesis, we study two distinct, yet equally fascinating, aspects of manybody physics. Synthetic quantum matter – engineered many-body systems with high controllability and tunability – serves as a cornerstone for the research. The combination of these features makes them incredibly suited for research at the crossroads of traditional statistical physics, new concepts coming from field theory (particularly gauge theories), and quantum information. In the first part, we delve into the quantum simulation of novel quantum phenomena with analog quantum simulators. In particular, recent advances in the synthetic platforms based on Rydberg atoms, i.e., neutral atoms trapped in optical tweezers, have opened up new avenues to explore exotic topological phases like quantum spin liquids, which hold potential applications in quantum computing. We employ a combination of numerical and analytical tools to address open questions regarding the realization of these phases in experiments. Our results show that the recently observed spin liquids can be directly connected to a Higgs-Ising lattice gauge theory, providing a clear-cut theoretical argument for its origin. Moreover, different setups with chiral interacting Hamiltonians offer new avenues to realize chiral spin liquids - a lattice version of the celebrated quantum Hall effect. In the second part, we explore the role of a quantum information quantity known as nonstabilizerness in quantum many-body systems. By now, the study of quantum information concepts in the realm of many-body theory represents a very active research effort, with a prominent example being entanglement, which has revolutionized our understanding of quantum systems. Similar to entanglement, nonstabilizerness has emerged as a crucial resource for achieving quantum advantage, although in a fundamentally different way. While entanglement is well-studied, the exploration of nonstabilizerness in many-body contexts is still in its very early stages. Its ongoing investigation holds significant promise for deepening our understanding of many-body systems and their complexity. We develop new methods based on tensor networks and Monte Carlo methods to enable the quantification of nonstabilizerness at large scales. We subsequently apply them to uncover the role of nonstabilizerness in various many-body settings, particularly in critical phenomena, gauge theory, and quantum circuits.
In this thesis, we study two distinct, yet equally fascinating, aspects of manybody physics. Synthetic quantum matter – engineered many-body systems with high controllability and tunability – serves as a cornerstone for the research. The combination of these features makes them incredibly suited for research at the crossroads of traditional statistical physics, new concepts coming from field theory (particularly gauge theories), and quantum information. In the first part, we delve into the quantum simulation of novel quantum phenomena with analog quantum simulators. In particular, recent advances in the synthetic platforms based on Rydberg atoms, i.e., neutral atoms trapped in optical tweezers, have opened up new avenues to explore exotic topological phases like quantum spin liquids, which hold potential applications in quantum computing. We employ a combination of numerical and analytical tools to address open questions regarding the realization of these phases in experiments. Our results show that the recently observed spin liquids can be directly connected to a Higgs-Ising lattice gauge theory, providing a clear-cut theoretical argument for its origin. Moreover, different setups with chiral interacting Hamiltonians offer new avenues to realize chiral spin liquids - a lattice version of the celebrated quantum Hall effect. In the second part, we explore the role of a quantum information quantity known as nonstabilizerness in quantum many-body systems. By now, the study of quantum information concepts in the realm of many-body theory represents a very active research effort, with a prominent example being entanglement, which has revolutionized our understanding of quantum systems. Similar to entanglement, nonstabilizerness has emerged as a crucial resource for achieving quantum advantage, although in a fundamentally different way. While entanglement is well-studied, the exploration of nonstabilizerness in many-body contexts is still in its very early stages. Its ongoing investigation holds significant promise for deepening our understanding of many-body systems and their complexity. We develop new methods based on tensor networks and Monte Carlo methods to enable the quantification of nonstabilizerness at large scales. We subsequently apply them to uncover the role of nonstabilizerness in various many-body settings, particularly in critical phenomena, gauge theory, and quantum circuits.
(English) In recent years, the evolution of quantum technologies has resulted in an unprecedented control over individual quantum particles and many-body systems. This remarkable progress has given rise to a new era, marked by the convergence of classical and quantum methodologies to investigate fundamental questions concerning the nature of quantum matter, improving our understanding of the role of entanglement in solid-state systems or the mechanisms behind high-energy physics. From analog quantum simulators to digital quantum computers, these advancements hold potential across diverse domains. This thesis explores the interplay between classical and quantum methods in understanding topological phases of matter. We concentrate on three distinct directions: non-conventional topological superconductors, interaction-induced topological phases in ultracold atom quantum simulators, and applications of variational quantum algorithms. Each trajectory relies on the combination of different techniques with the aim of understanding and characterizing topological phenomena in different settings. Exploring non-conventional topological superconductors involves extending the paradigmatic Kitaev chain model by incorporating additional terms in the Hamiltonian such as long-range interactions and quasi-periodic potentials. This investigation is relevant to better understand the impact of real-world conditions on the both the topological and localization properties of systems hosting non-local Majorana modes, which are promising candidates for topological quantum computation. In the realm of interacting systems, we explore the realization of interaction-induced topological phases in systems of ultracold atoms in optical lattices, both in one and two dimensions. The remarkable control and versatility of such platforms enable the simulation of both theoretical topological models and strongly correlated physics. Notably, the interplay between interactions and topology can give rise to intriguing phenomena, such as delocalized fractional charges and gapless topological phases,challenging existing intuition. We employ advanced numerical methods based on tensor networks to benchmark the experimental proposals that open the door to the realization and detection of novel many-body phases of matter, including topological quantum critical points and a higher-order topological Peierls insulator in Bose-Hubbard models with long-range interactions. Variational quantum algorithms, conversely, have the potential to efficiently tackle a wide range of problems, including ground state search, phase classification or accessing topological invariants. Despite current limitations in trainability and scalability, these hybrid classical-quantum algorithms provide practical insights into current quantum hardware capabilities and can inspire future architectures. We explore the application of variational quantum algorithms to shed light on topological phenomena, raising questions about their ability to discern topological phase transitions and compute topological invariants in situations where classical approaches fail. This thesis presents a comprehensive exploration of distinct approaches to topological quantum matter by leveraging quantum technologies and quantum-inspired classical algorithms. Our results not only advance our understanding of quantum systems but also pave the way for the realization and discovery of novel physics extending to quantum information processing, materials science, and beyond.
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.
Quantum many-body systems are comprised of complex networks of microscopic interactions that work together to produce novel collective phases and phenomena. \nFor strongly-interacting systems, the connection between the underlying atomic-scale behaviour and the resulting emergent observable phenomena is exponentially (and, in practice, infinitely) complicated such that it becomes impossible to fully understand the connection between the physics at the microscopic and macroscopic levels. \nThe most promising theoretical approaches to addressing this infinite complexity utilize microscopic coarse-graining along with powerful computational simulations and strategies. \nIn this thesis, we apply numerical methods to effective lattice models and field theories with the goal of shedding light onto universal critical behaviour and exotic low-temperature phases in condensed matter physics. \n \nWe start by exploring universal features of critical non-interacting systems, for which several analytical strategies are readily available for calculating certain observables. \nWe study the behaviour of the system's entanglement for various entangling geometries, and utilize numerical techniques in order to isolate new universal features of Gaussian fixed points, which provide insight into the underlying critical theory's renormalization group flow. \nWe then proceed to examine computational strategies and, in particular, Monte Carlo simulations for studying more general interacting models. \nWe focus on the application of such strategies to the field of high-temperature superconductivity, for which we develop a coarse-grained model, simulate macroscopic observables and compare our results with those of recent experiments. \nWe conclude that lattice field theories together with innovative computational methods offer new perspectives on both universality and emergent phenomena in quantum matter.
The many-body problem is ubiquitous in the theoretical description of physical phenomena, ranging from the behaviour of elementary particles to the physics of electrons in solids. Most of our understanding of many-body systems comes from analysing the symmetric properties of Hamiltonian and states: the most striking examples are gauge theories such as quantum electrodynamics, where a local symmetry strongly constrains the microscopic dynamics. The physics of such gauge theories is relevant for the understanding of a diverse set of systems, including frustrated quantum magnets and the collective dynamics of elementary particles within the standard model. In the last few years, several approaches have been put forward to tackle the complex dynamics of gauge theories using quantum information concepts. In particular, quantum simulation platforms have been put forward for the realisation of synthetic gauge theories, and novel classical simulation algorithms based on quantum information concepts have been formulated. In this review, we present an introduction to these approaches, illustrating the basics concepts and highlighting the connections between apparently very different fields, and report the recent developments in this new thriving field of research.
Interactions among atoms can lead to emergent phenomena, one of the most fascinating being topological order. Topological phases are important in understanding high-temperature superconductors and fault-tolerant quantum computing. Quantum spin liquids, a key example of topologically ordered phases, exhibit unique properties such as long-range entanglement in the ground state and support of anyonic excitations, making them an exciting focus in the study of strongly correlated systems. This thesis explores the detection and learning of topological phases in quantum many-body systems, including quantum spin models, cuprate superconductors and ultracold atomic platform of Rydberg atom arrays. Guided by the experiments on Kitaev materials, we begin by using spinon mean-field theory of quantum spin liquids to explain the transition from a quantized thermal Hall effect to an unquantized values. Making closer connections to experiments in cuprates, we include additional interactions with phonons present in the system. We examine how spinon-phonon interactions, characterized by symmetry analysis, give rise to a chiral phonon Hall viscosity and can also contribute to the thermal Hall effect. In the second part, we turn to Rydberg atom arrays, which have recently been shown to realize a Z2 quantum spin liquid phase—an elusive state in conventional solid-state materials. We study how Rydberg interactions and lattice geometries lead to the emergence of Z2 gauge theories and distinct classes of topological quantum spin liquids. Finally, we study a variational tensor network tomography method for learning topological states from randomized measurements on Rydberg atom arrays. We demonstrate its efficiency in characterizing complex two-dimensional states that could be realized in future experiments.
Certain classes of quantum many-body systems, including those supporting phenomena like high-Tc superconductivity and spin liquids, are believed to be fundamentally intractable to classical modeling. Quantum simulations, in which synthetic materials are engineered by inducing well-controlled quantum systems like ultracold atoms to obey many-body Hamiltonians of interest, are a promising new approach to study this type of physics. In this work, I present several advances toward this ultimate goal of large-scale, highly controllable quantum simulations of many-body spin physics. We simulate long-range Ising and XY spin models in the presence of transverse and longitudinal magnetic fields using chains of up to 18 ultracold 171Yb+ ions held in a linear Paul trap, where two hyperfine levels in each ion encode spin-1/2 states. The tunable spin-spin interactions and effective magnetic fields are engineered using laser fields, and the individual spin states are directly imaged with state-dependent fluorescence. The results in this thesis address several of the ongoing challenges in the development of synthetic quantum matter platforms. One such challenge is establishing more flexible capabilities in the sorts of Hamiltonians we can model. By observing suppression of the ground state spin ordering, we have demonstrated our ability to continuously tune the interaction range in a power-law interaction pattern, and hence the amount of frustration present in the spin system. We have additionally begun developing tools to study particles of higher spin, which could eventually be used to create and study topological phases of matter. Another challenge is the necessity of identifying problems that the next generation of experiments, with flexible (but not arbitrary) controls and classically intractable (but not infinitely large) system sizes, can feasibly shed new light on. We have made measurements of how the range of interaction affects dynamics of spin correlations propagating through the chain, and the excellent agreement between our observations and numerical simulations indicate that at larger sizes, our experiment can meaningfully contribute to the open question of the fundamental speed limit on the transfer of information through such a spin chain. Finally, for classically intractable system sizes, it will be crucial to have multiple techniques at our disposal for validating our understanding of the exact microscopic model being implemented. We have developed and demonstrated an MRI-like spectroscopic technique for probing the energies of the many-body Hamiltonian, which serves as a new method for validating quantum simulations of the transverse Ising model. Our experiments can potentially be scaled up in the near future to study fully connected lattice spin models with several tens of spins, where classical computation begins to fail, and the results described in this thesis contribute to the effort to build experiments that can break new ground in the study of quantum many-body physics.