The work in this thesis is written with the following question in mind: does the physics investigated in colliders like the one at CERN have its counterpart in phenomena like superfluidity and superconductivity? \nMore precisely, this thesis focuses on the condensed matter parallel of non-Abelian gauge theories exploited in high-energy physics. These theories capture the notion of parallel transport and offer the possibility of deconfinement, i.e., the existence of quarks and gluons. In the first part of the thesis, the notion of parallel transport in the context of spin-orbit coupled systems is studied, which have attracted much attention in recent years. The question arises whether hydrodynamic conservation laws for non-Abelian spin currents exist. The answer is yes, provided the spin system is in a superfluid state.\nIn the second part, the doped Mott insulator is studied, which is relevant for the problem of high-temperature superconductivity. A gauge theory considering the electron as being split up into spinons and holons (a condensed matter analogue of quarks and gluons) is studied. It is shown how this theory, when some subtle issues are treated correctly, leads to physical predictions which can be (and, in some cases, have been) confirmed experimentally.
The work in this thesis is written with the following question in mind: does the physics investigated in colliders like the one at CERN have its counterpart in phenomena like superfluidity and superconductivity? \nMore precisely, this thesis focuses on the condensed matter parallel of non-Abelian gauge theories exploited in high-energy physics. These theories capture the notion of parallel transport and offer the possibility of deconfinement, i.e., the existence of quarks and gluons. In the first part of the thesis, the notion of parallel transport in the context of spin-orbit coupled systems is studied, which have attracted much attention in recent years. The question arises whether hydrodynamic conservation laws for non-Abelian spin currents exist. The answer is yes, provided the spin system is in a superfluid state.\nIn the second part, the doped Mott insulator is studied, which is relevant for the problem of high-temperature superconductivity. A gauge theory considering the electron as being split up into spinons and holons (a condensed matter analogue of quarks and gluons) is studied. It is shown how this theory, when some subtle issues are treated correctly, leads to physical predictions which can be (and, in some cases, have been) confirmed experimentally.
Gauge theories are ubiquitous in physics. Many intriguing phenomena in condensed matter physics owe to the action of the electromagnetic field, which is an Abelian gauge theory. The numerical treatment of many-body systems is inherently complex due to the exponentially growing size of the Hilbert space. While in one dimension an area law guarantees that numerical methods on classical computers can deal with strongly correlated systems, in higher dimensions the quantum simulation comes as the panacea for the many-body problem. The present thesis comprises the elaboration of experimentally feasible methods for the quantum simulation of dynamical Abelian gauge fields with ultra-cold gases of neutral atoms and the theoretical analysis of the related model Hamiltonians. As neutral atoms do not interact with external vector potentials like charged particles would do, the gauge fields have to be artificially engineered. The elements of a gauge theory that need to be replicated on a quantum simulator vary depending on the subject of investigation. The key ingredient at the root of many condensed matter phenomena, from the quantum Hall effect to superconductivity and chiral topological insulators, is the Berry phase. Whilst artificial static gauge fields have been widely explored, much remains to do regarding the realization of artificial dynamical gauge fields. In Chapter 3 we present a method based on the amplitude modulation of a one-dimensional optical lattice, which allows for an unprecedented degree of control over a wide range of parameters. The method also comprises the generation of a density-dependent complex phase, fundamental to the creation of anyonic pseudo-particles. The anyons are amenable of observation through interferometric measurement, realizable with the same experimental set-up. With regard to gauge theories, the Berry phase is just the visible tip of the iceberg. Below the waterline, there is more to consider in order to comprehensively reproduce a gauge theory, like the electric and magnetic fields in quantum electrodynamics. Moreover, a full account for the inherent symmetry is crucial to investigate phenomena proper of non-Abelian gauge theories in the context of high-energy physics, such as confinement. For this collection of topics, one can turn to lattice gauge theories. In Chapter 5, we consider a class of lattice gauge theories particularly suitable for quantum simulation, the Quantum Link Model. The study of the Abelian U(1) Quantum Link Model on a ladder geometry reveals a highly non-trivial phase diagram, featuring a symmetry-protected topological phase. In both Chapters, innovative solutions for the experimental realization of the model Hamiltonians are designed and proposed. To gain numerical access to the ground-state properties and the dynamics of the systems investigated we make use of state-of-the-art numerical methods based on Tensor Networks. The elements of the numerical analysis carried out throughout this thesis are presented in Chapter 6. In the last part we offer an outlook on research perspectives related to the topics discussed in the thesis.
Abelian and non‐Abelian gauge theories are of central importance in many areas of physics. In condensed matter physics, Abelian U (1) lattice gauge theories arise in the description of certain quantum spin liquids. In quantum information theory, Kitaev's toric code is a lattice gauge theory. In particle physics, Quantum Chromodynamics (QCD), the non‐Abelian gauge theory of the strong interactions between quarks and gluons, is non‐perturbatively regularized on a lattice. Quantum link models extend the concept of lattice gauge theories beyond the Wilson formulation, and are well suited for both digital and analog quantum simulation using ultracold atomic gases in optical lattices. Since quantum simulators do not suffer from the notorious sign problem, they open the door to studies of the real‐time evolution of strongly coupled quantum systems, which are impossible with classical simulation methods. A plethora of interesting lattice gauge theories suggests itself for quantum simulation, which should allow us to address very challenging problems, ranging from confinement and deconfinement, or chiral symmetry breaking and its restoration at finite baryon density, to color superconductivity and the real‐time evolution of heavy‐ion collisions, first in simpler model gauge theories and ultimately in QCD.
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 thesis considers out-of-equilibrium dynamics of strongly interacting non-relativistic Fermi gases in several two and three dimensional geometries. The tools of second-order hydrodynamics and gauge-gravity duality will be utilized to address this system. Many of the themes of this work are motivated by the observed similarities in transport properties between strongly interacting Fermi gases and other very different strongly interacting quantum fluids such as the quark-gluon plasma, high temperature superconductors, and quantum field theories described by gauge-gravity duality. In particular, these systems all nearly saturate the conjectured lower bound on the ratio of shear viscosity to entropy density η/s≥ℏ/(4πkB) coming from the AdS/CFT correspondence. Among other things, this observation, in conjunction with current experiment and data analysis in atomic, condensed matter, and nuclear physics lends itself to the following questions: How perfect of a fluid is the strongly interacting Fermi gas, and can one find a more stringent constraint on η/s in Fermi gases? Do the similarities in transport properties among strongly interacting quantum systems extend beyond dynamics controlled by the hydrodynamical shear viscosity? In regards to the first question, by utilizing second-order hydrodynamics, it will be demonstrated that higher-order collective modes of a harmonically trapped Fermi gas may serve as a more sensitive probe of the shear viscosity. For the second question, both second-order hydrodynamics and a gravity dual theory are used to make predictions about dynamics occurring on short timescales where hydrodynamics is expected to break down. In particular the appearance of a class of "non-hydrodynamic" collective modes not contained within a Navier-Stokes description of the strongly interacting Fermi gas will be discussed.
The study of systems formed from ultracold atomic gases has emerged to become one of the most active research fields within the condensed matter landscape. These highly controllable macroscopic systems amalgamate ideas from many sub disciplines of physics, including the study of low temperatures, quantum optics and quantum information theory as well as the seemingly disparate field of high energy physics. The central concept of this thesis is gauge theories as applied to systems of bosonic atoms, which at temperatures close to absolute zero form Bose-Einstein condensates. To simulate the mathematical structure of a gauge theory, the geometric (Berry) phase formalism is adopted. This is in turn accomplished by considering the adiabatic following of the eigenstates of the light-matter coupling for an ensemble of atoms forming a Bose-Einstein condensate. These concepts are then applied to show how one can generate a spin-orbit coupling in a one-dimensional condensate, which additionally features a random mass term that allows us to study the physics of Anderson localization in an intriguing “quasi” relativistic regime. One of the features of light induced gauge potentials is that they are static; in the sense that there is no feedback between the light-matter interaction and the matter field. In the second part of this thesis it is demonstrated how such a feedback mechanism can be induced by the appropriate modification of the light-matter interaction. The consequences this has for the condensate are then described at the mean-field level, including the expected experimental signatures of the resulting ‘interacting’ gauge theory, in terms of the expansion of the condensate and also the structure of the solitons of this nonlinear system. Finally, this nonlinear model is applied to a double well system, from which the associated Bose-Hubbard model is derived and analysed; and the nonlinear Josephson problem studied.
In this thesis we investigate the phase structure of dense quark matter, the structure and stability of neutron and quark stars, and the phases of ultracold fermions in the presence of an artificial spin-orbit coupling. While spanning an extraordinary twenty orders of magnitude in energy scales, these systems exhibit some remarkable similarities including non-perturbative many-body interactions, perfect fluid behavior, the formation of Cooper pairs, and the possibility of BCS-BEC crossovers between weakly and strongly interacting regimes. Moreover, due to phenomenal advancements in laser cooling techniques and the ability to exert an unprecedented level of control over the interactions of ultracold atomic gases, the possibility of using these systems to simulate the complex behavior of systems not easily realized in the laboratory (e.g., non-Abelian gauge fields, quantum chromodynamics) is becoming increasingly real. Despite the widespread success of quantum chromodynamics (QCD), the theory of the strong nuclear force, much remains unknown about the properties of strongly-interacting quark matter. In large part, this continued ignorance is a result of the mathematical intractability of QCD and the limitations of current numerical techniques to very low densities. In the first part of this thesis, in order to gain some insight into the phase structure of dense quark matter we therefore apply an effective field theory which is built upon the symmetries of QCD, the Polyakov–Nambu–Jona-Lasinio (PNJL) model. We construct the QCD phase diagram for two and three quark flavors, giving special attention to the effect of the intermediate strange quark mass on the preferred quark pairing structure at intermediate to high density. In addition, we investigate the impact of the strange quark mass and axial anomaly on a recently proposed low temperature critical point, which may allow for a smooth crossover between hadronic and color superconducting matter. Finally, we investigate the impact of a local color neutrality constraint on phases of asymmetric quark pairing. While the Relativistic Heavy Ion Collider (RHIC) continues to probe the QCD phase diagram at ever-higher temperatures and researchers await the completion of the highly anticipated Facility for Antiprotons and Ion Research (FAIR), the only known “laboratories” in which low temperature dense quark matter is encountered are the cores of neutron stars. Fortunately, the structure of these astrophysical objects is highly dependent upon the properties of the dense matter in their cores, and observations of the mass-radius relationship of these stars impose constraints on the quark matter equation of state. In the second part of this thesis we investigate the possibility of realizing massive neutron stars, such as the recently observed PSRs J1614-2230, J0348+0432, and J1311-3430 with masses of two solar masses or larger, by including a flavor-symmetric vector coupling within an NJL model description of quark matter. By extracting the quark matter equation of state we show that in the absence of diquark pairing, a reasonable magnitude vector repulsion can to stabilize neutron stars against gravitational collapse up to 2.34 solar masses. We also investigate the possibility of realizing stable quark stars with densities much higher than those obtained in conventional neutron stars, but find that stars with central densities much greater than ten times nuclear density are always unstable to gravitational collapse. In the third part of this thesis we study the properties of ultracold atomic gases in the presence of artificial gauge fields. While neutral atoms do not naturally couple to the gauge fields in nature (e.g., magnetic fields, the strong nuclear force), recent advancements in laser techniques have led to the realization of synthetic gauge fields in which ultracold atoms behave as if they were charged. Combined with the Feshbach resonance, through which the two-body interactions which dominate these dilute gases can be arbitrarily tuned, these gases can be used to simulate a wide variety of systems, including those that occur naturally and those that do not. We study the phase structure of a two-species mixture of fermions in the presence of Rashba-Dresselhaus (RD) spin-orbit coupling, which is induced by a specific type of non-Abelian gauge field. In particular, we compute the dependence of the superfluid critical temperature on the RD coupling strength and the tunable two-body interaction. We also investigate the effects of the spin-orbit coupling on the crossover between weakly bound (BCS) atomic pairs and strongly bound (BEC) molecules and the effects of fluctuations on the stability of the superfluid phase.