Pairing and topological phases in cold atoms with long range interactions Chapter 1 Introduction The systematic study and classification of phase transitions has become a popular research area in physics over the past decades. The most familiar phenomena associated to the concept of phase transition are the ones giving rise to macroscopic changes in the properties of strongly correlated many-body systems due to thermal fluctuations. The most prominent examples of this kind of behaviour are the ferromagnetic-paramagnetic transition in lattice spin models and the liquid-gas transition. On the other hand, when dealing with quantum-mechanical systems changes of matter from one state to another can arise as the result of the variation of parameters other than temperature. In particular, when a given many-body quantum problem is studied at zero temperature, thereby suppressing thermal fluctuations, it is the interplay between kinetic energy and interactions that can cause the system to jump into different phases as the model parameters are varied. Phase transitions occurring at zero temperature are widely known as quantum phase transitions. In the framework of the physics of phase transitions, the role of dimensionality is a noticeable one. The major challenge in this perspective is the treatment of low dimensional systems, where standard mean-field treatments and ordinary perturbation theory are known to fail due to the enhancement of fluctuations. The goal of describing collective behaviours in reduced dimensionality is usually pursued in one-dimensional (1D) systems, which turn out to be simple enough to be studied in an effective way through both numerical and analytical techniques, while exhibiting a rich and deeply fascinating phenomenology. As a matter of fact, since particles confined to one dimension cannot avoid interactions with each other, the typical low energy excitations of the system are represented by collective density waves, which substitute nearly free quasiparticle excitations characterizing the behaviour of three-dimensional Fermi liquids in the description of the effects of the interplay between kinetic fluctuations and interactions. On the other hand, the tools employed in the activity of tackling 1D manybody problems are efficient and well developed. The analytical techniques aremainly based on the bosonization technique, which allows for the reformulation of strongly correlated 1D systems onto free bosonic theories, which can then be analyzed by means of standard path integral techniques. Meanwhile, the density matrix renormalization group (DMRG) algorithm represents the state-of-theart numerical machinery in the study of 1D physics, since it allows to efficiently extract the low energy properties of the model Hamiltonian and consequently characterize the corresponding phase diagram by means of the behaviour of properly chosen order parameters. As a final remark pointing out the relevance of the research effort in the direction of a better understanding of 1D many-body quantum systems, it is worth noticing that 1D systems do not represent only a playground for theorists, but have been experimentally realized by means of setups with cold atoms trapped in optical lattices, where the experimentalist is able to tune the model parameters by varying the characteristic features of the trapping laser beams and meanwhile has a comfortable access to the most relevant observables.
Pairing and topological phases in cold atoms with long range interactions Chapter 1 Introduction The systematic study and classification of phase transitions has become a popular research area in physics over the past decades. The most familiar phenomena associated to the concept of phase transition are the ones giving rise to macroscopic changes in the properties of strongly correlated many-body systems due to thermal fluctuations. The most prominent examples of this kind of behaviour are the ferromagnetic-paramagnetic transition in lattice spin models and the liquid-gas transition. On the other hand, when dealing with quantum-mechanical systems changes of matter from one state to another can arise as the result of the variation of parameters other than temperature. In particular, when a given many-body quantum problem is studied at zero temperature, thereby suppressing thermal fluctuations, it is the interplay between kinetic energy and interactions that can cause the system to jump into different phases as the model parameters are varied. Phase transitions occurring at zero temperature are widely known as quantum phase transitions. In the framework of the physics of phase transitions, the role of dimensionality is a noticeable one. The major challenge in this perspective is the treatment of low dimensional systems, where standard mean-field treatments and ordinary perturbation theory are known to fail due to the enhancement of fluctuations. The goal of describing collective behaviours in reduced dimensionality is usually pursued in one-dimensional (1D) systems, which turn out to be simple enough to be studied in an effective way through both numerical and analytical techniques, while exhibiting a rich and deeply fascinating phenomenology. As a matter of fact, since particles confined to one dimension cannot avoid interactions with each other, the typical low energy excitations of the system are represented by collective density waves, which substitute nearly free quasiparticle excitations characterizing the behaviour of three-dimensional Fermi liquids in the description of the effects of the interplay between kinetic fluctuations and interactions. On the other hand, the tools employed in the activity of tackling 1D manybody problems are efficient and well developed. The analytical techniques aremainly based on the bosonization technique, which allows for the reformulation of strongly correlated 1D systems onto free bosonic theories, which can then be analyzed by means of standard path integral techniques. Meanwhile, the density matrix renormalization group (DMRG) algorithm represents the state-of-theart numerical machinery in the study of 1D physics, since it allows to efficiently extract the low energy properties of the model Hamiltonian and consequently characterize the corresponding phase diagram by means of the behaviour of properly chosen order parameters. As a final remark pointing out the relevance of the research effort in the direction of a better understanding of 1D many-body quantum systems, it is worth noticing that 1D systems do not represent only a playground for theorists, but have been experimentally realized by means of setups with cold atoms trapped in optical lattices, where the experimentalist is able to tune the model parameters by varying the characteristic features of the trapping laser beams and meanwhile has a comfortable access to the most relevant observables.
The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.
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.
Condensed matter is the area of physics which deals with the phases of matter in condensed form with an aim to understand the properties of these phases and the transition between them based on the known laws of physics. This is usually done by figuring out some characteristic properties of the phase which sets it apart from the others, e.g., symmetry, topological order, entanglement, etc. The most common way of characterizing the transition between these phases is the broken symmetry. In this classification method, one finds the symmetry which phase breaks spontaneously during the transition and develops a local long-range order. This is the famous Landau’s symmetry breaking paradigm [ 1 ]. Landau’s symmetry breaking theory is a phenomenological theory which describes the transition from a symmetric phase (high-temperature phases) to a less symmetric/broken symmetry phase (phases at low temperature). A zero-temperature phase transition is also possible (by tuning the parameters of the Hamiltonian) and is known as quantum phase transition [ 2 ]. Although there were other paradigms of phase transition available, like Anderson localization [ 3 ], Berezinsky—Kosterlitz—Thouless transition [ 4 ] (broken symmetry but no long-range order), etc., Landau’s spontaneous symmetry breaking dominated over the others.
In this work we develop a new numerical method, the adaptive time-dependent density-matrix renormalization-group (adaptive t-DMRG), which turns out to be very well suited to investigate time-dependent phenomena in one-dimensional strongly correlated systems. The method is based on the original density-matrix renormalization group method by White and the time evolving block-decimation procedure by Vidal. We further show the applicability of the new method to different physical systems: bosonic, fermionic, and spin chains. We first discuss the consequences of the presence of an external trapping potential on the static properties of trapped cold bosonic atoms in an optical lattice using the Bose-Hubbard model. It is shown that a properly rescaled one-particle density matrix characterizes superfluid versus insulating states just as in the homogeneous system. We note that the superfluid to Mott-insulating transition is seen most directly in the half width of the interference peak. Then we turn to the evolution of density perturbations in these systems. In particular, we discuss the dependence of the velocity and the decay of the amplitude on density, interaction strength, and the extent and height of the perturbation in a numerically exact way, covering a wide range of interaction and perturbation strengths. By comparing our results for the sound velocity to theoretical predictions, we determine the limits of a Gross-Pitaevskii or Bogoliubov type description and the regime where repulsive one-dimensional Bose gases exhibit fermionic behaviour. As a second application of the adaptive t-DMRG, we investigate the phenomenon of spin-charge separation in the Hubbard model and propose an experimental setup for its observation in cold Fermi gases. We show the robustness of this separation beyond the low-energy regime by studying the time evolution of density wave packets of finite strength and at length scales down to a few lattice spacings. A striking signature of spin-charge separation is found in 1D cold Fermi gases in a harmonic trap using the different propagation properties of the liquid and Mott-insulating phases. The third application relates to the area of spintronics. A simplified model for the magnetization transport between two coupled reservoirs with opposite spin polarization is studied. We investigate whether a simple long time limit for the spin transport exists and if so, how it depends on the properties of the system such as integrability and criticality. Time-scales accessible to us are of the order of 100 units of time measured in 1/J while maintaining insignificant error in the observables.
The past decade has seen a substantial rejuvenation of interest in the study of quantum phase transitions, driven by experiments on cuprate superconductors, heavy fermion materials, organic conductors, and related compounds. Although quantum phase transitions in simple spin systems, like the Ising model in a transverse field, were studied in the early 1970s, much of the subsequent theoretical work examined a particular example: the metal–insulator transition. While this is a subject of considerable experimental importance, the greatest theoretical progress was made for the case of the Anderson transition of non-interacting electrons, which is driven by localization of the electronic states in the presence of a random potential. The critical properties of this transition of noninteracting electrons constituted the primary basis upon which most condensed matter physicists have formed their intuition on the behavior of the systems near a quantum phase transition. However, it is clear that strong electronic interactions play a crucial role in the systems of current interest noted earlier, and simple paradigms for the behavior of such systems near quantum critical points are not widely known. It is the purpose of this book to move interactions to center stage by describing and classifying the physical properties of the simplest interacting systems undergoing a quantum phase transition. The effects of disorder will be neglected for the most part but will be considered in the concluding chapters. Our focus will be on the dynamical properties of such systems at nonzero temperature, and it will become apparent that these differ substantially from the noninteracting case.
The research field of topological phases and materials is one of the most exciting frontiers in modern condensed matter physics [1]; [2]; [3]; [4]; [5]; [6] ; [7]. In contrast to the classification of conventional condensed matter phases according to their symmetries, different topological states of matter generally do not involve broken symmetries. Rather, they are associated with certain fundamental physical properties that are insensitive to smooth changes in materials parameters. Therefore, a topological state cannot change into another unless the system undergoes a quantum phase transition [6] ; [7].