PACS. 74.20.-z – Theories and models of superconducting state. PACS. 11.15.Ha – Lattice gauge theory. PACS. 71.10.Hf – Non-Fermi-liquid ground states, electron phase diagrams and phase transitions in model systems. Abstract. – We study topological aspects of a compact lattice superconductor, and show that the characteristic energy splitting, ∆, between almost degenerate ground states, is simply related to a novel order parameter ˜ W, which is closely related to large Wilson loops. Using Monte Carlo methods, we study the scaling properties of ˜ W close to the deconfining phase transition, and conclude that ∆ ∼ e −L/ξ, where L is the size of the system, thus giving quantitative support to the vortex tunneling scenario proposed by Wen. The concept of topological order was first introduced to describe the ground states of the quantum Hall system [1]. Although the present interest in topological order to large extent derives from the quest for exotic non-Fermi liquid states of relevance for the high-Tc superconductors [2–6], the concept is of much more general interest. Both spin liquids [7,8], believed to be of relevance for frustrated magnets, and ordinary BCS superconductors with dynamical
PACS. 74.20.-z – Theories and models of superconducting state. PACS. 11.15.Ha – Lattice gauge theory. PACS. 71.10.Hf – Non-Fermi-liquid ground states, electron phase diagrams and phase transitions in model systems. Abstract. – We study topological aspects of a compact lattice superconductor, and show that the characteristic energy splitting, ∆, between almost degenerate ground states, is simply related to a novel order parameter ˜ W, which is closely related to large Wilson loops. Using Monte Carlo methods, we study the scaling properties of ˜ W close to the deconfining phase transition, and conclude that ∆ ∼ e −L/ξ, where L is the size of the system, thus giving quantitative support to the vortex tunneling scenario proposed by Wen. The concept of topological order was first introduced to describe the ground states of the quantum Hall system [1]. Although the present interest in topological order to large extent derives from the quest for exotic non-Fermi liquid states of relevance for the high-Tc superconductors [2–6], the concept is of much more general interest. Both spin liquids [7,8], believed to be of relevance for frustrated magnets, and ordinary BCS superconductors with dynamical
We study topological aspects of a compact lattice superconductor, and show that the characteristic energy splitting, Δ, between almost degenerate ground states, is simply related to a novel order parameter , which is closely related to large Wilson loops. Using Monte Carlo methods, we study the scaling properties of close to the deconfining phase transition, and conclude that Δ ∼ e − L /ξ , where L is the size of the system, thus giving quantitative support to the vortex tunneling scenario proposed by Wen.
The Landau Fermi liquid theory and the Ginzburg-Landau phase transition theory stand as two pivotal cornerstones in traditional condensed matter physics, achieving significant success in addressing crucial physical phenomena such as BCS superconductors and liquid helium superfluids. However, marked by the discoveries of the quantum Hall effect and high-temperature superconductivity in the 1980s, it gradually became evident that for a broad class of novel quantum states, such as fractional quantum Hall states and quantum spin liquids, their properties transcend the Landau Fermi liquid theory and Ginzburg-Landau phase transition theory. Topological order and its related concepts of long-range many-body quantum entanglement and fractionalized excitation have become the key concepts to understand these exotic quantum states. Designing and identifying topologically ordered states of matter in quantum materials and quantum simulation systems, and probing and manipulating their fractionalized excitations, are important research directions in modern condensed matter physics. In recent years, great progress has been made in the quantum simulation and manipulation of topological order on highly controllable quantum simulation platforms, such as Rydberg atomic systems, superconducting quantum processors, and two-dimensional moiré superlattices. This article provides a brief overview of recent research advances and challenges in the study of topological order in traditional condensed matter systems and quantum simulation experimental platforms. It also provides prospects for the future developments of this field.
At zero temperature, a two-dimensional lattice of Majorana zero modes on mesoscopic superconducting islands exhibits a topologically ordered toric code phase. Recently, a Landau field theory was used to describe the different phases of the aforementioned system and the phase transitions separating them. While the field theory provides details on the properties of the system close to the phase transitions, signatures of topological ordering in the different phases have not been computed. This is the primary goal of this work. We describe a lattice gauge theory of the Majorana toric code in terms of U(1) matter fields coupled to an emergent Z2 gauge field. Subsequently, we use a generalized Wilson-loop order parameter, the equal-time Fredenhagen-Marcu order parameter, to characterize the topological ordering in the different phases. Our computation provides evidence of the toric code phase both in the Mott-insulator and the charge-2e superconductor phases, while showing that the toric code phase disappears in the charge-e superconductor phase. In addition, we perturbatively analyze the influence of Cooper pair tunneling on the topological gap of the toric code in the limit of strong charging energy and show that the toric code phase is, in fact, stabilized by the Cooper pair tunneling. Our results are relevant for experimental realizations of the Majorana toric code.
In this paper we show that BF topological superconductors (insulators) exibit phase transitions between different topologically ordered phases characterized by different ground-state degeneracy on manifold with non-trivial topology. These phase transitions are induced by the condensation of (or lack of) topological defects. We concentrate on the (2+1)-dimensional case where the BF model reduces to a mixed Chern-Simons term and we show that the superconducting phase has a ground-state degeneracy k and not k 2 . When the symmetry is U (1) × U (1), namely when both gauge fields are compact, the mixed Chern-Simons model is not equivalent to the sum of two Chern-Simons terms with opposite chirality (even if naively diagonalizable) since the U (1) symmetry requires an ultraviolet regularization that makes the diagonalization impossible. We analyze this aspect using a lattice regularization, where the gauge fields become angular variables. In addition, we will show that the phase in which both gauge fields are compact is not allowed dynamically.
Phase transitions in a number of models related to hightemperature superconductors are investigated, using scalingmethods and Monte Carlo simulations. This thesis considers twomain topics. The first topic is phase transitions, phase diagrams, andvortex motion in high temperature superconductors at finitetemperature, subject to magnetic fields and disorder. We studya vortex glass model at finite temperature, with stronguncorrelated vortex pinning and a magnetic field. We find thatthe vortex glass exists at finite temperature and calculate thecritical exponents of the transition. We also investigate hightemperature superconductors with columnar disorder in zero andapplied magnetic fields. Some of these studies are alsorelevant for the superfluid to Mott insulator transition ofbosons in two dimensions. We find that the unscreened Boseglass transition belongs to a new universality class. Wecalculate the critical exponents of the superconductingtransition with columnar defects in zero applied magneticfield. The transverse Meissner transition is studied, and wefind an exotic universality class with a correlation volumethat is infinitely anisotropic in all directions. The second topic is confinement-deconfinement transitions incompact Abelian Higgs models. We develop a new order parameter,related to a large Wilson loop for fractionalized charges, anduse it to study the concept of topological order. Thesetransitions may be relevant for strongly correlated electronsin two dimensions.
Topological phases of matter are purely quantum mechanical and have no classical analogue. Most phases in nature can be classified and studied classically through the concept of symmetry breaking and its theoretical description, Landau-Ginzburg field theory. In contrast to the general wisdom of Landau-Ginzburg field theory, the topological phases share the same symmetry as a trivial insulator and still are different phases. Having gapped spectrum in bulk, they support a metallic edge excition robust against symmetry-respecting perturbation or an emergent fractional excitation in bulk. Following after fractional quantum Hall fluids, many topologically orderd phases, such as spin liquids and topological insulators, have been found and studied. Spin liquids are disordered phases of frustrated antiferromagnets and do not freeze and order even at the lowest temperature. They support an electrically neutral spin-1/2 excitation, which does not exist in a microscopic scale, with emergent dynamical gauge field in bulk and do not have an adiabatic path to a trivial paramagnet phase. The topological insulators are time-reversal symmetric band insulators which cannot evolve smoothly into a trivial insulator with the symmetry, and they have been intensely studied theoretically and experimetally for the last decade. Though the topological insulators are inherently non-interacting systems, they have exotic gapless edge and surface states which demonstrate many interesting quantum phenomena such as fractionalization, axionic electromagnetism, and half quantum Hall effect. In this thesis, we study various quantum phenomena of the topological phases, mainly of the topological insulator and its close relatives in which the physics of spin liquids has been merged into. As they are intrinsically quantum many-body states, the quantum field theory is an invaluable tool to explore the venue of the phases and will be used thorougly in this work. First, we present a topological field theoretic description of the topological insulators and translational symmetric Z2 spin liquid. We demostrate that abelian BF theory is an effective field theory for the two and three dimensional topological insulators. We show that the phenomenologies of BF theory are consistent with those of the topological insulators such as gapless edge and surface excitations and response functions to external gauge field. The same form of the BF theory is also applicable to study the certain classes of Z2 spin liquids with the gapless edge states. We closely examine the possible gapless edge theory of the BF theory and the translational symmetric spin liquids, and we show that the effective BF theories capture the bulk properties as well as the gapless edge spectrum of the spin liquid. Also we discuss the zero-temperature phase diagram of a thin film three dimensional topological insulators with two competing mass terms: time-reversal symmetric exciton condensation and time-reversal symmetry breaking Zeeman effect. There are two phases, quantum spin Hall phase and quantum anomalous Hall phase, and both phases support fractional excitations. We derive an effective topological field theory for the fractional excitations and examine the origin of the fractional excitations. Next we consider the relatives of the topological insulators. First we discuss the proximate symmetry broken phases of topological Mott insulator, a U(1) spin liquid with fermionic spinons in the topological insulator state. The topologically non-trivial band structure of the spinon generates the axion term for the gauge field, and we show that the axion term changes the nature of the confined phase of the spin liquid. Contrast to the conventional confined phase which has only a bond order, the confined phase of the spin liquid is in general a coexisting phase of the two different orders : a current order and a bond order. We then consider another relative of the topological insulator: a bosonic symmetry protected topological phase in two dimensional space, with both PSU(N) and time reversal symmetry. We develop an effective field theory for the phase, which is a SU(N) principal chiral model with a theta-term, and reveal the physics of the topological phase in detail.Finally we consider a topological semimetal, namely a Weyl semimetal. We first demonstrate that the Weyl semimetal can be realized in the magnetically doped topological band insulators. We explicitly derive the low energy theory of Weyl points from the general continuum Hamiltonian and tight-binding model of the topological insulators. Then we study superconducting instabilities of doped Weyl semimetals. We consider a minimal model for a Weyl semimetal and study the superconducting instabilities induced by a short-ranged attractive interaction. With the interaction, we find two competing states: a fully gapped finite-momentum pairing state and a nodal even-parity pairing state. We show that, in a mean field approximateion, the finite-momentum pairing state wins energetically against the usual even-parity paired state. We also show that exotic modes are localized at the full and half quantum vortices of the finite-momentum pairing state.