Topological Phase Transitions and New Developments

The following sections are included:

• Preface
• Contents

We propose as key of superconductivity in (hole-doped) cuprates a novel excitation of magnetic origin, characteristic of two-dimensions and of purely quantum nature: the antiferromagnetic spin vortices. In this formalism the charge pairing arises from a Kosterlitz-Thouless-like attraction between such vortices centered on opposite Néel sub-lattices. This charge pairing induces also the spin pairing through the action of a gauge force generated by the no-double occupation constraint imposed in the t-J model of the CuO planes of the cuprates. Superconductivity arises from coherence of pairs of exci tations describing Zhang-Rice singlets and it is not of standard BCS type. We show that many experimental features of the cuprates can find a natural explanation in this formalism.

Topological acoustics is a subfield of topological physics that studies the construction and behavior of topological phases for sound propagation. Different from electrons and photons, sound waves travelling through air are essentially longitudinal waves that carry no intrinsic spin and do not respond to magnetic fields, thus lacking sufficient degrees of freedom. Here we review the recent progress in our group in constructing topological phases for sound. The emphasis is laid on the underlying insights as inspired from the corresponding development in condensed matter and photonic systems. The first exam ple is the acoustic quantum Hall effect in a phononic crystal that consists of spinning rods. This design follows Haldane and Raghu’s work in constructing the first topo logical photonic crystal. The second example is the acoustic Weyl nodes by stacking one-dimensional dimerized chains of acoustic resonators. It can be identified that its topological phase corresponds to the recently studied type-II Weyl semimetal phase. The last example is the acoustic Landau levels by introducing strain engineering to a two-dimensional aperiodic acoustic lattice. This offers a path to previously inaccessible magnetic-like effects in traditional periodic acoustic structures.

This contribution describes the latest developments in the field of light nuclei and explores how the tendency of nucleons to cluster into α-particle subunits influences their structure. In particular there is a focus on the structure of the nucleus ¹²C and in turn its link to nucleosynthesis and the formation of organic life.

Topological defects are typically quantified relative to ordered backgrounds. The importance of these defects to the understanding of physical phenomena including diverse equilibrium melting transitions from low temperature ordered to higher temperatures disordered systems (and vice versa) can hardly be overstated. Amorphous materials such as glasses seem to constitute a fundamental challenge to this paradigm. A long held dogma is that transitions into and out of an amorphous glassy state are distinctly different from typical equilibrium phase transitions and must call for radically different concepts. In this work, we critique this belief. We examine systems that may be viewed as simultaneous distribution of different ordinary equilibrium structures. In particular, we focus on the analogs of melting (or freezing) transitions in such distributed systems. The theory that we arrive at yields dynamical, structural, and thermodynamic behaviors of glasses and supercooled fluids that, for the properties tested thus far, are in qualitative and quantitative agreement with experiment. We arrive at a prediction for the viscosity and dielectric relaxations that is universally satisfied for all experimentally measured supercooled liquids and glasses over 15 decades.

Topological crystalline insulators support gapless surface states even in the presence of a bulk gap, in situations where the surface preserves some crystalline symmetry of the bulk. In most realizations this is a mirror symmetry. We study the effect of magnetic impurities in such a system, for which ordering at the surface lowers the electronic energy by spontaneous breaking the crystalline symmetry. This is shown to induce surface ferromagnetism, with favored directions and numbers of energetically equivalent groundstates that are sensitive to the precise density of electrons at the surface. In particular for a (111) surface of a SnTe model in the topological state, magnetic states will have either a two-fold or six-fold symmetry. We compute spin stiffnesses within the model to demonstrate the stability of ferromagnetic states, and consider their ramifications for thermal disordering. Possible experimental consequences of the surface magnetism are discussed.

In this Proceedings we report some recent results on Majorana quasi-particles published in Phys. Rev. Lett. 118, 200404 (2017). We show how angular momentum conservation can stabilise a symmetry-protected quasi-topological phase of matter supporting Majorana quasi-particles as edge modes in one-dimensional cold atom gases. We investigate a number-conserving four-species Hubbard model in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional reservoirs. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated by density-matrix-renormalization-group simulations. Our results pave the way toward the observation of Majorana edge modes with alkaline-earth-like fermions in optical lattices, where all basic ingredients for our recipe - spin-orbit coupling and strong inter-orbital interactions - have been experimentally realized over the last two years.

Destruction of superconductivity in thin films was thought to be a simple instance of Berezinskii-Kosterlitz-Thouless physics in which only two phases exist: a superconductor with algebraic long range order in which the vortices condense and an insulator where the vortex-antivortex pairs proliferate. However, since 1989 this view has been challenged as now a preponderance of experiments indicate that an intervening bosonic metallic state obtains upon the destruction of superconductivity. Two key features of the intervening metallic state are that the resistivity turns on continuously from the zero resistance state as a power law, namely ρBM ∝ (g – gc)^α and the Hall conductance appears to vanish. We review here a glassy model which is capable of capturing both of these features. The finite resistance arises from three features. First, the disordered insulator-superconductor transition in the absence of fermionic degrees of freedom (Cooper pairs only), is controlled by a diffusive fixed point rather than the critical point of the clean system. Hence, the relevant physics that generates the Bose metal should arise from a term in the action in which different replicas are mixed. We show explicitly how such physics arises in the phase glass. Second, in 2D (not in 3D) the phase stiffness of the glass phase vanishes explicitly as has been shown in extensive numerical simulations. Third, bosons moving in such a glassy environment fail to localize as a result of the false minima in the landscape. We calculate the conductivity explicitly using Kubo response and show that it turns on as a power law and has a vanishing Hall response as a result of underlying particle-hole symmetry. We show that when particle-hole symmetry is broken, the Hall conductance turns on with the same power law as does the longitudinal conductance. This prediction can be verified experimentally by applying a ground plane to the 2D samples.

What is a topological superfluid in a particle-number conserving system? Motivated by this question we report on a novel pathway to topological superfluidity with fermionic atoms in an optical lattice. We consider a situation where atoms in two internal states experience different lattice potentials: one species is localized and the other itinerant, and show how quantum fluctuations of the localized fermions give rise to an attraction and strong effective spin-orbit coupling in the itinerant band. At low temperature, these effects stabilize a topological superfluid of mobile atoms even if their bare interactions are repulsive. This emergent state can be engineered with ⁸⁷Sr atoms in a dimerized superlattice. To probe its unique properties we describe protocols that use high spectral resolution and controllability of the Sr clock transition, such as momentum-resolved spectroscopy and supercurrent response to a synthetic (laser-induced) magnetic field.

These notes are a brief tour in the realm of quantum interacting one dimensional systems, and their connection to topological phase transitions. Two examples are examined. The first one is the superfluid-Mott transition for interacting 1D bosons. Via bosonization such systems are directly described by the Berezinskii-Kosterlitz-Thouless (BKT) transition. Both the universal jump at the transition and the nature of topological excitations can be probed experimentally. The second example is disordered 1D interacting bosons, for which the superfluid - Bose glass transition occurring in such systems is also a variant of the BKT transition. Some theoretical aspects and experimental realizations are discussed.

This article provides an overview, primarily from an experimental perspective, of recent progress and future prospects in using helium to realize a range of quantum materials of generic interest, by “top-down” and “bottom-up” nanotechnology. We can grow model systems to realise new quantum states of matter, and explore key issues in condensed matter physics. In the language of cold atomic gases, two dimensional and confined ³He and ⁴He provide “quantum simulators”, with the potential to uncover new emergent quantum states. These include: strictly 2D Fermi system with Mott-Hubbard transition; interacting coupled 2D fermion-boson system; heavy fermion quantum criticality; ideal 2D frustrated ferromagnetism; 2D quantum spin liquid; intertwined superfluid and density wave order with emergent large symmetry; topological mesoscopic superfluidity (new materials and emergent excitations).

We develop a topological gauge theory of the superconductor-insulator transition (SIT) in the context of Josephson junction arrays (JJA) and disordered superconducting films. We show that the universal long-distance physics of these systems is governed by the BF topological gauge theory, which, in two dimensions, reduces to the doubled Chern-Simons gauge theory. This approach reveals that the quantum phase structure of the SIT is governed by the competition between three quantum orders: (i) Bose condensate of Cooper pairs, which is the superconductor (SC) with no Higgs field, (ii) its dual Bose condensate of magnetic monopoles, corresponding to the superinsulator (SI), a state having infinite resistance at finite temperatures, and (iii) No-condensate state, usually called a Bose (or quantum) metal (BM). We demonstrate that this BM state, described by the purely topological BF term, is a topological insulator (TI), in which the large topological mass suppresses bulk currents and the electronic transport in the self-dual approximation is carried only by the edge excitations. We derive the quantitative criterion for the direct vs the transition via the intermediate metallic state SIT (SMIT) and find the critical strength of quantum fluctuations necessary for the SMIT scenario. The developed approach establishes a deep connection between the exotic phases emerging in the vicinity of the SIT and QED.

I present a brief overview of work done with my students and collaborators during a period of several years after the seminal publications by Berezinskii-Kosterlitz-Thouless (BKT) [1–3], and the José-Kadanoff-Kirkpatrick-Nelson (JKKN) paper [4]. We proceeded to study the stability of the BKT scenario and the impact in the KT topological phase structure against the presence of different types of disorder [5–7], classical and quantum fluctuations, with and without constant external magnetic fields [8–23], including also quasi-particle tunneling dissipation [13]. The models considered have different types classical and quantum duality symmetries plus gauge invariances. I will also discuss the interesting case of having two capacity coupled JJA layers, one dominated by quantum charge fluctuations and the other by thermal vortex fluctuations. In these studies, we used different analytical and numerical approaches: e.g. the Replica Trick [5, 35], Classical and Quantum Renormalization Group calculations [5–7], WKB and variational calculations plus Classical and Quantum Monte Carlo simulations [8–12]. Adding these extra perturbations produced small but also significant changes to the BKT phase transition structure in particular at low temperatures, including reentrant putative Quantum Induced Transitions (QUIT) [8–23]. Good experimental representation of these model variations are the Josephson Junction arrays (JJA) and granular superconducting films [24–31].