Topological Phase Transitions and New Developments

Lars Brink, Mike Gunn;Jorge V José;John Michael Kosterlitz;Kok Khoo Phua

• Publisher: WSPC
• Year: 2018
• eBook ISBN: 9789813271357
• Topic: Physical Sciences
• Subtopic: Condensed Matter

Geometry and topology have been a fascination in physics since the start of the 20th century. A leading example is Einstein's geometrical theory of gravity. At the beginning of the 1970s, topological ideas entered areas of condensed matter physics. These advances were driven by new seminal ideas resolving a serious contradiction between experiment and the standard interpretation of a rigorous mathematical theorem which led to the study of new exotic topological phases of matter. Topological defect driven phase transitions in thin, two dimensional films of superfluids, superconductors and crystals have provided great insight into the mechanism governing these topological phases present in those physical systems. Moreover, many of these topological properties remain 'protected' against disorder and topological distortion perturbations. An example of possible applications of such robustness to perturbations is in the search for encoding information in quantum computers, potentially providing the platform for fault-tolerant quantum computations.

In the past four decades, the discovery of topological phases engendered great interest in condensed matter physics. It also attracted the attention of researchers working on quantum information, quantum materials and simulations, high energy physics and string theory. This unique volume contains articles written by some of the most prominent names in the field, including Nobel Laureate John Michael Kosterlitz and Professor Jorge V José. They originate from talks and discussions by leading experts at a recent workshop. They review previous works as well as addressing contemporary developments in the most pressing and important issues on various aspects of topological phases and topological phase transitions.

Contents:

• Preface
• Topological Defects and Phase Transitions (J M Kosterlitz)
• Geometry of Flux Attachment in the Fractional Quantum Hall Effect States (F D M Haldane)
• The Attraction Between Antiferromagnetic Quantum Vortices as Origin of Superconductivity in Cuprates (P A Marchetti)
• Some Topological Phases for Sound (B Zhang)
• Influences of Geometry and Topology in Nuclei (M Freer)
• The "Glass Transition" as a Topological Defect Driven Transition in a Distribution of Crystals and a Prediction of a Universal Viscosity Collapse (Z Nussinov, N B Weingartner and F S Nogueira)
• Surface Ferromagnetism in a Topological Crystalline Insulator (S Reja, H A Fertig, S Zhang and L Brey)
• Majorana Quasiparticles in Ultracold One-Dimensional Gases (F Iemini, L Mazza, L Fallani, P Zoller, R Fazio and M Dalmonte)
• Bose Metal as a Disruption of the Berezinskii–Kosterlitz–Thouless Transition in 2D Superconductors (P W Phillips)
• Topological Superfluidity with Repulsive Fermionic Atoms (G Ortiz, L Isaev, A Kaufman and A M Rey)
• Clean and Dirty Bosons in 1D Lattices (T Giamarchi)
• Realizing Quantum Materials with Helium: Helium Films at Ultralow Temperatures, from Strongly Correlated Atomically Layered Films to Topological Superfluidity (J Saunders)
• Topological Gauge Theory of the Superconductor-Insulator Transition (M C Diamantini, C A Trugenberger and V M Vinokur)
• BKT Stability Against Disorder, External Magnetic Fields, Classical and Quantum Fluctuations and Quasi-Particle Tunneling Dissipation (J V José)
• Superfluidity, Phase Transitions, and Topology (J Reppy)
• Topological Phenomena in the Moiré Pattern of Van der Waals Heterostructures (W Yao)
• Heterogeneous Interfaces for Teasing Out the Physics of Embedded Surface States (A Hebard)
• Emergent Particle-Hole Symmetry in Spinful Bosonic Quantum Hall Systems (N Regnault)
• Dynamical Signatures of Quantum Spin Liquids (F Pollmann)
• Getting the Jump in the Kosterlitz–Thouless Transition (C Lobb)
• Phase Transitions: From Josephson Junction Arrays to Flowing Granular Matter (S Teitel)
• Topological Phase Transitions in Photonic Lattices (Y Chong)
• Instabilities and Solitary Waves of Light and Atoms in Photonic Crystal Fibres (M Gunn)
• Spin Topology Architectures in Low Dimensional Magnets (C Panagopoulos)
• Realizing and Manipulating Topological Metals and their Exotic Properties (Z Wang)
• Tuning Magnetism and Topology in Topological Insulators with Broken Time Reversal Symmetry (Y Wang)
• Anomalous Collective Modes in Topological Matter (J Song)

Readership: Graduate students and researchers in condensed matter and theoretical physics.

Topological defects and phase transitions

J. Michael Kosterlitz

Brown University
E-mail: [email protected]

This talk reviews some of the applications of topology and topological defects in phase transitions in two-dimensional systems for which Kosterlitz and Thouless split half the 2016 Physics Nobel Prize. The theoretical predictions and experimental verification in two-dimensional superfluids, superconductors and crystals will be reviewed as they provide very convincing quantitative agreement with topological defect theories.

Full paper found at https://www.nobelprize.org/nobel_prizes/physics/laureates/2016/kosterlit-zlecture.pdf.

References

1.J. M. Kosterlitz, Int. J. Mod. Phys. B, 32, 1830005 (2018) https://doi.org/10.1142/S02179792/8300050.

Geometry of flux attachment in the fractional quantum hall effect states

F. Duncan M. Haldane

Princeton University
E-mail: [email protected]

The unexpected experimental discovery of the topologically-ordered Fractional Quantum Hall (FQH) states showed that the powerful diagrammatic perturbation theoretic methods of the time were only useful for a subclass of problems adiabatically related to free-particle problems, and instead, Laughlin’s discovery of a model state that describes “flux attachment” to form composite particles has been the source of most subsequent understanding of the effect. In recent years, it has become apparent that “flux attachment” has important sort-distance geometrical properties as well as long-distance topological entanglement properties. I will describe geometric analogies between the unit cell of a solid and the “composite boson” which is the elementary unit of incompressible FQH liquids, and the place for “composite fermions” in their description.

The attraction between antiferromagnetic quantum vortices as origin of superconductivity in cuprates

P. A. Marchetti

Dipartimento di Fisica e Astronomia, Universitá di Padova, INFN,
Padova, I-35131, Italy
E-mail: [email protected]

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 excitations 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.

Keywords: Superconductivity, Cuprates, Vortices, Gauge field theories.

1.Introduction

Thirty years after the discovery of the first high-T_c superconducting cuprate¹, the microscopical mechanism behind superconductivity in this class of materials is still not understood, despite constant experimental advances. It is commonly believed that antiferromagnetism (AF) is a key ingredient for the superconductivity in cuprates, then a natural pairing glue would be provided by the spin fluctuations, i.e. antiferromagnetic spin-waves (see e.g. Ref. 2). Their action would be enhanced by nesting of the Fermi surface (FS), but evidence for this is not so clear. We propose as pairing glue another excitation still emerging from AF, but of purely quantum origin: antiferromagnetic spin vortices. In the antiferromagnetic phases the spin group SU(2) is broken to U(1), the quotient SU(2)/U(1) is isomorphic to the 2-sphere S² whose points label the directions of the magnetization. Their fluctuations are described by spin waves. The unbroken U(1) group describes unphysical gauge fluctuations. However in two dimensions (2D) one can consider vortices of Aharonov-Bohm type in this U(1); due to AF the vortices have opposite chirality when centered in two different Néel sublattices, hence we dub them antiferromagnetic spin vortices. Lowering the temperature such gas of vortices in 2D undergoes a Kosterlitz-Thouless-like transition, with the formation of a finite density of vortex-antivortex pairs. If the vortices are centered on charges, this induces a new form of charge-pairing, again due to AF, but different from the spin-fluctuation pairing. As discussed later this pairing finally leads to superconductivity.

To present how this is realized in the (hole-doped) cuprates is the fundamental aim of this paper. One discovers that through this key idea many structural features of the phase diagram of the cuprates could be understood and many physical properties successfully computed. The derivations are based on well-defined conjectures and approximations and many experimental consequences are consistent with availabla data, as sketched in the final section. Let us stress that, due to their structure, the antiferromagnetic vortices are specific to quasi-2D doped antiferromagnets and some phenomena they give rise to, like the charge pairing discussed above, the induced spin-pairing and the metal-insulator crossover of in-plane resistivity, are peculiar to this approach and do not appear in this form in other approaches to the physics of cuprates.

Whenever we succeed we present the intuitive ideas at the beginning of the section, giving later on a brief summary of their formal implementation and deferring to the references for the explicit proofs.

This paper reviews the results on a mechanism for superconductivity arising from a long joint project on a spin-charge gauge approach to the physics of cuprates initiated with Z.-B. Su and L. Yu, to whom I express a deep gratitude for all the knowledge and ideas shared with me. Along the way we profit of fundamental contributions from many researcher, among which a leading role have been played by F. Ye and G. Bighin, but it is also a pleasure to cite as co-authors J.-H. Dai, L. De Leo, G. Orso, A. Ambrosetti, M. Gambaccini and last but not least to gratefully acknowledge the original insight by J. Froehlich.

2.Model and phase diagram

The high T_c cuprates share a layered structure incorporating one or more copper–oxygen (CuO) planes; an excellent recent summary of their properties can be found in Ref. 3. There is no consensus yet on the theoretical interpretation of their low-energy physics, but an agreement has been achieved that superconductivity is due to formation of Cooper pairs with principal locus the CuO planes and that the order parameter is a spin ...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Preface
• Contents
• 1. Topological Defects and Phase Transitions*
• 2. Geometry of Flux Attachment in the Fractional Quantum Hall Effect States*
• 3. The Attraction Between Antiferromagnetic Quantum Vortices as Origin of Superconductivity in Cuprates
• 4. Some Topological Phases for Sound
• 5. Influences of Geometry and Topology in Nuclei
• 6. The “Glass Transition” as a Topological Defect Driven Transition in a Distribution of Crystals and a Prediction of a Universal Viscosity Collapse
• 7. Surface Ferromagnetism in a Topological Crystalline Insulator
• 8. Majorana Quasiparticles in Ultracold One-Dimensional Gases
• 9. Bose Metal as a Disruption of the Berezinskii–Kosterlitz–Thouless Transition in 2D Superconductors
• 10. Topological Superfluidity with Repulsive Fermionic Atoms
• 11. Clean and Dirty Bosons in 1D Lattices
• 12. Realizing Quantum Materials with Helium: Helium Films at Ultralow Temperatures, from Strongly Correlated Atomically Layered Films to Topological Superfluidity
• 13. Topological Gauge Theory of the Superconductor-Insulator Transition
• 14. BKT Stability Against Disorder, External Magnetic Fields, Classical and Quantum Fluctuations and Quasi-Particle Tunneling Dissipation
• 15. Superfluidity, Phase Transitions, and Topology*
• 16. Topological Phenomena in the Moiré Pattern of Van der Waals Heterostructures*
• 17. Heterogeneous Interfaces for Teasing Out the Physics of Embedded Surface States*
• 18. Emergent Particle-Hole Symmetry in Spinful Bosonic Quantum Hall Systems*
• 19. Dynamical Signatures of Quantum Spin Liquids*
• 20. Getting the Jump in the Kosterlitz–Thouless Transition*
• 21. Phase Transitions: From Josephson Junction Arrays to Flowing Granular Matter*
• 22. Topological Phase Transitions in Photonic Lattices*
• 23. Instabilities and Solitary Waves of Light and Atoms in Photonic Crystal Fibres*
• 24. Spin Topology Architectures in Low Dimensional Magnets*
• 25. Realizing and Manipulating Topological Metals and their Exotic Properties*
• 26. Tuning Magnetism and Topology in Topological Insulators with Broken Time Reversal Symmetry*
• 27. Anomalous Collective Modes in Topological Matter*