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The Physics of Metals and Metallography

Orbital Magnetic Correlations, Topological Order, and Superconductivity in Strongly Correlated Systems

V. Yu. Irkhin, Yu. N. Skryabin

Published Feb 1, 2020• 2 citations
Publisher

Abstract

Abstract This paper considers modern ideas about the topological nature of superconductivity and the Mott–Hubbard state in strongly correlated systems. These systems are described in terms of spin and charge separation, i.e., deconfinement of spinons and holons in a gauge field. The role of orbital degrees of freedom is discussed. Analogies with topological insulators, quantum Hall effect, and Landau quantization are considered. The importance of the Chern–Simons term for the correct description of the Hubbard splitting is demonstrated.

DOI
10.1134/s0031918x20020076

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Abstract This paper considers modern ideas about the topological nature of superconductivity and the Mott–Hubbard state in strongly correlated systems. These systems are described in terms of spin and charge separation, i.e., deconfinement of spinons and holons in a gauge field. The role of orbital degrees of freedom is discussed. Analogies with topological insulators, quantum Hall effect, and Landau quantization are considered. The importance of the Chern–Simons term for the correct description of the Hubbard splitting is demonstrated.
In this work, we present a topological characterization of superconductivity in a prototype electron fractionalization model for doped Mott insulators. In this model, spinons and holons are coupled via the mutual Chern-Simons gauge fields. We obtain a low-lying effective description of the collective current fluctuations by integrating out the matter fields, which replaces the conventional Ginzburg-Landau action to describe the generalized rigidity of superconductivity. The superconducting phase coherence is essentially characterized by a topological order parameter related to a Gaussian linking number, and an experiment is proposed to probe this topological property. We further show that a gauge-neutral fermionic mode can naturally emerge in this model, which behaves like a Bogoliubov quasiparticle.
We discuss the appearance of quantum orders in the Hubbard model for interacting electrons, at half-filling. Such phases do not have local order parameters and need to be characterized by the quantum mechanical properties of their ground state. On one hand, we study the Mott transition from a metal to a spin liquid insulator in two dimensions, of potential relevance to some layered organic compounds. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the insulator, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We focus on the transport properties near the quantum critical point and find that the emergent gauge fluctuations play a key role in determining the universal scaling. Second, motivated by a class of three-dimensional transition metal oxides, the pyrochlore iridates, we study the interplay of non-trivial band topology and correlations. Building on the strong spin orbit coupling in these compounds, we construct a general microscopic Hubbard model and determine its mean-field phase diagram, which contains topological insulators, Weyl semimetals, axion insulators and various antiferromagnets. We also discuss the effects many-body correlations on theses phases. We close by examining a fractionalized topological insulator that combines the two main themes of the thesis: fractionalization and non-trivial band topology. Specifically, we study how the two-dimensional protected surface states of a topological Mott insulator interact with a three-dimensional emergent gauge field. Various correlation effects on observables are identified.
We study superfluidity in strongly correlated systems by focusing three features as follows. First, interplay between superfluidity and other order parameters near quantum critical points is studied. The onset positions of the order parameters with or without superconductivity determines how they interact with each other. For the spin density wave (SDW) transition, we find that the quantum critical point shifts by order |∆|, where ∆ is pairing amplitude, so that the region of SDW order is smaller in the superconductor than in the metal. This shift is larger than the |∆|2 shift predicted by theories of competing orders which ignore Fermi surface effects. Second, we study a doped antiferromagnetic Mott insulator motivated by copper based high temperature superconductors. Based on spin and charge fractionalization, the algebraic charge liquid phase whose bosonic spinons have energy gap but fermionic holons form Fermi surfaces and its superconductivity are studied. We show fluctuations of local SDW order orientation lead to d-wave superconductivity, and its quantum critical pairing mechanisms are investigated. Also, a theory of bound states between spinons and holons is studied. We show that the fractionalization induces a doubling electron-like quasiparticles, which naturally induces the fractionalized Fermi liquid. The bound state theory describes exotic properties of the underdoped Cuprates such as “nodal-antinodal dichotomy” at a mean field level description. At last, we study superfluidity of fermions in optical lattices near the Feshbach resonance. Due to divergence of scattering length at the resonance, universality is observed and characteristics of superfluidity has been studied. We extend such universal properties by considering superfluid-insulator transition with an even integer number of fermions per unit cell. The resulting insulator interpolates between a band insulator of fermions and a Mott insulator of fermion pairs.
The present state of the gauge theory of high-temperature superconductivity in strongly-correlated two-dimensional spin systems is reviewed. Basic ideas on the statistics of elementary excitations in spatially two-dimensional systems are presented and are used in an analysis of the structure of the energy spectrum and the form of the wave function of a set of anyon quasiparticles in both the long-wave continuous and lattice limits. The continuous and lattice theories are used to classify the phase states, and the hierarchy of phase transitions is described in terms of the topological quantum field theory. Thermodynamic and electrodynamic properties of anyon systems are described, and the experimental consequences of the Chern–Simons theory of high-Tc superconductivity are discussed.
In the presence of symmetry-protection, topological invariants of topological phases of matter in free fermion systems, e.g., topological band insulators, can be directly computed via the properties of band structure. Nevertheless, it is usually difficult to extract topological invariants in strongly-correlated topological phases of matter in which band structure is not well-defined. One typical example is the fractional quantum Hall effect whose low-energy physics is governed by Chern-Simons topological gauge theory and Hall conductivity plateaus involve extremely fruitful physics of strong correlation. In this article, we focus on intrinsic topological order (iTO), symmetry-protected topological phases (SPT), and symmetry-enriched topological phases (SET) in boson and spin systems. Through gauge field-theoretical approach, we review some research progress on these topological phases of matter from the aspects of projective construction, low-energy effective theory and topological response theory.
The concept of topological excitations and the related ground state degeneracy are employed to establish an effective theory of the superconducting state evolving from the Mott insulator for high-Tc{T}_{c}Tc​ cuprates. The theory includes the effects of the relevant energy scales with the emphasis on the Coulomb interaction UUU governed by the electromagnetic U(1) compact group. The results are obtained for the layered $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}{t}_{\ensuremath{\perp}}\ensuremath{-}U\ensuremath{-}J$ system of strongly correlated electrons relevant for cuprates. Casting the Coulomb interaction in terms of composite-fermions via the gauge flux attachment facility, we show that instanton events in the Matsubara ``imaginary time,'' labeled by topological winding numbers, are essential configurations of the phase field dual to the charge. This provides a nonperturbative concept of the topological quantization and displays the significance of discrete topological sectors in the theory governed by the global characteristics of the phase field. We show that for topologically ordered states these quantum numbers play the role of an order parameter in a way similar to the Landau order parameter for conventionally ordered states. In analogy to the usual phase transition that is characterized by a sudden change of the symmetry, the topological phase transitions are governed by a discontinuous change of the topological numbers signaled by the divergence of the zero-temperature topological susceptibility. This defines a quantum criticality ruled by topologically conserved numbers rather than the Landau principle of the symmetry breaking. We show that in the limit of strong correlations topological charge is linked to the average electronic filling number and the topological susceptibility to the electronic compressibility of the system. We exploit the impact of these nontrivial U(1) instanton phase field configurations for the cuprate phase diagram which displays the ``hidden'' quantum critical point covered by the superconducting lobe in addition to a sharp crossover between a compressible normal ``strange metal'' state and a region characterized by a vanishing compressibility, which marks the Mott insulator. Finally, we argue that the existence of robust quantum numbers explains the stability against small perturbation of the system and attributes to the topological ``quantum protectorate'' as observed in strongly correlated systems.