Topological order, emergent gauge fields, and Fermi surface reconstruction

Abstract

This review describes how topological order associated with the presence of emergent gauge fields can reconstruct Fermi surfaces of metals, even in the absence of translational symmetry breaking. We begin with an introduction to topological order using Wegner's quantum [Formula: see text] gauge theory on the square lattice: the topological state is characterized by the expulsion of defects, carrying [Formula: see text] magnetic flux. The interplay between topological order and the breaking of global symmetry is described by the non-zero temperature statistical mechanics of classical XY models in dimension D = 3; such models also describe the zero temperature quantum phases of bosons with short-range interactions on the square lattice at integer filling. The topological state is again characterized by the expulsion of certain defects, in a state with fluctuating symmetry-breaking order, along with the presence of emergent gauge fields. The phase diagrams of the [Formula: see text] gauge theory and the XY models are obtained by embedding them in U(1) gauge theories, and by studying their Higgs and confining phases. These ideas are then applied to the single-band Hubbard model on the square lattice. A SU(2) gauge theory describes the fluctuations of spin-density-wave order, and its phase diagram is presented by analogy to the XY models. We obtain a class of zero temperature metallic states with fluctuating spin-density wave order, topological order associated with defect expulsion, deconfined emergent gauge fields, reconstructed Fermi surfaces (with 'chargon' or electron-like quasiparticles), but no broken symmetry. We conclude with the application of such metallic states to the pseudogap phase of the cuprates, and note the recent comparison with numerical studies of the Hubbard model and photoemission observations of the electron-doped cuprates. In a detour, we also discuss the influence of Berry phases, and how they can lead to deconfined quantum critical points: this applies to bosons on the square lattice at half-integer filling, and to quantum dimer models.

Unsupervised identification of topological phase transitions using predictive models

“…Wegner's Ising gauge theory (IGT) is a spin model defined on a N×N square lattice with spins placed on the lattice bonds [8][9][10]13]. It is described by the Hamiltonian where J is a coupling constant, p refers to plaquettes on the lattice (see figure 1), and s i z is the Pauli matrix describing a single spin-1/2.…”

“…The IGT constrained phase then has the property that all these loops are closed. Whenever the constraint is violated it results in an open loop [10,16,13], see figure 1.…”

“…We now turn to a concrete model of a quantum phase transition in a system without a local order parameter. The obvious generalization of (1) is the application of a transverse field [9,13,27]…”

Tunable Flat Band in Large‐Scale Kagome Lattice of Single Layer (BETS)2GaCl4

“…In recent years, there has been a flurry of research activities on 2D materials targeting exotic topological states which might emerge in electronically frustrated systems or at the interface between completely different topological orders . Some examples have already been materialized in lattices such as Kagome, twisted bilayer grapheme, and Lieb or at interfaces such as the one between ferromagnetic chains and conventional superconductors .…”

“…In recent years, there has been a flurry of research activities on 2D materials targeting exotic topological states which might emerge in electronically frustrated systems or at the interface between completely different topological orders. [1][2][3][4][5][6][7] Some examples have already been materialized in lattices such as Kagome, [8] twisted bilayer grapheme, [9] and Lieb [10] or at interfaces such as the one between ferromagnetic chains and conventional superconductors. [11,12] Emphasis was placed on geometrically driven localization to test the theoretical models [13,14] that have been proposed as well as to control the ground-state properties by either varying the twist angle in bilayer graphene or by design such as in atomically engineered lattice as well as epitaxial self-assembled monolayers.…”