We present quantum dimer models in two dimensions which realize metallic ground states with Z2 topological order. Our models are generalizations of a dimer model introduced in [PNAS 112, 9552-9557 (2015)] to provide an effective description of unconventional metallic states in hole-doped Mott insulators. We construct exact ground state wave functions in a specific parameter regime and show that the ground state realizes a fractionalized Fermi liquid. Due to the presence of Z2 topological order the Luttinger count is modified and the volume enclosed by the Fermi surface is proportional to the density of doped holes away from half filling. We also comment on possible applications to magic-angle twisted bilayer graphene.
We present an exact ground state solution of a quantum dimer model introduced by Punk, Allais, and Sachdev [Quantum dimer model for the pseudogap metal, Proc. Natl. Acad. Sci. U.S.A. 112, 9552 (2015).PNASA60027-842410.1073/pnas.1512206112], which features ordinary bosonic spin-singlet dimers as well as fermionic dimers that can be viewed as bound states of spinons and holons in a hole-doped resonating valence bond liquid. Interestingly, this model captures several essential properties of the metallic pseudogap phase in high-T_{c} cuprate superconductors. We identify a line in parameter space where the exact ground state wave functions can be constructed at an arbitrary density of fermionic dimers. At this exactly solvable line the ground state has a huge degeneracy, which can be interpreted as a flat band of fermionic excitations. Perturbing around the exactly solvable line, this degeneracy is lifted and the ground state is a fractionalized Fermi liquid with a small pocket Fermi surface in the low doping limit.
This article is based on a talk by S.S. at the Nambu Memorial Symposium at the University of Chicago. We review ideas on the nature of the metallic states of the hole-doped cuprate high temperature superconductors, with an emphasis on the connections between the Luttinger theorem for the size of the Fermi surface, topological quantum field theories (TQFTs), and critical theories involving changes in the size of the Fermi surface. We begin with the derivation of the Luttinger theorem for a Fermi liquid, using momentum balance during a process of flux-insertion in a lattice electronic model with toroidal boundary conditions. We then review the TQFT of the Z2 spin liquid, and demonstrate its compatibility with the toroidal momentum balance argument. This discussion leads naturally to a simple construction of `topological' Fermi liquid states: the fractionalized Fermi liquid (FL*) and the algebraic charge liquid (ACL). We present arguments for a description of the pseudogap metal of the cuprates using Z2-FL* or Z2-ACL states with Ising-nematic order. These pseudogap metal states are also described as Higgs phases of a SU(2) gauge theory. The Higgs field represents local antiferromagnetism, but the Higgs-condensed phase does not have long-range antiferromagnetic order: the magnitude of the Higgs field determines the pseudogap, the reconstruction of the Fermi surface, and the Ising-nematic order. Finally, we discuss the route to the large Fermi surface Fermi liquid via the critical point where the Higgs condensate and Ising nematic order vanish, and the application of Higgs criticality to the strange metal.
Insulating quantum spin liquids can undergo a confinement transition to a valence bond solid via the condensation of topological excitations of the associated gauge theory. We extend the theory of such transitions to fractionalized Fermi liquids (FL*): These are metallic doped spin liquids in which the Fermi surfaces only have gauge neutral quasiparticles. Using insights from a duality transform on a doped quantum dimer model for the U(1)-FL* state, we show that projective symmetry group of the theory of the topological excitations remains unmodified, but the Fermi surfaces can lead to additional frustrating interactions. We propose a theory for the confinement transition of
Z2-FL* states via the condensation of visons. A variety of confining, incommensurate density wave states are possible, including some that are similar to the incommensurate
d-form factor density wave order observed in several recent experiments on the cuprate superconductors.
This review describes how topological order 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
Z2 gauge theory on the square lattice: the topological state is characterized by the expulsion of defects, carrying
Z2 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. The phase diagrams of the
Z2 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, 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. 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.
A
Z2 fractionalized Fermi liquid (FL*) is a novel state of strongly correlated quantum matter. Although it is metallic, it violates Luttinger's theorem on the volume enclosed by the Fermi surface obeyed by conventional metals; this is possible due to the presence of emergent gauge excitations. The authors study here the superconducting states that can arise out of instabilities of such a topological metal. In particular, they focus on a FL* topological metal that has favorable energetics on the square lattice with nematic order, relevant to the cuprate superconductors. They find that a Higgs transition out of this FL* results in confinement of the anyonic degrees of freedom, and the resulting state is a superconductor with broken translation symmetry or time-reversal invariance. In the process, they also establish a complete mapping between bosonic and fermionic descriptions of time-reversal invariant gapped insulating
Z2 spin liquid states on the rectangular lattice, which, on doping with charge carriers, can give rise to FL* phases. They also note a possible connection to the recent observation of pair density waves in the superconducting state of the underdoped cuprates.