Abstract
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.
Highlights
Nambu’s early papers [1,2,3] laid down the close connection between fundamental questions in superconductivity and high energy physics
We described a connection between the size of the Fermi surface and the odd/even nature of the ‘symmetry enriched’ [34] topological quantum field theories (TQFTs) describing the Z2 topological order
A central mystery in the study of cuprate superconductors concerns the nature of the strange metal without quasiparticle excitations and its relation to an underlying quantum critical point
Summary
Nambu’s early papers [1,2,3] laid down the close connection between fundamental questions in superconductivity and high energy physics. By adiabatic continuity from a free electron ground state, the Luttinger theorem states that we should obtain a metal with a Fermi surface of size equivalent to 1 + p holes. A small part of the discussion in Sections III and IV overlaps with a separate, less technical, recent article by one of us [7]
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