Thermal SU(2) lattice gauge theory for the pseudogap and the transition to $d$-wave superconductivity in the cuprates

Abstract: We study a square lattice SU(2) gauge theory with link variables $U$, formulated to describe the intermediate temperature pseudogap metal phase of underdoped cuprates of hole density $p$, and its evolution into nodal $d$-wave superconductivity upon cooling. The theory features two fermionic matter sectors: one consists of SU(2)-gauge-neutral, electron-like quasiparticles forming hole pockets; the other is a critical quantum spin liquid composed of electrically neutral spinons, carrying spin $S = 1/2$, Dirac nodes, and transforming in the fundamental representation of the SU(2) gauge group. These sectors interact via a Yukawa coupling to a spinless Higgs boson $B$, which carries electric charge $+e$ and also transforms under the fundamental representation of SU(2). The theory arises from a canonical transformation of the square lattice Hubbard model and a pair of ancilla qubits on each site, along with a specific choice of the spin liquid. We employ a Born-Oppenheimer-type approximation: thermal fluctuations of the bosons $B$ and $U$ are treated classically, while the fermions are fully quantum. Monte Carlo simulations of an effective boson energy functional are consistent with a Kosterlitz-Thouless transition involving $h/(2e)$ vortices, driving the onset of $d$-wave superconductivity. Each vortex is surrounded by a halo of charge order, consistent with scanning tunneling microscopy observations by Hoffman et al. (arXiv:cond-mat/0201348). Above the transition, the electron spectral function exhibits "Fermi arcs", in agreement with ARPES data by Norman et al. (arXiv:cond-mat/9710163) and Shen et al. (Science 307, 901 (2005)). Our model also displays quantum oscillations with area $p/8$ under a magnetic field, aligning with recent evidence via the Yamaji effect by Chan et al. (arXiv:2411.10631).