A model ofd-wave superconductivity, antiferromagnetism, and charge order on the square lattice

Abstract

We describe the confining instabilities of a proposed quantum spin liquid underlying the pseudogap metal state of the hole-doped cuprates. The spin liquid can be described by a SU(2) gauge theory of N f = 2 massless Dirac fermions carrying fundamental gauge charges—this is the low-energy theory of a mean-field state of fermionic spinons moving on the square lattice with π -flux per plaquette in the ℤ 2 center of SU(2). This theory has an emergent SO(5) f global symmetry and is presumed to confine at low energies to the Néel state. At nonzero doping (or smaller Hubbard repulsion U at half-filling), we argue that confinement occurs via the Higgs condensation of bosonic chargons carrying fundamental SU(2) gauge charges also moving in π ℤ 2 -flux. At half-filling, the low-energy theory of the Higgs sector has N b = 2 relativistic bosons with a possible emergent SO(5) b global symmetry describing rotations between a d -wave superconductor, period-2 charge stripes, and the time-reversal breaking “ d -density wave” state. We propose a conformal SU(2) gauge theory with N f = 2 fundamental fermions, N b = 2 fundamental bosons, and a SO(5) f ×SO(5) b global symmetry, which describes a deconfined quantum critical point between a confining state which breaks SO(5) f and a confining state which breaks SO(5) b . The pattern of symmetry breaking within both SO(5)s is determined by terms likely irrelevant at the critical point, which can be chosen to obtain a transition between Néel order and d -wave superconductivity. A similar theory applies at nonzero doping and large U , with longer-range couplings of the chargons leading to charge order with longer periods.

Emergence of nodal Bogoliubov quasiparticles across the transition from the pseudogap metal to the d-wave superconductor

“…In this work, we will consider starting from a normal state where only Φ is condensed such that the Fermi surface has hole-density 1 p and ask how the electronic spectrum evolves as a d-wave superconductor sets in when B condenses on top of this normal state. As discussed in previous work [24]f o rt h ec a s eo ft h e⇡-flux spin liquid and in [26,28,29]f o rt h ec a s eo ft h eU ( 1 ) staggered flux spin liquid, the different ways in which the two components of B condense can break different symmetries corresponding to distinct orders which may be inherited by the physical electrons if Φ is also condensed. Expanding about the two band minima of the ⇡-flux dispersion, we have the following expression for the chargon B in terms of B a+ and B a ,t h ec o n t i n u u m degrees of freedom associated with the + and minima of the ⇡-flux mean field dispersion:…”

“…We have written the saddle point ⇡-flux spin liquid in the second layer of spins in the gauge previously used in [24,49]w h e r ee i,j = e j,i , e i,i+x =1, e i,i+y =( 1) x . After a mean field decoupling the model can be written in the following form:…”

“…In all of our analysis on the electron-doped side of the phase diagram, we have assumed t f2 = .14.H o w e v e r , changing the sign of t f2 will result in qualitatively different behavior in the number of nodes similar to the hole-doped case as shown in the Supplement [51]. However, as was shown in Appendix 3 of [24], only the former sign of t f2 corresponds to a chargon potential which favors the continuum superconductor ansatz we have taken here.…”

“…We thank Patrick Lee for probing questions on previous work [24], Z.-X. Shen for informing us about ongoing photoemission experiments in the electron-doped cuprates [11], Steve Kivelson for discussions on Refs.…”

“…We will employ a theory [24]o ft h ep s e u d o g a pm e t a l with fermionic spinons coupled to to an SU(2) gauge field moving in a background of ⇡-flux [25][26][27][28]. The fermionic spinons are coupled to physical electrons which carry the doping via a charge e boson B [26,28,29] which transforms under the same gauge SU(2) symmetry as the spinons.…”