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.
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.
We consider a SU(2) lattice gauge theory on the square lattice, with a single fundamental complex fermion and a single fundamental complex boson on each lattice site. Projective symmetries of the gauge-charged fermions are chosen so that they match with those of the spinons of the π -flux spin liquid. Global symmetries of all gauge-invariant observables are chosen to match with those of the particle-hole symmetric electronic Hubbard model at half-filling. Consequently, both the fundamental fermion and fundamental boson move in an average background π -flux, their gauge-invariant composite is the physical electron, and eliminating gauge fields in a strong gauge-coupling expansion yields an effective extended Hubbard model for the electrons. The SU(2) gauge theory displays several confining/Higgs phases: a nodal d -wave superconductor, and states with Néel, valence-bond solid, charge, or staggered current orders. There are also a number of quantum phase transitions between these phases that are very likely described by ( 2 + 1 ) -dimensional deconfined conformal gauge theories, and we present large flavor expansions for such theories. These include the phenomenologically attractive case of a transition between a conventional insulator with a charge gap and Néel order, and a conventional d -wave superconductor with gapless Bogoliubov quasiparticles at four nodal points in the Brillouin zone. We also apply our approach to the honeycomb lattice, where we find a bicritical point at the junction of Néel, valence bond solid (Kekulé), and Dirac semimetal phases.
Quantum fluctuations of the Néel state of the square lattice antiferromagnet are usually described by a CP 1 theory of bosonic spinons coupled to a U(1) gauge field, and with a global SU(2) spin rotation symmetry.Such a theory also has a confining phase with valence bond solid (VBS) order, and upon including spin-singlet charge-2 Higgs fields, deconfined phases with Z 2 topological order possibly intertwined with discrete broken global symmetries.We present dual theories of the same phases starting from a mean-field theory of fermionic spinons moving in π flux in each square lattice plaquette.Fluctuations about this π-flux state are described by (2 þ 1)-dimensional quantum chromodynamics (QCD 3 ) with a SU(2) gauge group and N f ¼ 2 flavors of massless Dirac fermions.It has recently been argued by Wang et al. [Deconfined Quantum Critical Points: Symmetries and Dualities, Phys. Rev. X 7, 031051 (2017).]that this QCD 3 theory describes the Néel-VBS quantum phase transition.We introduce adjoint Higgs fields in QCD 3 and obtain fermionic dual descriptions of the phases with Z 2 topological order obtained earlier using the bosonic CP 1 theory.We also present a fermionic spinon derivation of the monopole Berry phases in the U(1) gauge theory of the VBS state.The global phase diagram of these phases contains multicritical points, and our results imply new boson-fermion dualities between critical gauge theories of these points.
We present a quantum Monte Carlo study of the transition between the insulating Néel state and the nodal d -wave superconductor on the square lattice at half-filling. We access a regime of frustrated magnetic order without a sign problem using a parton representation of the electron in terms of fermionic spinons and bosonic chargons. Both partons move in a background π -flux (so the electron experiences no net flux) and are coupled to a quantum fluctuating SU(2) lattice gauge field. In contrast to earlier studies directly on the electronic degrees of freedom, we find evidence for a second-order deconfined quantum phase transition at which both the Néel and d -wave superconductivity orders vanish continuously. We compute correlators of the spinon-chargon composite with the same quantum numbers as the electron: we find a gapless Dirac dispersion inside the d -wave superconductor, turning into a gapped dispersion in the antiferromagnet.
Abstract We review a theoretical framework for the cuprate superconductors, rooted in a fractionalized Fermi liquid (FL*) description of the intermediate-temperature pseudogap phase at low doping. The FL* theory predicted hole pockets each of fractional area p / 8 at hole doping p , in contrast to the area p / 4 in spin density wave theory. Magnetotransport measurements, including observation of the Yamaji angle, show clear evidence of hole pocket quasiparticles which can tunnel coherently between square lattice layers, and are consistent with the FL* description. The FL* phase of a single-band model is described using a layer construction with a pair of ancilla qubits on each site: the Ancilla layer model (ALM). Its mean field theory yields hole pockets of area p / 8 , and matches the gapped photoemission spectrum in the anti-nodal region of the Brillouin zone. Fluctuations are described by the SU(2) gauge theory of a background spin liquid with critical Dirac spinons. A Monte Carlo study of the thermal SU(2) gauge theory transforms the hole pockets into Fermi arcs in photoemission. One route to confinement of FL* upon lowering temperature yields a d -wave superconductor via a Kosterlitz–Thouless transition of h / ( 2 e ) vortices, with nodal Bogoliubov quasiparticles featuring anisotropic velocities and vortices surrounded by charge order halos. An alternative route yields a charge-ordered metallic state that has quantum oscillations consistent with observations. These confinement transitions are driven by the condensation of a SU(2) fundamental Higgs field, which also provides a fractionalized description of intertwined orders. Increasing doping from the FL* phase in the ALM drives a transition to a conventional FL at large doping, passing through an intermediate strange metal regime. We formulate a theory of the FL*-FL metal-metal transition without a symmetry-breaking order parameter, using a critical quantum ‘charge’ liquid of mobile electrons in the presence of disorder, developed via an extension of the Sachdev–Ye–Kitaev model to two spatial dimensions. At low temperatures, and across optimal and over doping, we address the regimes of extended non-FL behavior by Griffiths effects near quantum phase transitions in disordered metals. Partly based on lectures by S S at Boulder School 2025, Dynamics of Strongly Correlated Electrons , 14–18 July. Lecture videos . Joint ICTP-WE Heraeus School and Workshop on Advances in Quantum Matter: Pushing the Boundaries , ICTP, Trieste, 4, 6 August 2025. Lecture videos . School on Quantum Dynamics of Matter, Light and Information , ICTP, Trieste, 18, 19 August 2025. Lecture videos . Croucher Advanced Study Institute for Fractional Chern Insulators , University of Hong Kong, 4, 5 September 2025. Lecture slides . Advanced School and Conference on Quantum Matter , ICTP Trieste, 1–12 December 2025. Lecture Notes . Lecture videos .
Triangular-lattice quantum spin-liquid insulators are observed to undergo transitions to superconductivity under pressure or doping, and exhibit enhanced terahertz conductivity when driven by mid-infrared light. We present a general theoretical framework for the emergence of superconducting and charge-ordered phases from a U(1) Dirac spin liquid with fermionic spinons, as well as from its gapped Z2 and chiral descendants. Numerical studies have provided substantial evidence for these spin-liquid states. The spin-liquid phase hosts fractionalized Dirac spinons coupled to an emergent gauge field, whereas the superconducting and charge-ordered phases are conventional, with neither fractionalized excitations nor emergent gauge dynamics. The transition between these phases is driven by the Higgs condensation of spinless charge-e bosonic chargons (``doublons'' and ``holons''). We show that the projective symmetry group of the Dirac spinons uniquely determines the symmetry and dispersion of the chargons, allowing us to construct an effective low-energy theory near the chargon band minima. Gauge-invariant composites of the chargon Higgs fields provide the order parameters characterizing the phases. The resulting phase diagram contains a rich variety of ordered states, including d+id superconductivity, charge-density waves, bond-density waves, and pair-density waves.
The cuprate pseudogap phase displays Fermi arc spectral weight in photoemission and scanning tunneling microscopy, while recent magnetotransport observations yield evidence for the existence of hole pockets of fractional area p /8, where p is the doping density. We present a Monte Carlo study of a thermal SU(2) lattice gauge theory which can reconcile these observations. Our simulation includes the SU(2) gauge field U of a π -flux spin liquid, and a SU(2) fundamental charge e Higgs boson B . There is a Yukawa coupling between B , the fermionic spinons of the spin liquid, and the hole pockets of a fractionalized Fermi liquid. At the higher temperatures of the pseudogap, the finite-doping sign problem is evaded by including only thermal fluctuations of B and U , while the fermions are diagonalized exactly for each boson background. Our study also yields a fractionalized description of intertwined orders at lower temperatures, including the onset of d -wave superconductivity by the expulsion of vortices with flux h / ( 2 e ) , each with charge-order halos. We discuss conditions under which quantum oscillations in the density of states from hole pockets of area p / 8 could be observable in clean under-hole-doped cuprates.