We examine spectral properties of a recently proposed theory of the intermediate temperature pseudogap metal phase of the cuprates. We show that this theory can be obtained from the familiar paramagnon theory of nearly antiferromagnetic metals by fractionalizing the paramagnon into two ``hidden'' layers of $S=1/2$ spins. The first hidden layer of spins hybridizes with the electrons as in a Kondo lattice heavy Fermi liquid, whereas the second hidden layer of spins forms a spin liquid with fractionalized spinon excitations. We compute the imaginary part of the electronic self-energy induced by the spinon excitations. The energy and momentum dependence of the photoemission spectrum across the Brillouin zone provides a good match to observations by He et al. [Science 331, 1579 (2011)] in $({\mathrm{Pb}}_{x},{\mathrm{Bi}}_{2\ensuremath{-}x})({\mathrm{La}}_{y}{\mathrm{Sr}}_{2\ensuremath{-}y}){\mathrm{CuO}}_{6+\ensuremath{\delta}}$ and by Chen et al. [Science 366, 1099 (2019)] in ${(\mathrm{Bi},\mathrm{Pb})}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$.
We examine spectral properties of a recently proposed theory of the intermediate temperature pseudogap metal phase of the cuprates. We show that this theory can be obtained from the familiar paramagnon theory of nearly antiferromagnetic metals by fractionalizing the paramagnon into two ``hidden'' layers of S=1/2 spins. The first hidden layer of spins hybridizes with the electrons as in a Kondo lattice heavy Fermi liquid, whereas the second hidden layer of spins forms a spin liquid with fractionalized spinon excitations. We compute the imaginary part of the electronic self-energy induced by the spinon excitations. The energy and momentum dependence of the photoemission spectrum across the Brillouin zone provides a good match to observations by He et al. [Science 331, 1579 (2011)] in (Pbx,Bi2\ensuremath−x)(LaySr2\ensuremath−y)CuO6+\ensuremathδ and by Chen et al. [Science 366, 1099 (2019)] in (Bi,Pb)2Sr2CaCu2O8+\ensuremathδ.
The pseudogap metal phase of hole-doped cuprates can be described by small Fermi surfaces of electronlike quasiparticles, which enclose a volume violating the Luttinger relation. This violation requires the existence of additional fractionalized excitations which can be viewed as fractionalized remnants of the paramagnon. We fractionalize the paramagnon into the bosonic spinons of the spin liquid described by the CP1 U(1) gauge theory, and we present a gauge theory of the bosonic spinons, a Higgs field, and an ancilla layer of fermions coupled to the original electrons. Along with the small Fermi surface pseudogap metal, this theory displays conventional phases: the large Fermi surface Fermi liquid with a low-energy paramagnon mode, and phases with spin density wave order. We describe the evolution of the electronic photoemission spectrum across these quantum phase transitions. We consider both the two-sublattice N\'eel and incommensurate spin density wave phases, and we find that the latter has spiral spin correlations.
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 .
We study the electron spectral function of various zero-temperature spin-charge separated phases in two dimensions. In these phases, the electron is not a fundamental excitation of the system, but rather ``decays'' into a spin-1/2 chargeless fermion (the spinon) and a spinless charge e boson (the chargon). Using low-energy effective theories for the spinons (d-wave pairing plus possible N\'eel order) and the chargons (condensed or quantum-disordered bosons), we explore three phases of possible relevance to the cuprate superconductors: (1) AF∗, a fractionalized antiferromagnet where the spinons are paired into a state with long-ranged N\'eel order and the chargons are 1/2-filled and (Mott) insulating; (2) the nodal liquid, a fractionalized insulator where the spinons are d-wave paired and the chargons are uncondensed; and (3) the d-wave superconductor, where the chargons are condensed and the spinons retain a d-wave gap. Working within the Z2 gauge theory of such fractionalized phases, our results should be valid at scales below the energy gap of the vison---the basic vortex excitation in the theory. However, on a phenomenological level, our results should apply to any spin-charge separated system where the excitations have these low-energy effective forms. Comparison with angle-resolved photoemission spectroscopy data in the undoped, pseudogapped, and superconducting regions is made.
Two of the iconic phases of the hole-doped cuprate materials are the intermediate temperature pseudogap metal and the lower temperature d-wave superconductor. Following the suggestion of P. W. Anderson, there were early theories of these phases as doped quantum spin liquids. However, these theories have had difficulties with two prominent observations: (i) angle-dependent magnetoresistance measurements (ADMR) in the pseudogap metal, including observation of the Yamaji effect, present convincing evidence of small hole pockets which can tunnel coherently between square lattice layers (Fang et al., Nature Physics 18, 558 (2022); Chan et al., Nature Physics 21, 1753 (2025)) and (ii) the velocities of the nodal Bogoliubov quasiparticles in the d-wave superconductor are highly anisotropic, with vF≫vΔ (Chiao et al., Phys. Rev. B 62, 3554 (2000)). These notes review how the fractionalized Fermi Liquid (FL*) state, which dopes quantum spin liquids with gauge-neutral electron-like quasiparticles, resolves both difficulties. Theories of insulating quantum spin liquids employing fractionalization of the electron spin into bosonic or fermionic partons are discussed. Doping the bosonic parton theory leads to a holon metal theory: while not appropriate for the cuprate pseudogap, this theory is argued to apply to ultracold atom experiments on the Lieb lattice. Doping the fermionic parton theory leads to a d-wave superconductor with nearly isotropic quasiparticle velocities. The construction of the FL* state in a single band Hubbard-type model is described using a quantum dimer model, followed by a more realistic description using the Ancilla Layer Model (ALM), which is then used to obtain the theory of the pseudogap and the d-wave superconductor. The ALM also leads to a wavefunction for the FL* state of the Hubbard model, which agrees with ultracold atom observations.
We consider a two-dimensional Fermi liquid in the vicinity of a spin-density-wave transition to a phase with commensurate antiferromagnetic long-range order. We assume that near the transition, the Fermi surface is large and crosses the magnetic Brillouin zone boundary. We show that under these conditions, the self-energy corrections to the dynamical spin susceptibility, \ensuremath{\chi}(q,\ensuremath{\omega}), and to the quasiparticle spectral function, A(k,\ensuremath{\omega}), are divergent near the transition. We identify and sum the series of most singular diagrams, and obtain a solution for \ensuremath{\chi}(q,\ensuremath{\omega}) and an approximate solution for A(k,\ensuremath{\omega}). We show that (i) A(k) at a given, small \ensuremath{\omega} has an extra peak at k=kF+\ensuremath{\pi} (shadow band), and (ii) the dispersion near the crossing points is much flatter than for free electrons. The relevance of these results to recent photoemission experiments in YBa2Cu3${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ and Bi2Sr2CaCu2O8 systems is discussed.
Many properties of the copper oxide high-temperature superconductors are still poorly understood, despite more than thirty years of research. This is true in particular for the metallic ``pseudogap'' state, out of which the well-known superconducting phase develops upon increasing the concentration of hole-like charge carriers. Here, the authors analyze the single-electron spectral function of a recently introduced quantum dimer model, which features a very interesting metallic ground state with topological order: a so-called fractionalized Fermi liquid. Earlier works have argued that this model describes several key properties of the pseudogap metal, such as Fermi-arc like features in photoemission experiments. Using a combination of numerical and analytical methods, the authors show that this model indeed features a sizable antinodal pseudogap with an angular dependence deviating from a simple d-wave form, in accordance with various experimental results on underdoped cuprates.