We show that in the $SU(2)\ifmmode\times\else\texttimes\fi{}U(1)$ spin-charge gauge approach we developed earlier one can attribute consistently an exclusion statistics with parameter 1/2 to the spinless charge carriers of the $t\text{\ensuremath{-}}J$ model in two dimensions, as it occurs in one dimension. Like the one-dimensional case, the no-double occupation constraint is at the origin of this fractional exclusion statistics. With this statistics we recover a large Fermi volume of holes at high dopings, close to that of the tight binding approximation. Furthermore, the composite nature of the hole, made of charge and spin carriers only weakly bounded, can provide a natural explanation of many unusual experimental features of the hole-doped cuprates.
We show that in the SU(2)\ifmmode×\else\texttimes\fiU(1) spin-charge gauge approach we developed earlier one can attribute consistently an exclusion statistics with parameter 1/2 to the spinless charge carriers of the t\ensuremath-J model in two dimensions, as it occurs in one dimension. Like the one-dimensional case, the no-double occupation constraint is at the origin of this fractional exclusion statistics. With this statistics we recover a large Fermi volume of holes at high dopings, close to that of the tight binding approximation. Furthermore, the composite nature of the hole, made of charge and spin carriers only weakly bounded, can provide a natural explanation of many unusual experimental features of the hole-doped cuprates.
We show that we can interpret the exact solution of the one-dimensional t-J model in the limit of small J in terms of charge carriers with both exchange (braid) and exclusion (Haldane) statistics with parameter 1/2. We discuss an implementation of the same statistics in the two-dimensional t-J model, emphasizing similarities and differences with respect to one dimension. In both cases, the exclusion statistics is a consequence of the no-double occupation constraint. We argue that the application of this formalism to hole-doped high Tc cuprates and the derived composite nature of the hole give a hint to grasp many unusual properties of these materials.
Electrical and magnetic properties of underdoped cuprates in the spin gap phase, a precursor to high Tc superconducting state, is riddled with puzzles. We propose a novel reference state to study this phase, where each hole dopant adds one real holon and one real spinon, with Haldane exclusion statistics gh=1 and gs=1/2, to a fairly inert (pseudo gaped) spin liquid vacuum, resulting in a low density spin-charge liquid. Spins and charges interact and form a novel collective state, a \textit{3/2-Fermi liquid}. A holon pairs with a spinon; this hole like fermion composite carries charge +e and a spin-\hlf moment and novel exclusion statistics ghole=3/2. We explain an anomalous expansion of Fermi sea area by 3/2, seen in recent quantum oscillation experiments at two different dopings.
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.
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 critically look at Senthil Fisher constructions of quantum number fractionization in cuprates. The first construction, while mathematically correct is of limited relevance to cuprates. The second one has a missing local U(1) symmetry. Once this aspect is repaired, it reveals itself as the U(1) RVB gauge theory construction of quantum number fractionization. An approximate local Z2 symmetry arises in the spin sector for a reason very different from Senthil and Fisher's. The charge sector continues to have the local U(1) symmetry and the metallic spin gap phase of the 2d cuprates may, in principle, carry (irrational) fractional charge e∗=e obeying Haldane's fractional exclusion statistics like the 1d models.
This thesis deals with different aspects of high temperature superconductivity in hole-doped cuprates. We assume the t - J model to describe the CuO2 planes of cuprates and we use a spin-charge gauge approach with spin-charge separation to describe holes in terms of a spinless fermion carrying the charge (holon) and a neutral boson carrying spin 1/2 (spinon), coupled by a slave-particle gauge field. In this framework we consider the effects of the presence of a finite density of incoherent holon pairs in the normal state as a precursor of superconductivity. We show that it is possible to take such pairs into account through a strongly direction dependent spectral weight for holons, which suppresses quasi-particles starting from the anti-nodal directions, and a wave function renormalization for the slave particle gauge field. In this way we prove that, when temperature is reduced, the formation of holon pairs causes both the deviation from linearity of the resistivity and the deviation from the constant value of the Knight shift. Moreover we point out the need for a negative next-nearest-neighbor hopping term ((t - t' - J model) to get a good continuum limit and to evaluate the Knight shift. These results are obtained through a Green's function, which follows naturally from the formalism and analytically interpolates between a Fermi liquid-like behavior and a d-wave superconductor when the coherence length of the holon pair order parameter is increased. The system preserves a finite Fermi surface until the superconducting transition, where it reduces to four nodes. Finally when, at lower temperatures, both the holon pairs and the RVB spinon pairs condense we enter the superconducting phase. In this phase we study the magnetic excitations showing how to extend to spin waves the hourglass-shape dispersion, found for spinons near the antiferromagnetic vector, if a suitable local mechanism of attraction between spinon and anti-spinon is assumed. The resulting spin wave Green's function, whose imaginary part is directly comparable with experiments, generates the hourglass with a finite gap between the two branches. Since the U(1) slave-particle gauge field gains mass in the superconducting phase via Anderson-Higgs mechanism, we propose the necessary mechanism of attraction comes from the unbroken Z2 subgroup of the U(1) gauge group.