Inspired by the recent discovery of superconductivity in the nickelate ${\mathrm{Nd}}_{1\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Ni}{\mathrm{O}}_{2}$, we study a generalized $t\text{\ensuremath{-}}J$ model to investigate the correlated phases induced by doping spin-1 ${\mathrm{Ni}}^{2+}$ into a spin-$1/2$ Mott insulator formed by ${\mathrm{Ni}}^{1+}$. Based on a three-fermion parton mean-field analysis, we identify a robust fractional Fermi liquid (FL*) phase for almost every doping level. The FL* state is characterized by a small Fermi pocket on top of a spin liquid, which violates the Luttinger theorem of a conventional Fermi liquid and is an example of a symmetric pseudogap metal. Furthermore, we verify our theory in one dimension through density matrix renormalization group simulations on both the generalized $t\text{\ensuremath{-}}J$ model and a two-orbital Hubbard model. The fractional Fermi liquid reduces to a fractional Luttinger liquid (LL*) in one dimension, which is connected to the conventional Luttinger liquid through a continuous quantum phase transition by tuning interaction strength. Our findings offer insights into correlated electron phenomena in nickelate superconductors and other multiorbital transition-metal oxides with a spin-triplet ${d}^{8}$ state.
Inspired by the recent discovery of superconductivity in the nickelate Nd1\ensuremath−xSrxNiO2, we study a generalized t\ensuremath-J model to investigate the correlated phases induced by doping spin-1 Ni2+ into a spin-1/2 Mott insulator formed by Ni1+. Based on a three-fermion parton mean-field analysis, we identify a robust fractional Fermi liquid (FL*) phase for almost every doping level. The FL* state is characterized by a small Fermi pocket on top of a spin liquid, which violates the Luttinger theorem of a conventional Fermi liquid and is an example of a symmetric pseudogap metal. Furthermore, we verify our theory in one dimension through density matrix renormalization group simulations on both the generalized t\ensuremath-J model and a two-orbital Hubbard model. The fractional Fermi liquid reduces to a fractional Luttinger liquid (LL*) in one dimension, which is connected to the conventional Luttinger liquid through a continuous quantum phase transition by tuning interaction strength. Our findings offer insights into correlated electron phenomena in nickelate superconductors and other multiorbital transition-metal oxides with a spin-triplet d8 state.
Symmetric pseudogap metal in a generalized t−J model
In this article we study a generalized t−J model in which we dope spin-one doublons into a spin 1/2 Mott insulator. We generalize this model to SU(N) case and obtain the ground state based on a three-fermion parton mean field calculation. Surprisingly, for almost every doping we find a fractional Fermi liquid (FL*) phase , which is a symmetric pseudogap metal in the sense that the total Fermi volume is equal to x, instead of 1+x as required by Luttinger constraint of a Fermi liquid. We then verify our theory in one dimension through DMRG simulation of the t−J model. Indeed we find that the doped doublon forms a small Fermi on top of the spinon Fermi surface (SU(2)1 CFT) inherited from the Mott insulator. This phase is clearly a one dimension analog of the FL* phase and we dub it as fractional Luttinger liquid (LL*). We also discover a continuous quantum phase transition from this LL* phase with small Fermi to conventional Luttinger liquid phase with large Fermi by tuning interaction strength in a two-orbital Hubbard model at fixed density. Lastly we discuss the experimental realizations of this generalized t−J model and comment on possible connection to the recently discovered nickelate superconductor Nd1−xSrxNiO2.
Abstract In this article, we review some recent theoretical developments on potential high-temperature superconductors and unconventional metallic states that can arise from doping a spin-one Mott insulator in the d 8 valence. These studies are particularly relevant—though not limited—to the recently discovered bilayer nickelate superconductor La 3 Ni 2 O 7 . We focus on a ferromagnetic Kondo lattice model with mobile electrons in the d x 2 − y 2 orbital coupled to the localized spin moments in d z 2 orbital through a large Hund’s coupling J H . In the large J H limit, the model reduces to the type II t − J model with a mixture of spin-half singlon states and spin-one doublon states. We summarize density matrix renormalization group results on the Luther-Emery liquid in one dimensional chain and two-leg ladder. Then we mainly focus on bilayer square lattice and show that a large inter-layer coupling J ⊥ of d z 2 orbital can induce strong inter-layer pairing of d x 2 − y 2 orbital. In the strong J ⊥ limit, a kinetic-energy driven high T c superconductivity is demonstrated in an ideal model with only a single hopping term. Furthermore, the model predicts a symmetric pseudogap metal—dubbed ‘second Fermi liquid (FL)’—in the underdoped regime, yielding a phase diagram analogous to that of hole-doped cuprates. The bilayer Kondo model therefore presents a promising platform for both realizing higher- T c superconductors and exploring non-FL physics. We also comment on the possible limitations of the current models for the bilayer nickelate material and point out some future directions.
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 .
A Z2 fractionalized Fermi liquid (FL*) is a novel state of strongly correlated quantum matter. Although it is metallic, it violates Luttinger's theorem on the volume enclosed by the Fermi surface obeyed by conventional metals; this is possible due to the presence of emergent gauge excitations. The authors study here the superconducting states that can arise out of instabilities of such a topological metal. In particular, they focus on a FL* topological metal that has favorable energetics on the square lattice with nematic order, relevant to the cuprate superconductors. They find that a Higgs transition out of this FL* results in confinement of the anyonic degrees of freedom, and the resulting state is a superconductor with broken translation symmetry or time-reversal invariance. In the process, they also establish a complete mapping between bosonic and fermionic descriptions of time-reversal invariant gapped insulating Z2 spin liquid states on the rectangular lattice, which, on doping with charge carriers, can give rise to FL* phases. They also note a possible connection to the recent observation of pair density waves in the superconducting state of the underdoped cuprates.
Doping a Mott-insulating Z2 spin liquid can lead to a fractionalized Fermi liquid (FL*). Such a phase has several favorable features that make it a candidate for the pseudogap metal for the underdoped cuprates. We focus on a particular, simple Z2-FL* state which can undergo a confinement transition to a spatially uniform superconductor which is smoothly connected to the ``plain vanilla'' BCS superconductor with d-wave pairing. Such a transition occurs by the condensation of bosonic particles carrying +e charge but no spin (``chargons''). We show that modifying the dispersion of the bosonic chargons can lead to confinement transitions with charge density waves and pair density waves at the same wave vector K, coexisting with d-wave superconductivity. We also compute the evolution of the Hall number in the normal state during the transition from the plain vanilla FL* state to a Fermi liquid, and argue, following Coleman, Marston, and Schofield [Phys. Rev. B 72, 245111 (2005)], that it exhibits a discontinuous jump near optimal doping. We note the distinction between these results and those obtained from models of the pseudogap with fermionic chargons.
Heavy fermion criticality has been a long-standing problem in condensed matter physics. Here we study a one-dimensional Kondo lattice model through numerical simulation and observe signatures of local criticality. We vary the Kondo coupling J_K J K at fixed doping x. At large positive J_K J K , we confirm the expected conventional Luttinger liquid phase with 2k_F=\frac{1+x}{2} 2 k F = 1 + x 2 (in units of 2\pi 2 π ), an analogue of the heavy Fermi liquid (HFL) in the higher dimension. In the J_K ≤ 0 J K ≤ 0 side, our simulation finds the existence of a fractional Luttinger liquid (LL \star ⋆ ) phase with 2k_F=\frac{x}{2} 2 k F = x 2 , accompanied by a gapless spin mode originating from localized spin moments, which serves as an analogue of the fractional Fermi liquid (FL \star ⋆ ) phase in higher dimensions. The LL \star ⋆ phase becomes unstable and transitions to a spin-gapped Luther-Emery (LE) liquid phase at small positive J_K J K . Then we mainly focus on the “critical regime” between the LE phase and the LL phase. Approaching the critical point from the spin-gapped LE phase, we often find that the spin gap vanishes continuously, while the spin-spin correlation length in real space stays finite and small. For a certain range of doping, in a point (or narrow region) of J_K J K , the dynamical spin structure factor obtained through the time-evolving block decimation (TEBD) simulation shows dispersion-less spin fluctuations in a finite range of momentum space above a small energy scale (around 0.035 J 0.035 J ) that is limited by the TEBD accuracy. All of these results are unexpected for a regular gapless phase (or critical point) described by conformal field theory (CFT). Instead, they are more consistent with exotic ultra-local criticality with an infinite dynamical exponent z=+ z = + . The numerical discovery here may have important implications on our general theoretical understanding of the strange metals in heavy fermion systems. Lastly, we propose to simulate the model in a bilayer optical lattice with a potential difference.