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arXiv:2606.17159

Superconductivity from emergent dipolar interactions in a fractionalized Fermi liquid

Mina-Lou Schleith, Arthur Bril, Nick Bultinck

Published Jun 15, 2026• 0 citations
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Abstract

Starting from the spin-fermion model or Hertz-Millis theory describing electrons coupled to anti-ferromagnetic spin fluctuations we develop a theory to describe the transition from a fractionalized Fermi liquid into a $d_{x^2-y^2}$ superconductor. We focus on small electron doping on top of the half-filled state. The doped electrons enter the system as spinon-chargon bound states, which form a small, reconstructed Fermi surface. The bound states are neutral under the emergent U(1) gauge symmetry of the fractionalized Fermi liquid, but interact via a dipolar two-body potential. We show that because of the projective action of translation symmetry on the spinons and chargons, the Fourier components of this repulsive dipolar interaction are peaked at the anti-ferromagnetic wave vector, thereby providing a robust microscopic mechanism for $d_{x^2-y^2}$ pairing in a fractionalized metal.

DOI
10.48550/arxiv.2606.17159

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Starting from the spin-fermion model or Hertz-Millis theory describing electrons coupled to anti-ferromagnetic spin fluctuations we develop a theory to describe the transition from a fractionalized Fermi liquid into a dx2−y2d_{x^2-y^2}dx2−y2​ superconductor. We focus on small electron doping on top of the half-filled state. The doped electrons enter the system as spinon-chargon bound states, which form a small, reconstructed Fermi surface. The bound states are neutral under the emergent U(1) gauge symmetry of the fractionalized Fermi liquid, but interact via a dipolar two-body potential. We show that because of the projective action of translation symmetry on the spinons and chargons, the Fourier components of this repulsive dipolar interaction are peaked at the anti-ferromagnetic wave vector, thereby providing a robust microscopic mechanism for dx2−y2d_{x^2-y^2}dx2−y2​ pairing in a fractionalized metal.
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\mathbb{Z}_2Z2​ 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-eee 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+idd+idd+id superconductivity, charge-density waves, bond-density waves, and pair-density waves.
Doping a Mott-insulating Z2{\mathbb{Z}}_{2}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{\mathbb{Z}}_{2}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 ddd-wave pairing. Such a transition occurs by the condensation of bosonic particles carrying +e+e+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\mathbf{K}K, coexisting with ddd-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.
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{\mathbb{CP}}^{1}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.
We study superfluidity in strongly correlated systems by focusing three features as follows. First, interplay between superfluidity and other order parameters near quantum critical points is studied. The onset positions of the order parameters with or without superconductivity determines how they interact with each other. For the spin density wave (SDW) transition, we find that the quantum critical point shifts by order |∆|, where ∆ is pairing amplitude, so that the region of SDW order is smaller in the superconductor than in the metal. This shift is larger than the |∆|2 shift predicted by theories of competing orders which ignore Fermi surface effects. Second, we study a doped antiferromagnetic Mott insulator motivated by copper based high temperature superconductors. Based on spin and charge fractionalization, the algebraic charge liquid phase whose bosonic spinons have energy gap but fermionic holons form Fermi surfaces and its superconductivity are studied. We show fluctuations of local SDW order orientation lead to d-wave superconductivity, and its quantum critical pairing mechanisms are investigated. Also, a theory of bound states between spinons and holons is studied. We show that the fractionalization induces a doubling electron-like quasiparticles, which naturally induces the fractionalized Fermi liquid. The bound state theory describes exotic properties of the underdoped Cuprates such as “nodal-antinodal dichotomy” at a mean field level description. At last, we study superfluidity of fermions in optical lattices near the Feshbach resonance. Due to divergence of scattering length at the resonance, universality is observed and characteristics of superfluidity has been studied. We extend such universal properties by considering superfluid-insulator transition with an even integer number of fermions per unit cell. The resulting insulator interpolates between a band insulator of fermions and a Mott insulator of fermion pairs.
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 Ni2+{\mathrm{Ni}}^{2+}Ni2+ into a spin-1/21/21/2 Mott insulator formed by Ni1+{\mathrm{Ni}}^{1+}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 $t\text{\ensuremath{-}}J$ 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 d8{d}^{8}d8 state.
Insulating quantum spin liquids can undergo a confinement transition to a valence bond solid via the condensation of topological excitations of the associated gauge theory. We extend the theory of such transitions to fractionalized Fermi liquids (FL*): These are metallic doped spin liquids in which the Fermi surfaces only have gauge neutral quasiparticles. Using insights from a duality transform on a doped quantum dimer model for the U(1)-FL* state, we show that projective symmetry group of the theory of the topological excitations remains unmodified, but the Fermi surfaces can lead to additional frustrating interactions. We propose a theory for the confinement transition of Z2{\mathbb{Z}}_{2}Z2​-FL* states via the condensation of visons. A variety of confining, incommensurate density wave states are possible, including some that are similar to the incommensurate ddd-form factor density wave order observed in several recent experiments on the cuprate superconductors.