Fractionalized Fermi Liquid (FL*)

• Fractionalized Fermi Liquid (FL*) is a metallic phase where a small Fermi surface of charged quasiparticles coexists with a gapped, topologically ordered spin sector featuring emergent anyons and gauge fields.
• It modifies the conventional Luttinger theorem by subtracting fractionalized density contributions that are bound in the SET sector, explaining the discrepancy between microscopic filling and observed Fermi volume.
• Microscopic realizations span heavy-fermion systems, cuprate pseudogap metals, and Kitaev–Kondo lattice models, highlighting pathways to instabilities such as density-wave and unconventional superconducting states.

Fractionalized Fermi liquid (FL*) denotes a metallic phase in which a conventional Fermi surface of charged quasiparticles coexists with a symmetry-enriched topologically ordered sector. In contrast to an ordinary Fermi liquid, where the total Fermi volume is tied directly to the microscopic filling, an FL* permits a “small” Fermi surface because part of the microscopic density is carried by a fractionalized sector with emergent anyons and gauge fields; the charged quasiparticles then count only the residual density not sequestered in topological order (Bonderson et al., 2016). In heavy-fermion language, the local moments do not form part of the Fermi surface but instead form a spin liquid, while in cuprate-oriented constructions the same logic underlies pseudogap metals with pocket Fermi surfaces and arc-like spectral weight (Thomson et al., 2015, Chowdhury et al., 2014).

1. Definition and distinguishing characteristics

In a conventional interacting Fermi liquid with conserved $U(1)$ charge and translational invariance, Luttinger’s theorem fixes the total Fermi volume $V_F$ by the microscopic filling $\nu$ per unit cell: $\nu = \frac{V_F}{(2\pi)^D} \mod 1.$ Here $V_F$ already includes spin and band degeneracies. An FL* departs from this relation without abandoning metallicity: it contains a conventional Fermi surface of charged quasiparticles with Fermi-liquid-like kinematics, but also a gapped topologically ordered sector with emergent anyons and gauge fields, and the global symmetries act in a fractionalized way on the anyons (Bonderson et al., 2016).

The standard physical picture is that part of the microscopic density is “bound up” in the topologically ordered sector and therefore does not appear as conventional charge-$1$ quasiparticles at the Fermi surface. In heavy-fermion constructions this is the small-versus-large Fermi-surface distinction: in the ordinary heavy Fermi liquid, localized moments are Kondo-screened and counted in the Fermi volume, whereas in the FL* phase the localized moments form a spin liquid and the Fermi surface counts only the conduction electrons (Thomson et al., 2015). In cuprate-oriented formulations, the corresponding statement is that the pocket Fermi surfaces enclose a volume associated with the doped holes $x$, not $1+x$, even though translational symmetry remains unbroken (Moon et al., 2010).

Across the literature, the term FL* consistently refers to a metallic state with electron-like or gauge-neutral charge carriers coexisting with a fractionalized spin sector, but the detailed character of the charged sector varies. Some constructions emphasize sharp charged quasiparticles and an otherwise Fermi-liquid-like metal (Bonderson et al., 2016), whereas others place FL* in a broader family of non-Fermi-liquid metals or use it as the parent state for density-wave or superconducting instabilities (Seifert et al., 2017, Chatterjee et al., 2016).

2. Topological enrichment of Luttinger’s theorem

The most precise formulation of FL* is the topologically enriched Luttinger theorem. For a two-dimensional FL* whose low-energy degrees of freedom decompose into a Fermi-liquid sector and an SET sector, the relation becomes

$$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$

or equivalently

$$V_F = (2\pi)^2 \bigl(\nu - \nu_{\mathrm{topo}}\bigr) \quad (\mathrm{mod}\,(2\pi)^2).$$

The subtraction term $V_F$0 is a topological filling fraction fixed entirely by the SET sector, not by the charged quasiparticles (Bonderson et al., 2016).

In two dimensions, $V_F$1 is identified with the $V_F$2 charge $V_F$3 carried by the background anyon $V_F$4 per unit cell: $V_F$5 The derivation combines Oshikawa’s flux-threading argument with the statement that threading a $V_F$6 $V_F$7 flux is equivalent, in the topological sector, to creating a vison loop around the torus. The microscopic momentum shift from flux insertion is matched to the emergent momentum shift produced by braiding that vison around the background anyonic flux $V_F$8, and the result is the modified count above (Bonderson et al., 2016).

This formulation immediately yields selection rules on admissible topological orders. For a measured filling $V_F$9 and Fermi volume $\nu$0, any compatible FL* phase must satisfy

$\nu$1

Since $\nu$2, only SET orders admitting an Abelian anyon $\nu$3 with that charge are allowed. A concrete example is provided by pure Ising topological order: its Abelian anyons $\nu$4 and $\nu$5 have trivial mutual statistics among themselves, so any Abelian background flux $\nu$6 or $\nu$7 has $\nu$8. Consequently, a two-dimensional Ising SET with $\nu$9 charge conservation and translations is forbidden at non-integer filling (Bonderson et al., 2016).

The same logic extends beyond two dimensions in special cases. For a three-dimensional $\nu = \frac{V_F}{(2\pi)^D} \mod 1.$0 example studied explicitly, the enriched relation takes the form

$\nu = \frac{V_F}{(2\pi)^D} \mod 1.$1

showing that the topological subtraction survives, although a general higher-dimensional classification is not yet available (Bonderson et al., 2016).

3. Symmetry fractionalization and the SET sector

The topological subtraction term is determined by symmetry fractionalization. For a 2D topological order with symmetry group $\nu = \frac{V_F}{(2\pi)^D} \mod 1.$2, the action of a symmetry element $\nu = \frac{V_F}{(2\pi)^D} \mod 1.$3 on a state with anyons $\nu = \frac{V_F}{(2\pi)^D} \mod 1.$4 factorizes into localized operators near each anyon,

$\nu = \frac{V_F}{(2\pi)^D} \mod 1.$5

and these localized operators generally form projective representations: $\nu = \frac{V_F}{(2\pi)^D} \mod 1.$6 Consistency with fusion and braiding implies that the phases are encoded in an Abelian anyon-valued 2-cocycle $\nu = \frac{V_F}{(2\pi)^D} \mod 1.$7, with

$\nu = \frac{V_F}{(2\pi)^D} \mod 1.$8

Thus symmetry fractionalization classes are classified by the cohomology class $\nu = \frac{V_F}{(2\pi)^D} \mod 1.$9 (Bonderson et al., 2016).

For $V_F$0, one has $V_F$1. A convenient representative is labeled by an Abelian anyon $V_F$2, interpreted as the vison created by threading a $V_F$3 $V_F$4 flux. The fractional $V_F$5 charge of an anyon $V_F$6 is then fixed by its mutual statistics with $V_F$7: $V_F$8 For translations, $V_F$9, and the gauge-invariant datum is the background anyon per unit cell,

$1$0

Transporting anyon $1$1 around a unit cell yields the phase $1$2, exactly as if a flux $1$3 sat in every unit cell (Bonderson et al., 2016).

For the combined symmetry $1$4, the Künneth decomposition allows the $1$5 fractionalization class $1$6 and the translational fractionalization class $1$7 to be specified independently. The enriched Luttinger theorem then states that the topological filling fraction is fixed by this symmetry-fractionalization data: $1$8 In this sense, the Fermi volume of an FL* is a probe not only of charge density but also of the symmetry action on the anyon content (Bonderson et al., 2016).

4. Microscopic realizations and model classes

Heavy-fermion and Kondo-lattice settings supplied the earliest systematic arena for FL*. In the conventional heavy Fermi liquid of a Kondo lattice, Oshikawa’s argument yields

$1$9

so the local-moment sector contributes to the large Fermi surface. In an FL* phase where the localized spins form a $x$0 spin liquid, the spin-summed relation becomes

$x$1

so the Fermi surface counts only the conduction electrons; this is the precise version of the “small Fermi surface” associated with Kondo breakdown (Bonderson et al., 2016). A microscopic Kondo–Heisenberg realization with unbroken translational symmetry and gapped spinons was later constructed nonperturbatively, yielding sharp quasiparticles coexisting with a topologically nontrivial spin liquid and ordering temperatures parametrically smaller than the quasiparticle Fermi energy (Tsvelik, 2016).

Several later models make the FL* structure explicit in concrete spin liquids. On the honeycomb lattice, the Kitaev–Kondo lattice realizes FL* phases in which itinerant electrons coexist with a Kitaev $x$2 spin liquid of local moments. There the charged sector forms a small Fermi surface, while the neutral sector consists of fractionalized Majorana excitations and gauge fluxes inherited from the Kitaev spin liquid (Seifert et al., 2017). On the surface of a topological Kondo insulator, suppressed near-surface hybridization can produce a surface FL$x$3, or SFL$x$4, where conduction-electron surface states coexist with a spin-chain or spin-ladder spin liquid of decoupled local moments (Thomson et al., 2015).

Cuprate-oriented constructions place FL* at the center of pseudogap phenomenology. One line of work describes a metal near a quantum fluctuating antiferromagnet in which gauge-neutral fermions form small hole pockets of total area proportional to the doped holes $x$5, while the spin sector remains fractionalized and topologically ordered (Chowdhury et al., 2014). A closely related theory models underdoped cuprates with local antiferromagnetic order fluctuating in orientation but not magnitude, so that no long-range antiferromagnetism survives, a topological order remains, and the resulting FL* has pocket Fermi surfaces enclosing total area $x$6 and supports a nodal–antinodal dichotomy once $x$7-wave pairing is introduced (Moon et al., 2010). In ancilla-layer and ancilla-qubit formulations, the pseudogap metal becomes an FL* with small hole pockets whose quasiparticle weight is large only on “Fermi arcs,” while hidden layers support fermionic spinons carrying the charges of emergent gauge fields (Zhang et al., 2020).

More recent square-lattice constructions sharpen the same picture. A tractable model based on conduction electrons coupled to a Yao–Lee-type $x$8 spin liquid with a Majorana Fermi surface exhibits a small-Fermi-surface phase with analytically derived momentum-dependent coherence factors responsible for Fermi arcs à la Yang–Rice–Zhang (Coleman et al., 7 Apr 2026). Related reviews emphasize that FL* resolves two central cuprate difficulties simultaneously: the existence of small hole pockets compatible with angle-dependent magnetoresistance and coherent interlayer tunneling, and the strongly anisotropic nodal velocities $x$9 in the $1+x$0-wave superconductor (Sachdev, 30 Dec 2025).

5. Instabilities and descendant phases

FL* phases are not terminal endpoints; they commonly sit near density-wave, superconducting, or confinement transitions. In a square-lattice FL* motivated by the cuprates, weak-coupling analysis of particle–hole instabilities shows that the leading density-wave tendency occurs at axial wavevectors $1+x$1 and $1+x$2 with a predominantly $1+x$3-form factor, in contrast to the diagonal ordering vectors favored by a nearly antiferromagnetic metal with a large Fermi surface (Chowdhury et al., 2014). This distinction is one of the principal reasons FL* has been used as a parent state for the charge-ordered pseudogap regime.

Confinement transitions out of FL* often generate superconductivity. In a doped $1+x$4 spin liquid with Ising-nematic order, the FL* phase contains small Fermi pockets of electron-like quasiparticles coexisting with $1+x$5 topological order and visons. Condensation of the bosonic chargon $1+x$6 Higgses the $1+x$7 gauge field, destroys the topological order, and produces a superconducting state that is usually of Fulde–Ferrell–Larkin–Ovchinnikov or pair-density-wave type, often accompanied by bond-density-wave order (Chatterjee et al., 2016). In this framework, superconductivity is the confined descendant of a deconfined metallic FL*.

The Kitaev–Kondo lattice supplies a different descendant route. There the quantum phase transition between FL* and conventional Fermi liquid is masked by triplet superconducting phases whose pairing structure is inherited from the Kitaev spin liquid; the paper characterizes this as superconductivity driven by “Majorana glue” (Seifert et al., 2017). In surface FL$1+x$8 constructions for topological Kondo insulators, the main instability discussed is not superconductivity but surface Kondo breakdown itself, which yields lighter surface quasiparticles than in a fully hybridized topological Kondo insulator and coexists with neutral spin-liquid excitations on the surface (Thomson et al., 2015).

In cuprate-oriented gauge theories, the relation between FL* and $1+x$9-wave superconductivity is especially central. Thermal fluctuations above an FL* or holon metal have been proposed as the appropriate description of the pseudogap and the strange-metal fan, while a confinement crossover at lower temperatures connects these higher-temperature descriptions to $$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$0-wave superconductivity and related ordered states (Sachdev, 27 Jan 2025). In the ancilla-layer formulation reviewed later, doping fermionic partons produces a $$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$1-wave superconductor with nearly isotropic quasiparticle velocities, whereas the FL* construction yields a $$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$2-wave superconductor with realistic nodal anisotropy after confinement (Sachdev, 30 Dec 2025).

6. Diagnostics, related phases, and limitations

The most direct diagnostic of FL* is a mismatch between the microscopic filling and the measured Fermi volume that cannot be attributed to filled bands or symmetry breaking. The enriched theorem makes this quantitative: once $$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$3 is measured from quantum oscillations or ARPES and $$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$4 is fixed from stoichiometry or Hall measurements, the difference

$$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$5

determines $$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$6, and hence constrains the allowed SET orders through the charge of the background anyon $$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$7 (Bonderson et al., 2016). In heavy-fermion realizations with gapless fermionic spinons, thermodynamics provide additional handles: in sufficiently clean samples and an appropriate temperature window, thermal transport can be dominated by spinons, producing a characteristic maximum in the Wiedemann–Franz ratio, while the spin susceptibility is dominated by the spinon contribution and in three dimensions acquires a logarithmic enhancement relative to the Fermi-liquid result (Hackl et al., 2011). In the tractable square-lattice model tied to pseudogap phenomenology, quantum and thermal fluctuations yield a strong diamagnetic response and a logarithmically divergent Sommerfeld coefficient at the onset of the pseudogap (Coleman et al., 7 Apr 2026).

Several related phases clarify the conceptual boundaries of FL*. The orthogonal metal is explicitly described as not equal to the canonical FL*, but as a minimal “baby” version: a $$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$8-fractionalized metal with a Fermi surface of charge-carrying $$\nu - \nu_{\mathrm{topo}} = \frac{V_F}{(2\pi)^2} \mod 1,$$9 fermions, conventional thermodynamics and transport, but a hard gap in the electron spectral function (Nandkishore et al., 2012). Holographic metals near charged AdS black holes were argued to correspond closely to FL* phases of the lattice Anderson model, with a small conduction-electron Fermi surface coexisting with a spin-liquid sector whose infrared correlators resemble AdS$$V_F = (2\pi)^2 \bigl(\nu - \nu_{\mathrm{topo}}\bigr) \quad (\mathrm{mod}\,(2\pi)^2).$$0 physics (Sachdev, 2010). More recently, an alternative “geometric” orthogonal-metal or geometric FL$$V_F = (2\pi)^2 \bigl(\nu - \nu_{\mathrm{topo}}\bigr) \quad (\mathrm{mod}\,(2\pi)^2).$$1 scenario has been proposed in which hidden antiferromagnetic order obscured by fluctuating domain walls replaces a conventional spin-liquid background, although the resulting metal still has a small Fermi surface and a $$V_F = (2\pi)^2 \bigl(\nu - \nu_{\mathrm{topo}}\bigr) \quad (\mathrm{mod}\,(2\pi)^2).$$2 topological sector (Schlömer et al., 2024).

The established FL* results also come with sharp limitations. The enriched Luttinger theorem in its cleanest form assumes translational invariance, global $$V_F = (2\pi)^2 \bigl(\nu - \nu_{\mathrm{topo}}\bigr) \quad (\mathrm{mod}\,(2\pi)^2).$$3 charge conservation, a well-defined Fermi surface in the charged sector, a gapped topological sector, and effective decoupling of the Fermi-liquid and SET sectors in a strong quasi-topological phase (Bonderson et al., 2016). Extensions to non-Fermi-liquid metals without sharp Fermi surfaces, gapless gauge-field spin liquids, strongly entangled “weak quasi-topological” metals, general higher-dimensional topological orders, and non-Abelian topological orders remain subtler or open (Bonderson et al., 2016). For cuprates specifically, current work places FL* within a broader phase diagram that also includes holon metals, SDW metals, disorder-broadened Griffiths regimes, and confinement crossovers, so the precise low-energy fate of a putative FL* may depend sensitively on temperature, disorder, and competing orders (Sachdev, 27 Jan 2025).

FL* therefore occupies a specific and technically constrained position in correlated-electron theory: it is neither an ordinary reconstructed Fermi liquid nor merely a spin liquid plus spectators, but a metallic phase in which the Fermi volume itself becomes an observable of symmetry fractionalization and topological order. In that sense, the concept replaces the traditional identification of Fermi volume with band filling by a more general statement: the Luttinger count of a metal can be topologically enriched, and the “missing” Fermi volume of an FL* is not missing at all, but carried by hidden fractionalized structure (Bonderson et al., 2016).