Orthogonal Fermi Liquid: Fractionalized Metal

• Orthogonal Fermi Liquid is a fractionalized metallic state defined by a separation between coherent f-fermion charge transport and a lack of quasiparticle poles in the electron spectral function.
• Slave-spin and Z2 gauge theory formulations show that disordering the slave spins preserves a gapless Fermi surface while causing a vanishing single-particle overlap.
• Numerical simulations and analytical models confirm that despite missing visible electron quasiparticles, thermodynamic and transport properties mimic those of conventional metals.

Searching arXiv for relevant papers on Orthogonal Fermi Liquid / Orthogonal Metal. Orthogonal Fermi liquid denotes a fractionalized metallic state in which metallic transport and thermodynamics coexist with anomalous one-electron properties. In the formulation introduced through slave Ising spins, the electron operator is written as $c_{i\sigma}=f_{i\sigma}\tau_i^x$, with a local $Z_2$ gauge redundancy under $f_{i\sigma}\to -f_{i\sigma}$ and $\tau_i^x\to -\tau_i^x$ (Nandkishore et al., 2012). In the phase where the slave spins are disordered, the $f$-fermions can still form a gapless Fermi surface and thus sustain compressibility and conductivity, while the overlap with the physical electron vanishes so that single-particle probes cease to exhibit ordinary quasiparticle poles (Nandkishore et al., 2012). Closely related lattice realizations in $Z_2$ gauge theories establish orthogonal metal and deconfined Fermi-liquid regimes, clarify their connection to confinement and Higgs phenomena, and provide sign-problem-free quantum Monte Carlo evidence for metallic phases with either no electron Fermi surface or a large gauge-neutral Fermi surface coexisting with topological order (Chen et al., 2019, Gazit et al., 2019).

1. Concept and defining characteristics

The essential defining feature is a separation between charge transport and electron quasiparticle coherence. In the slave-spin representation, the $f$-fermions carry the full physical charge $e$ and spin-$\tfrac12$ of an electron, while the $\tau$-spins carry no electromagnetic quantum numbers (Nandkishore et al., 2012). Consequently, when the $Z_2$0-spins are paramagnetic, the system remains metallic because the $Z_2$1-fermions retain a gapless Fermi surface, but the one-electron overlap $Z_2$2 vanishes and single-particle probes see a hard gap (Nandkishore et al., 2012).

This construction was introduced as a phase “indistinguishable from the Fermi liquid in conductivity and thermodynamics, but is sharply distinct in one electron properties, such as the electron spectral function” (Nandkishore et al., 2012). The same source emphasizes that disordering the slave spins does not generically produce a Mott insulator; rather, it yields the Orthogonal Metal (Nandkishore et al., 2012). That point directly addresses a recurrent misconception in earlier literature.

In later lattice gauge-theory studies, related distinctions emerge in more than one form. In the “Metal to Orthogonal Metal Transition” model, the orthogonal metal conducts electricity but acquires no Fermi surface or quasiparticles in the physical single-particle spectral function, while still responding to magnetic probes like a Fermi liquid (Chen et al., 2019). In the “Fermi-surface reconstruction without symmetry breaking” model, a deconfined Fermi liquid appears in which a large Luttinger-volume Fermi surface of gauge-neutral $Z_2$3-fermions coexists with $Z_2$4 topological order and fractionalized excitations (Gazit et al., 2019). This usage suggests that “orthogonal Fermi liquid” has been employed in two nearby but not identical senses: one emphasizing vanishing single-particle overlap, the other emphasizing a gauge-deconfined Fermi liquid of gauge-neutral electrons coexisting with topological order.

2. Slave-spin and gauge-theory formulations

A generic starting point is the slave–Ising-spin representation

$Z_2$5

together with the local constraint

$Z_2$6

which projects onto the physical subspace (Nandkishore et al., 2012). Under mean-field factorization of a Hubbard-type model,

$Z_2$7

one obtains coupled effective Hamiltonians for $Z_2$8-fermions and Ising spins, with renormalized hoppings $Z_2$9 and Ising couplings determined by fermion bilinears (Nandkishore et al., 2012). The phase distinction is then direct: $f_{i\sigma}\to -f_{i\sigma}$0 gives an ordinary Fermi liquid, while $f_{i\sigma}\to -f_{i\sigma}$1 gives an orthogonal metal (Nandkishore et al., 2012).

Unbiased lattice realizations reformulate the same structure as fermions and Ising matter coupled to dynamical $f_{i\sigma}\to -f_{i\sigma}$2 gauge fields. A representative Hamiltonian is

$f_{i\sigma}\to -f_{i\sigma}$3

with

$f_{i\sigma}\to -f_{i\sigma}$4

$f_{i\sigma}\to -f_{i\sigma}$5

$f_{i\sigma}\to -f_{i\sigma}$6

and gauge-invariant physical fermion $f_{i\sigma}\to -f_{i\sigma}$7 (Chen et al., 2019). In the simulations quoted there, $f_{i\sigma}\to -f_{i\sigma}$8, $f_{i\sigma}\to -f_{i\sigma}$9, $\tau_i^x\to -\tau_i^x$0, and $\tau_i^x\to -\tau_i^x$1 is the tuning parameter (Chen et al., 2019).

A second gauge-theory construction introduces both gauge-charged orthogonal fermions and gauge-neutral physical electrons: $\tau_i^x\to -\tau_i^x$2 with Ising gauge field $\tau_i^x\to -\tau_i^x$3, Higgs matter $\tau_i^x\to -\tau_i^x$4, spinful orthogonal fermions $\tau_i^x\to -\tau_i^x$5, and physical electron $\tau_i^x\to -\tau_i^x$6 (Gazit et al., 2019). The model is studied in the “odd” gauge sector with Gauss law

$\tau_i^x\to -\tau_i^x$7

on every site $\tau_i^x\to -\tau_i^x$8 (Gazit et al., 2019). A related doped model uses

$\tau_i^x\to -\tau_i^x$9

with Gauss’s law

$f$0

and an additional gauge-neutral hopping term

$f$1

that directly tunes propagation of the composite electron (Chen et al., 2020).

3. Electron Green’s function, spectral function, and the meaning of “orthogonal”

The orthogonality is most clearly expressed in the single-particle Green’s function of the physical, gauge-neutral electron. In the gauge-theory lattice realization,

$f$2

with real-frequency continuation

$f$3

and spectral function

$f$4

(Chen et al., 2019). In a conventional Fermi liquid,

$f$5

so $f$6 has a $f$7-peak at $f$8 with weight $f$9 (Chen et al., 2019). In the orthogonal metal, by contrast, the Monte Carlo data show that $Z_2$0 continuously as $Z_2$1 crosses a critical $Z_2$2, and $Z_2$3 develops a full single-particle pseudogap with no pole or Fermi-surface singularity at $Z_2$4 (Chen et al., 2019).

The same structure already appears in the original slave-spin treatment. Because $Z_2$5, the physical Green’s function is a convolution of spin and fermion correlators: $Z_2$6 and similarly for the spectral function (Nandkishore et al., 2012). On the Fermi-liquid side, $Z_2$7 develops a zero-frequency condensate and $Z_2$8 with residue $Z_2$9, where $f$0 (Nandkishore et al., 2012). On the orthogonal-metal side, the spin spectral function vanishes for $f$1, implying $f$2 for $f$3, even though the $f$4-fermions remain gapless and metallic (Nandkishore et al., 2012).

The same diagnostic is used numerically via the low-temperature proxy

$f$5

or $f$6 (Chen et al., 2019, Gazit et al., 2019, Chen et al., 2020). In the orthogonal semi-metal, this quantity shows four peaks at $f$7 but no continuous Fermi surface; as the electron-like hopping parameter $f$8 is increased through the transition, these peaks broaden and connect into a diamond-shaped large Fermi surface centered at $f$9 (Gazit et al., 2019).

4. Phase structure, deconfinement, and transitions without symmetry breaking

The physical interpretation of orthogonal metallicity is closely tied to confinement and deconfinement of emergent $e$0 gauge fields. In the normal-metal to orthogonal-metal transition, small $e$1 orders the Ising matter $e$2, “Higgses” the $e$3 gauge field, confines the gauge topology, and locks $e$4, producing a conventional Fermi liquid with a sharp Fermi surface that obeys Luttinger’s theorem (Chen et al., 2019). Large $e$5 disorders $e$6, restoring a deconfined $e$7 gauge phase in which the $e$8-fermions remain mobile but are orthogonal to the gauge-neutral $e$9-operators; the $\tfrac12$0-fermions then lose coherence and their Fermi surface disappears from $\tfrac12$1, even though gauge-invariant two-particle and current correlators remain gapless (Chen et al., 2019). The transition occurs at $\tfrac12$2 and is continuous; it is described there as a Higgs-type quantum critical point in an Ising gauge theory (Chen et al., 2019).

The phase diagram of the model studied by Gazit, Assaad and Sachdev contains three zero-temperature phases: an orthogonal semi-metal, a deconfined Fermi liquid, and a confined Fermi liquid (Gazit et al., 2019). The orthogonal semi-metal has deconfined $\tfrac12$3 gauge field, four two-component Dirac cones of $\tfrac12$4-fermions at $\tfrac12$5, $\tfrac12$6 topological order, and gapped physical $\tfrac12$7-fermions (Gazit et al., 2019). The deconfined Fermi liquid retains deconfined gauge structure and gapped visons, but $\tfrac12$8 and $\tfrac12$9 bind into $\tau$0, and the $\tau$1-fermions form a large Luttinger-volume Fermi surface of volume $\tau$2 per spin (Gazit et al., 2019). The confined Fermi liquid has vison condensation and gauge confinement, yet the same large gauge-neutral Fermi surface survives (Gazit et al., 2019). The transition from orthogonal semi-metal to deconfined Fermi liquid occurs by condensation of the bound state $\tau$3 without confining the gauge field, while the transition from deconfined to confined Fermi liquid is a confinement transition without Fermi-surface change (Gazit et al., 2019).

The doped model extends this structure away from half filling. At fixed doping, the $\tau$4 phase diagram in the $\tau$5–$\tau$6 plane contains a Fermi-arc/pseudogap phase for $\tau$7 and $\tau$8, a deconfined Fermi liquid for $\tau$9 and $Z_2$00, and a confined Fermi liquid for $Z_2$01 (Chen et al., 2020). The deconfined region is characterized by $Z_2$02 topological order and either broken visible Fermi surface arcs or a full large Fermi surface, depending on $Z_2$03 (Chen et al., 2020). This establishes explicitly that Fermi-surface reconstruction can occur without symmetry breaking and can be governed purely by gauge dynamics (Gazit et al., 2019).

5. Luttinger count, hidden Fermi surfaces, and two-particle probes

A central issue is the relation between visible electron spectral weight and Luttinger counting. In the original orthogonal metal proposal, the physical electron spectral function is gapped while the $Z_2$04-fermions remain gapless and form the actual metallic subsystem (Nandkishore et al., 2012). In the “Metal to Orthogonal Metal Transition” study, the orthogonal metal has no detectable quasiparticle Fermi surface in $Z_2$05, yet remains compressible and conducts both charge and spin; this violates naïve Luttinger counting unless one includes the volume of a “hidden” $Z_2$06-fermion Fermi surface in the deconfined $Z_2$07 sector (Chen et al., 2019).

The hidden Fermi surface is revealed by two-particle observables. In that model, the magnetic susceptibility

$Z_2$08

shows no sharp change at the nesting vector $Z_2$09 across the normal-metal to orthogonal-metal transition (Chen et al., 2019). The orthogonal metal retains the same magnetic instability, with peak at $Z_2$10, as the normal metal, demonstrating that magnetic and other two-particle probes still see the $Z_2$11-fermion Fermi surface (Chen et al., 2019).

The same logic underlies the doped Fermi-arc phase. There, the visible $Z_2$12-fermion Fermi surface consists only of arcs and encloses a much smaller volume than the filling $Z_2$13, which appears to violate Luttinger’s theorem (Chen et al., 2020). However, the hidden $Z_2$14-fermion Fermi surface has pockets whose total area matches the density of $Z_2$15-fermions, equivalent to the $Z_2$16 filling, and produces magnetic response ring patterns near $Z_2$17 identical to those of doped Dirac-cone fermions (Chen et al., 2020). This provides a concrete mechanism for apparent Luttinger violation in the physical electron sector without symmetry breaking.

By contrast, in the deconfined and confined Fermi liquids of the Gazit–Assaad–Sachdev model, the gauge-neutral $Z_2$18-fermions are themselves gapless on a large Fermi surface enclosing half the Brillouin zone per spin,

$Z_2$19

which satisfies the Luttinger theorem (Gazit et al., 2019). The novelty there is not a hidden Fermi surface replacing the physical one, but the coexistence of a conventional large electron-like Fermi surface with deconfined $Z_2$20 topological order (Gazit et al., 2019).

6. Transport, thermodynamics, and response functions

One of the most distinctive aspects of the orthogonal metal is that thermodynamic and transport observables need not track the dramatic changes seen in one-electron spectroscopy. In the original construction, gauging the physical $Z_2$21 charge couples only to the $Z_2$22-fermion hoppings, so the lattice current operator is built from $Z_2$23-fermions, and if they form a Fermi surface the zero-temperature d.c. conductivity is nonzero (Nandkishore et al., 2012). Similarly, the low-temperature specific heat $Z_2$24 and the compressibility are identical to those of a Fermi liquid because all such responses are carried by the $Z_2$25-fermions (Nandkishore et al., 2012).

The unbiased lattice simulations confirm this logic. In the orthogonal metal phase of the 2019 square-lattice model, the optical conductivity

$Z_2$26

has finite d.c. conductivity $Z_2$27 despite the absence of a single-particle pole (Chen et al., 2019). The density $Z_2$28 versus $Z_2$29 is smooth in the orthogonal metal at $Z_2$30, demonstrating compressibility and hence metallic response despite the missing quasiparticle pole (Chen et al., 2019).

Thermodynamic and transport signatures have also been analyzed in the broader fractionalized Fermi-liquid setting relevant to heavy-fermion systems. In that framework, below a crossover $Z_2$31, the hybridization boson is gapped and the low-energy theory contains gauge-neutral conduction electrons and neutral spinons coupled to an emergent $Z_2$32 gauge field (Hackl et al., 2011). The state is described there as an orthogonal or FL* Fermi liquid with a “small” charged Fermi surface of conduction electrons and a “ghost” neutral Fermi surface of spinons (Hackl et al., 2011). The predicted signatures include spinon-dominated thermal conductivity in an intermediate temperature regime, a pronounced maximum in the Wiedemann–Franz ratio, and a spin susceptibility in which the spinon contribution dominates the conduction-electron Pauli term (Hackl et al., 2011). In three dimensions, the fluctuation correction to the spinon susceptibility has a $Z_2$33 form, while in two dimensions it scales as $Z_2$34 (Hackl et al., 2011). These results pertain to FL* rather than the slave-spin orthogonal metal in the narrow sense, but they reinforce the general theme that fractionalized Fermi liquids can resemble ordinary metals in charge transport while retaining anomalous neutral-sector thermodynamics (Hackl et al., 2011).

7. Numerical realizations, wavefunctions, and open issues

A major advance after the original proposal was the construction of sign-problem-free lattice models with unbiased quantum Monte Carlo access. In the normal-metal to orthogonal-metal study, the simulations use an imaginary-time path integral with $Z_2$35, $Z_2$36, and $Z_2$37 up to 24 (Chen et al., 2019). Tracing over $Z_2$38 and $Z_2$39 bases yields bosonic Boltzmann weights times $Z_2$40, which is manifestly nonnegative, allowing local updates with fast Sherman–Morrison determinant updates (Chen et al., 2019). Measured observables include $Z_2$41, $Z_2$42, energy derivatives, and string correlators

$Z_2$43

to detect Higgs ordering (Chen et al., 2019).

Representative results from that work show a sharp diamond Fermi surface in the normal metal at $Z_2$44, loss of spectral weight near the quantum critical point $Z_2$45, and complete disappearance of the single-particle Fermi surface in the orthogonal metal at $Z_2$46 (Chen et al., 2019). Simultaneously, $Z_2$47 retains its peak at $Z_2$48, the string correlator changes from long-range order to exponential decay, and the quasiparticle weight $Z_2$49 vanishes continuously at the critical point (Chen et al., 2019). The authors state that no simple exponent $Z_2$50 or $Z_2$51 has yet been extracted from these data and that establishing the universality class remains an open problem (Chen et al., 2019).

The original work also proposed explicit wavefunction constructions. A prototypical Fermi-liquid wavefunction is of Slater–Jastrow form,

$Z_2$52

with bosonic Jastrow factor $Z_2$53 approaching a constant at long distance (Nandkishore et al., 2012). The orthogonal-metal wavefunction is obtained by replacing the Jastrow factor with a boson-pairing wavefunction

$Z_2$54

where $Z_2$55 at large distance, leading to

$Z_2$56

(Nandkishore et al., 2012). This construction encodes strong electron-pair correlations without single-particle coherence.

Exactly soluble models sharpen the conceptual picture further. In one family, a nonlocal mapping decouples electrons coupled to Ising gauge fields into free $Z_2$57-fermions plus a transverse-field Ising model, so that the paramagnet-to-ferromagnet transition of the Ising sector is exactly the orthogonal-metal to Fermi-liquid transition (Nandkishore et al., 2012). The same paper argues that the transition can, in some circumstances, provide a continuous destruction of a Fermi surface with a critical Fermi surface appearing at the critical point (Nandkishore et al., 2012). At mean field, the slave-spin critical point has $Z_2$58, and the physical spectral function retains a sharply defined critical Fermi surface despite the absence of quasiparticles (Nandkishore et al., 2012). Beyond mean field, the coupling of slave-spin energy fluctuations to fermion density can be irrelevant in the continuum model with long-range Coulomb interactions, but on a lattice in two dimensions it is described as weakly relevant, suggesting either a weak first-order transition or flow to a new universality class except in fine-tuned soluble models (Nandkishore et al., 2012).

Taken together, these developments establish orthogonal Fermi liquid and orthogonal metal physics as a concrete route beyond Landau’s paradigm. Depending on the realization, the resulting phase may exhibit a full gap or pseudogap in the electron spectral function, a hidden $Z_2$59-fermion Fermi surface visible only to two-particle probes, or a large gauge-neutral Fermi surface coexisting with deconfined $Z_2$60 topological order. The common thread is that metallicity survives fractionalization, while the relation between electron quasiparticles, Fermi-surface volume, and gauge structure is fundamentally reorganized (Nandkishore et al., 2012, Chen et al., 2019, Gazit et al., 2019, Chen et al., 2020).