Orthogonal Metal: Fractionalized Metallic Phase

• Orthogonal metals are compressible phases where emergent fermions form a Fermi surface that carries charge and spin, while physical electrons are gapped.
• They employ a slave-spin formulation with a Z₂ gauge structure, decoupling the microscopic electron from the low-energy excitations.
• Their spectral, thermodynamic, and transport properties mimic Fermi liquids, showing metallic conductivity alongside a suppressed quasiparticle pole in ARPES.

Orthogonal metal denotes a fractionalized metallic phase in which the low-energy charge carriers are not adiabatically connected to the microscopic electrons. In the canonical construction, the physical electron is expressed as a product of a fermionic parton and an Ising slave spin, so that charge and spin are carried by emergent fermions while the physical single-electron excitation is gapped or strongly suppressed. The resulting state is compressible and conducting, with thermodynamic and transport properties close to those of an ordinary metal or semimetal, yet it lacks an electron quasiparticle pole and can exhibit a vanishing electron quasiparticle residue $Z$ (Nandkishore et al., 2012, Zhong et al., 2013).

1. Definition and distinguishing characteristics

An orthogonal metal is a compressible metallic phase with thermodynamic and transport properties very similar to an ordinary Landau Fermi liquid, but with gapped single-particle excitations of the physical electron and no conventional electron Fermi surface. In the original formulation, the phase is metallic because emergent fermions form a Fermi surface and carry the physical $U(1)$ charge and spin, whereas the electron operator has zero overlap with these low-energy carriers; in this sense the low-energy quasiparticles are “orthogonal” to the microscopic electrons (Nandkishore et al., 2012).

This definition separates orthogonal metals from both Fermi liquids and Mott insulators. A Fermi liquid has a pole in the electron Green’s function and a nonzero quasiparticle residue $Z>0$. A Mott insulator has a charge gap and insulating transport. By contrast, an orthogonal metal retains metallic conductivity, compressibility, and Fermi-surface-controlled thermodynamics, but the electron spectral function lacks a low-energy pole and may show a hard gap or strong pseudogap structure (Nandkishore et al., 2012, Hohenadler et al., 2019).

The phase has also been described as one of the “simplest non-Fermi liquids” because it preserves much of the phenomenology usually associated with a Fermi surface while sharply violating the Landau identification between electron quasiparticles and the actual low-energy charge carriers (Zhong et al., 2014).

2. Slave-spin formulation and $Z_2$ gauge structure

The standard microscopic representation writes the electron operator as

$c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$

where $f_{i\sigma}$ is a fermionic parton and $\tau_i^x$ is a Pauli operator acting on an auxiliary Ising slave spin. In the single-band Hubbard setting, the physical Hilbert space is enforced by a local constraint; for the honeycomb-lattice formulation one convenient form is

$\tau_i^z + 1 = 2(n_i - 1)^2.$

This decomposition has a local $Z_2$ redundancy,

$$f_{i\sigma}^{(\dagger)} \to \epsilon_i f_{i\sigma}^{(\dagger)}, \qquad \tau_i^x \to \epsilon_i \tau_i^x, \qquad \epsilon_i=\pm1,$$

under which the physical electron remains invariant (Zhong et al., 2013).

The crucial point is that the slave spin is electrically neutral. Under a global electromagnetic $U(1)$0, the charged transformation acts on $U(1)$1, not on $U(1)$2, because $U(1)$3 is a real operator. Hence the $U(1)$4-fermion carries the physical charge and spin, while the slave spin carries only $U(1)$5 gauge charge. Mean-field decoupling then yields a Fermi-liquid-like $U(1)$6 sector and a transverse-field Ising model for the slave spins. In the ordered slave-spin phase, $U(1)$7, the electron overlaps with $U(1)$8 and the system is an ordinary metal with

$U(1)$9

In the disordered slave-spin phase, $Z>0$0, the slave-spin sector is gapped, the electron quasiparticle residue vanishes, and the system enters the orthogonal-metal regime (Nandkishore et al., 2012).

At the level of wavefunctions, the original construction also provided explicit orthogonal-metal trial states by replacing the bosonic factor of a conventional Slater–Jastrow metal with a paired-boson superfluid factor, giving

$Z>0$1

which is orthogonal to the conventional Slater–Jastrow Fermi-liquid state in the thermodynamic limit (Nandkishore et al., 2012).

3. Spectral, thermodynamic, and transport properties

The defining observable of an orthogonal metal is the separation between single-particle spectroscopy and collective metallic response. In mean field, the electron Green’s function factorizes into slave-spin and parton correlators, so that the electron spectral function is a convolution of a gapped slave-spin spectrum with a gapless fermionic spectrum. In the ordered phase, the slave-spin correlator contributes a zero-momentum condensate and one obtains a conventional quasiparticle pole. In the disordered phase, the slave-spin spectral function vanishes below a gap $Z>0$2, and therefore the electron spectral function also vanishes below threshold even though the $Z>0$3-fermions remain metallic (Nandkishore et al., 2012).

For generic orthogonal metals with a Fermi surface, Zhong and Luo derived the electron spectral function at the Fermi momentum,

$Z>0$4

so $Z>0$5 for $Z>0$6 and spectral weight turns on continuously only above the slave-spin gap. This makes the underlying Fermi surface of the emergent fermions invisible in the electron spectral function (Zhong et al., 2013).

Thermodynamics and transport are nevertheless governed by the $Z>0$7-fermions. The current operator couples to the charged partons, so conductivity and compressibility remain metallic. In the original proposal, observables such as specific heat, compressibility, dc conductivity, and quantum oscillations were expected to look essentially Fermi-liquid-like because they are controlled by the $Z>0$8-Fermi surface rather than by electron quasiparticles (Nandkishore et al., 2012).

Sign-problem-free quantum Monte Carlo studies supplied explicit non-mean-field realizations of this phenomenology. In the two-dimensional SU(2) Falicov–Kimball model, a fractionalized metal was found with a single-particle gap but gapless spin and charge excitations and a nonsaturating resistivity; the phase is exactly dual to an unconstrained slave-spin theory and was classified as a fractionalized or orthogonal metal (Hohenadler et al., 2018). In the Hubbard model with liberated slave spins, the strong-coupling regime likewise exhibited a single-particle gap in $Z>0$9, gapless spin and charge susceptibilities, and a finite low-frequency optical conductivity with no superfluid stiffness, matching the gauge-invariant signature set expected of an orthogonal metal (Hohenadler et al., 2019).

4. Lattice realizations and major variants

Several distinct microscopic settings realize or generalize the orthogonal-metal idea.

Setting	Framework	Principal outcome
Honeycomb Hubbard model	$Z_2$0 slave-spin mean field	Orthogonal Dirac semimetal
SU(2) Falicov–Kimball model	Exact duality + sign-free QMC	Fractionalized metal with single-particle gap
2D $Z_2$1 gauge lattice model	Fermionic and bosonic matter + dynamic gauge field + QMC	Metal-to-orthogonal-metal transition
Doped $Z_2$2 orthogonal metal	Gauge field + composite hopping + QMC	Fermi arcs, pseudogap, confinement-induced superconductivity
Extended Anderson lattice	Slave-spin theory	Orbital-selective orthogonal metal

On the honeycomb lattice at half filling, the slave-fermion sector retains a Dirac spectrum while the slave-spin sector can disorder. The resulting phase, termed the orthogonal Dirac semimetal, has Dirac-semimetal-like thermodynamics and transport but a gapped electron spectral function; ARPES would therefore see a gap at the Dirac points even though charge and spin are transported by gapless Dirac fermions (Zhong et al., 2013).

In the SU(2) Falicov–Kimball model, an exact mapping to an unconstrained slave-spin Hamiltonian,

$Z_2$3

allowed unbiased QMC access to a metallic phase with a true electron single-particle gap, gapless spin and charge excitations, and metallic transport. In that model, the ordered slave-spin regime corresponds to a Fermi liquid, while the disordered slave-spin regime realizes the orthogonal metal (Hohenadler et al., 2018).

A distinct sign-free QMC construction with spinful $Z_2$4-fermions, Ising matter $Z_2$5, and dynamic $Z_2$6 gauge fields on a square lattice showed a continuous transition from a conventional metal with an electron Fermi surface to an orthogonal metal with no electron Fermi surface and no electron quasiparticles, while the magnetic response continued to display Fermi-surface nesting structure. In that setting the orthogonal metal coincides with the deconfined side of a $Z_2$7 Higgs transition (Chen et al., 2019).

At generic filling, a doped orthogonal-metal lattice model produced Fermi arcs and pseudogap in the single-particle spectrum without symmetry breaking. The deconfined $Z_2$8 phase supported a hidden $Z_2$9-fermion Fermi surface, while the gauge-neutral electron spectrum displayed broken Fermi surfaces. The same study found that confinement of the gauge field triggered an $c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$0-wave superconducting instability and that increasing the hopping of gauge-neutral fermions restored a large electron Fermi surface (Chen et al., 2020).

In heavy-fermion language, the concept extends to an orbital-selective orthogonal metal in the extended Anderson lattice. There the correlated orbital fractionalizes as $c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$1, and the disordered slave-spin phase is metallic in the $c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$2 sector but has a gapped $c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$3-electron spectral function. This state was proposed as an alternative Kondo-breakdown mechanism distinct from the conventional fractionalized-Fermi-liquid scenario (Zhong et al., 2012).

5. Phase transitions, criticality, and dimensional constraints

The simplest orthogonal-metal transition is the ordering transition of the slave spins. In the original slave-spin description, the Fermi liquid corresponds to the ordered phase $c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$4, whereas the orthogonal metal corresponds to the disordered phase $c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$5. Nandkishore, Metlitski, and Senthil emphasized that this transition can provide a continuous destruction of a Fermi surface with a critical Fermi surface at the transition, where the electron quasiparticle pole is replaced by power-law spectral weight rather than disappearing abruptly (Nandkishore et al., 2012).

On the honeycomb lattice, the corresponding transition between a conventional Dirac semimetal and an orthogonal Dirac semimetal is controlled at mean field by the slave-spin ordering transition, i.e. a $c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$6-dimensional quantum Ising critical point. On the ordered side, the electron Green’s function is

$c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$7

whereas on the disordered side the slave-spin sector acquires a gap and the electron spectral function becomes gapped although the $c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$8-Dirac fermions remain gapless (Zhong et al., 2013).

In the extended Anderson lattice, however, the critical theory is more involved because the slave-spin order parameter couples directly to conduction electrons through a hybridization term $c_{i\sigma} = f_{i\sigma}\,\tau_i^x,$9. Zhong and coauthors argued that this coupling is relevant in $f_{i\sigma}$0 and marginal in $f_{i\sigma}$1, producing multiscale quantum criticality. Most of the critical regime is dominated by a $f_{i\sigma}$2 Landau-damped mode, while below a small mismatch scale $f_{i\sigma}$3 the theory crosses to a $f_{i\sigma}$4 regime. This makes the orbital-selective orthogonal-metal transition an alternative Kondo-breakdown critical point rather than a naive decoupled Ising transition (Zhong et al., 2012).

Strictly one-dimensional realizations appear to be more constrained. Bosonization of the one-dimensional Penson–Kolb–Hubbard model with pair hopping found no featureless orthogonal metal: when single-particle excitations are gapped, the phase is a pair-density wave with clear order, and when exotic pair-superconducting correlations dominate, single-particle excitations remain gapless. This led to the conclusion that one should go at least to quasi-one-dimensional multi-leg ladders to obtain the desired orthogonal metallic phase (Zhong et al., 2014).

A later non-equilibrium extension combined orthogonal-metal fractionalization with kinetic constraints in one dimension and constructed orthogonal quantum many-body scars. In that model the physical correlator factorized as

$f_{i\sigma}$5

allowing persistent oscillations with infinite lifetime to coexist with rapid volume-law entanglement growth (Zhao et al., 2021).

6. Experimental relevance, common misconceptions, and terminological ambiguity

The most direct experimental signature of an orthogonal metal is the mismatch between single-particle probes and collective metallic response. ARPES or tunneling should see a hard gap, a sharp suppression, or in doped settings a pseudogap and Fermi arcs in the electron spectral function. At the same time, transport, compressibility, and magnetic response should remain metallic because they are controlled by the hidden parton Fermi surface. In the orthogonal Dirac semimetal, this implies Dirac-like thermodynamics and transport coexisting with a gapped electron spectrum at the Dirac points (Zhong et al., 2013). In the 2D gauge-lattice realization, magnetic susceptibility retained the same Fermi-surface-nesting structure across the metal-to-orthogonal-metal transition even after the electron Fermi surface disappeared (Chen et al., 2019). In the doped orthogonal metal, the visible electron arcs coexisted with a background deconfined $f_{i\sigma}$6 gauge field and a hidden Fermi surface inferred from spin response (Chen et al., 2020).

A recurrent misconception in earlier slave-spin literature was to identify the disordered slave-spin phase with a Mott insulator. The orthogonal-metal construction corrected this point: disordering the slave spins does not by itself gap the charged $f_{i\sigma}$7-fermions, because the partons, not the slave spins, carry the physical charge and spin. A true Mott insulator requires the charged fermions themselves to localize or gap out; a disordered slave-spin phase with a metallic $f_{i\sigma}$8 sector is instead an orthogonal metal (Nandkishore et al., 2012).

A second conceptual pitfall is to equate any metal with suppressed single-particle weight to an orthogonal metal. The one-dimensional pair-hopping study showed that a gapped electron spectrum can arise from more conventional Luther–Emery or ordered phases, such as a pair-density wave, without producing a featureless orthogonal metal. The distinguishing ingredient is not merely a pseudogap or gap in $f_{i\sigma}$9, but fractionalization together with a deconfined gauge structure and metallic collective response (Zhong et al., 2014).

The term also has an unrelated usage in the Anderson-localization literature. There, “orthogonal metal” can denote a diffusive metallic phase in symmetry classes with time-reversal symmetry $\tau_i^x$0, including AI and its particle-hole or chiral extensions BDI and CI. In that context the term refers to Wigner–Dyson or Altland–Zirnbauer symmetry classification, not to slave-spin fractionalization. Three-dimensional studies reported distinct Anderson-transition exponents, including $\tau_i^x$1 in class CI and $\tau_i^x$2 in class BDI (Luo et al., 2019). This terminological overlap is substantive but conceptually separate from the many-body orthogonal metal discussed above.