--- title: 'Orthogonal Fermi Liquid: Fractionalized Metal' url: https://www.emergentmind.com/topics/orthogonal-fermi-liquid type: topic --- # Orthogonal Fermi Liquid: Fractionalized Metal 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 \(\tau_i^x\to -\tau_i^x\) [1201.5998]. 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 [1201.5998]. 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 [1904.12872] [1906.11250]. ## 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 [1201.5998]. Consequently, when the \(\tau\)-spins are paramagnetic, the system remains metallic because the \(f\)-fermions retain a gapless Fermi surface, but the one-electron overlap \(Z=\langle\tau^x\rangle^2\) vanishes and single-particle probes see a hard gap [1201.5998]. 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” [1201.5998]. The same source emphasizes that disordering the slave spins does not generically produce a Mott insulator; rather, it yields the Orthogonal Metal [1201.5998]. 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 [1904.12872]. 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 \(c\)-fermions coexists with \(Z_2\) topological order and fractionalized excitations [1906.11250]. 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 \[ c_{i\sigma}=f_{i\sigma}\tau_i^x, \] together with the local constraint \[ \tau_i^z=-\bigl(1-2f^\dagger_{i\uparrow}f_{i\uparrow}\bigr)\bigl(1-2f^\dagger_{i\downarrow}f_{i\downarrow}\bigr), \] which projects onto the physical subspace [1201.5998]. Under mean-field factorization of a Hubbard-type model, \[ H=-\sum_{ij,\sigma} t_{ij} c^\dagger_{i\sigma}c_{j\sigma} + \sum_{ij} V_{ij} n_i n_j - \mu\sum_i n_i, \] one obtains coupled effective Hamiltonians for \(f\)-fermions and Ising spins, with renormalized hoppings \(t'_{ij}=\langle\tau_i^x\tau_j^x\rangle t_{ij}\) and Ising couplings determined by fermion bilinears [1201.5998]. The phase distinction is then direct: \(\langle\tau^x\rangle\neq 0\) gives an ordinary Fermi liquid, while \(\langle\tau^x\rangle=0\) gives an orthogonal metal [1201.5998]. Unbiased lattice realizations reformulate the same structure as fermions and Ising matter coupled to dynamical \(Z_2\) gauge fields. A representative Hamiltonian is \[ H=H_f+H_z+H_g, \] with \[ H_f=-t\sum_{\langle i,j\rangle,\alpha}\bigl(f^\dagger_{i,\alpha}\sigma^z_{\langle i,j\rangle}f_{j,\alpha}+h.c.\bigr)-\mu\sum_{i,\alpha} f^\dagger_{i,\alpha}f_{i,\alpha}, \] \[ H_z=-J\sum_{\langle i,j\rangle} S_i^z \sigma^z_{\langle i,j\rangle} S_j^z-h\sum_i S_i^x, \] \[ H_g=-K\sum_{\square}\prod_{b\in \square}\sigma_b^z-g\sum_b \sigma_b^x, \] and gauge-invariant physical fermion \(c_{i,\alpha}\equiv f_{i,\alpha}S_i^z\) [1904.12872]. In the simulations quoted there, \(t=J=K=1\), \(\mu=0\), \(g=0.5\), and \(h\) is the tuning parameter [1904.12872]. A second gauge-theory construction introduces both gauge-charged orthogonal fermions and gauge-neutral physical electrons: \[ H=H_Z+H_\tau+H_f+H_c, \] with Ising gauge field \(\sigma\), Higgs matter \(\tau\), spinful orthogonal fermions \(f\), and physical electron \(c_{r,\alpha}=\tau_r^z f_{r,\alpha}\) [1906.11250]. The model is studied in the “odd” gauge sector with Gauss law \[ G_r\equiv (-1)^{n_r^f}\tau_r^x \prod_{b\in +_r}\sigma_b^x=-1 \] on every site \(r\) [1906.11250]. A related doped model uses \[ c_{i,\alpha}\equiv f_{i,\alpha}S_i^z, \] with Gauss’s law \[ G_i\equiv (-1)^{n_i^f} S_i^x \prod_{b\in +\,i}\sigma_b^x = +1, \] and an additional gauge-neutral hopping term \[ H_c=-t\sum_{\langle i,j\rangle}\bigl[f^\dagger_{i,\alpha}S_i^z f_{j,\alpha}S_j^z+h.c.\bigr] \] that directly tunes propagation of the composite electron [2007.05543]. ## 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, \[ G(k,\tau)\equiv \langle T_\tau c_k(\tau)c_k^\dagger(0)\rangle, \] with real-frequency continuation \[ G(k,\omega)=\int_0^\beta d\tau\, e^{i\omega\tau} G(k,\tau)\big|_{i\omega\to \omega+i0^+}, \] and spectral function \[ A(k,\omega)\equiv -\frac{1}{\pi}\,\mathrm{Im}\, G(k,\omega+i0^+) \] [1904.12872]. In a conventional Fermi liquid, \[ G(k,\omega)\simeq \frac{Z_{k_F}}{\omega-\epsilon_k^*+i0^+}+\mathrm{regular}, \] so \(A(k_F,\omega)\) has a \(\delta\)-peak at \(\omega=0\) with weight \(Z_{k_F}>0\) [1904.12872]. In the orthogonal metal, by contrast, the Monte Carlo data show that \(Z_{k_F}\to 0\) continuously as \(h\) crosses a critical \(h_c\), and \(A(k,\omega)\) develops a full single-particle pseudogap with no pole or Fermi-surface singularity at \(\omega=0\) [1904.12872]. The same structure already appears in the original slave-spin treatment. Because \(c_i(\tau)=\tau_i^x(\tau) f_i(\tau)\), the physical Green’s function is a convolution of spin and fermion correlators: \[ G_c(q,i\omega)=\int\frac{d^d k}{(2\pi)^d}\int_{-\infty}^{+\infty}\frac{d\Omega}{2\pi}\, G_{\rm spin}(q-k,i\omega-i\Omega)\,G_f(k,i\Omega), \] and similarly for the spectral function [1201.5998]. On the Fermi-liquid side, \(G_{\rm spin}\) develops a zero-frequency condensate and \(A_c(q,\omega)=m^2\delta(\omega-E_q)\) with residue \(Z=m^2\), where \(m=\langle\tau^x\rangle\) [1201.5998]. On the orthogonal-metal side, the spin spectral function vanishes for \(|\omega|<\Delta\), implying \(A_c(q,\omega)=0\) for \(|\omega|<\Delta\), even though the \(f\)-fermions remain gapless and metallic [1201.5998]. The same diagnostic is used numerically via the low-temperature proxy \[ A(k,\omega=0)\simeq \beta\, G(k,\beta/2) \] or \(\tilde Z(k)\simeq \beta G(k,\beta/2)\) [1904.12872] [1906.11250] [2007.05543]. In the orthogonal semi-metal, this quantity shows four peaks at \(k=(\pm\pi/2,\pm\pi/2)\) but no continuous Fermi surface; as the electron-like hopping parameter \(t\) is increased through the transition, these peaks broaden and connect into a diamond-shaped large Fermi surface centered at \((0,0)\) [1906.11250]. ## 4. Phase structure, deconfinement, and transitions without symmetry breaking The physical interpretation of orthogonal metallicity is closely tied to confinement and deconfinement of emergent \(Z_2\) gauge fields. In the normal-metal to orthogonal-metal transition, small \(h\) orders the Ising matter \(S_i^z\), “Higgses” the \(Z_2\) gauge field, confines the gauge topology, and locks \(c\simeq f\), producing a conventional Fermi liquid with a sharp Fermi surface that obeys Luttinger’s theorem [1904.12872]. Large \(h\) disorders \(S^z\), restoring a deconfined \(Z_2\) gauge phase in which the \(f\)-fermions remain mobile but are orthogonal to the gauge-neutral \(c\)-operators; the \(c\)-fermions then lose coherence and their Fermi surface disappears from \(A(k,\omega)\), even though gauge-invariant two-particle and current correlators remain gapless [1904.12872]. The transition occurs at \(h_c\simeq 2.2(2)\) and is continuous; it is described there as a Higgs-type quantum critical point in an Ising gauge theory [1904.12872]. 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 [1906.11250]. The orthogonal semi-metal has deconfined \(Z_2\) gauge field, four two-component Dirac cones of \(f\)-fermions at \(k=(\pm\pi/2,\pm\pi/2)\), \(Z_2\) topological order, and gapped physical \(c\)-fermions [1906.11250]. The deconfined Fermi liquid retains deconfined gauge structure and gapped visons, but \(f\) and \(\tau\) bind into \(c=\tau f\), and the \(c\)-fermions form a large Luttinger-volume Fermi surface of volume \( \tfrac12\,{\rm BZ}\) per spin [1906.11250]. The confined Fermi liquid has vison condensation and gauge confinement, yet the same large gauge-neutral Fermi surface survives [1906.11250]. The transition from orthogonal semi-metal to deconfined Fermi liquid occurs by condensation of the bound state \(c\) without confining the gauge field, while the transition from deconfined to confined Fermi liquid is a confinement transition without Fermi-surface change [1906.11250]. The doped model extends this structure away from half filling. At fixed doping, the \(T\to 0\) phase diagram in the \(t\)–\(g\) plane contains a Fermi-arc/pseudogap phase for \(gt_c\), and a confined Fermi liquid for \(g>g_c\) [2007.05543]. The deconfined region is characterized by \(Z_2\) topological order and either broken visible Fermi surface arcs or a full large Fermi surface, depending on \(t\) [2007.05543]. This establishes explicitly that Fermi-surface reconstruction can occur without symmetry breaking and can be governed purely by gauge dynamics [1906.11250]. ## 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 \(f\)-fermions remain gapless and form the actual metallic subsystem [1201.5998]. In the “Metal to Orthogonal Metal Transition” study, the orthogonal metal has no detectable quasiparticle Fermi surface in \(A(k,\omega)\), yet remains compressible and conducts both charge and spin; this violates naïve Luttinger counting unless one includes the volume of a “hidden” \(f\)-fermion Fermi surface in the deconfined \(Z_2\) sector [1904.12872]. The hidden Fermi surface is revealed by two-particle observables. In that model, the magnetic susceptibility \[ \chi(q)=\int_0^\beta d\tau\, \langle M_z(q,\tau)M_z(-q,0)\rangle, \quad M_z(q)=\sum_i e^{iq\cdot r_i}(n_{i\uparrow}-n_{i\downarrow}), \] shows no sharp change at the nesting vector \((\pi,\pi)\) across the normal-metal to orthogonal-metal transition [1904.12872]. The orthogonal metal retains the same magnetic instability, with peak at \((\pi,\pi)\), as the normal metal, demonstrating that magnetic and other two-particle probes still see the \(f\)-fermion Fermi surface [1904.12872]. The same logic underlies the doped Fermi-arc phase. There, the visible \(c\)-fermion Fermi surface consists only of arcs and encloses a much smaller volume than the filling \(n\approx 1.1\), which appears to violate Luttinger’s theorem [2007.05543]. However, the hidden \(f\)-fermion Fermi surface has pockets whose total area matches the density of \(f\)-fermions, equivalent to the \(c\) filling, and produces magnetic response ring patterns near \((\pi,\pi)\) identical to those of doped Dirac-cone fermions [2007.05543]. 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 \(c\)-fermions are themselves gapless on a large Fermi surface enclosing half the Brillouin zone per spin, \[ A_{\rm FS}=\frac12 (2\pi)^2, \] which satisfies the Luttinger theorem [1906.11250]. 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 \(Z_2\) topological order [1906.11250]. ## 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 \(U(1)\) charge couples only to the \(f\)-fermion hoppings, so the lattice current operator is built from \(f\)-fermions, and if they form a Fermi surface the zero-temperature d.c. conductivity is nonzero [1201.5998]. Similarly, the low-temperature specific heat \(C_v\approx \gamma T\) and the compressibility are identical to those of a Fermi liquid because all such responses are carried by the \(f\)-fermions [1201.5998]. The unbiased lattice simulations confirm this logic. In the orthogonal metal phase of the 2019 square-lattice model, the optical conductivity \[ \sigma(\omega)=\frac{1}{\omega}\,\mathrm{Im}\,\langle J;J\rangle_{\omega+i0^+}, \quad J=\sum_k v_k c_k^\dagger c_k, \] has finite d.c. conductivity \(\sigma(\omega\to 0)\) despite the absence of a single-particle pole [1904.12872]. The density \(\langle n\rangle\) versus \(\mu\) is smooth in the orthogonal metal at \(h=4\), demonstrating compressibility and hence metallic response despite the missing quasiparticle pole [1904.12872]. 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 \(T