Non-Fermi Liquids

A non-Fermi liquid is a gapless fermionic state or regime that cannot be organized by the long-lived quasiparticles and regular forward-scattering functional of Landau Fermi-liquid theory. The label identifies a failure of one infrared description. It does not, by itself, identify the replacement theory.

That distinction matters. A one-dimensional Luttinger liquid, a metal at an Ising-nematic quantum critical point, a spinon Fermi surface coupled to an emergent gauge field, an orthogonal metal, and a large-$N$ SYK model can all lack an electron-like Landau pole. Their dimensions, conserved charges, low-energy operators, transport laws, and degrees of theoretical control are nevertheless different.

The strongest diagnosis therefore has two stages:

1. establish which Landau criteria fail in a controlled low-energy limit;
2. identify a replacement mechanism that predicts several independent observables.

A resistivity exponent, a broad photoemission line, or a large heat-capacity coefficient is evidence to explain, not a complete diagnosis.

Required background. Fermi Liquid Theory Preview supplies the Landau quasiparticle and low-energy response criteria tested here. Green Functions in Many-Body QM supplies the propagator and self-energy language. Spectral Functions supplies transition weights and continuum thresholds, while Lifetime and Spectral Weight distinguishes pole residue, decay width, and the window in which a quasiparticle is well defined.

Phases, Critical Trajectories, and Crossovers

The phrase non-Fermi liquid is used for at least three logically different claims.

An infrared phase

The strongest claim concerns the limiting ground-state theory. At fixed Hamiltonian parameters and after taking the thermodynamic limit, the state remains metallic as

$$T,\lvert E\rvert,\lvert k_\perp\rvert \longrightarrow 0,$$

where $E$ is measured from the chemical potential and $k_\perp$ is the wavevector displacement normal to a candidate Fermi surface, but no Landau quasiparticle description emerges. A Luttinger liquid is a controlled example. A fractionalized metal can be another, although the electron and emergent-fermion spectra must be distinguished.

A quantum-critical trajectory

At an isolated critical coupling $g=g_c$, quasiparticles may be destroyed only on the critical trajectory. Detuning can restore a Fermi liquid below a crossover scale $T_{\mathrm{FL}}(g)$. The finite-temperature region above that scale can be wide, but it is not automatically a distinct thermodynamic phase.

A finite-window regime

Experiments and numerics access finite temperature, frequency, size, time, and resolution. A system may look incoherent over that window and still cross over to a Fermi liquid, an ordered state, a superconductor, or a localized state at lower energy. A trustworthy statement therefore reports the window and says whether the extrapolated zero-energy behavior is measured, inferred, or unknown.

Fermi-Liquid Baseline

Near a regular interacting Fermi surface, the retarded electron Green function has a pole contribution

$$G^{\mathrm R}(\mathbf k,E) \simeq \frac{Z_{\mathbf k}} {E-\hbar v_{\mathrm F}^{\star}k_\perp+i\gamma_E(\mathbf k,E,T)} +G_{\mathrm{inc}}^{\mathrm R}.$$

Here $k_\perp$ measures wavevector normal to the Fermi surface, $v_{\mathrm F}^{\star}$ is a physical velocity, $Z_{\mathbf k}>0$ is the pole residue, and $\gamma_E$ is the energy half-width. The residue measures how strongly adding an electron excites this mode; the width measures how quickly the mode decays. The following sharpness test concerns intrinsic inelastic damping in the clean limit; a residual elastic width must be treated separately. The decisive low-energy condition is not merely a narrow-looking peak at one resolution. It is asymptotic sharpness:

$$\lim_{E,T\to0} \frac{\gamma_E(\mathbf k_{\mathrm F},E,T)} {\max(\lvert E\rvert,k_{\mathrm B}T)} =0.$$

For a conventional Fermi liquid, phase-space restrictions give a quadratic rate, with logarithmic qualifications in two dimensions,

$$\gamma_E \propto E^2+(\pi k_{\mathrm B}T)^2.$$

Finite residue, asymptotic sharpness, a Fermi-surface occupation discontinuity, regular Landau parameters, and mutually consistent thermodynamics and response form a package. Losing one electron spectral feature does not imply that every fermionic or collective excitation has disappeared.

What Can Fail

The pole can lose asymptotic sharpness

If $\gamma_E/\lvert E\rvert$ approaches a nonzero constant or diverges, an excitation does not complete parametrically many oscillations before decaying. A peak may remain visible over a finite window, but a controlled particle expansion around it is absent.

The residue can vanish

The electron overlap with the low-energy excitation can scale to zero,

$$Z(E) \longrightarrow 0.$$

This can turn a pole into a branch-point singularity or a threshold continuum. It is an operator-specific statement: an electron residue can vanish while neutral spinons, composite fermions, or collective bosons remain sharp in their own channels.

Only part of the Fermi surface can become singular

An ordering wavevector may connect discrete hot spots or hot lines. Those regions can have singular self-energies while cold regions retain quasiparticles. By contrast, a zero-momentum nematic fluctuation or a transverse gauge field can couple singularly to an extended Fermi surface. The phrase “the quasiparticle is destroyed” must therefore specify where in momentum space.

Thermodynamics can survive a spectral failure

Some fractionalized metals have compressibility, heat capacity, and electrical conductivity resembling a Fermi liquid even though the physical-electron spectral function has no low-energy pole. Conversely, a large $C/T$ may arise from neutral excitations, rare regions, or a small coherence scale rather than from charged electron quasiparticles.

An instability can preempt the putative fixed point

The same boson that degrades quasiparticles can mediate pairing or drive another order. A non-Fermi-liquid scaling regime may then be well defined above the ordering scale but not survive as the ground state. The order of limits and the accessible normal-state window are part of the claim.

Self-Energy Diagnostics

Write Dyson’s equation near a candidate Fermi surface as

$$\left[G^{\mathrm R}(\mathbf k,E)\right]^{-1} = E-\xi_{\mathbf k}-\Sigma^{\mathrm R}(\mathbf k,E),$$

with $\operatorname{Im}\Sigma^{\mathrm R}\le0$ for a passive retarded channel. When the self-energy is differentiable at the pole,

$$Z_{\mathbf k} = \left[ 1- \left. \frac{\partial\operatorname{Re}\Sigma^{\mathrm R}} {\partial E} \right|_{E=0} \right]^{-1}.$$

This formula is a diagnostic only when the derivative and pole expansion exist. Applying it mechanically to a branch cut or a nonanalytic self-energy can manufacture a meaningless “residue.”

Power-law self-energy

A useful particle–hole-symmetric parametrization is

$$C^{-1} \left[ -\operatorname{Im}\Sigma^{\mathrm R}(E) \right] = \Lambda^{1-\alpha} \lvert E\rvert^\alpha, \qquad C>0, \quad 0<\alpha<1,$$

with $\Lambda$ a microscopic energy. Causality fixes a corresponding odd, nonanalytic real part through the Kramers–Kronig relation; the stated even power law fixes its leading nonanalytic coefficient, while the ultraviolet completion also supplies analytic terms. The imaginary part produces

$$\frac{-\operatorname{Im}\Sigma^{\mathrm R}(E)} {\lvert E\rvert} \propto \left( \frac{\Lambda}{\lvert E\rvert} \right)^{1-\alpha},$$

which grows toward low energy. The derivative of the real part is also singular, so the electron pole residue vanishes. The exponent alone does not identify the mechanism: gauge-field patch theories, critical order-parameter fluctuations, and locally critical models can produce different power laws with different momentum structures.

Marginal behavior

At the boundary between self-energy damping that vanishes faster than energy and damping that dominates it, a schematic zero-temperature marginal self-energy is

$$\Sigma^{\mathrm R}(E) \simeq -\lambda E \ln\!\left( \frac{E_c}{\lvert E\rvert} \right) - i\frac{\pi\lambda}{2}\lvert E\rvert.$$

The self-energy ratio $-\operatorname{Im}\Sigma^{\mathrm R}/\lvert E\rvert=\pi\lambda/2$ is constant. It differs from the dressed pole-width ratio. The local derivative estimate gives

$$\begin{aligned} Z(E)&\simeq\left[1+\lambda\ln\!\left(\frac{E_c}{\lvert E\rvert}\right)-\lambda\right]^{-1},\\ \frac{\gamma_E}{\lvert E\rvert}&\simeq\frac{\pi\lambda}{2}Z(E)\longrightarrow0. \end{aligned}$$

Thus a marginal pole can sharpen logarithmically while its residue vanishes; this still fails the finite-residue Landau criterion. These are scale-dependent pole estimates, not a regular expansion at $E=0$. At finite temperature, replacing $\lvert E\rvert$ by an informal $\max(\lvert E\rvert,k_{\mathrm B}T)$ is useful for scaling intuition but is not a complete causal Green function. A quantitative fit should use an analytic finite-temperature scaling function and include instrumental convolution.

Frequency–temperature scaling

At a scale-invariant trajectory, one may test

$$-\operatorname{Im}\Sigma^{\mathrm R}(\mathbf k,E,T) = \Lambda^{1-\alpha} (k_{\mathrm B}T)^\alpha \Phi_{\mathbf k}\!\left( \frac{E}{k_{\mathrm B}T} \right).$$

Here $\Phi_{\mathbf k}$ is dimensionless and $\Lambda$ is a fixed microscopic energy. The test is stronger than fitting separate powers of $E$ and $T$. It requires one exponent, one scaling function over overlapping windows, a declared background, and stability under changes of fitting range. Momentum dependence must be retained rather than averaged into a single apparent exponent.

A narrower excitation need not recover a finite electron residue. The upper panel evaluates the marginal example with $\lambda=0.5$: $Z=[1+\lambda\ln(E_c/\lvert E\rvert)-\lambda]^{-1}$ and the local dressed-width estimate $\gamma_E/\lvert E\rvert=(\pi\lambda/2)Z$. Both decrease toward lower energy, although the self-energy ratio stays constant. These are low-energy estimates, not an exact finite-temperature line shape. Below, the dark regions distinguish isolated hot spots from an extended singular surface; the circles are schematic, not material Fermi surfaces.

Spectra: What to Measure

For a single electron,

$$A(\mathbf k,E) = -\frac{1}{\pi} \operatorname{Im}G^{\mathrm R}(\mathbf k,E).$$

Angle-resolved photoemission accesses the occupied part multiplied by matrix elements and the Fermi function; tunneling usually integrates over momentum with its own matrix-element weighting. A credible non-Fermi-liquid spectral analysis should therefore:

1. map the full momentum dependence rather than only one cut;
2. separate intrinsic linewidth from energy and momentum resolution;
3. enforce causality between real and imaginary self-energy components;
4. verify spectral-weight and sum-rule accounting;
5. test whether fitted exponents persist as temperature and energy decrease;
6. compare electron spectra with thermodynamic and two-particle probes.

An apparent absence of a peak can result from a matrix-element zero, band overlap, a gap, surface disorder, or insufficient resolution. Conversely, a finite-temperature peak can survive even when its width scales as fast as its energy and no asymptotic quasiparticle exists.

Quantum oscillations provide a complementary but subtle constraint. Oscillations imply coherent orbital quantization over a field- and temperature-dependent window. They can reveal an extremal momentum-space area and an effective cyclotron mass, but do not by themselves prove a zero-field Landau pole over the whole Fermi surface. Magnetic field can suppress competing order, change scattering, or create a distinct low-energy regime.

Transport Is Not a Self-Energy Meter

Conductivity probes a current correlation function, including current vertices and contact terms; it is not the inverse of a single-particle linewidth. The Kubo Formula distinguishes the regular optical response from a possible zero-frequency Drude delta function. In a clean Galilean-invariant fluid, current is proportional to conserved momentum: electron–electron collisions can broaden a single-particle spectrum while leaving dc resistivity zero. A finite resistivity requires a way to relax the electrical current, such as disorder or lattice Umklapp scattering.

Resistivity exponents are nonunique

A measured law

$$\rho(T) = \rho_0+A T^n$$

with $n\ne2$ is incompatible with the simplest isotropic Fermi-liquid transport model over that window. It is not a unique fingerprint of quasiparticle destruction. Phonons, disorder, Umklapp geometry, multiband conduction, hot and cold regions, fluctuating order, and crossover between rates can all change the fitted exponent.

For independent parallel channels at zero Hall response, along a principal conductivity axis,

$$\sigma_{xx} = \sum_a \sigma_{xx}^{(a)}, \qquad \rho_{xx} = \frac{1}{\sigma_{xx}},$$

so the longest-lived current-carrying sector can short-circuit a strongly scattered one. A singular hot-spot self-energy does not automatically determine the bulk resistivity. With a Hall response, the full conductivity tensor must be inverted instead. A worked two-channel example appears in Quantum Criticality.

What transport measurements constrain

Probe	Information and limitation
dc resistivity	Current-relaxation law and crossover scales; not a direct electron linewidth.
Optical conductivity	Frequency-dependent weight and coherence; a fitted rate depends on mass, vertices and interband subtraction.
Thermal transport	Mobile entropy carriers; phonons and neutral excitations can contribute.
Hall response	Carrier geometry and scattering; multiband correlations obstruct a one-band density interpretation.
Thermopower	Particle–hole asymmetry of charge transport; not a direct residue measurement.
Wiedemann–Franz ratio	Relation between heat and charge conduction; a finite-temperature deviation alone does not establish fractionalization.

Transport becomes compelling when its crossover scales, anisotropy, field dependence, and spectral redistribution agree with a mechanism that also explains single-particle and thermodynamic data.

Candidate Infrared Mechanisms

One-dimensional collective fixed points

Generic gapless interacting systems in one dimension flow to collective bosonic theories. Fermion insertion creates many density waves, the momentum-distribution jump disappears, and spectral functions develop power-law thresholds. This is a controlled replacement of Landau theory, not merely a badly broadened Fermi liquid. The full construction belongs to Luttinger Liquid Preview.

A Fermi surface coupled to a critical boson

Near a metallic ordering transition, low-energy fermions and an order-parameter field must be treated together. In this patch description only, use $\hbar=k_{\mathrm B}=1$, with velocities, curvature coefficients, fields, and couplings normalized accordingly. A two-patch action has the schematic form

$$\begin{aligned} S={}& \sum_{s=\pm}\int_k \psi_s^\dagger(k) \left( -i\omega+s v_{\mathrm F}k_\perp +\kappa k_\parallel^2 \right) \psi_s(k) \\ &+ \frac12\int_q D_0^{-1}(q)\lvert\phi(q)\rvert^2 +\sum_{s=\pm}g_s\int_{k,q} \phi(q)\psi_s^\dagger(k+q)\psi_s(k). \end{aligned}$$

The index $s=\pm$ labels antipodal patches; $k_\parallel$ lies tangent to them and $\kappa$ is their curvature coefficient. The coupling has the same sign on the two patches for an inversion-even nematic mode, and opposite signs for a transverse gauge field coupled to antiparallel currents (Mross and collaborators, 2010). The Fermi surface Landau-damps the boson, while the boson generates a singular fermion self-energy. For a common two-dimensional nematic or transverse-gauge scaling structure, the Matsubara forms are

$$D^{-1}(\mathbf q,i\Omega_n) \sim q_\parallel^2 +\gamma\frac{\lvert\Omega_n\rvert}{\lvert q_\parallel\rvert}, \qquad \Sigma(i\omega_n) \sim -i\,\operatorname{sgn}(\omega_n) \lvert\omega_n\rvert^{2/3},$$

up to coefficients, analytic terms, and corrections that depend on the model and control scheme. After causal continuation, $-\operatorname{Im}\Sigma^{\mathrm R}(E)\propto\lvert E\rvert^{2/3}$. For a finite-wavevector spin-density-wave transition, only hot spots connected by the ordering vector are directly singular at leading order. For a zero-momentum nematic mode, an extended Fermi surface can be affected.

These theories are active research subjects. Large-$N$, dimensional, codimensional, and dynamical-exponent expansions each control selected quantities, but extrapolating to a physical two-dimensional metal can be subtle. Pairing and other instabilities may intervene.

Emergent gauge fields and fractionalization

A neutral spinon Fermi surface coupled to a gapless U(1) gauge field has the same patch-level tension: gapless fermions damp the gauge field, and gauge fluctuations destroy ordinary spinon quasiparticles. The physical electron may be gapped because charge is localized, so this example can be an electrical insulator: electron photoemission and neutral thermal response probe different operators.

More generally, fractionalization can produce an orthogonal metal whose electrical conductivity and thermodynamics resemble those of a Fermi liquid while the physical-electron spectral function is gapped. This is a sharp warning against defining a Fermi liquid from transport and heat capacity alone.

Kondo destruction and multichannel screening

An overscreened multichannel Kondo impurity with equivalent screening channels can flow to a non-Fermi-liquid boundary fixed point. Relevant channel asymmetry or a local magnetic field can destabilize that limit. In a lattice, however, impurity exponents cannot simply be copied: coherence, intersite exchange, symmetry breaking, disorder, and Fermi-surface reconstruction enter. At some heavy-fermion critical points, collapse of Kondo entanglement is proposed to destroy quasiparticles across an extended Fermi surface. Kondo Lattices and Quantum Criticality develop the lattice physics and material comparisons.

Local criticality and SYK-like models

The large-$N$ SYK family gives a controlled strongly interacting Green function without a quasiparticle pole. For the particle–hole-symmetric or Majorana case, the zero-temperature long-time conformal form is

$$G(\tau) \propto \frac{\operatorname{sgn}(\tau)} {\lvert\tau\rvert^{2\Delta}},$$

with $\Delta$ fixed by the interaction order. At finite temperature the power law is replaced by a thermal scaling function, and a charge-asymmetric complex-fermion model has different positive- and negative-time amplitudes. The baseline model has no spatial momentum, so it is not by itself a theory of electrical resistivity or a material Fermi surface. Transport appears only after specifying coupled dots, lattice structure, conserved charge, and momentum relaxation.

Disorder and rare regions

Broad distributions of local Kondo or coherence scales can generate power-law thermodynamics and anomalous transport. For example, a low-scale distribution

$$P(T_0) \propto T_0^{\lambda-1}, \qquad 0<\lambda<1,$$

makes rare regions with very small $T_0$ dominate low-temperature averages. This can mimic clean quantum-critical exponents while producing strong sample dependence, spatial inhomogeneity, and broad local-relaxation distributions. The disorder ensemble and typical-versus-average distinction must be measured, not treated as a nuisance parameter.

Relation to Quantum Field Theory

A Fermi surface is an unusual field-theory object because low-energy modes live near a codimension-one manifold rather than near a finite set of momenta. Renormalization rescales energy and momentum normal to each patch while leaving the Fermi-surface label as a continuous index. Couplings that look irrelevant by naive point-particle counting can remain important because there are infinitely many low-energy patches.

The patch action above is therefore a quantum field theory, but not usually a relativistic one. It has anisotropic scaling, Landau damping, a finite density, and a preferred frame. Ward identities still tie self-energies to conserved-current vertices, while the Fermi surface supplies nonlocal response and a large phase space of gapless particle–hole states. Why Many-Body QM Leads to QFT gives the broader bridge, and Renormalization Group Preview owns the general fixed-point language.

What holography computes

Gauge/gravity duality supplies nonperturbative correlators for selected large-$N$, strongly coupled quantum field theories at finite density. In early holographic finite-density constructions, a fermionic operator near a momentum $k_{\mathrm F}$ has, at positive energy and real infrared exponent $\nu>0$, a Green function of the schematic form (Faulkner and collaborators, 2011)

$$G^{\mathrm R}(k_\perp,E) \simeq \frac{h_1} {k_\perp-E/(\hbar v_{\mathrm F})-h_2e^{i\theta}E^{2\nu}},$$

Here $v_{\mathrm F}$ is a physical velocity and $h_2$ carries the units needed to make the denominator a wavevector; continuation to negative or complex energy must use the causal branch. The exponent $\nu$ is set by the operator’s effective scaling dimension in the near-horizon sector. For $0<\nu<1/2$, the nonanalytic term dominates the linear-frequency term and the excitation is not Landau-like. At $\nu=1/2$, logarithms produce marginal behavior. This realizes a calculable family of critical Fermi-surface correlators.

What it does not establish is equally important. A holographic Green function is an exact or controlled result only for its declared boundary theory and gravity limit. Similarity to a material spectrum does not identify the material’s microscopic degrees of freedom, momentum-relaxation mechanism, finite-$N$ corrections, or unique dual geometry. Holography is a laboratory for possible strongly coupled infrared structures, not evidence by itself that a particular crystal has a gravity dual.

Exercises

1. Classify a power-law self-energy

Suppose at $T=0$

$$-\operatorname{Im}\Sigma^{\mathrm R}(E) = C\Lambda^{1-\alpha}\lvert E\rvert^\alpha, \qquad C>0.$$

For which values of $\alpha$ is the excitation asymptotically sharp if residue effects remain finite? Classify $\alpha=2$, $\alpha=1$, and $\alpha=2/3$.

Solution

The relevant ratio is

$$\frac{-\operatorname{Im}\Sigma^{\mathrm R}(E)}{\lvert E\rvert} = C\left( \frac{\lvert E\rvert}{\Lambda} \right)^{\alpha-1}.$$

It vanishes for $\alpha>1$, remains constant for $\alpha=1$, and diverges for $0<\alpha<1$. With a finite residue these would respectively imply a sharp excitation, a finite relative width, and a diverging relative width. However, a causal power law extending to zero energy with $0<\alpha\le1$ does not generally permit that finite-residue assumption. The marginal example in Exercise 2 explicitly separates the two ratios; for $0<\alpha<1$ the full nonanalytic Green function is needed.

2. Residue in a marginal self-energy

For positive $E$ well below $E_c$, take

$$\operatorname{Re}\Sigma^{\mathrm R}(E) = -\lambda E\ln\!\left(\frac{E_c}{E}\right).$$

Find the scale-dependent derivative estimate for $Z(E)$ and determine its low-energy trend. If $-\operatorname{Im}\Sigma^{\mathrm R}=\pi\lambda E/2$, distinguish its damping ratio from the dressed pole-width ratio.

Solution

Differentiation gives

$$\frac{\partial\operatorname{Re}\Sigma^{\mathrm R}}{\partial E} = -\lambda \left[ \ln\!\left(\frac{E_c}{E}\right)-1 \right].$$

With the sign convention written in the question,

$$Z(E) \simeq \left[ 1+\lambda\ln\!\left(\frac{E_c}{E}\right)-\lambda \right]^{-1}.$$

The logarithm makes the derivative singular, so $Z(E)$ tends to zero slowly as $E\to0$. Although $-\operatorname{Im}\Sigma^{\mathrm R}/E=\pi\lambda/2$ is constant, the local pole estimate gives $\gamma_E/E\simeq Z(E)\pi\lambda/2\to0$. Vanishing residue still excludes an ordinary Landau quasiparticle. The formula is a scale-dependent pole estimate; sufficiently close to the singular point, the full causal Green function is more informative than a constant-residue expansion.

3. Why the marginal logarithm is required

Assume over a broad symmetric low-energy window that

$$\operatorname{Im}\Sigma^{\mathrm R}(E) \propto -\lvert E\rvert.$$

Use the Kramers–Kronig structure to explain why the real part cannot simply be proportional to $E$ with a constant coefficient.

Solution

The retarded real part is a principal-value transform,

$$\operatorname{Re}\Sigma^{\mathrm R}(E) = \frac{1}{\pi} \mathcal P \int dE' \frac{\operatorname{Im}\Sigma^{\mathrm R}(E')}{E'-E}.$$

For an even cusp $-\lvert E'\rvert$ cut off at $E_c$, subtracting the value at $E=0$ leaves an odd term proportional to

$$E\ln\!\left(\frac{E_c}{\lvert E\rvert}\right)$$

plus an analytic term proportional to $E$. The logarithm is therefore required by causality. Fitting the imaginary part to a linear cusp while using an unrelated constant-slope real part is internally inconsistent.

4. Phase or finite-window regime?

A material shows $\rho-\rho_0\propto T^{1.2}$ from $2$ to $20\,\mathrm K$ and a broad photoemission peak at $10\,\mathrm K$. Below $1.5\,\mathrm K$ it becomes superconducting. List four measurements or tuning tests needed before claiming a non-Fermi-liquid ground state.

Solution

Useful tests include: suppressing superconductivity with a tuning parameter while checking that the field or pressure does not create a different normal state; tracking the intrinsic linewidth-to-energy ratio toward lower $E$ and $T$; mapping momentum dependence to distinguish hot and cold regions; and locating any critical coupling through vanishing crossover scales on both sides. One should also test sample-quality dependence, optical spectral weight, thermodynamics, and competing order. The stated observations establish an anomalous finite window, not a ground-state non-Fermi liquid.

5. An orthogonal diagnostic

Two metallic phases have the same leading heat capacity, compressibility, and dc conductivity. In phase A, electron photoemission has a Fermi-surface pole. In phase B, the electron spectrum is gapped, but charge transport remains metallic. Can thermodynamics and dc transport alone distinguish the phases? What observation does?

Solution

No. By construction, those bulk observables agree at leading order. The phases differ in the operator that overlaps with the gapless charged fermion. Momentum-resolved electron addition or removal, tunneling, or another probe explicitly sensitive to the physical-electron spectral function distinguishes a pole in phase A from a gap in phase B. This is the defining lesson of an orthogonal metal: conventional-looking transport and thermodynamics do not guarantee an electron Landau quasiparticle.

Further Connections

Fractionalization explains why electron and emergent-fermion spectra can disagree. Emergent Gauge Fields develops the constraint and transverse response behind the gauge-coupled example. Strange Metals compares material transport phenomenology, which is broader than a demonstrated failure of electron quasiparticles.

References

• I. Affleck and A. W. W. Ludwig, “Critical theory of overscreened Kondo fixed points,” Nuclear Physics B 360, 641–696 (1991). Boundary conformal-field-theory solution of multichannel Kondo non-Fermi-liquid fixed points.
• B. L. Altshuler, L. B. Ioffe, and A. J. Millis, “Low-energy properties of fermions with singular interactions,” Physical Review B 50, 14048–14064 (1994). Singular gauge interaction and nonanalytic fermionic response.
• E. C. Andrade, E. Miranda, and V. Dobrosavljević, “Electronic Griffiths phase of the Mott transition,” Physical Review Letters 102, 206403 (2009). Spatially resolved rare-region mechanism near a disordered Mott transition.
• T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, “Emergent quantum criticality, Fermi surfaces, and AdS$_2$,” Physical Review D 83, 125002 (2011). Near-horizon scaling dimensions and the low-energy Green-function classification.
• F. D. M. Haldane, “Luttinger liquid theory of one-dimensional quantum fluids,” Journal of Physics C 14, 2585–2609 (1981). Harmonic-fluid fixed point and power-law one-dimensional correlations.
• B. I. Halperin, P. A. Lee, and N. Read, “Theory of the half-filled Landau level,” Physical Review B 47, 7312–7343 (1993). Composite fermions coupled to an emergent gauge field.
• L. D. Landau, “The theory of a Fermi liquid,” Soviet Physics JETP 3, 920–925 (1957). Foundational quasiparticle and energy-functional construction.
• S.-S. Lee, “Low-energy effective theory of Fermi surface coupled with U(1) gauge field in 2+1 dimensions,” Physical Review B 80, 165102 (2009). Patch theory and large-$N$ difficulties for gauge-coupled fermions.
• H. Liu, J. McGreevy, and D. Vegh, “Non-Fermi liquids from holography,” Physical Review D 83, 065029 (2011). Finite-density holographic fermion spectra and critical Fermi surfaces.
• J. Maldacena and D. Stanford, “Remarks on the Sachdev–Ye–Kitaev model,” Physical Review D 94, 106002 (2016). Large-$N$ conformal dynamics and four-point functions in SYK.
• M. A. Metlitski and S. Sachdev, “Quantum phase transitions of metals in two spatial dimensions. I. Ising-nematic order,” Physical Review B 82, 075127 (2010). Patch field theory for a critical Fermi surface coupled to nematic order.
• M. A. Metlitski, D. F. Mross, S. Sachdev, and T. Senthil, “Cooper pairing in non-Fermi liquids,” Physical Review B 91, 115111 (2015). Pairing instabilities of critical and gauge-coupled Fermi surfaces.
• E. Miranda and V. Dobrosavljević, “Localization-induced Griffiths phase of disordered Anderson lattices,” Physical Review Letters 86, 264–267 (2001). Disorder-generated distributions of local scales and non-Fermi-liquid thermodynamics.
• D. F. Mross, J. McGreevy, H. Liu, and T. Senthil, “Controlled expansion for certain non-Fermi-liquid metals,” Physical Review B 82, 045121 (2010). Combined expansion for gauge-field and nematic non-Fermi liquids.
• R. Nandkishore, M. A. Metlitski, and T. Senthil, “Orthogonal metals: The simplest non-Fermi liquids,” Physical Review B 86, 045128 (2012). Fractionalized metal with conventional thermodynamics and transport but a gapped electron spectrum.
• V. Oganesyan, S. A. Kivelson, and E. Fradkin, “Quantum theory of a nematic Fermi fluid,” Physical Review B 64, 195109 (2001). Metallic nematic criticality and singular fermion dynamics.
• O. Parcollet and A. Georges, “Non-Fermi-liquid regime of a doped Mott insulator,” Physical Review B 59, 5341–5360 (1999). Controlled large-degeneracy doped-Mott crossover with a low coherence scale.
• S. Sachdev and J. Ye, “Gapless spin-fluid ground state in a random quantum Heisenberg magnet,” Physical Review Letters 70, 3339–3342 (1993). Large-symmetry random-interaction ancestor of SYK-like local criticality.
• T. Senthil, “Critical Fermi surfaces and non-Fermi liquid metals,” Physical Review B 78, 035103 (2008). Scaling framework for a sharp Fermi surface without Landau quasiparticles.
• C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, and A. E. Ruckenstein, “Phenomenology of the normal state of Cu-O high-temperature superconductors,” Physical Review Letters 63, 1996–1999 (1989); erratum 64, 497 (1990). Marginal-Fermi-liquid phenomenology.