Holographic Fermions and Spectral Functions

The most direct experimental signature of a metal is not the DC resistivity. It is the single-particle spectral function. In angle-resolved photoemission, one tries to see whether low-energy electron spectral weight is concentrated near a Fermi surface, broadened into an incoherent continuum, gapped, or split into several bands. Holography gives a clean theoretical analogue of this diagnostic: solve a bulk Dirac equation in a charged black-brane geometry and read off the retarded Green function of a fermionic boundary operator.

That sentence hides several subtleties. A holographic fermion is usually not the microscopic electron of a material. It is a gauge-invariant fermionic operator $\mathcal O_\Psi$ in a large-$N$ quantum field theory. Its spectral function is nevertheless extremely useful because it asks the right universal question:

$$\boxed{ \text{Can a compressible, strongly coupled state have Fermi-surface-like response without Landau quasiparticles?} }$$

The answer is yes. A probe bulk spinor in a charged black brane can show sharp Fermi-surface poles, marginal-Fermi-liquid-like behavior, broad non-Fermi-liquid spectral weight, algebraic pseudogaps, and log-periodic oscillations. The best way to understand all of these is to combine three ideas from earlier pages:

1. finite density is radial electric flux,
2. the extremal charged horizon contains an $AdS_2\times\mathbb R^{d_s}$ throat,
3. retarded correlators are computed by infalling boundary conditions at the future horizon.

Throughout this page, $d_s$ is the number of boundary spatial dimensions. The boundary spacetime dimension is $d=d_s+1$, and the bulk dimension is $d+1=d_s+2$. We use a radial coordinate $r$ for which the asymptotic AdS boundary is at $r\to\infty$.

The boundary observable

For a fermionic operator $\mathcal O_\Psi$, the retarded Green function is defined with an anticommutator:

$$G^R_{\mathcal O\mathcal O^\dagger}(t,\vec x) = -i\theta(t)\left\langle \{\mathcal O_\Psi(t,\vec x),\mathcal O_\Psi^\dagger(0,0)\} \right\rangle.$$

After Fourier transform, $G^R(\omega,\vec k)$ is a matrix in spinor indices. The spectral function is

$$A(\omega,\vec k) = -2\,\operatorname{Im}\operatorname{Tr}G^R(\omega,\vec k),$$

up to harmless convention choices. It is the holographic analogue of the photoemission intensity, modulo matrix elements and the fact that $\mathcal O_\Psi$ need not be the physical electron.

A Fermi surface is diagnosed by a low-frequency singularity at nonzero momentum. Operationally, one looks for

$$\det G_R^{-1}(\omega=0,k_F)=0.$$

This criterion is more robust than the existence of a long-lived quasiparticle. A Fermi liquid has such a zero together with a narrow pole of the form

$$G^R_{\rm FL}(\omega,k) \simeq \frac{Z}{\omega-v_F(k-k_F)+i\Gamma(\omega,T)}, \qquad \Gamma\ll |\omega|.$$

A non-Fermi liquid can still have a Fermi momentum $k_F$, but the pole may be strongly broadened, the residue may vanish, or the low-energy self-energy may dominate over the bare dispersion.

A holographic fermion spectral function is obtained by solving a charged bulk Dirac equation with infalling horizon boundary conditions. A Fermi momentum $k_F$ appears when the infalling bulk solution is normalizable at $\omega=0$, equivalently when the boundary source coefficient vanishes. The width of the spectral peak is controlled by leakage into the deep IR throat.

The bulk Dirac problem

The minimal bottom-up bulk action for a charged spinor is

$$S_\Psi = -i\int d^{d_s+2}x\sqrt{-g}\, \bar\Psi\left(\Gamma^M D_M-m\right)\Psi,$$

with

$$D_M = \partial_M+\frac{1}{4}\omega_{MAB}\Gamma^{AB}-iqA_M.$$

Here $m$ is the bulk spinor mass, $q$ is its bulk $U(1)$ charge, $A_M$ is the Maxwell field dual to the boundary current, and $\omega_{MAB}$ is the spin connection. The curved gamma matrices obey

$$\{\Gamma^M,\Gamma^N\}=2g^{MN}.$$

The simplest finite-density backgrounds are diagonal and translationally invariant:

$$ds^2 = g_{tt}(r)dt^2+g_{rr}(r)dr^2+g_{xx}(r)d\vec x^2, \qquad A=A_t(r)dt,$$

with $g_{tt}<0$. We take momentum in the $x$ direction,

$$\Psi(t,x,r)=e^{-i\omega t+ikx}\,\Psi(r).$$

A useful field redefinition removes the spin-connection term for diagonal metrics:

$$\Psi=(-g g^{rr})^{-1/4}\psi,$$

where $g=\det g_{MN}$. The Dirac equation becomes a radial first-order equation.

After choosing a gamma-matrix basis adapted to the radial direction, the spinor splits into two independent two-component sectors, labelled by $\alpha=1,2$. A representative form is

$$\left[ \sqrt{\frac{g_{xx}}{g_{rr}}}\,\sigma^3\partial_r -m\sqrt{g_{xx}}\,\sigma^2 +i\sqrt{\frac{g_{xx}}{-g_{tt}}}(\omega+qA_t)\sigma^1 -(-1)^\alpha k \right]\psi_\alpha=0.$$

The precise signs depend on the gamma-matrix basis, but the structure is universal. The spinor experiences an effective local frequency

$$\omega_{\rm loc}(r)=\sqrt{-g^{tt}(r)}\,[\omega+qA_t(r)]$$

and a local momentum

$$k_{\rm loc}(r)=\sqrt{g^{xx}(r)}\,k.$$

Thus the chemical potential does not just shift the boundary frequency. It creates a radial electrostatic environment in which a charged bulk fermion can form quasi-bound states.

The physical picture

The bulk Dirac equation is a one-dimensional scattering problem in the radial direction. The boundary is where one sources and measures the fermionic operator. The horizon is an absorber. A sharp spectral peak is a quasi-bound spinor wave that lingers in the bulk before leaking into the horizon.

Source and response for spinors

Spinors differ from scalars because the Dirac equation is first order. One does not freely specify all components at the boundary. Half of the spinor components are sources; the conjugate half are responses.

In asymptotically $AdS_{d_s+2}$,

$$ds^2\sim \frac{r^2}{L^2}(-dt^2+d\vec x^2)+\frac{L^2}{r^2}dr^2, \qquad r\to\infty.$$

For the rescaled spinor $\psi_\alpha$, the near-boundary expansion takes the schematic form

$$\psi_\alpha(r) = a_\alpha \begin{pmatrix}0\\1\end{pmatrix} r^{mL} + b_\alpha \begin{pmatrix}1\\0\end{pmatrix} r^{-mL} +\cdots.$$

For $mL\ge0$ in standard quantization, $a_\alpha$ is the source and $b_\alpha$ is the response. The dimension of the dual fermionic operator is

$$\boxed{ \Delta_+=\frac{d_s+1}{2}+mL. }$$

When $|mL|<1/2$, an alternative quantization is also allowed, in which the roles of source and response are exchanged and

$$\Delta_- = \frac{d_s+1}{2}-mL.$$

For most holographic-fermion applications one chooses standard quantization and writes

$$b_\alpha(\omega,k)=G^R_\alpha(\omega,k)\,a_\alpha(\omega,k),$$

with possible normalization factors depending on convention. In matrix notation,

$$\mathbf b=G_R\mathbf a.$$

The retarded prescription is imposed in the interior. At a nonextremal horizon $r=r_h$,

$$\psi_\alpha(r) \sim (r-r_h)^{-i\omega/(4\pi T)}\psi_\alpha^{\rm in},$$

which is the spinor version of the infalling boundary condition. The computation is now well-posed: start with the infalling solution at the horizon, integrate to the boundary, and read off $a_\alpha$ and $b_\alpha$.

Fermi surfaces as normalizable zero modes

At fixed $k$, the infalling boundary condition determines a unique solution up to overall normalization. Near the boundary this solution has coefficients $a_\alpha(\omega,k)$ and $b_\alpha(\omega,k)$. Poles of the retarded Green function occur when the source coefficient vanishes:

$$a_\alpha(\omega,k)=0.$$

A holographic Fermi surface is therefore a normalizable, zero-frequency, finite-momentum bulk spinor mode:

$$\boxed{ a_\alpha(0,k_F)=0. }$$

This statement is simple and important. The bulk problem looks like finding a bound state at zero energy. The boundary interpretation is a pole in the fermionic Green function at a Fermi momentum.

There is no guarantee that such a $k_F$ exists. It depends on the background geometry and on the spinor data $(m,q)$, plus possible nonminimal couplings. A large charge-to-mass ratio tends to favor Fermi-surface-like poles because the electrostatic potential can trap the charged spinor. A large mass or small charge tends to suppress such quasi-bound states, leaving the deep IR continuum more visible.

A practical flow equation

For numerical work, it is convenient to avoid tracking both spinor components separately. Write

$$\psi_\alpha= \begin{pmatrix}y_\alpha\\z_\alpha\end{pmatrix}, \qquad \xi_\alpha=\frac{y_\alpha}{z_\alpha}.$$

The Dirac equation becomes a first-order nonlinear flow equation for $\xi_\alpha$:

$$\sqrt{\frac{g_{xx}}{g_{rr}}}\,\partial_r\xi_\alpha = -2m\sqrt{g_{xx}}\,\xi_\alpha + \left[u(r)+(-1)^\alpha k\right] + \left[u(r)-(-1)^\alpha k\right]\xi_\alpha^2,$$

where, up to gamma-matrix conventions,

$$u(r)=\sqrt{\frac{g_{xx}}{-g_{tt}}}\,[\omega+qA_t(r)].$$

The horizon condition is usually a simple constant, for example $\xi_\alpha(r_h)=i$ for $\omega\ne0$ in a common basis. Then

$$G^R_\alpha(\omega,k) \propto \lim_{r\to\infty}r^{2mL}\xi_\alpha(r;\omega,k).$$

This flow-equation method is the workhorse for plotting holographic fermion spectral functions. It is also a good sanity check: the result must be insensitive to small changes in the radial cutoff and to the overall normalization of the infalling solution.

Conventions warning

Different papers place the radial gamma matrix, Pauli matrices, and spinor components in different orders. The source/response ratio may be $b/a$, $a/b$, or differ by factors of $i$ and $\gamma^t$. The invariant statements are: impose infalling conditions, identify source and response by the variational principle, and locate poles by vanishing source.

The charged black brane and the $AdS_2$ throat

For the Reissner—Nordström AdS black brane, the zero-temperature near-horizon geometry is

$$AdS_2\times\mathbb R^{d_s}.$$

This region controls the low-frequency fermion response. The spatial momentum $k$ is a parameter from the point of view of $AdS_2$; it contributes to an effective mass in the $AdS_2$ Dirac equation. The IR scaling dimension is

$$\delta_k=\frac{1}{2}+\nu_k,$$

where, schematically,

$$\boxed{ \nu_k = \sqrt{m_{\rm eff}^2(k)L_2^2-q_{\rm eff}^2} }$$

with

$$m_{\rm eff}^2(k) \sim m^2+\frac{k^2}{g_{xx}(r_h)}.$$

The constants $L_2$ and $q_{\rm eff}$ depend on the near-horizon electric field and on the normalization of the Maxwell field. Nonminimal spinor couplings can modify $m_{\rm eff}(k)$ and $q_{\rm eff}$.

The retarded Green function of the IR $AdS_2$ problem has the universal small-frequency form

$$\mathcal G_k^R(\omega) \propto \omega^{2\nu_k},$$

with a complex phase fixed by infalling boundary conditions. This is the origin of semi-local quantum criticality in the fermion spectral function: time scales, but space mostly enters as a label through $\nu_k$.

At finite temperature, the $AdS_2$ result is thermally rounded:

$$\mathcal G_k^R(\omega,T) = T^{2\nu_k}\,\Phi_{\nu_k,q_{\rm eff}}\!\left(\frac{\omega}{T}\right),$$

where $\Phi$ is a known ratio of Gamma functions in the pure $AdS_2$ region. The important point is the scaling form, not the precise normalization.

Matching: from the IR throat to the UV boundary

The full Green function is obtained by matching an inner $AdS_2$ solution to an outer solution in the asymptotic AdS region. At small $\omega$, the result has the form

$$G^R(\omega,k) = \frac{ b_+^{(0)}(k)+\omega b_+^{(1)}(k)+\cdots +\mathcal G_k^R(\omega)\left[b_-^{(0)}(k)+\omega b_-^{(1)}(k)+\cdots\right] }{ a_+^{(0)}(k)+\omega a_+^{(1)}(k)+\cdots +\mathcal G_k^R(\omega)\left[a_-^{(0)}(k)+\omega a_-^{(1)}(k)+\cdots\right] }.$$

The coefficients $a_\pm^{(n)}$ and $b_\pm^{(n)}$ are determined by the UV-to-IR geometry. They are analytic in $\omega$ near $\omega=0$. The nonanalytic physics is isolated in $\mathcal G_k^R(\omega)$.

If a Fermi surface exists, then

$$a_+^{(0)}(k_F)=0.$$

Expanding near $k_F$ gives the canonical holographic-fermion form

$$\boxed{ G^R(\omega,k) \simeq \frac{h_1}{k_\perp-\omega/v_F-h_2\,\mathcal G_{k_F}^R(\omega)} }$$

where

$$k_\perp=k-k_F.$$

Since

$$\mathcal G_{k_F}^R(\omega)\sim\omega^{2\nu_{k_F}},$$

the exponent $\nu_{k_F}$ determines the character of the spectral peak.

Three regimes of holographic Fermi surfaces

Fermi-liquid-like poles: $\nu_{k_F}>1/2$

When $\nu_{k_F}>1/2$, the analytic $\omega/v_F$ term dominates the denominator at low frequency. The pole has approximately linear dispersion,

$$\omega\simeq v_F k_\perp,$$

and the width scales as

$$\Gamma(\omega)\sim \omega^{2\nu_{k_F}}.$$

Therefore

$$\frac{\Gamma}{\omega}\sim \omega^{2\nu_{k_F}-1}\to0, \qquad \omega\to0.$$

The excitation becomes sharp at low energy. It is Fermi-liquid-like in the sense of having a long-lived pole, but it is not automatically an ordinary Landau quasiparticle. The exponent need not be $2$, and the operator is a large-$N$ gauge-invariant fermion rather than a microscopic electron.

Marginal case: $\nu_{k_F}=1/2$

At $\nu_{k_F}=1/2$, the analytic $\omega$ term and the IR self-energy compete. In many conventions the self-energy becomes

$$\Sigma(\omega) \sim \omega\log\omega-i c\omega.$$

This resembles marginal Fermi liquid phenomenology: the decay rate is of the same order as the energy, up to logarithms. The spectral peak is neither a stable quasiparticle nor a fully featureless continuum.

Non-Fermi-liquid poles: $0<\nu_{k_F}<1/2$

When $0<\nu_{k_F}<1/2$, the nonanalytic IR term dominates:

$$G^R(\omega,k) \simeq \frac{h_1}{k_\perp-h_2\omega^{2\nu_{k_F}}}.$$

The dispersion is nonlinear,

$$\omega\sim k_\perp^{1/(2\nu_{k_F})},$$

and the width is of the same order as the energy. There is still a low-energy singularity at $k_F$, but it is not a Landau quasiparticle.

The punchline is compact:

$$\boxed{ \nu_{k_F}\text{ controls whether a holographic Fermi surface is sharp, marginal, or incoherent.} }$$

Algebraic pseudogaps and no Fermi surface

Not every holographic fermion has a Fermi momentum. If the bulk spinor does not form a quasi-bound state, then the $AdS_2$ continuum can dominate directly:

$$A(\omega,k) \sim \omega^{2\nu_k}$$

at small $\omega$ and fixed $k$, up to phases and matrix factors. This is sometimes called an algebraic pseudogap. The spectral weight vanishes as a power rather than being exponentially gapped.

This behavior is one of the cleanest fingerprints of local quantum criticality. It says that low-frequency fermionic spectral weight is controlled by an IR scaling dimension that depends on momentum. Unlike a Fermi liquid, there is no special shell of low-energy excitations around $k_F$; the whole momentum dependence enters through $\nu_k$.

Log-oscillatory regions and IR instabilities

The exponent $\nu_k$ can become imaginary:

$$\nu_k=i\lambda_k, \qquad \lambda_k\in\mathbb R.$$

Then

$$\mathcal G_k^R(\omega) \sim \omega^{2i\lambda_k} = \exp\!\left(2i\lambda_k\log\omega\right),$$

which is periodic in $\log\omega$. This is the log-oscillatory region.

The mathematical reason is an $AdS_2$ analogue of violating a stability bound. The physical interpretation is that the charged near-horizon electric field is strong enough to produce an instability in the probe fermion sector. The simple RN-AdS background should then not be trusted as the final ground state. One expects the bulk charge to reorganize, for example into fermionic matter outside the horizon. This is the doorway to electron stars, Dirac hair, and Luttinger-count questions, which are the subject of the next page.

Do not overinterpret log oscillations

Log-periodic fermion response is not merely exotic scaling. It is usually a warning that the chosen background and probe approximation are missing important backreaction or an instability channel.

Schrödinger potential intuition

Squaring the Dirac equation gives a second-order radial equation that can be viewed as a Schrödinger problem,

$$-\partial_{r_*}^2\varphi+V_{\rm eff}(r;k,\omega)\varphi=0,$$

where $r_*$ is a tortoise-like radial coordinate. The precise potential is basis-dependent, but the intuition is robust.

The geometry has three regions:

• the UV AdS boundary, where source and response are defined,
• an intermediate domain wall controlled by finite density,
• the IR horizon or $AdS_2$ throat, which absorbs probability.

For suitable $(m,q)$, the effective potential develops a well in the intermediate region. A quasi-bound state in this well is the bulk origin of the boundary Fermi-surface pole. Because the inner barrier is not perfectly reflecting, the state can tunnel into the horizon. That tunneling is the imaginary part of the boundary self-energy.

This explains why holographic Fermi surfaces can look familiar and unfamiliar at the same time. The spectral function may have a peak dispersing through $k_F$, but the decay channel is not phonons, impurities, or particle-hole excitations of a Fermi liquid. It is the strongly coupled IR sector represented by the horizon.

Semi-holographic interpretation

The matched Green function has a useful boundary interpretation. Near a Fermi momentum, it is as if an emergent fermion $c$ is coupled to a locally critical large-$N$ bath:

$$S_{\rm eff} = \int_{\omega,k} c^\dagger(\omega,k) \left(\omega-v_F k_\perp\right)c(\omega,k) + \lambda\int_{\omega,k} \left(c^\dagger\mathcal O_{\rm IR}+\mathcal O_{\rm IR}^\dagger c\right) +S_{\rm IR}.$$

The bath correlator is

$$\langle\mathcal O_{\rm IR}\mathcal O_{\rm IR}^\dagger\rangle = \mathcal G_{k_F}^R(\omega) \sim \omega^{2\nu_{k_F}}.$$

Integrating out the bath gives

$$G_c^R(\omega,k) = \frac{1}{\omega-v_F k_\perp-\lambda^2\mathcal G_{k_F}^R(\omega)}.$$

This is exactly the structure produced by the bulk matching calculation. The large-$N$ nature of the IR sector suppresses many corrections that would ordinarily complicate an interacting electron problem. This is why the simple self-energy form is controlled in holography, even though the same expression would be much more phenomenological in a microscopic condensed-matter model.

Nonminimal couplings and model dependence

The minimal Dirac action is not the only possible fermion model. Bottom-up holography often adds Pauli or dipole couplings such as

$$S_{\rm Pauli} \sim p\int d^{d_s+2}x\sqrt{-g}\, \bar\Psi\Gamma^{MN}F_{MN}\Psi.$$

Such terms can shift Fermi momenta, change the IR exponent, create zeros of the Green function, or open gap-like features. This is useful for model-building, but it reduces universality. Unless the coupling is fixed by a top-down truncation or symmetry, one should treat it as phenomenological.

The same warning applies to the choice of background. RN-AdS, electron stars, probe brane systems, Q-lattices, and hyperscaling-violating geometries can all give different fermion spectral functions. Holography is not a single prediction for ARPES; it is a controlled framework for producing and organizing strongly coupled spectral functions.

What is trustworthy?

The most trustworthy statements are structural:

• Retarded fermion correlators come from infalling solutions of a bulk Dirac equation.
• The Fermi momentum is a normalizable zero mode of the radial Dirac problem.
• In an $AdS_2$ throat, the IR Green function scales as $\omega^{2\nu_k}$.
• The matched Green function naturally has a semi-holographic self-energy.
• Imaginary $\nu_k$ signals an IR instability or at least a breakdown of the naive probe background.

Less universal are the numerical values of $k_F$, the number of Fermi surfaces, the existence of zeros, and the detailed resemblance to a particular material.

The ARPES analogy has limits

A holographic spectral function is like ARPES because it measures fermionic spectral weight as a function of $(\omega,k)$. It is unlike ARPES if the operator $\mathcal O_\Psi$ is not the microscopic electron. Agreement with a broad spectral shape is suggestive; a convincing comparison should involve constrained predictions for transport, thermodynamics, optical conductivity, Hall response, and the evolution of poles.

Common pitfalls

Mistaking every pole for a Landau quasiparticle. A pole with width comparable to its energy is not a long-lived quasiparticle. The exponent $\nu_{k_F}$ matters.

Forgetting the large-$N$ limit. Probe fermion spectral functions are leading large-$N$ observables. They do not automatically include the full quantum dynamics of a finite-density electron fluid.

Reading $k_F$ as the full charge density. The probe calculation can find Fermi momenta, but it does not by itself prove a Luttinger theorem. Horizon charge may carry part of the density.

Ignoring instabilities. Log-oscillatory behavior is not merely exotic scaling. It often signals that the RN-AdS background is not the correct ground state.

Overfitting materials. Holographic fermions can resemble features of cuprate or heavy-fermion spectra, but resemblance is not identification. The operator dictionary and model status must be stated honestly.

Exercises

Exercise 1: Spinor dimension from the near-boundary Dirac equation

In asymptotic $AdS_{d_s+2}$, the rescaled spinor behaves as

$$\psi\sim a\,r^{mL}+b\,r^{-mL}.$$

Use the usual factor relating the full spinor $\Psi$ to the rescaled spinor to argue that the two possible operator dimensions are

$$\Delta_\pm=\frac{d_s+1}{2}\pm mL.$$

Solution

Near the boundary,

$$\Psi=(-g g^{rr})^{-1/4}\psi.$$

For

$$ds^2\sim \frac{r^2}{L^2}dx_d^2+\frac{L^2}{r^2}dr^2,$$

one has

$$(-g g^{rr})^{-1/4}\sim r^{-(d_s+1)/2}.$$

Thus

$$\Psi\sim a\,r^{-\frac{d_s+1}{2}+mL} +b\,r^{-\frac{d_s+1}{2}-mL}.$$

For a spinor operator, the coefficient multiplying the non-normalizable falloff is the source for an operator whose scaling dimension is

$$\Delta_+=\frac{d_s+1}{2}+mL$$

in standard quantization. When alternative quantization is allowed, the other choice gives

$$\Delta_- = \frac{d_s+1}{2}-mL.$$

Exercise 2: Why a Fermi surface is a normalizable zero mode

Suppose the infalling bulk spinor solution has boundary coefficients $a(\omega,k)$ and $b(\omega,k)$, with

$$G^R(\omega,k)=\frac{b(\omega,k)}{a(\omega,k)}.$$

Show that $a(0,k_F)=0$ gives a pole in $G^R$ at $\omega=0$, $k=k_F$. Explain why this is the holographic Fermi-surface condition.

Solution

If

$$a(0,k_F)=0$$

while $b(0,k_F)$ is finite and nonzero, then

$$G^R(0,k)=\frac{b(0,k)}{a(0,k)}$$

has a pole at $k=k_F$. The vanishing of $a$ means that the source term is absent. The bulk solution is therefore normalizable with infalling interior boundary conditions.

A pole of the retarded Green function at zero frequency and finite momentum is precisely the operational definition of a Fermi surface, even when there is no long-lived Landau quasiparticle.

Exercise 3: Width of a holographic Fermi-surface pole

Near $k_F$, assume

$$G^R(\omega,k) \simeq \frac{h_1}{k_\perp-\omega/v_F-h_2\omega^{2\nu}},$$

where $\nu$ is real and positive. For $\nu>1/2$, show that the excitation is sharp at low energy.

Solution

For $\nu>1/2$, the analytic term $\omega/v_F$ dominates over $\omega^{2\nu}$ at sufficiently small $\omega$, because

$$\omega^{2\nu}\ll \omega.$$

The leading dispersion is therefore

$$\omega\simeq v_F k_\perp.$$

The imaginary part of $h_2\omega^{2\nu}$ gives the width,

$$\Gamma(\omega)\sim \omega^{2\nu}.$$

Hence

$$\frac{\Gamma}{\omega}\sim \omega^{2\nu-1}\to0$$

as $\omega\to0$. The pole becomes sharp. It is Fermi-liquid-like, although it is only an ordinary Fermi-liquid pole when the detailed exponent and residue structure match the Landau case.

Exercise 4: Non-Fermi-liquid dispersion

For $0<\nu<1/2$, the nonanalytic self-energy dominates. Estimate the dispersion relation of the spectral peak.

Solution

When $0<\nu<1/2$,

$$\omega^{2\nu}\gg \omega$$

at small $\omega$. The denominator is approximately

$$k_\perp-h_2\omega^{2\nu}.$$

The peak occurs when this nearly vanishes:

$$k_\perp\sim h_2\omega^{2\nu}.$$

Thus

$$\omega\sim k_\perp^{1/(2\nu)}.$$

Because the phase of $h_2\omega^{2\nu}$ is complex, the width is generally of the same order as the energy. The excitation is not a long-lived quasiparticle.

Exercise 5: Log oscillations from imaginary $\nu_k$

Let

$$\nu_k=i\lambda, \qquad \lambda>0.$$

Show that the IR Green function is periodic in $\log\omega$. What does this suggest physically?

Solution

The IR Green function scales as

$$\mathcal G_k^R(\omega)\sim \omega^{2\nu_k}.$$

If $\nu_k=i\lambda$,

$$\mathcal G_k^R(\omega) \sim \omega^{2i\lambda} = \exp\!\left(2i\lambda\log\omega\right).$$

This is periodic under

$$\log\omega\to \log\omega+\frac{\pi}{\lambda}.$$

The oscillation in logarithmic frequency indicates that the IR scaling dimension has become complex. In the $AdS_2$ description, this is analogous to violating an IR stability bound. Physically it warns that the probe fermion sector or the RN-AdS background is unstable and that backreaction or a new ground state must be considered.

Further reading

The standard research papers on holographic fermions include the early work by Henningson and Sfetsos, Mueck and Viswanathan, Iqbal and Liu, Lee, Liu, McGreevy and Vegh, Cubrovic, Zaanen and Schalm, and Faulkner, Liu, McGreevy and Vegh. For broad treatments, see Hartnoll, Lucas and Sachdev, Holographic Quantum Matter, especially the sections on fermions in compressible phases; and Zaanen, Liu, Sun and Schalm, Holographic Duality in Condensed Matter Physics, chapter 9. The next page continues from the probe calculation to backreacted fermion charge, electron stars, Dirac hair, and Luttinger counts.