Holographic Fermions and Spectral Functions

A holographic fermion spectrum is obtained from a radial boundary-value problem, not from a normal-mode plot alone. One must specify the spinor action and charge normalization, identify source and response components at the AdS boundary, impose a future-horizon condition for the retarded function, and state the contact-term and quantization conventions. The resulting poles and spectral weight characterize the chosen large-$N$ state.

Required background. Euclidean and Real-Time Access at Finite Density supplies the finite-density analytic continuation. Lorentzian Holographic Correlators and Infalling Conditions supplies the causal horizon prescription.

Helpful background. Thermal Propagators and Spectral Representations fixes spectral-function conventions. Bulk Fields and Boundary Operators supplies the near-boundary source–response map.

The radial Dirac problem

For a probe spinor of mass $m$ and charge $q$,

$$S_\psi=\mathcal N_\psi\int d^{d+1}x\sqrt{\lvert g\rvert}\, \bar\psi\left(i\Gamma^aD_a-m\right)\psi+S_{\mathrm{bdy}}, \qquad D_a=\nabla_a-iqA_a .$$

The normalization $\mathcal N_\psi$, the normalization of $A_a$, and the boundary term are part of the observable definition. Fourier transform as $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$ and rescale away the spin connection. Because the radial orthonormal direction is spacelike in the inherited $(+,-,\ldots,-)$ convention,

$$(\Gamma^{\underline z})^2=-1, \qquad P_\pm=\frac12\left(1\pm i\Gamma^{\underline z}\right), \qquad \Gamma^{\underline z}\psi_\pm=\mp i\psi_\pm.$$

These are genuine complementary projectors; $(1\pm\Gamma^{\underline z})/2$ would not be idempotent. For non-half-integer $mL$, the asymptotic components take the schematic form

$$\psi_-=z^{d/2-mL}\,A+\cdots, \qquad \psi_+=z^{d/2+mL}\,B+\cdots .$$

In standard quantization $A$ is the source and $B$ is the response. Solving a basis of ingoing solutions gives matrices $A(\omega,\mathbf k)$ and $B(\omega,\mathbf k)$; after the conventional gamma-matrix factor and local counterterms,

$$G_R(\omega,\mathbf k)=\mathcal N_\psi\,\mathcal S_\Gamma\,B A^{-1}+G_{\mathrm{local}} .$$

This matrix construction avoids assigning a Green function to a component before the boundary variational problem has been diagonalized. The detailed prescription and its Euclidean continuation are derived in Iqbal and Liu 2009, §§ 3–4.

Causality at the horizon

Near a nonextremal future horizon,

$$\psi\sim(r-r_h)^{-i\omega/(4\pi T)}\psi_{\mathrm{in}}$$

in an outgoing-singular Schwarzschild coordinate. This is the solution regular in ingoing Eddington–Finkelstein coordinates. The opposite exponent computes an advanced response. At $\omega=0$, the prescription is defined by analytic continuation from $\operatorname{Im}\omega>0$, which removes an otherwise ambiguous choice of real static solutions.

The spectral density convention used here is

$$\mathcal A(\omega,\mathbf k)=-2\,\operatorname{Im}\operatorname{Tr}G_R(\omega,\mathbf k).$$

For a Hermitian operator it is nonnegative at positive frequency after the appropriate spinor projection. Violating that property is a useful diagnostic for a sign, horizon, or gamma-matrix error; a local real contact term cannot repair a negative absorptive part.

Tracking a Fermi pole

As a first application, place the probe spinor in a charged black-brane background and scan the zero-frequency source matrix. A Fermi momentum is a real $k_F$ satisfying

$$\det A(0,k_F)=0,$$

with a normalizable ingoing solution. Near an isolated pole one may fit

$$G_R(\omega,k)\simeq \frac{Z}{k-k_F-v_F^{-1}\omega-\Sigma(\omega)}+G_{\mathrm{reg}}.$$

The fit must be made in a frequency window where a single pole dominates and checked against direct complex-frequency pole finding. In charged AdS black holes this procedure produces sharp Fermi surfaces and non-Landau self-energies for ranges of $(m,q)$ Liu, McGreevy, and Vegh 2011. The pole is evidence for a fermionic excitation in that holographic state, not evidence that the boundary state contains weakly coupled electrons.

Quantization and horizon adversaries

Three controlled changes test the interpretation.

1. Boundary quantization. When alternate quantization is allowed, exchange source and response. Poles and zeros are interchanged up to local terms, so an asserted Fermi surface must name the quantization.
2. Counterterms. Add admissible finite local terms. They can shift the analytic background but cannot create an absorptive pole. A feature that disappears under a local redefinition was not a robust excitation.
3. Horizon condition. Replace infalling by outgoing data. The imaginary part and pole half-plane reverse; if a code returns the same answer, it has not implemented causal response.

Backreaction is also a physical boundary: a probe spectrum is reliable only when the fermion sector’s stress tensor and charge density are parametrically negligible relative to the background.

Exercises

Show that a pole of $G_R=B A^{-1}$ becomes a zero under the formal exchange $A\leftrightarrow B$.

Solution

Under the exchange, the nonlocal part becomes $A B^{-1}=G_R^{-1}$ up to normalization and local terms. Hence a zero eigenvalue of $A$, which yields a pole of $G_R$, yields a zero eigenvalue of the exchanged response.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Iqbal, Nabil, and Hong Liu. “Real-Time Response in AdS/CFT with Application to Spinors.” Fortschritte der Physik 57, 367–384 (2009). DOI.
• Liu, Hong, John McGreevy, and David Vegh. “Non-Fermi Liquids from Holography.” Physical Review D 83, 065029 (2011). DOI.