Thermal Spectral Function in Holographic CFTs

• Thermal spectral functions in holographic CFTs encode the absorptive part of retarded correlators, linking boundary quantum dynamics with bulk quasinormal spectra.
• They are computed using the Son–Starinets prescription, which reveals the pole structure, duality constraints, and transport coefficients in strongly coupled systems.
• Analytic product formulas and numerical methods enable a detailed exploration of hydrodynamic modes, non-equilibrium dynamics, and thermalization processes.

The thermal spectral function of holographic conformal field theories (CFTs) encodes the absorptive part of retarded two-point correlators at finite temperature and serves as a bridge between boundary quantum field theory dynamics and the quasinormal spectra of classical black holes or black branes in the bulk gravitational theory. In holography, the analytic structure, pole content, and duality properties of these spectral functions are precisely dictated by the bulk geometry and its characteristic symmetries, such as electric–magnetic duality and S-duality. The spectral function is a central object in the study of strongly-coupled quantum matter, providing direct access to transport coefficients, hydrodynamic modes, and the mechanism of thermalisation after energy injection.

1. Definition and General Properties of Thermal Spectral Functions

For a Hermitian operator $O$ in a $d$-dimensional CFT at temperature $T=1/\beta$, the retarded Green’s function is defined by

$$G_R(\omega, \mathbf{k}) = \int_{0}^\infty dt \int d^{d-1}x\, e^{i\omega t - i\mathbf{k}\cdot \mathbf{x}}\, \langle [O(t, \mathbf{x}), O(0, 0)] \rangle_\beta.$$

The thermal spectral function is

$$\rho(\omega, \mathbf{k}) = -2\,\mathrm{Im}\,G_R(\omega, \mathbf{k}),$$

encoding the density of states in response to external perturbations. In equilibrium, $\rho(\omega, \mathbf{k})$ satisfies the fluctuation–dissipation theorem and KMS condition: $$G_{12}(\omega, \mathbf{k}) = e^{-\omega/T} G_{21}(\omega, \mathbf{k}) \implies \rho(-\omega, \mathbf{k}) = -e^{-\omega/T} \rho(\omega, \mathbf{k})$$ which ensures causality and thermal detailed balance (Banerjee et al., 2016).

In large-$N$ holography, $G_R(\omega, \mathbf{k})$ is a meromorphic function of $\omega$ with simple poles in $\mathrm{Im}\,\omega < 0$, corresponding to the bulk quasinormal modes (QNMs) of the dual black hole spacetime (Grozdanov et al., 22 Sep 2025, Grozdanov et al., 28 Jun 2024).

2. Holographic Calculation: Bulk Setup and Prescription

The standard method for obtaining $G_R$ and $\rho$ proceeds via the Son–Starinets prescription (Banerjee et al., 2016). The black brane metric typically takes the form

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2 d\vec{x}^2,$$

with $f(r)$ characteristic of AdS-Schwarzschild (and generalizations to charged/axionic/rotational backgrounds). Linearized perturbations (scalars, vectors, tensors) in this background satisfy wave equations with ingoing boundary conditions at the horizon and Dirichlet/mixed conditions at the AdS boundary.

Near the boundary ($z \to 0$ in Fefferman–Graham coordinates),

$$\Phi(z) \simeq A(\omega, k) z^{\Delta_-} + B(\omega, k) z^{\Delta_+},$$

where $\Delta_\pm$ are related to mass/spin/dimension. The retarded correlator is

$$G_R(\omega, k) \propto \frac{B(\omega, k)}{A(\omega, k)},$$

and the spectral function reads

$\rho(\omega, k) = -2 \mathrm{Im}\,G_R(\omega, k).$

Exact analytic forms are available in certain limits (e.g., AdS$_3$/BTZ, hyperbolic AdS$_5$), while in higher dimensions and generic backgrounds, semi-analytic (WKB, Heun function), or numerical techniques are required (Jia et al., 9 Aug 2024, Bhatta et al., 2023).

QNMs $\omega_n(k)$ are found by shooting methods or spectral techniques (Grozdanov et al., 20 May 2025). Their distribution determines the complete pole structure and thus the analytic and transport properties of $\rho$.

3. Product Structure, Duality, and Thermal Spectral Constraints

The meromorphic structure permits representations of $G_R$ and $\rho$ as infinite products over the QNM spectrum. The thermal product formula states (Grozdanov et al., 22 Sep 2025, Grozdanov et al., 28 Jun 2024): $$\rho(\omega) = \frac{\mu}{\sinh(\beta\omega/2)} \prod_n \left(1-\frac{\omega^2}{\omega_n^2} \right),$$ where $\mu$ is fixed by small-$\omega$ expansion and OPE data.

Duality relations—arising from bulk electric-magnetic (Darboux) symmetry, S-duality, or Legendre transform in double-trace flows—constrain the spectra of pairs of correlators (e.g., longitudinal/transverse, UV/IR) (Grozdanov et al., 20 May 2025, Grozdanov et al., 28 Jun 2024, Grozdanov et al., 22 Sep 2025). For instance,

$$G_+(\omega, k) G_-(\omega, k) = \frac{\omega^2}{\omega_*^2(k)} - 1$$

where $\omega_*(k)$ is the algebraically special frequency from the bulk. The combined spectra are captured by

$$S(\omega) = (1+\omega/\omega_*) \prod_n (1-\omega/\omega_n^+) (1+\omega/\omega_n^-),$$

with the spectral duality constraint

$$S(\omega) - S(-\omega) = 2i \lambda(k) \sinh \frac{\omega}{2T}$$

for some $\lambda(k)$ (Grozdanov et al., 20 May 2025). These nontrivial constraints ensure precise relations between excitation spectra, pole-skipping phenomena, and transport coefficients.

4. Analytic Structure, Quasinormal Modes, and Physical Features

Poles of $G_R$—the QNMs—lie strictly in the lower half $\omega$-plane; their residues set the height, width, and decay rate of resonances in $\rho$. Example (AdS$_5$ planar black brane, scalar of dimension $\Delta$) (Bhatta et al., 2023): $$\rho(\omega) = 2 \sin(\pi\Delta)\, (2\pi T)^{2\Delta-4} \frac{\left|\Gamma(\frac{\Delta}{2} - \frac{i\omega}{4\pi T}) \right|^2}{\left|\Gamma(2 - \frac{\Delta}{2} - \frac{i\omega}{4\pi T}) \right|^2}.$$ Poles occur at $2-\Delta/2 - (i\omega)/(4\pi T) = -n$, yielding the QNM spectrum $\omega_n = -i 4\pi T (n + 2 - \Delta/2)$ (Bhatta et al., 2023).

Branch cuts may arise from continuum contributions or thermal sums, notably in the large-$N$ limit, but closed-form product representations remain valid under the meromorphicity and the KMS condition (Burić et al., 15 May 2025).

Hydrodynamic modes manifest as low-frequency poles, with transport parameters (e.g., diffusion constant $D$, sound attenuation $\Gamma$) determined by near-origin behavior and duality-imposed sum rules (Grozdanov et al., 20 May 2025, Banerjee et al., 2016).

Pole-skipping refers to loci in the complex $\omega$-plane (e.g., Matsubara frequencies $\omega = -2\pi i n T$) where analytic structure allows simultaneous zeroes and poles in $G_R$ for different channels, reflecting deep gravitational features (Grozdanov et al., 20 May 2025).

5. Non-Equilibrium Dynamics, Thermalisation, and Time-Dependent Spectra

In settings with explicit time dependence—e.g., energy injection via AdS–Vaidya quench—time–translation invariance is broken, requiring the use of the Wigner transform to define a time-dependent spectral function (Bernamonti et al., 2013, Balasubramanian et al., 2012): $$G_R(\omega, k; T) = \int_{-\infty}^{+\infty} dt\, e^{i\omega t}\int dx\, e^{-ikx}\, G_R(T + t/2, T - t/2; x),$$

$$\rho(\omega, k; T) = -2\,\mathrm{Im}\,G_R(\omega, k; T).$$

Numerical studies show that $\rho(\omega, k; T)$ interpolates between the vacuum and thermal forms: at early times it exhibits a gap, at late times it mirrors the thermal continuum, with spectral weight transferred to low frequencies as thermalisation progresses (Balasubramanian et al., 2012). The process is typically oscillatory for fixed $\omega$, reflecting quasinormal ringing, with exponential decay rates governed by the lowest QNM (Bernamonti et al., 2013).

Effective temperature $\theta_{\rm eff}(T)$, defined via a fluctuation–dissipation relation,

$$\theta_{\rm eff}(T) = \frac{1}{2} \frac{\mathrm{Im}\,G_F(\omega=0; T)}{\left.\partial_\omega\,\mathrm{Im}\,G_R(\omega; T)\right|_{\omega=0}},$$

tracks the system’s progression from initial to equilibrium temperature, with non-trivial behavior around quench events and for large operator dimension $\Delta$ (Bernamonti et al., 2013).

6. Universal Asymptotics, Operator Product Expansion, and Lightcone Physics

At high frequency $(\omega \gg T)$, the spectral function universally exhibits power-law growth set by the operator dimension,

$\rho(\omega, k) \sim \omega^{2\Delta - d - 1}$

reflecting the OPE short-distance singularity (Banerjee et al., 2016, Bhatta et al., 2023). Near the lightcone ($q^+ \gg T \gg q^-$), spectral functions for stress-tensor correlators decompose into universal functions $f_{\rm scalar}(\alpha)$, $f_{\rm shear}(\alpha)$, $f_{\rm sound}(\alpha)$, with perturbative and non-perturbative contributions extracted via holography and OPE methods (Esper et al., 2023).

Non-perturbative imaginary parts in $f(\alpha)$ produce exponentially small tails in $\rho$ near the lightcone, with higher-derivative corrections (e.g., Gauss–Bonnet couplings) entering only in the argument $\alpha_r$ of the universal functions, leaving the functional form invariant (Esper et al., 2023).

7. Summary Table: Spectral Function Representations

Channel / Context	Formula / Representation	Reference
Scalar, planar AdS$_{d+1}$	$\rho(\omega, k) = 2\sin(\pi\Delta)\ldots$	(Bhatta et al., 2023)
Product over QNMs	$$\rho(\omega) = \mu/\sinh(\beta\omega/2)\prod_n(1-\omega^2/\omega_n^2)$$	(Grozdanov et al., 22 Sep 2025)
Duality constraint	$G_+G_- = \omega^2/\omega_*^2 - 1$, $S(\omega) - S(-\omega) = 2i\lambda\sinh(\omega/2T)$	(Grozdanov et al., 20 May 2025, Grozdanov et al., 28 Jun 2024)
Time-dependent (Vaidya)	$$\rho(\omega, k; T) = -2\mathrm{Im}\,G_R(\omega, k; T)$$ (Wigner)	(Balasubramanian et al., 2012, Bernamonti et al., 2013)
Near-lightcone stress tensor	$$\rho^{\mu\nu,\alpha\beta}(\omega, q) = -2\,\mathrm{Im}\, G_R^{\mu\nu,\alpha\beta}(\omega, q)$$	(Esper et al., 2023)

Holographic thermal spectral functions thus emerge as highly structured, analytically tractable objects, dictated by bulk geometry, QNM spectra, duality symmetries, and equilibrium/non-equilibrium dynamics. Notably, the infinite set of constraints from duality relations and the thermal product formula ensures the possibility of reconstructing the spectrum of one channel from its dual, encoding fundamental transport properties and offering a robust framework for the exploration of strongly coupled quantum systems.