High-Temperature Holographic CFTs

• High-temperature holographic CFTs are strongly coupled field theories whose thermal states are dual to static black brane or black hole solutions, exhibiting universal entropy and transport properties.
• They employ gauge/gravity duality to map thermodynamic observables like stress tensors, finite‑temperature correlators, and conductivity onto classical gravitational phenomena, yielding precise scaling laws.
• Key insights include detailed predictions for anomalous currents, operator growth with chaotic scrambling, and phase structures, with implications supported by both analytic models and experimental setups.

High-temperature holographic conformal field theories (CFTs) constitute a central arena for the quantitative paper of strongly coupled quantum many-body dynamics at finite temperature, exploiting gauge/gravity duality to relate thermal field theories to classical gravitational systems. The high-temperature regime is particularly tractable because the dual gravitational description simplifies to static black hole solutions, and many thermodynamic and dynamical observables exhibit universal behavior controlled by geometry and topology. Key aspects span finite-temperature correlators, anomalous transport, operator growth and chaos, conformal order, and microscopic spectrum and entropy.

1. Holographic Dictionary at High Temperature

At high temperature, a d-dimensional CFT with a classical AdS gravity dual is described by a thermal state associated with a static black brane or black hole background, with the field theory temperature $T$ identified with the Hawking temperature $T = \kappa/2\pi$ of the bulk horizon ($\kappa$ is the surface gravity). On flat space, the planar AdS black brane metric

$$ds^2 = \ell^2\left[-r^2 f(r) dt^2 + \frac{dr^2}{r^2 f(r)} + r^2 d\mathbf{x}^2\right],\quad f(r)=1-(r_h/r)^d$$

has horizon radius $r_h$ fixed by $T = d r_h/(4\pi)$ (Marolf et al., 2013). In curved backgrounds, such as $S^1_\beta\times S^{d-1}_\ell$, the AdS-Schwarzschild or other black holes deliver a precise temperature–horizon map.

The renormalized stress tensor $\langle T_{\mu\nu}\rangle$ at high $T$ (after subtracting Casimir terms) scales extensively,

$$\langle T_{\mu\nu} \rangle \sim c_{\mathrm{eff}}\, T^d\left[(d-1)u_\mu u_\nu + \eta_{\mu\nu}\right],$$

where $c_{\mathrm{eff}}\sim\ell^{d-1}/G_{d+1}$ is the effective central charge. This leading behavior reflects the conformal equation of state.

2. Thermodynamics, Entropy, and Density of States

The equilibrium thermodynamics of high-temperature holographic CFTs is governed by the Bekenstein–Hawking area law, which for a planar black brane yields entropy density

$$s = \frac{\mathrm{Area}_{\rm hor}}{4G_{d+1}\, V_{d-1}} \propto c_{\mathrm{eff}}\, T^{d-1}$$

(Marolf et al., 2013, Benjamin et al., 2023). The free energy density $f$ and internal energy $e$ similarly scale with $T^d$, with small corrections arising from finite-coupling (higher-derivative) terms or subleading geometric effects. For instance, on $S^1_\beta\times S^{d-1}$,

$\log Z \sim f\, T^{d-1} \mathrm{Vol}(S^{d-1}),$

so the density of high-energy states is

$$\rho^{(d)}(\Delta, J) \sim \exp\left[ d(d-1)^{-1} \big( f\, \mathrm{Vol}(S^{d-1})\big)^{1/d} a^{(d-1)/d}\right],$$

where $a$ is a saddle-point auxiliary variable determined by the Laplace transform (Benjamin et al., 2023). In $d=2$, this specializes to the Cardy formula.

3. Finite-temperature Correlators, Conductivity, and OPE Structure

Time-ordered and retarded correlators at $T>0$ in holographic CFTs capture the transport and response properties of strongly coupled plasmas. For conserved currents $J_\mu$ in $2+1$D, the frequency-dependent conductivity $\sigma(\omega)/\sigma_Q$ at high frequency exhibits a universal expansion extracted from the OPE,

$$\sigma(i\omega_n)/\sigma_Q = \sigma_\infty + b_1 (T/\omega_n)^{\Delta_g} + b_2 (T/\omega_n)^3 + \cdots,$$

where $\Delta_g$ is the scaling dimension of the leading thermal operator acquiring an expectation value at $T>0$ (Katz et al., 2014). The coefficients can be matched between OPEs, quantum Monte Carlo, and holography (Einstein–Maxwell–scalar theory in AdS$_4$), yielding parameter-free predictions for models such as the $O(2)$ Wilson–Fisher CFT.

Boundary anomalous currents in holographic BCFTs at high $T$ exhibit universal nontrivial scaling: $$J \sim T^{d-4}\quad(d \neq 4),\qquad J \sim -2b\,\ln(\pi T)\quad(d=4),$$ contrasting the linear-$T$ behavior of free theories and signaling strong coupling and finite resistivity in the bulk dual (Liu et al., 2021).

Finite-$T$ two- and three-point functions in experimental table-top settings such as hyperbolic lattices engineered with electric circuits have been shown to reproduce holographic predictions, including BTZ black hole temperature and spatial-spectral structure (Dey et al., 3 Apr 2024).

4. Scrambling, Chaos, and Operator Growth

Strongly coupled high-$T$ holographic CFTs display rapid quantum chaos with a maximal Lyapunov exponent $\lambda_L=2\pi T$ and associated scrambling time

$$t_* \sim \frac{1}{\lambda_L} \log N_{\mathrm{eff}},$$

diagnosed by the decay of out-of-time-ordered correlators (OTOCs) (Hernandez et al., 5 Nov 2025).

Precise analytic formulas relate differences in scrambling times $\Delta t_*$ to excitation kinematics (energy $E$ and angular momentum $J$) both in the bulk (BTZ, AdS-Schwarzschild black holes) and on the boundary via smeared operator construction and CFT OTOC evaluation. In $d=2$, all local perturbations eventually scramble. In higher dimensions $(d>2)$, null geodesics with $|J|/E$ above a critical value do not reach the horizon—the angular momentum barrier implies stalling rather than scrambling:

• For $J<J_{\rm crit}$, $\Delta t_*(J)$ increases logarithmically with $J$.
• For $J\geq J_{\rm crit}$, there is no leading-order scrambling:

$$t_*(J) \rightarrow \infty \text{ as } J \to J_{\rm crit}^{-}.$$

This phase transition in operator growth structure is a direct prediction for operator evolution in high-temperature holographic CFTs (Hernandez et al., 5 Nov 2025).

5. High-Temperature Phases, Order, and Instabilities

Beyond the standard deconfined (plasma) phase, finite-temperature holographic CFTs can realize metastable or unstable ordered phases:

• Conformal order (thermally translation invariant states with nonzero operator vevs) exists for all temperatures in bottom-up models with higher-derivative corrections but is always subdominant to the “symmetric” black brane phase (Buchel, 2023).
• The dimensionless coefficient $\kappa$ controlling the free energy density $F/T^4$ in these phases satisfies $\kappa < 1$: ordered solutions are metastable and persist for all $T$ and allowed higher-derivative couplings but are never entropically favored or dynamically stable.
• There is generally no true phase transition: the ordered and symmetric (deconfined) branches coexist at all $T$ in the allowed coupling range, and the equilibrium state remains the deconfined plasma. A plausible implication is that strongly coupled holographic plasmas at high $T$ universally maximize entropy and degrade all order parameters.

Distinct spatial structures appear for CFTs on nontrivial manifolds and with compactifications: “plasma balls” (droplets of deconfined plasma in a confining vacuum) and “funnels/droplets” in the presence of boundary black holes (Marolf et al., 2013). These phases are governed by critical dimensionless ratios (e.g., $R T$) and marked by the topology of the dual bulk horizon.

6. Microscopic Spectrum, OPE Data, and Universal Asymptotics

The thermal effective action of a CFT at high $T$ determines the asymptotic form of the spectrum and operator product coefficients:

• The partition function on $S^1_\beta\times S^{d-1}$, utilizing a thermal derivative expansion, yields explicit formulas for $\log \rho(\Delta, J_i)$ at large $(\Delta, J_i)$, matching the Bekenstein–Hawking entropy of rotating AdS black holes. Subleading corrections stem from higher-derivative terms and nonperturbative saddle points (Benjamin et al., 2023).
• Universal asymptotics for heavy-heavy-heavy OPE coefficients are extracted via genus-2 "hot spot" geometries, leading to exponential growth behaviors related to multi-black-hole wormhole amplitudes.
• Thermal one-point functions for primary operators with spin $J$ obey scaling laws fixed by the thermal effective field theory and the underlying AdS geometry, in direct analogy with hair on Kerr–AdS black holes.

Thermal observables such as the free energy and specific heat on $S^1\times{\mathbb R}^2$ and $S^1\times S^2$ at large $T$ are computable via supersymmetric localization and holography for ABJM and similar M2-brane theories. The large-$T$ expansion takes the universal form: $$F(T) = a_3 N^{3/2} T^3 + a_1 N^{1/2} T + a_{-1} N^{-1/2}T^{-1} + \dots,$$ where $a_i$ can be fixed via localization or supergravity computations (Bobev et al., 2023).

7. Universality, Strong Coupling Effects, and Model Realizations

The high-temperature scaling laws in holographic CFTs are signatures of strong coupling:

• Anomalous currents in holographic BCFTs grow more slowly than in weakly coupled models ($T^{d-4}$ or $\ln T$ as opposed to $T$) (Liu et al., 2021).
• Transport observables, OPE coefficients, and spectrum asymptotics derived holographically display independence of microscopic boundary details in the high-$T$ regime.
• Experimental simulations using hyperbolic lattices and nonlinear electric circuits reproduce CFT correlator scaling, confirming the universality of the AdS/CFT thermal correspondence at the tabletop scale (Dey et al., 3 Apr 2024).

Strong coupling corrections—via higher-derivative (e.g., Riemann$^2$, Weyl$^4$) terms or finite ’t Hooft coupling/large-$N$ effects—enter at subleading order in $1/T$ or $1/c_\mathrm{eff}$ and may affect stability, transport gaps, or phase structure, but the universal high-$T$ scaling persists.

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In conclusion, high-temperature holographic CFTs exhibit a robust and universal structure controlled by their gravitational duals: thermal states (black branes/holes), hydrodynamics, entropy, anomalous and transport phenomena, chaos, and operator product asymptotics are all computable and display distinctive strong-coupling behavior. These phenomena admit direct theoretical (and increasingly, experimental) probes of the correspondence between quantum many-body systems and emergent gravitational dynamics.