Bekenstein-Hawking Entropy and Strange Metals

TL;DR

The paper presents a detailed correspondence between a Sachdev-Ye-Kitaev–like infinite-range fermion model and holographic theories with AdS$_2$ horizons. It shows that the low-energy fermionic correlators exhibit a spectral asymmetry parameter $\omega_{\mathcal{S}}$ tied to the zero-temperature entropy via $\omega_{\mathcal{S}} = \frac{q T}{\hbar} \frac{\partial \mathcal{S}}{\partial \mathcal{Q}}$, and demonstrates that the same relation emerges from the near-horizon AdS$_2$ geometry of charged black holes through the AdS/CFT correspondence. A Maxwell/Wald-type thermodynamic identity, $\frac{\partial \mathcal{S}_{\rm BH}}{\partial \mathcal{Q}} = 2\pi \mathcal{E}$, matches the field-theoretic derivative, providing a precise quantitative link between microscopic entropy and Bekenstein-Hawking entropy. The work strengthens the case for a gravity dual of the SY state with an AdS$_2$ horizon and clarifies how universal low-energy data are encoded in black-hole thermodynamics, while highlighting the role of UV completion in determining nonuniversal constants.

Abstract

We examine models of fermions with infinite-range interactions which realize non-Fermi liquids with a continuously variable U(1) charge density $\mathcal{Q}$, and a non-zero entropy density $\mathcal{S}$ at vanishing temperature. Real time correlators of operators carrying U(1) charge $q$ at a low temperature $T$ are characterized by a $\mathcal{Q}$-dependent frequency $ω_{\mathcal{S}} = (q \, T/\hbar) (\partial \mathcal{S}/\partial{\mathcal{Q}})$ which determines a spectral asymmetry. We show that the correlators match precisely with those of the AdS$_2$ horizons of extremal charged black holes. On the black hole side, the matching employs $\mathcal{S}$ as the Bekenstein-Hawking entropy density, and the laws of black hole thermodynamics which relate $(\partial{\mathcal{S}}/\partial{\mathcal{Q}})/(2 π)$ to the electric field strength in AdS$_2$. The fermion model entropy is computed using the microscopic degrees of freedom of a UV complete theory without supersymmetry.

Figures (2)

• Figure 1: Plots of the Green's functions in Eq. (\ref{['GR']}) for $\Delta = 1/4$, $q=1$, $T=1$, $A=1$, $\mathcal{E} = 1/4$ with $\hbar=k_B=1$. Note that while neither $\hbox{Im} G^R (\omega)$ or $\hbox{Re} G^R (\omega)$ have any definite properties under $\omega \leftrightarrow - \omega$, the product $G^R (\omega) G^A (\omega)$ becomes an even function of $\omega$ after a shift by $\omega_{\mathcal{S}} = 2 \pi q \mathcal{E} T = \pi/2$.
• Figure 2: Summary of the properties of the SY state (Section \ref{['sec:inf']}) and planar charged black holes (Section \ref{['sec:holo']}) at $T=0$. The spatial co-ordinate $\vec{x}$ has $d$ dimensions. All results apply also to spherical black holes considered in Appendix \ref{['app:sphere']}. The AdS$_2 \times R^d$ metric has unimportant prefactors noted in Eq. (\ref{['factor']}) which are not displayed above. The fermion mass $m$ has to be adjusted to obtain the displayed power-law. The spectral asymmetry parameter $\mathcal{E}$ appears in the fermion correlators and in the AdS$_2$ electric field. As the charge $\mathcal{Q}$ is increased, the horizon moves closer to the boundary, and its area, $\mathcal{A}_h$, increases. In black hole thermodynamics, the Bekenstein-Hawking entropy density, $\mathcal{S}_{\rm BH}$ is related to area of the horizon via $\mathcal{S}_{\rm BH} = \mathcal{A}_h/(4 G_N \mathcal{A}_b)$, where $G_N$ is Newton's constant.