Bekenstein-Hawking Entropy and Strange Metals

We examine models of fermions with infinite-range interactions that realize non-Fermi liquids with a continuously variable U(1) charge density $\mathcal{Q}$ and a nonzero 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 ${\omega }_{\mathcal{S}}=(qT/\hslash )(\partial \mathcal{S}/\partial \mathcal{Q})$, which determines a spectral asymmetry. We show that the correlators match precisely with those of the two-dimensional anti–de Sitter (${\mathrm{AdS}}_2$) 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 that relate $(\partial \mathcal{S}/\partial \mathcal{Q})/(2\pi )$ to the electric field strength in ${\mathrm{AdS}}_2$. The fermion model entropy is computed using the microscopic degrees of freedom of a UV complete theory without supersymmetry.

Popular Summary

The combination of general relativity and quantum theory led Jacob Bekenstein and Stephen Hawking to the startling result that black hole horizons obey thermodynamics and possess an entropy and a temperature. A reasonable interpretation of the black hole entropy is that it is associated with quantum degrees of freedom at the very small Planck length scale. Thus far, this interpretation has only been verified using models that derive from string theory. Here, we examine a class of models long-studied as simple descriptions of the “strange metal” phase of high-temperature superconductors and other correlated electron compounds. We show that their quantum correlations and thermodynamic properties match precisely with those of the horizons of “extremal” charged black holes, including Bekenstein-Hawking entropy.

We focus on planar, charged black holes described by the Maxwell-Einstein theory, although our findings can also be applied to a large class of charged black holes. The models of fermions that we consider interact over an infinite range, and their entropy is nonzero even at zero temperature. We mathematically show that the quantum and thermal properties of the strange metal state, i.e., a quantum state without quasiparticle excitations proposed by Sachdev and Ye in 1993 (and similar to observations in cuprates), map to analogous properties of planar, charged black holes. In particular, the Sachdev and Ye state and the Bekenstein-Hawking entropy can be described by analogous equations.

We expect that our findings will increase our understanding of quantum states of matter and the dynamics of black hole horizons.

Figure 1

Plots of the Green functions in Eq. (3) for $\mathrm{\Delta }=1/4$, $q=1$, $T=1$, $A=1$, $\mathcal{E}=1/4$, with $\hslash =k_B=1$. Note that while neither $\mathrm{Im}{G}^R(\omega )$ nor $\mathrm{Re}{G}^R(\omega )$ 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 (Sec. 2) and planar charged black holes (Sec. 3) at $T=0$. The spatial coordinate $\overrightarrow{x}$ has $d$ dimensions. All results also apply to spherical black holes considered in Appendix pp2. The ${\mathrm{AdS}}_2\times R^d$ metric has unimportant prefactors noted in Eq. (55), which are not displayed in the figure. 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 ${\mathrm{AdS}}_2$ 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 ${\mathcal{S}}_{\mathrm{BH}}$ is related to the area of the horizon via ${\mathcal{S}}_{\mathrm{BH}}={\mathcal{A}}_h/(4G_N{\mathcal{A}}_b)$, where $G_N$ is Newton’s constant.