Black holes and non-relativistic quantum systems

TL;DR

The paper constructs a $(d{+}3)$-dimensional gravitational model with a scalar and a massive vector to realize a $z=2$ non-relativistic scale-invariant system holographically. It derives black-hole solutions whose thermodynamics reproduce the expected scaling for a $d{+}1$-dimensional scale-invariant theory, computes the grand potential and derived quantities, and demonstrates a universal shear-viscosity-to-entropy ratio $\eta/s = \hbar/(4\pi)$ via holographic methods. Key findings include the scaling form $\Omega(T,\mu) \propto T^{(d+2)/2}(T/|\mu|)^{(d+2)/2}$ and the relation $2\epsilon = d p$, consistent with scale invariance, as well as a specific energy-per-particle $E/N = \frac{d}{d+2}|\mu|$. The work generalizes previous $d=2$ results to arbitrary $d\ge2$, offering a tractable holographic framework for strongly interacting non-relativistic quantum systems and highlighting universal transport properties in this class of theories.

Abstract

We describe black holes in d+3 dimensions, whose thermodynamic properties correspond to those of a scale invariant non-relativistic d+1 dimensional quantum system with dynamical exponent z=2. The gravitational model involves a massive abelian vector field and a scalar field, in addition to the metric. The energy per particle in the dual theory is $|μ| d/(d+2)$, exactly as in a non-interacting Fermi gas, while the ratio of shear viscosity to entropy density is $\hbar/4π$.