Schrödinger Holography with and without Hyperscaling Violation

TL;DR

This work extends non-relativistic holography to Schrödinger-type spacetimes with general dynamical exponent $z$ and hyperscaling violation exponent $\theta$, establishing a codimension-2 holographic framework where a spectator direction $\xi$ plays a crucial role. It analyzes scalar correlation functions and their scaling dimensions, and develops a minimal-surface prescription for entanglement entropy that reveals rich area-law violations, including logarithmic and volume-like behavior, across a range of $z$ and $\theta$ and even without hyperscaling violation. The results identify novel phases in the dual field theories, such as potential Fermi surfaces, via entanglement entropy diagnostics, and they explore string-theory realizations and the AdS in light-cone limit ($\beta=0$) as a cross-check. Overall, the paper maps the interplay between $z$, $\theta$, entanglement structure, and holographic correlation functions, highlighting conditions under which area laws fail and new states of matter may emerge in non-relativistic holography. The findings have implications for condensed matter applications and guide future investigations into finite-temperature completions and top-down embeddings.

Abstract

We study the properties of the Schrödinger-type non-relativistic holography for general dynamical exponent z with and without hyperscaling violation exponent θ. The scalar correlation function has a more general form due to general z as well as the presence of θ, whose effects also modify the scaling dimension of the scalar operator. We propose a prescription for minimal surfaces of this "codimension 2 holography," and demonstrate the (d-1) dimensional area law for the entanglement entropy from (d+3) dimensional Schrödinger backgrounds. Surprisingly, the area law is violated for d+1 < z < d+2, even without hyperscaling violation, which interpolates between the logarithmic violation and extensive volume dependence of entanglement entropy. Similar violations are also found in the presence of the hyperscaling violation. Their dual field theories are expected to have novel phases for the parameter range, including Fermi surface. We also analyze string theory embeddings using non-relativistic branes.

Figures (3)

• Figure 1: The allowed parameter space of $(z, \theta)$ for the spatial field theory dimensions, $d=2$ and $d=3$, with $D=d+1$ from the null energy condition. As we increase $d$, the allowed regions around the point $(z, \theta) =(0, 0)$ are pushed further for negative $z$ and positive $\theta$.
• Figure 2: The parameter ranges of $(z, \theta)$ for the novel phases are plotted for $d=2$ and $d=3$. The plot assumes $D=d+1$. The novel phases lie in the region between the black dashed lines. The blue background is allowed regions from the null energy condition.
• Figure 3: The parameter ranges of $(z, \theta)$ for the novel phases in the case of ACLF with $\beta=0$ are plotted for $d=2$ and $d=3$. The plot assumes $D=d+1$. The novel phases lie in the region between the black dashed lines. The blue background is allowed regions from the null energy condition.