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Holographic Entanglement Entropy and Fermi Surfaces

Edgar Shaghoulian

TL;DR

The paper investigates holographic realizations of the log-violating entanglement entropy associated with Fermi surfaces by analyzing IR bulk geometries. It shows that AdS$_3\times\mathbb{R}^{d-1}$-type IR regions can produce the desired $S_A$ scaling, while many standard IR geometries fail to generate the logarithmic term, narrowing focus to the OTU geometry. A detailed singularity analysis reveals a null curvature singularity in the OTU background except for a special parameter set where the IR remains nonsingular and admits an analytic extension; in this nonsingular case the four-dimensional exponents are $\theta=1$ and $\gamma=3/2$. The authors construct an explicit Einstein–Maxwell–dilaton embedding for the nonsingular IR that keeps the dilaton constant in the IR, offering a consistent $T=0$ model for a theory with a Fermi surface and providing a framework for future investigations into zero-temperature holographic Fermiology and potential string-theoretic resolutions.

Abstract

The entanglement entropy in theories with a Fermi surface is known to produce a logarithmic violation of the usual area law behavior. We explore the possibility of producing this logarithmic violation holographically by analyzing the IR regions of the bulk geometries dual to such theories. The geometry of Ogawa, Takayanagi, and Ugajin is explored and shown to have a null curvature singularity for all values of parameters, except for dynamical critical exponent 3/2 in four dimensions. The results are extended to general hyperscaling violation exponent. We explore strings propagating through the singularity and show that they become infinitely excited, suggesting the singularity is not resolved by stringy effects and may become a full-fledged "stringularity." An Einstein-Maxwell-dilaton embedding of the nonsingular geometry is exhibited where the dilaton asymptotes to a constant in the IR. The unique nonsingular geometry in any given number of dimensions is proposed as a model to study the T=0 limit of a theory with a Fermi surface.

Holographic Entanglement Entropy and Fermi Surfaces

TL;DR

The paper investigates holographic realizations of the log-violating entanglement entropy associated with Fermi surfaces by analyzing IR bulk geometries. It shows that AdS-type IR regions can produce the desired scaling, while many standard IR geometries fail to generate the logarithmic term, narrowing focus to the OTU geometry. A detailed singularity analysis reveals a null curvature singularity in the OTU background except for a special parameter set where the IR remains nonsingular and admits an analytic extension; in this nonsingular case the four-dimensional exponents are and . The authors construct an explicit Einstein–Maxwell–dilaton embedding for the nonsingular IR that keeps the dilaton constant in the IR, offering a consistent model for a theory with a Fermi surface and providing a framework for future investigations into zero-temperature holographic Fermiology and potential string-theoretic resolutions.

Abstract

The entanglement entropy in theories with a Fermi surface is known to produce a logarithmic violation of the usual area law behavior. We explore the possibility of producing this logarithmic violation holographically by analyzing the IR regions of the bulk geometries dual to such theories. The geometry of Ogawa, Takayanagi, and Ugajin is explored and shown to have a null curvature singularity for all values of parameters, except for dynamical critical exponent 3/2 in four dimensions. The results are extended to general hyperscaling violation exponent. We explore strings propagating through the singularity and show that they become infinitely excited, suggesting the singularity is not resolved by stringy effects and may become a full-fledged "stringularity." An Einstein-Maxwell-dilaton embedding of the nonsingular geometry is exhibited where the dilaton asymptotes to a constant in the IR. The unique nonsingular geometry in any given number of dimensions is proposed as a model to study the T=0 limit of a theory with a Fermi surface.

Paper Structure

This paper contains 16 sections, 76 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Area Law Violation
  3. AdS$_3\times\mathbb{R}^{d-1}$
  4. Other Fermi Surface Geometries
  5. The OTU Geometry
  6. Singularities of the OTU geometry
  7. Diverging Tidal Forces
  8. Plane Wave Approximation
  9. A Stringularity
  10. Effective Gravity Embedding of Nonsingular Geometry
  11. Conclusions
  12. Holographic Entanglement Entropy Computations
  13. Entanglement Entropy in 2D CFT
  14. Entanglement Entropy in Higher Dimensions
  15. Comparison to Flat Space Scaling
  16. ...and 1 more sections

Figures (4)

  • Figure 1: $\phi(z)=2\sqrt{1-k/z_F^2}F\left[\arctan(z/\sqrt{k}) |1-k/z_F^2\right]$, with $k=1$ and $z_F=2$. The scalar field now asymptotes to a constant in the IR, whereas in the case $m>p=1$ it diverges logarithmically.
  • Figure 2: $V(\phi)-6=\frac{\mathrm{sc}[\phi/(2 \sqrt{c_1}); c_1]^2 (c_2 +c_3 \mathrm{sc}[\phi/(2 \sqrt{c_1}); c_1]^2 + c_4 \mathrm{sc}[\phi/(2 \sqrt{c_1}); c_1]^4 + c_5 \mathrm{sc}[\phi/(2 \sqrt{c_1}); c_1]^6)}{(1 + \mathrm{sc}[\phi/(2 \sqrt{c_1}); c_1]^2)^2 (z_F^2 + k \mathrm{sc}[\phi/(2 \sqrt{c_1}); c_1]^2)^2}-6$, with $k=1$ and $z_F=2$. The potential vanishes in the IR.
  • Figure 3: $Z(\phi)=\frac{2 k^2 \mathrm{sc}[\phi/(2 \sqrt{c}); c]^2 (1 + \mathrm{sc}[\phi/(2 \sqrt{c}); c]^2)^2 (z_F^2 + k \mathrm{sc}[\phi/(2 \sqrt{c}); c]^2)^2}{z_F^2 (z_F^2 + 2 (k + 2 z_F^2) \mathrm{sc}[\phi/(2 \sqrt{c}); c]^2 + 5 k \mathrm{ sc}[\phi/(2 \sqrt{c}); c]^4)}$, with $k=1$ and $z_F=2$. The divergence is simply indicating that the gauge coupling, which scales as $Z(\phi)^{-1}$, is vanishing in the IR.
  • Figure 4: $a(z)=\frac{Az_F\sqrt{k}}{2z(k+z^2)^{3/2}\sqrt{z^2+z_F^2}}$