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Conformal Field Theory on the Fermi Surface

Brian Swingle

TL;DR

The paper tackles the problem of understanding anomalous entanglement entropy and fluctuations in Fermi liquids by modeling the Fermi surface as a collection of 1+1D chiral conformal field theories, one per momentum-space patch. The core method is the patch construction, which yields the 1+1D entanglement entropy formula $S_{\rm patch} = \frac{c_L+c_R}{6} \ln\left(\frac{L}{\epsilon}\right)$ and, upon summing over patches, the total entropy scales as $S_L \sim L^{d_s-1} \ln L$ (the Widom formula). The approach extends to finite temperature and weak interactions, reproducing the usual 2D fermion entropy at high temperature and showing that forward scattering preserves the patch structure with a renormalized Fermi velocity; crucially, the entanglement structure depends only on the geometry of the Fermi surface, suggesting a universal low-energy entanglement. The results provide a unified framework for calculating entanglement and fluctuations in Fermi liquids and offer a path to understanding non-Fermi-liquid phases via similar chiral-CFT constructions on the Fermi surface.

Abstract

The Fermi surface may be usefully viewed as a collection of 1+1 dimensional chiral conformal field theories. This approach permits straightforward calculation of many anomalous ground state properties of the Fermi gas including entanglement entropy and number fluctuations. The 1+1 dimensional picture also generalizes to finite temperature and the presence of interactions. Finally, I argue that the low energy entanglement structure of Fermi liquid theory is universal, depending only on the geometry of the interacting Fermi surface.

Conformal Field Theory on the Fermi Surface

TL;DR

The paper tackles the problem of understanding anomalous entanglement entropy and fluctuations in Fermi liquids by modeling the Fermi surface as a collection of 1+1D chiral conformal field theories, one per momentum-space patch. The core method is the patch construction, which yields the 1+1D entanglement entropy formula and, upon summing over patches, the total entropy scales as (the Widom formula). The approach extends to finite temperature and weak interactions, reproducing the usual 2D fermion entropy at high temperature and showing that forward scattering preserves the patch structure with a renormalized Fermi velocity; crucially, the entanglement structure depends only on the geometry of the Fermi surface, suggesting a universal low-energy entanglement. The results provide a unified framework for calculating entanglement and fluctuations in Fermi liquids and offer a path to understanding non-Fermi-liquid phases via similar chiral-CFT constructions on the Fermi surface.

Abstract

The Fermi surface may be usefully viewed as a collection of 1+1 dimensional chiral conformal field theories. This approach permits straightforward calculation of many anomalous ground state properties of the Fermi gas including entanglement entropy and number fluctuations. The 1+1 dimensional picture also generalizes to finite temperature and the presence of interactions. Finally, I argue that the low energy entanglement structure of Fermi liquid theory is universal, depending only on the geometry of the interacting Fermi surface.

Paper Structure

This paper contains 6 sections, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Chiral Fermions on the Fermi Surface
  3. Number Fluctuations at Zero Temperature
  4. Generalization to Finite Temperature and Interactions
  5. Discussion and Conclusions
  6. Acknowledgements

Figures (2)

  • Figure 1: A sketch of a circular Fermi surface in $d_s = 2$ dimensions. The angle $\theta$ parameterizes the Fermi surface and labels the local patches. The Fermi velocity for a particular $\theta$ is shown.
  • Figure 2: A sketch of a circular real space region $A$ of radius $L$. The Fermi velocity of a particular patch on the Fermi surface is shown superimposed on the circle. The effective length $2 L \cos{\theta}$ is shown as a function of angle $\theta$ relative to the direction defined by $v_F$.