Entanglement entropy in Galilean conformal field theories and flat holography (1410.4089v1)

Abstract: We present the analytical calculation of entanglement entropy for a class of two dimensional field theories governed by the symmetries of the Galilean conformal algebra, thus providing a rare example of such an exact computation. These field theories are the putative holographic duals to theories of gravity in three-dimensional asymptotically flat spacetimes. We provide a check of our field theory answers by an analysis of geodesics. We also exploit the Chern-Simons formulation of three-dimensional gravity and adapt recent proposals of calculating entanglement entropy by Wilson lines in this context to find an independent confirmation of our results from holography.

• The paper derives an exact analytical expression for entanglement entropy in Galilean conformal field theories using novel limiting techniques.
• It adapts the Ryu-Takayanagi prescription to flat holography by employing geodesic lengths in a dual 3D gravitational framework.
• The findings provide a robust basis for exploring non-relativistic quantum theories and their connections to quantum gravity and information science.

Entanglement Entropy in Galilean Conformal Field Theories and Flat Holography

The paper presents an analytical exploration into entanglement entropy (EE) for a specific class of two-dimensional field theories governed by the Galilean conformal algebra. These theories, known as Galilean Conformal Field Theories (GCFTs), are notable for their rare exact computation of EE and propose a theoretical framework aligning them as potential holographic duals to theories of gravity in three-dimensional asymptotically flat spacetimes.

Analytical Calculation of Entanglement Entropy

The paper systematically derives the EE for GCFTs, denoting a successful example of exact analytical progress in complex quantum field theories (QFTs). The authors employ a limiting technique from conventional two-dimensional conformal field theories (CFTs), exploiting the Galilean conformal symmetries, akin to how infinite dimensional symmetries simplify calculations in CFTs.

The EE for a GCFT is formulated, bearing structural resemblance to its more commonly studied CFT counterpart, though with modifications attributable to the unique Galilean symmetries: $$S_{\text{EE}} = \frac{c_L}{6}\, \ln \frac{\ell_x}{a} + \frac{c_M}{6} \,\frac{\ell_y}{\ell_x}$$ where $c_L$ and $c_M$ are central charges of the Galilean conformal algebra, while $\ell_x$ and $\ell_y$ represent spatial separations, and $a$ is a cutoff scale.

Holographic Calculation Using Flat Space Geodesics

The paper further extends its purview into flat holography by validating the GCFT EE findings through a holographic framework involving 3D gravity in flat spacetimes. This is accomplished using the Ryu-Takayanagi prescription adapted for Galilean symmetries, where EE can be understood as the length of geodesics in a dual gravitational description.

In scenarios where both central charges ($c_L$ and $c_M$) are non-zero, the paper transcends Einstein's gravity assumption and adapts a Chern-Simons formalism involving holographic EE derivations via Wilson lines. It thereby outlines a robust check by independently confirming EE results consistent with those obtained from the GCFT calculations.

Implications and Future Prospects

This work significantly contributes to the theoretical understanding of non-relativistic field theories and their potential interplay with gravity through holography. By providing precise EE formulations in GCFTs, it paves pathways for the exploration of similar exact methods in higher-dimensional extensions and other non-relativistic symmetries.

Practically, the findings may impact quantum information science, especially in systems aligned with Galilean dynamics like certain condensed matter systems. Theorists interested in flat spacetime holography, quantum gravity, and non-relativistic symmetries would find this research particularly relevant. Future inquiries might explore the relation to asymptotic symmetries, delve into non-linear corrections in holography, or extend these methodologies to additional classes of lower-dimensional field theories.