[Paper Review] How to construct a gravitating quantum electron star

Andrea Allais, John McGreevy|DSpace@MIT (Massachusetts Institute of Technology)|Jun 25, 2013

Black Holes and Theoretical Physics2 references8 citations

TL;DR

This paper constructs a gravitating quantum electron star by solving Einstein gravity coupled to quantum fermions and gauge fields in asymptotically AdS spacetime, using a saddle-point approximation for metric and gauge fields while treating fermions quantum mechanically. The key result is a self-consistent solution with finite electron density, non-Fermi liquid behavior, and a non-trivial geometry that avoids the extremal black hole limit, providing a holographic model for strongly correlated electron systems with a well-defined Fermi surface and short-lived quasiparticles.

ABSTRACT

Motivated by the holographic study of Fermi surfaces, we develop methods to solve Einstein gravity coupled to fermions and gauge fields, with AdS boundary conditions and a chemical potential.

Motivation & Objective

• To construct a self-consistent, finite-density solution of Einstein gravity coupled to quantum fermions and gauge fields in asymptotically AdS spacetime.
• To model a system with a well-defined Fermi surface and non-Fermi liquid behavior, avoiding the limitations of the large-N limit where fermion backreaction is neglected.
• To develop a method that treats fermions quantum mechanically while treating gravity and gauge fields classically via the saddle-point approximation.
• To avoid unphysical approximations such as large fermion mass or strong magnetic fields, which lead to classical or one-dimensional behavior.
• To provide a holographic toy model for exotic quantum materials with strong correlations and short-lived quasiparticles.

Proposed method

• Uses the saddle-point approximation for metric and gauge fields, equivalent to the Hartree-Fock approximation at large N, while treating the fermionic path integral quantum mechanically.
• Applies an adiabatic expansion to compute fermionic currents in a curved, charged background, with careful regularization and renormalization to handle UV divergences.
• Diagonalizes the Dirac Hamiltonian in momentum space for each mode, reducing the 4D problem to a 1+1 dimensional effective Dirac equation with a k-dependent mass.
• Employs a non-trivial background geometry with periodic functions for the metric components and vanishing electrostatic potential, ensuring a well-defined compact spatial direction.
• Uses a lattice regularization of the Dirac operator, but identifies a mobility edge separating extended and localized states, showing that localized modes are essential for completeness.
• Transforms the Dirac Hamiltonian into a form without metric derivatives via a non-unitary similarity transformation, enabling stable numerical implementation on a lattice.

Experimental results

Research questions

• RQ1 Can a self-consistent solution be constructed for a finite density of gravitating, charged fermions in asymptotically AdS spacetime, with quantum fermions and classical gravity?
• RQ2 What is the resulting geometry and fermionic spectrum when fermion backreaction is included without the large-N or large-mass approximations?
• RQ3 How does the presence of a Fermi surface manifest in the fermionic Green’s function when the system is not in a Fermi liquid state?
• RQ4 What are the implications of lattice regularization for the spectrum of the Dirac operator, particularly regarding the localization of eigenstates?
• RQ5 Can a non-extremal, non-trivial geometry emerge that avoids the AdS2 near-horizon limit of the Reissner-Nordström black hole?

Key findings

• The self-consistent solution yields a non-extremal, finite-density geometry with a non-trivial profile for the metric and gauge field, avoiding the AdS2 near-horizon limit.
• The fermionic Green’s function exhibits non-Fermi liquid behavior, with a singularity over a Fermi surface and a finite width that does not vanish at the Fermi surface.
• Lattice regularization reveals a mobility edge in the spectrum, separating extended (continuum-like) modes from localized modes, with the latter essential for completeness.
• The Inverse Participation Ratio (IPR) confirms that a significant fraction of eigenstates are localized, and their removal would break the ability to resolve the identity.
• The transformation to a derivative-free Hamiltonian form enables stable numerical computation and confirms isospectrality with the original Dirac Hamiltonian.
• The method successfully avoids unphysical approximations such as large fermion mass or strong magnetic fields, yielding a more realistic model for non-Fermi liquid systems.