Pacifying the Fermi-liquid: battling the devious fermion signs

TL;DR

This paper explores the fermion sign problem in path integrals, analyzing Fermi liquids, reviewing Ceperley's constrained path integral approach, and proposing a holographic conjecture linking Fermi liquids to one-dimensional bosonic systems.

Contribution

It introduces a geometric constraint framework for fermionic path integrals, analyzes low-dimensional systems, and proposes a novel holographic relation between Fermi liquids and bosonic systems.

Findings

Nodal hypersurface constraints elucidate 1D physics.

Fermi gas analogy to Mott insulators in traps.

Topological properties of nodal cells suggest new holographic links.

Abstract

The fermion sign problem is studied in the path integral formalism. The standard picture of Fermi liquids is first critically analyzed, pointing out some of its rather peculiar properties. The insightful work of Ceperley in constructing fermionic path integrals in terms of constrained world-lines is then reviewed. In this representation, the minus signs associated with Fermi-Dirac statistics are self consistently translated into a geometrical constraint structure (the {\em nodal hypersurface}) acting on an effective bosonic dynamics. As an illustrative example we use this formalism to study 1+1-dimensional systems, where statistics are irrelevant, and hence the sign problem can be circumvented. In this low-dimensional example, the structure of the nodal constraints leads to a lucid picture of the entropic interaction essential to one-dimensional physics. Working with the path integral in momentum space, we then show that the Fermi gas can be understood by analogy to a Mott insulator in a harmonic trap. Going back to real space, we discuss the topological properties of the nodal cells, and suggest a new holographic conjecture relating Fermi liquids in higher dimensions to soft-core bosons in one dimension. We also discuss some possible connections between mixed Bose/Fermi systems and supersymmetry.