[Paper Review] Berry Phase Physics in Free and Interacting Fermionic Systems

Jingyuan Chen|arXiv (Cornell University)|Jan 1, 2016

Topological Materials and Phenomena44 references3 citations

TL;DR

This dissertation develops a unified framework for Berry phase physics in free and interacting fermionic systems, deriving kinetic theories for both non-interacting (Berry Fermi gas) and interacting (Berry Fermi liquid) fermions from quantum field theory. The key contribution is the microscopic derivation of anomalous transport effects—such as the chiral magnetic effect and chiral vortical effect—via momentum-space Berry curvature, revealing topological contributions to Hall conductivity that are robust against interactions.

ABSTRACT

Berry phase plays an important role in many non-trivial phenomena over a broad range of many-body systems. In this thesis we focus on the Berry phase due to the change of the particles' momenta, and study its effects in free and interacting fermionic systems. We start with reviewing the semi-classical kinetic theory with Berry phase for a non-interacting ensemble of fermions -- a Berry Fermi gas -- which might be far-from-equilibrium. We particularly review the famous Berry phase contribution to the anomalous Hall current. We then provide a concrete and general path integral derivation for the semi-classical theory. Then we turn to the specific example of Weyl fermion, which exhibits the profound quantum phenomenon of chiral anomaly; we review how this quantum effect, and its closely related chiral magnetic effect and chiral vortical effect, arise from Berry phase in the semi-classical kinetic theory. We also discuss how Lorentz symmetry in the kinetic theory of Weyl fermion, seemly violated by the Berry phase term, is realized non-trivially; we provide a physical interpretation for this non-trivial realization, and discuss its mathematical foundation in Wigner translation. Next, we turn towards interacting fermionic systems. We consider Fermi liquid near equilibrium, and propose the Berry Fermi liquid theory -- the extension to Landau Fermi liquid theory incorporating Berry phase (and other) effects. In our proposed Berry Fermi liquid theory, we can show the Berry phase is a Fermi surface property, qualitatively unmodified by interactions. But there also arise new effects from interactions, most notably the emergent electric dipole moment which contributes to the anomalous Hall current in addition to the usual Berry phase contribution. We prove our proposed Berry Fermi liquid theory from quantum field theory to all orders in Feynman diagram expansion under very general assumptions.

Motivation & Objective

• To establish a microscopic foundation for Berry phase effects in fermionic systems beyond the free-particle limit.
• To resolve the puzzle of Lorentz symmetry breakdown in chiral kinetic theory by identifying frame-dependent anomalies in collision terms.
• To extend Landau Fermi liquid theory to include Berry curvature effects, linking quasiparticle dynamics to topological invariants.
• To investigate whether the anomalous Hall conductivity contains a topologically protected, interaction-independent contribution.
• To explore the possibility of defining Berry curvature in non-Fermi liquid systems and its implications for topological transport.

Proposed method

• Formalism based on symplectic geometry and path integral quantization to derive semi-classical dynamics with Berry phase.
• Derivation of the Boltzmann equation with Berry curvature corrections for both free and interacting fermions.
• Use of Cutkosky cutting rules and imaginary-time field theory to compute self-energies and collision terms in interacting systems.
• Construction of the stress-energy tensor and entropy current in chiral kinetic theory to analyze physical consistency.
• Identification of the momentum-space Berry curvature from the eigenvector of the full fermion propagator, independent of the eigenvalue.
• Application of Wigner translation to relate frame dependence in chiral kinetic theory to physical observables.

Experimental results

Research questions

• RQ1 How does the inclusion of Berry phase modify the kinetic theory of free fermions, and what are the resulting anomalous transport effects?
• RQ2 Why does Lorentz invariance appear to be violated in chiral kinetic theory, and how can this be reconciled with physical consistency?
• RQ3 Can the anomalous Hall conductivity in interacting Fermi liquids contain a topological, interaction-independent contribution?
• RQ4 How can Berry phase physics be generalized beyond Fermi liquids, particularly in non-Fermi liquid states?
• RQ5 What is the role of momentum-space Berry curvature defects in higher-dimensional systems and their topological implications?

Key findings

• The anomalous Hall conductivity in the Berry Fermi liquid contains a constant, topological contribution σμνλ⁰ that is independent of the Fermi energy and potentially robust against interactions.
• The chiral magnetic effect and chiral vortical effect emerge naturally from the chiral kinetic theory derived from the path integral with Berry phase.
• Lorentz symmetry is not fundamentally violated in chiral kinetic theory, but its realization is frame-dependent due to non-local collision terms, which are corrected by the Wigner translation.
• The stress-energy tensor and entropy current in chiral kinetic theory are consistent with conservation laws only when the frame dependence is properly accounted for.
• The momentum-space Berry curvature can be defined even in non-Fermi liquids, suggesting a broader applicability of Berry phase physics beyond quasiparticle Fermi liquid theory.
• The theory predicts that the Hall conductivity may exhibit jumps at discrete chemical potentials where the Fermi surface topology changes, signaling a possible quantum phase transition.