Quantizing Bosonized Fermi Surfaces

TL;DR

This work casts the quantum dynamics of extended Fermi surfaces as a noncommutative, coadjoint-orbit problem that exacts a dual description in the $N\to\infty$ limit of a $U(N)_1$ WZW model. The authors show that a nonperturbative, strongly coupled, yet solvable, dynamics along the Fermi surface eliminates the naive overcounting of degrees of freedom and removes the need for patching, unifying flat and general Fermi-surface treatments. They derive leading actions for flat and general 2d Fermi surfaces in magnetic coordinates, reproduce the Lindhard continuum, and develop a systematic power-law expansion to compute corrections to observables such as the specific heat and density correlators. The framework connects Fermi-liquid dynamics to noncommutative geometry and 1+1d CFT, providing a robust nonperturbative tool to study beyond-Luttinger behavior and nonanalytic responses in higher dimensions, with potential applications to non-Fermi liquids and weak-field phenomena.

Abstract

Bosonization describes Fermi surface dynamics in terms of a collective field that lives on a part of phase space. While sensible semiclassically, the challenge of treating such a field quantum mechanically has prevented bosonization from providing as powerful a nonperturbative tool as in one dimension. We show that general Fermi surfaces can be exactly described by a particular $N\to \infty$ limit of a $U(N)_1$ WZW model, with a tower of irrelevant corrections. This matrix-valued description encodes the noncommutative nature of phase space, and its (solvable) strongly coupled dynamics resolves the naive overcounting of degrees of freedom of the collective field without the need to cut the Fermi surface into patches. This approach furthermore provides a quantitative tool to systematically study power-law corrections to Fermi surface dynamics.

Figures (5)

• Figure 1: (a) Fermi surface (FS) and its bosonized degree of freedom. (b) The patch prescription separates the smooth FS in discrete patches of size $\Lambda$, and cuts off the momentum of particle hole excitations along the FS $q<\Lambda$. (c) Strongly coupled dynamics caused by the noncommutative phase-space (grid) along the FS obviates patches.
• Figure 2: Relation between various approaches to Fermi surface bosonization. $N$ can play the role of spin or fermion flavors in 1d, number of wires for flat Fermi surfaces, or number of magnetic flux for general 2d Fermi surfaces.
• Figure 3: (a) $N$ wires, each containing a chiral fermion. Fermion bilinears $\psi^\dagger_i\psi_j$ form the $u(N)$ algebra, which becomes the Moyal $w_\infty$ algebra in the continuum limit $N\to \infty$. (b) Corresponding flat Fermi surface. The patch prescription can be derived by abelianizing our approach, using the $u(1)^N$ subalgebra spanned by the bilinears $f(q,p) = \psi^{\dagger}_{p-\frac{q}{2}} \psi_{p+\frac{q}{2}}$ satisfying \ref{['eq_u1N_patch']}.
• Figure 4: Power-law corrections to local density response can be obtained similarly to 1d bosonization. The diagrams in the first line come from corrections to the dispersion relation and have direct analogs in 1d. Those in the second line do not: they arise from the geometry of the Fermi surface.
• Figure 5: A single bosonic propagator produces the particle-hole continuum $\omega\leq v_F q$, Eq. \ref{['eq_rhorho_leading']}. Landau interactions \ref{['eq_Landau_int']} lead to nonzero spectral densities everywhere due to the two-particle-hole continuum.