Chiral Wigner crystal phases induced by Berry curvature

Abstract

We consider the impact of Berry phase on the Wigner crystal (WC) state of a two-dimensional electron system. We consider first a model of Bernal bilayer graphene with a perpendicular displacement field, and we show that Berry curvature leads to a new kind of WC state in which the electrons acquire a spontaneous orbital angular momentum when the displacement field exceeds a critical value. We determine the phase boundary of the WC state in terms of electron density and displacement field at low temperature. We then derive the general effective Hamiltonian that governs the ordering of the physical electron spin. We show that this Hamiltonian includes a chiral term that can drive the system into chiral spin-density wave or spin liquid phases. The phenomena we discuss are relevant for the valley-polarized Wigner crystal phases observed in multilayer graphene.

Figures (3)

• Figure 1: (a) Schematic illustration of the conduction band dispersion relation $\varepsilon\left(p\right)$ for electrons in BBG with a perpendicular displacement field [Eq. (\ref{['eq: MHdispersion']})]. (b) Schematic depiction of the WC, which can be described as a triangular lattice of nearly-independent quantum harmonic oscillators, each in the ground state of the confining potential created by its neighbors. (c) The ground state energy of an electron in the WC as a function of its angular momentum $\ell$ and the Berry flux $\Phi$ through the interior of its wave function [see Eq. (\ref{['eq:spectrum_wi']})]. When $|\Phi| > \pi$, the ground state acquires finite angular momentum, $\ell = \pm 1$.
• Figure 2: The phase diagram of the WC in BBG in the space of electron density $(n)$ and displacement field $(U)$. The blue lines represent the phase boundaries between the $\ell = 0$ WC, $\ell = \pm 1$ WC, and Fermi Liquid (FL) phases, calculated by numerical solution of the Schrodinger equation. The dashed red lines correspond to analytical approximations derived in the Supplemental Material. The inset shows the jump in the magnetization per electron as $U$ passes through the critical value $U_c$, calculated along the line cut indicated by the black dotted line in the main figure. The solid green line corresponds to the analytical result of Eq. (\ref{['eq:mag']}), and the dots are results from inserting the numerical solution of the wave function into Eq. (\ref{['eq:mag_full']}).
• Figure 3: Schematic depiction of a three-particle exchange process $J_3$ among neighboring electrons in the WC. Arrows indicate least-action, semiclassical tunneling trajectories. The real-space trajectories are depicted in (a) (with black dots indicating the Wigner lattice positions), while (b) shows the same trajectories in reciprocal space.