$λ$-Jellium Model for the Anomalous Hall Crystal

Abstract

The jellium model is a paradigmatic problem in condensed matter physics, exhibiting a phase transition between metallic and Wigner crystal phases. However, its vanishing Berry curvature makes it ill-suited for studying recent experimental platforms that combine strong interactions with nontrivial quantum geometry. These experiments inspired the anomalous Hall crystal (AHC) -- a topological variant of the Wigner crystal. The AHC spontaneously breaks continuous translation symmetry but has a nonzero Chern number. In this work, we introduce $λ-$jellium, a minimal extension of the two-dimensional jellium model. Its Berry curvature distribution is controlled by a single parameter, $λ$, where $λ=0$ corresponds to the standard jellium model. This setup facilitates the systematic exploration of Berry curvature's impact on electron crystallization. The phase diagram of this model, established using self-consistent Hartree Fock calculations, reveals several interesting features: (i) The AHC phase occupies a large region of the phase diagram. (ii) Two distinct Wigner crystal phases, the latter enabled by quantum geometry, and two distinct Fermi liquid phases are present. (iii) A continuous phase transition separates the AHC and one of the WC phases. (iv) In some parts of the AHC phase, the lattice geometry is non-triangular, unlike in the classical Wigner crystal. In addition to elucidating the physics of correlated electrons with nonzero Berry curvature, we expect that the simplicity of the model makes it an excellent starting point for more advanced numerical methods.

Figures (8)

• Figure 1: (a) Quadratic dispersion and (b) Berry curvature of the lower single-particle band of the topological electron gas model, Eq. \ref{['eq:ham']}. (c) Mean-field phase diagram. The limit $\lambda = 0$ is identical to the spinless 2DEG. Two significant Fermi liquid (FL) regions are present: one at low interaction strengths, and one at large interaction strength and Berry curvature concentration. Under strong interactions, a Wigner crystal (WC) appears that undergoes a first-order transition to an anomalous Hall crystal (AHC) at $\lambda \approx \frac{1}{2}$. A putative second order transition back to a WC appears at larger $\lambda$. A significant region of AHC is present, with the competition between triangular and square AHCs shown by light pink (AHC($\triangle$)) when the triangular AHC has lower energy than the square, and dark pink (AHC($\square$)) when the reverse is true. Within the AHC phase, a square unit cell is preferred to a triangular unit cell in a region near $\lambda = 2/3$.
• Figure 2: Fermi liquid phases of $\lambda-$jellium assuming continuous translation symmetry. (a) Self-consistent inner radius $k_m^*$ of the Fermi surface. (b) The Fermi liquid develops a strong peak at $\boldsymbol{k} =0$ at large $\lambda,r_s$ due to exchange (Fock) interactions, leading to annular Fermi surfaces. Arrows show the pseudospin on the Bloch sphere, $(\braket{\sigma_x}, \braket{\sigma_z})$ along the $k_x$-axis. A localized "skyrmion core" is present at $\lambda=3$.
• Figure 3: Occupations of plane wave states in the lower band for different Wigner crystals. (a) Normal Wigner crystal at $(r_s,\lambda) = (20,0)$. (b) Halo Wigner crystal at $(r_s,\lambda) = (20,2.5)$. The $\boldsymbol{k}=0$ region is depleted due to the Fock interactions at large $\lambda$.
• Figure 4: Continuous phase transition driven by $r_s$ at $\lambda=4/3$. (a) Direct charge gap near the phase boundary between AHC and the halo WC. (b) The Berry curvature of an AHC ($r_s = 19.25$) and WC ($r_s = 19.5$) near the transition. It is strongly peaked around the $\Gamma$ point, saturating the color scale. (c,d) self-consistent Hartree-Fock band structures before and after the transitions. Coloring corresponds to $z_a(\boldsymbol{p})$, with red corresponding to occupying the first component. Clearly, the $r_s$ tuned transition corresponds to a band inversion transition.
• Figure 5: Construction of the $\lambda$-jellium dispersion. (a) Dirac Hamiltonian. (b) Adding a momentum-dependent mass makes the lower band exactly flat with Berry curvature. (c) Adding an spinor-isotropic kinetic term gives a quadratic dispersion with an independently adjustable Berry curvature distribution.