Quantum Geometry Driven Crystallization: A Neural-Network Variational Monte Carlo Study

TL;DR

This work addresses how Berry curvature and quantum geometry reshape electron crystallization in a tunable two-band λ-jellium framework. It develops a neural-network variational Monte Carlo approach with a spinor Slater-Jastrow-backflow ansatz to go beyond mean-field and map the ground-state phase diagram, identifying LIQ, WC, AHC, and HWC phases and showing that the anomalous Hall crystal remains stable under quantum fluctuations. A key finding is that quantum geometry dramatically lowers crystallization thresholds (e.g., $r_s^\star$ dropping from $\gtrsim 25$ in polarized jellium to as low as $\approx 8$ for the AHC at $\lambda=0.72$), while enhancing the robustness of crystalline order via larger $dE_0/dr_s$ across transitions. These results reveal a rich interplay between quantum fluctuations, quantum geometry, and crystallization, offering guidance for experiments and opening avenues to fractional AHCs and chiral superconductors; they also demonstrate the efficacy of neural-network quantum states for exploring band-geometry problems in continuum systems.

Abstract

Wigner crystals are a paradigmatic form of interaction driven electronic order. A key open question is how Berry curvature and, more generally, quantum geometry reshape crystallization. The discovery of two-dimensional materials with relatively flat bands and pronounced Berry curvature has added fresh urgency to this question. Recent mean-field studies have proposed a topological variant of the Wigner crystal, the anomalous Hall crystal (AHC), with non-zero Chern number. However it remains unclear whether the AHC survives beyond the mean-field approximation. Here, we map out the ground-state phase diagram of the $λ$-jellium model - a simple model whose interaction strength and Berry curvature are independently tunable - using state-of-the-art neural-network variational Monte Carlo. The AHC is found to remain stable against quantum fluctuations. Surprisingly, quantum geometric effects are found to dramatically enhance crystallization. Both the AHC and the standard Wigner Crystal are stabilized at densities up to an order of magnitude above the critical density in the absence of quantum geometry, yet still significantly below the threshold predicted by mean-field theory. These striking results highlight the rich interplay between quantum fluctuations, quantum geometry, and crystallization, providing concrete guidance for experiments and enabling future explorations of fractionalized crystals and chiral superconductors.

Figures (13)

• Figure 1: (a) Phase diagram of the $\lambda$-jellium model, computed by neural-network VMC. The ground state is shown as a function of the two independent parameters, the interaction strength and quantum geometry. At $\lambda=0$, the Wigner crystal (WC) competes with a Fermi liquid (LIQ), with a crystallization transition at $r_s^\star \gtrsim 25$. The dashed region indicates a region in which the ground state starts showing first crystalline features (for more details, see App. \ref{['app:boundaries']}). As topology is added with $\lambda$, the critical $r_s^\star$ for WC transition decreases significantly. At sufficient $\lambda$, a "halo Wigner crystal" (HWC) appears at large $r_s$, and an anomalous Hall crystal (AHC) is stabilized at intermediate $r_s$. Symbols indicate numerical datapoints in each phase, with filled symbols indicating data points for which we plot the observables in Fig \ref{['fig:phase_identification']}. Dashed line indicate interpolated phase boundaries (see App. \ref{['app:boundaries']}). (b) Quadratic dispersion of $\lambda$-jellium's lower band at $\lambda=0.72$. The line color shows the Berry curvature, which is concentrated at $\boldsymbol{k}=0$. (c) The skyrmionic spinor texture of the lower band of $\lambda$-jellium along $k_x$ (taking $k_y=0$). The skyrmion core shrinks with increasing $\lambda$, concentrating the Berry curvature.
• Figure 2: The delicate balance of the energetics in different phases, and the effect from correlation and quantum geometry. (a) In the usual 2D jellium (polarized), the energy of the WC phase becomes lower than the LIQ phase at $r_s^\star \gtrsim 25$ (see Appendix \ref{['app:boundaries']} for more details on the $\lambda=0$ transition, here we mark the literature value $r_s^\star \approx 28$Drummond_Phase_Diagram with a black arrow). Mean-field yields a transition at a much smaller $r_s$, indicated by the crossing of the dashed lines around $r_s^\star \approx 2$. Correlation energy is much larger in the LIQ phase. (b-c) Quantum geometry lowers the critical value of the liquid-crystal transition. These have the same setup as in (a), but are for LIQ and WC at $\lambda=0.33$ and LIQ and AHC at $\lambda=0.72$. The black arrows mark the VMC transitions estimated via our simulations. (d) The total energies are given as a function of the Berry curvature concentration, for a fixed $r_s=8$.
• Figure 3: Properties of the different phases from the variational NQS ground states. Columns correspond to filled symbols in Fig. \ref{['fig:phase_diagram']}, at parameters $(r_s,\lambda)$ of $(4,0.72)$ LIQ, $(12,0.72)$ AHC, $(25,0.33)$ WC, and $(25,0.9)$ HWC. Row 1: charge density $\rho(\boldsymbol{r})$ of each state, normalized by the average charge density $\rho_0$. Row 2: static structure factor $S(\boldsymbol{q})$. The six sharp Bragg peaks indicate crystallization to a triangular lattice with one electron per unit cell. Row 3: momentum space occupations $n(\boldsymbol{k})$ for each phase versus unrestricted momentum $\left| \boldsymbol{k} \right|/k_F$. The liquid has a sharp Fermi surface (dotted line at $k_F$). The HWC has small maximal occupations in a "halo" peaked near $0.6k_F$ in this case, and a minium at $\left| \boldsymbol{k} \right|=0$. Error bars show statistical error. Insets: Monte Carlo estimates of $\braket{\Psi|C_3|\Psi} = e^{i2\pi C/3}$ where $C$ is the Chern number (see text). The WC and HWC have $C=0 \pmod 3$ while the AHC has $C=-1 \pmod 3$.
• Figure 4: (a-b) Relative energy per electron of liquid and AHC states for different twisted boundary conditions at $\lambda=0.72$ for (a)$N_e=16$ and (b)$N_e=36$. Here $\Delta E$ is defined relative to the twist-averaged AHC energy in the $N_e=16$ case, and relative to the un-twisted AHC energy in the $N_e=36$ case. (c) Energies of the liquid and AHC with spinor dependent backflow (spinor-NQS) and with spinor independent backflow (NQS) for $N_e=16$.
• Figure 5: Variational energy per electron for liquid and AHC states at $r_s =6, \lambda=0.72, N_e = 16$, for five variational ansätze with increasing expressive power. The ansätze, described in the text, are: self-consistent Hartree Fock (HF), Slater-Jastrow (SJ), SJ with backflow (SJBF), SJBF with a spinor-independent neural quantum state (NQS), and SJBF with a spinor-dependent NQS (spinor-NQS). The main text uses the NQS ansatz. Inset: energy differences between the liquid and AHC states.