Spin-triplet paired Wigner crystal stabilized by quantum geometry

TL;DR

This work shows that in a two-dimensional electron gas at low density, Berry curvature and quantum geometry can stabilize a Wigner crystal that localizes electrons into spin-triplet pairs with relative orbital angular momentum $m=-1$. Using variational, second-quantized and first-quantized (Hartree) approaches, the authors map the problem to an effective two-electron quantum dot with Berry curvature and derive a phase diagram where a spin-triplet, $m=-1$ paired Wigner crystal becomes favorable at large $r_s$ and Berry curvature. The key mechanism is the geometric coupling between the pair’s relative motion and the band topology, which lowers the energy of $m=-1$ configurations compared to $m=0$ or singlet states; this is captured by an effective Schrödinger equation for the relative coordinate with Berry-modified form factors. The results point to a purely electronic, strong-coupling route to local spin-triplet pairing and potentially to unconventional superconductivity in graphene- and moiré-based two-dimensional materials where quantum geometry is significant.

Abstract

We have used variational states to analyze the effects of band geometry on the two-dimensional Wigner crystal with one and two electrons per unit cell. At sufficiently low electron densities, we find that increasing Berry curvature drives a transition into a crystalline state composed of spin-triplet pairs carrying relative orbital angular momentum $m=-1$. The essential features of this transition are captured by an effective two-electron quantum dot problem in the presence of Berry curvature. Our results point to a purely electronic, strong-coupling mechanism for local spin-triplet pairing in correlated two-dimensional electron systems with quantum geometry.

Figures (3)

• Figure 1: Variational ground state phase diagram as a function of interaction strength $r_s$ and Berry curvature $\omega$. Only crystalline phases are shown. At sufficiently large $r_s$ and $\omega$, a paired Wigner crystal (PWC) composed of orbital angular momentum $m=-1$, spin-triplet electron pairs is energetically favored. Competing crystalline phases are the monatomic Wigner crystal (WC) and PWC with $m=0$, spin-singlet pairs.
• Figure 2: (a) Lowest-energy eigenvalues of the Schrödinger equation \ref{['eq:Schrodinger_momentum']} as a function of Berry curvature $\omega_{r}$ for $r_s=80$ and different orbital angular momenta $m$. To highlight the difference between the curves we subtract the energy $\epsilon_g$ (see text). Dashed lines are the perturbative result \ref{['eq:de_pert']}. (b) Derivative of the lowest-energy eigenvalues with respect to $\omega_r$ as a function of $r_s$ for $m=\pm 1$. Dashed lines show the asymptotic large-$r_s$ behavior \ref{['eq:de_pert']}. Derivatives are multiplied by $r_s^2$ for visual clarity.
• Figure 3: Berry curvature dependence of lowest-energy wave functions $u(r)$ of the effective quantum dot problem \ref{['eq:Schrodinger_momentum']}, for angular momenta (a) $m=0$, (b) $m=1$, and (c) $m=-1$. Here $r_s = 80$ and $r_0 = (e^2/K)^{1/3}$ is the classical distance between electrons in the pair; see Eq. \ref{['eq:rad_eq']}.