🔬 mesoscale physics

The influence of quantum geometry on the phase boundary and collective excitations of electron liquids and crystals

This study employs time-dependent Hartree-Fock theory on the $\lambda$ -jellium model to demonstrate that quantum geometry promotes electron crystallization at higher densities, suppresses Friedel oscillations and plasmon dispersion in the liquid phase via spectral weight transfer, and gives rise to a breathing mode in the crystal phase corresponding to a real-space pseudospin skyrmion lattice.

Original authors: Paul Froese, Mark R. Hirsbrunner, Yong Baek Kim

Published 2026-09-07

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CC BY 4.0

Original authors: Paul Froese, Mark R. Hirsbrunner, Yong Baek Kim

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the subatomic world, electrons do not merely bounce around like tiny billiard balls; they form a collective fluid that can behave in surprisingly complex ways. Under certain conditions, this fluid can freeze into a rigid, ordered pattern known as a crystal, a state where electrons lock themselves into a grid to minimize their mutual repulsion. For decades, physicists have studied this transition from a flowing liquid to a solid crystal, but a new layer of complexity has recently emerged. This layer is called quantum geometry. It is not a physical shape you can see, but rather a hidden property of the space the electrons inhabit, describing how their internal states twist and turn as they move. Think of it as a subtle curvature in the landscape of their existence, distinct from the physical terrain. When this quantum geometry is strong, it changes the rules of the game, potentially making it easier or harder for electrons to crystallize. Understanding this interplay is crucial for deciphering the behavior of exotic materials, such as stacked sheets of carbon atoms, where these effects are now being observed in the laboratory.

A team of researchers at the University of Toronto has taken a deep dive into this phenomenon, using powerful computer simulations to map out exactly how quantum geometry reshapes the boundary between electron liquids and crystals. They focused on a theoretical model that mimics the behavior of electrons in these advanced materials, allowing them to tweak the strength of the quantum geometry and watch what happens. Their work confirms a recent discovery: the presence of this hidden geometric structure strongly favors the formation of crystals. In their simulations, as the quantum geometry became more pronounced, the electrons required a much higher density to remain in a liquid state before they were forced to freeze into a solid arrangement. This shift suggests that the quantum geometry makes the liquid state energetically expensive, effectively pushing the electrons to organize themselves into a crystal at densities where they would otherwise remain fluid.

The researchers did not just look at the final state of the system; they also examined the unstable moments leading up to the transition. They discovered a new type of instability that arises only when quantum geometry is present. In a standard electron gas, the system is prone to a specific kind of fluctuation where electrons gather in waves. However, with the added quantum geometry, a second, distinct instability appears. This new fluctuation involves electrons being pushed away from the center of their momentum space toward the edges. The timing of this instability is significant; it emerges right before the liquid turns into a crystal, suggesting that these specific fluctuations are the driving force that tips the balance, making the liquid state unstable and encouraging the electrons to lock into a crystalline pattern.

Beyond the transition itself, the study revealed how quantum geometry alters the way these electron liquids respond to external forces. When the researchers probed the liquid phase, they found that the quantum geometry acts as a suppressor. It dampens the natural ripples and waves that usually travel through the electron fluid, specifically reducing the speed of a collective vibration known as the plasmon. Perhaps most strikingly, at a specific setting of the quantum geometry, the material completely stops responding to a particular type of static disturbance. In a normal electron gas, a single impurity would cause the surrounding charge to ripple in a predictable pattern, a phenomenon known as Friedel oscillations. In this simulated system, however, those ripples vanished entirely. The researchers traced this silence to a cancellation effect: the electrons in different internal states began to oscillate out of step with one another, effectively canceling out the total movement of charge even though the individual parts were still moving.

This same cancellation effect was found to persist even after the electrons had frozen into a crystal. In the solid phases, the researchers identified a unique mode of vibration where the electrons in different internal states breathed in and out of phase with each other. While the total amount of charge at any given spot remained constant, the charge shifted back and forth between the internal states. By analyzing the structure of the crystal, they realized this breathing motion corresponded to a specific, complex pattern in the arrangement of the electrons' internal states, resembling a lattice of tiny magnetic textures known as skyrmions. This finding connects the abstract mathematics of quantum geometry to a tangible, physical vibration within the crystal, revealing that the hidden geometry leaves a clear fingerprint on how these materials vibrate and respond to the world around them.

Technical Summary: The Influence of Quantum Geometry on the Phase Boundary and Collective Excitations of Electron Liquids and Crystals

Problem Statement
Recent experiments on rhombohedral $N$ -layer graphene (RNG) aligned with hexagonal boron nitride (hBN) have revealed a rich landscape of topological phases, including anomalous Hall crystals (AHCs) and Wigner crystals (WCs). These systems are characterized by the coexistence of strong electron-electron interactions and non-trivial quantum geometry (specifically, non-uniform Berry curvature). While mean-field Hartree-Fock (HF) theories have been applied to these systems, they are known to fail quantitatively in conventional two-dimensional electron gases (2DEG), often overestimating the stability of crystalline phases. Recent variational Neural Quantum State (NQS) calculations on the $\lambda$ -jellium model (a minimal model incorporating quantum geometry) suggested that quantum geometry favors electron crystallization, shifting the liquid-crystal transition to higher densities. However, the microscopic mechanisms driving this transition and the specific modifications to collective excitations in both liquid and crystal phases remain to be fully understood beyond static variational wavefunctions.

Methodology
This work employs the Time-Dependent Hartree-Fock (TDHF) method combined with the Quasi-Boson Approximation (QBA) applied to the $\lambda$ -jellium model.

Model: The system is described by a band-projected Hamiltonian where the interaction term includes form factors $F(k, k')$ derived from a pseudospinor texture. This texture introduces a momentum-dependent Berry curvature controlled by a length scale parameter $\lambda$ . The density is controlled by the Wigner-Seitz radius $r_s$ .
TDHF Formalism: The authors construct collective mode operators as superpositions of particle-hole excitations. By solving the equations of motion under the QBA, they derive a generalized eigenvalue problem involving a stability matrix $S_q$ .
Analysis:
1. Phase Diagram: The ground state energy is computed as the sum of the HF energy and the correlation energy derived from the zero-point energy of the collective modes. This allows for the construction of a liquid-crystal phase diagram.
2. Stability Analysis: The eigenvalues of the stability matrix $S_q$ are analyzed to identify instabilities in the Fermi liquid (FL) ground state.
3. Response Functions: The density-density ( $\chi_{\rho\rho}$ ) and transverse current-current ( $\chi_{j_\perp j_\perp}$ ) response functions are calculated. These are further decomposed into orbital-resolved components to probe the internal structure of collective excitations.
4. Crystal Phases: The method is extended to symmetry-broken crystalline states (WC, AHC, and Halo WC) to analyze their phonon and exciton spectra.

Key Contributions and Results

Phase Boundary and Crystallization:
- The TDHF calculations confirm the trend observed in recent NQS studies: the presence of quantum geometry (finite $\lambda$ ) favors electron crystallization, shifting the critical density ( $r_s$ ) for the liquid-crystal transition to significantly higher values.
- While TDHF is a perturbative technique and yields correlation energies more negative than exact Quantum Monte Carlo (QMC) results (limiting quantitative precision), the qualitative agreement with NQS results regarding the shift in the phase boundary is robust.
Novel Instabilities in the Fermi Liquid:
- The authors identify two distinct instabilities in the FL ground state as $r_s$ $r_{s}$ and $\lambda$ $λ$ increase:
  - A standard $2k_F$ instability (charge density wave) associated with the sharp Fermi surface, which persists regardless of $\lambda$ .
  - A novel instability emerging at $q = k_F$ that appears only in the presence of strong quantum geometry (large $\lambda$ ). This instability reflects a tendency for interactions to eject electrons from the Brillouin zone center (where Berry curvature is concentrated) toward the Fermi surface.
- The coincidence of this $k_F$ instability with the liquid-crystal transition suggests that fluctuations driving this specific mode play a crucial role in the competition between the liquid and crystal phases.
Suppression of Collective Modes in the Liquid Phase:
- Plasmon Dispersion: Quantum geometry reduces the dispersion of the plasmon mode. Analytic arguments using the Random Phase Approximation (RPA) link this reduction to the trace of the quantum metric integrated over the Fermi sea.
- Friedel Oscillations: In the static limit, Friedel oscillations are completely suppressed at $\lambda = 0.5$ . This occurs because the form factor connecting backscattering states ( $k \to -k$ ) vanishes when the quantum geometry renders these states orthogonal.
- Orbital-Resolved Response: The suppression of the total density response is not due to a lack of fluctuations but rather a redistribution of spectral weight. The authors resolve the response into orbital components and find that the total density suppression is accompanied by the emergence of a strong out-of-phase inter-orbital mode. In this mode, charge oscillates between the two pseudospin orbitals while the total density remains fixed.
Collective Modes in Crystalline Phases:
- The study characterizes three crystalline phases: the conventional Wigner Crystal (WC), the Anomalous Hall Crystal (AHC), and the Halo Wigner Crystal (HWC).
- The authors identify an analogous out-of-phase inter-orbital mode in the crystal phases, appearing specifically at the $\Gamma$ point.
- By computing the real-space pseudospin texture, they demonstrate that this mode corresponds to the breathing mode of an emergent real-space pseudospin skyrmion lattice. The strength of this mode correlates with the presence of a skyrmion texture in the real-space density matrix.

Significance
The paper provides a complementary perspective to recent variational NQS studies, utilizing TDHF to access the dynamic properties and instabilities of the system. The primary significance lies in:

Mechanistic Insight: Identifying a specific $k_F$ instability driven by quantum geometry that likely facilitates the crystallization transition.
Dynamic Signatures: Establishing that quantum geometry fundamentally alters the response of electron liquids, specifically by suppressing Friedel oscillations and plasmon dispersion through the generation of out-of-phase inter-orbital fluctuations.
Unified Picture: Connecting the suppression of density response in the liquid phase to the breathing modes of skyrmion lattices in the crystal phase, suggesting a continuous evolution of collective excitations across the phase transition driven by the underlying quantum geometric texture.

The work underscores that quantum geometry is not merely a static band structure feature but actively reshapes the energetic competition and dynamic response of strongly correlated electron systems.

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