Theory of two-dimensional Wigner crystals with defects: Interactions, melting transitions and collective modes

The gist: This work constructs a comprehensive framework using duality techniques to study the contribution of various lattice defects—vacancies, interstitials, dislocations, and disclinations—to the effective action of two-dimensional charged crystals.

Theory and Methodology

The research employs duality techniques, specifically elasticity-gauge duality or fracton-elasticity duality, to derive fundamental aspects of defect physics from first principles In this framework, different defects are viewed as sources of electromagnetic and elastic gauge fields The study is conducted within the paradigmatic jellium model, which treats conduction electrons as residing within a static uniform positively charged background while neglecting spin The action incorporates both the electron-electron interaction mediated by an electrostatic potential and terms describing lattice deformations and mobile charge carriers like vacancies and interstitials

Defect Interactions

The paper systematically computes the electrostatic potential and stress fields produced by isolated, static defects, including point charges, dislocations, and disclinations For a point charge with charge Q in the presence of gates at distance d, the long-range asymptotic behavior of the electrostatic potential A(r) is found to be screened by a characteristic inverse length given by k∗ ≡ s/(d + lsc)d 2lsc The energy of a single vacancy or interstitial is calculated as Evac = Ecore + d/(d + lsc) squared CsQ 2/12πe 2n 0a-2 core, which suggests that the energy of a single vacancy/interstitial can be separated into the energy of the core Ecore and the energy of this long-range displacement

Collective Modes and Melting

The analysis investigates collective modes in different phases, such as dislocation proliferation leading to a hexatic Wigner crystal In the context of the hexatic phase, two low-energy modes are found: a longitudinal plasmon mode with dispersion ωplasm = s/C˜b(k) 2n 0m k similar to Eq. (7) and a transverse mode with linear dispersion ωtrans = s gψ 2 0 m1 k The longitudinal plasmon mode is of a purely electronic origin and is only slightly affected by the loss of shear resistance, while the transverse mode is charge-neutral and thus not affected by the electronic character of the hexatic

Vacancy Proliferation in Metallic Wigner Crystals

The proliferation of charged defects, specifically vacancies or interstitials, leads to a metallic state where these holes form a Fermi liquid at low densities The hydrodynamic equations for this ideal Fermi liquid are encoded in the low-energy action after adding a new dynamical scalar field ϕ The longitudinal phonon spectrum of the theory produces two modes, with ω plasm(k) ≈ e 2/(2ε(2/3)ρv + n eff n 0m tanh(kd))k and ω ph(k) ≈ Cb + Cs + 4πħ 2n 0/mh 2n 0m + 2n effmh/ρv k squared By measuring the speed of sound cph, one can estimate ρv for different experimental parameters, and at a specific density ρ∗v, the density of states of the vacancy fluid mh/(2πħ 2) can be extracted

Discussion and Implications

The research confirms that thermal melting in the classical regime can be described by the standard KTHNY theory with an effectively infinite bulk elastic modulus Self-doping of vacancies or interstitials is predicted to lead to a metallic state, consistent with experimental observations Notably, the speed of sound of the longitudinal acoustic phonon in the metallic Wigner crystal depends on the density and effective mass of the vacancies or interstitials, suggesting an experimental way to measure them The utility of this framework extends to any charged system where translational symmetry is spontaneously broken, including charge density waves and anomalous Hall crystals

Acknowledgments

The authors acknowledge financial support by the European Research Council (ERC) under grant QuantumCUSP (Grant Agreement No. 101077020). P.M. acknowledges support by the Harry Bloomfield International Scholarship and a postdoctoral fellowship from the Azrieli Foundation. The paper also cites numerous foundational works on Wigner crystals, melting transitions, and duality techniques.

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The gist: This work constructs a comprehensive framework using duality techniques to study the contribution of various lattice defects—vacancies, interstitials, dislocations, and disclinations—to the effective action of two-dimensional charged crystals.

How it works

The research employs duality techniques, specifically elasticity-gauge duality or fracton-elasticity duality, to derive fundamental aspects of defect physics from first principles In this framework, different defects are viewed as sources of electromagnetic and elastic gauge fields The study is conducted within the paradigmatic jellium model, which treats conduction electrons as residing within a static uniform positively charged background while neglecting spin

Discussion and Implications

The research confirms that thermal melting in the classical regime can be described by the standard KTHNY theory with an effectively infinite bulk elastic modulus Self-doping of vacancies or interstitials is predicted to lead to a metallic state, consistent with experimental observations <ref:2607.

1. Increased Predictive Accuracy for Wigner Crystal Melting Temperatures: The improved AI can accurately predict melting temperatures by incorporating vacancy proliferation effects, as suggested by finding that the predicted melting temperature of the Wigner crystal to drop upon entering the metallic phase.

2. Enhanced Measurement of Vacancy Properties via Sound Speed: The system can estimate parameters like density of states of the vacancy fluid mh/(2πħ2/m) by comparing measured longitudinal phonon speed cph with theoretical curves, as shown in Fig. 3, and determining both m and mh.

3. Discovery of Novel Defect Interaction Energies: The AI can calculate analytical formulas for interactions between defect pairs other than two dislocations, specifically finding interaction energies for point charges, dislocations, and disclinations as detailed in Table IV.