Two-Dimensional Coulomb Systems

• Two-dimensional Coulomb systems are ensembles of interacting particles confined to a plane by logarithmic potentials, exhibiting unique screening, fluctuation, and phase transition behaviors.
• Mathematical analysis employs selfadjoint extensions, angular momentum decomposition, and operator product expansions to rigorously manage singular potentials and boundary effects.
• These systems underpin applications in classical plasmas, Wigner crystals, and quantum Hall physics, with dielectric environment tuning enabling control over screening and correlation phenomena.

Two-dimensional Coulomb systems are ensembles of interacting particles confined to a plane and interacting via the Coulomb potential, which, in two dimensions, is typically logarithmic, i.e., $-\ln r$. These systems serve as paradigmatic models for a broad spectrum of phenomena ranging from classical plasmas and condensed matter realizations (such as Wigner crystals in semiconductor heterostructures) to random matrix theory and quantum Hall physics. The interplay of long-range interactions, reduced dimensionality, singular potentials, and boundary effects leads to mathematically rich structures and a diversity of physical behaviors specific to the 2D context.

1. Mathematical Frameworks and Operator Theory

A core challenge of two-dimensional Coulomb systems is the proper definition of Hamiltonians with singular (logarithmic or Coulomb-like) potentials. The standard example is the Hamiltonian

$H_C = -\Delta - \frac{Z}{\sqrt{x^2 + y^2}}$

in $L^2(\mathbb{R}^2)$ for $Z>0$, which exhibits a singularity at the origin. The potential is shown to be "form-bounded" with respect to the Laplacian via a Kato-type inequality, ensuring that the quadratic form

$$q[f] = \int |\nabla f|^2 \,dx\,dy - Z \int \frac{|f(x,y)|^2}{\sqrt{x^2 + y^2}} \,dx\,dy$$

is closed and bounded below, with the form domain $H^1(\mathbb{R}^2)$. Selfadjoint realization via the Friedrichs extension is thus well-posed (Duclos et al., 2010).

Decomposition into angular momentum sectors via polar coordinates reveals that while $m \neq 0$ channels yield essentially selfadjoint Hamiltonians, the $m=0$ (rotationally invariant) sector exhibits deficiency indices $(1,1)$. This necessitates a one-parameter family of selfadjoint extensions parametrized by a boundary condition at the origin: $$f_1 = \kappa f_0, \quad f_0 = \lim_{\rho\to0_+}(-\ln\rho)^{-1}f(\rho),\quad f_1 = \lim_{\rho\to0_+} \left[f(\rho) + f_0\ln\rho\right].$$ The parameter $\kappa$ encodes a central point (delta-like) interaction.

In momentum space, the construction proceeds via a Whittaker-type transformation, leading to an alternative boundary parameter $\hat\kappa$ related to $\kappa$ by an explicit shift: $\kappa = \hat\kappa - \ln Z - \gamma + 3\ln 2,$ with $\gamma$ the Euler constant. The coordinate and momentum representations are thus unitarily equivalent (Duclos et al., 2010).

2. Statistical Mechanics and Correlation Regimes

Two-dimensional Coulomb systems display distinctive statistical mechanics, with the logarithmic potential leading to unconventional screening properties and long-range order phenomena.

Weak Coupling (Poisson–Boltzmann Regime)

For small coupling parameter $\Gamma=\beta e^2$, the mean-field Poisson–Boltzmann (PB) equation is valid. For a single charged line in the plane, the equilibrium counter-ion density is (Samaj et al., 2010): $$n(x) = \frac{1}{\pi\Gamma}\frac{1}{(x+\mu)^2},\quad \mu = \frac{1}{\pi\Gamma\sigma}.$$ At large $x$, $n(x) \sim 1/x^2$, independent of the surface charge, signifying a universal decay.

Strong Coupling (Wigner Crystal Regime)

As $\Gamma\to\infty$, counter-ions localize into a 1D Wigner crystal near the surface, and the energy change upon displacing a particle is, in rescaled units,

$$-\beta\delta E(x, y) \sim -\tilde{x} + \frac{1}{6\Gamma}(\tilde{x}^2 - \tilde{y}^2).$$

The density profile and pressure can be computed perturbatively; notably, an attractive regime (negative pressure) emerges between like-charged lines at sufficiently strong coupling, a phenomenon purely of correlation origin and forbidden by PB theory (Samaj et al., 2010).

Intermediate Coupling and Exact Solvability

For $\Gamma=2\gamma$ $(\gamma\in \mathbb{N})$, exact solution techniques based on Grassmann variables become available. At $\Gamma=2$, the decay $n(x)\sim 1/x^2$ persists; for $\Gamma=6$, the decay exponent increases ($n(x)\sim x^{-2-\nu}, \nu\approx 1.45$), signaling a fundamental change in asymptotic behavior (Samaj et al., 2010).

3. Quantum and Correlation Properties

Quantum short-range singularities critically influence two-dimensional Coulomb systems. Universal high-momentum tails in observables are dictated by operator product expansions (OPE):

• Momentum distribution:

$$n(\mathbf{q}) \sim \left(\frac{2\pi}{a_0}\right)^2 \frac{n^2 g(0)}{q^6}$$

• Static structure factor:

$$S(\mathbf{q}) - 1 \sim -\frac{4\pi}{a_0} \frac{n g(0)}{q^3}$$

where $g(0)$ is the "contact"—the pair distribution function at zero separation. These power-law behaviors are universal, independent of state or temperature, and vanish in the classical limit as $g(0)$ becomes exponentially small (Hofmann et al., 2013).

In few-body 2D Coulomb systems, multipole moments possess anomalous dimensional dependence: spherically symmetric states in 2D carry non-zero quadrupole and higher moments, leading to long-range effective interactions dominating as $-1/R^3$ (rather than $-1/R^6$ as in 3D). This modifies the asymptotic and stability properties of trions, and stability diagrams for trion binding exhibit pronounced 2D anomalies (Simenog et al., 2014).

4. Phase Transitions and Fluctuation Phenomena

The two-dimensional Coulomb gas exhibits a Kosterlitz–Thouless (KT) transition, separating a “plasma” regime of Debye screening from a “dipole” regime of bound pairs with power-law correlations.

• Rigorous renormalization group (RG) analysis proves that the pressure (thermodynamic limit) exists and is analytic along the KT transition line $\beta = \Sigma(z)$, where the effective flow equations for couplings close to the separatrix characterize the critical manifold. Along this line, the pressure exhibits finite, analytic behavior while the correlation length diverges exponentially:

$$\xi(\beta,z) \sim \exp \left[c(z) |\beta - \Sigma(z)|^{-1/2}\right]$$

(Falco, 2011).

Disordered 2D Coulomb systems, such as Coulomb glasses, display disorder-driven first order transitions; for example, at zero temperature a critical disorder $W_c = 0.2413$ demarcates a jump in staggered magnetization, with finite size scaling exponents $\beta=0$ and $\nu=1.0$ indicating a sharp, discontinuous phase change (Bhandari et al., 2016).

5. Geometric and Boundary Effects

Confinement and boundary conditions strongly impact 2D Coulomb gas properties:

• Hard wall constraints induce a singular component in the equilibrium measure—a delta function at the boundary—and introduce universal Laplace-type kernels in the edge correlations. Specifically, for the determinantal case (e.g., $\beta=1$), the local correlations near the wall are governed by

$$K_a(z,w) = (2\operatorname{Re}z)^{\frac{a+1}{2}} (2\operatorname{Re}w)^{\frac{a+1}{2}} \frac{1}{\Gamma(a+1)} \int_0^1 e^{-t(z+w)} t^a dt$$

This kernel matches that found for truncated unitary matrices in the weak non-Hermitian regime, illustrating universality across models (Seo, 2020).

• In geometries such as an elliptic annulus, the orthogonal polynomial approach (with Chebyshev polynomials of multiple kinds) enables explicit analysis of correlation kernels. In the thin annulus limit, the kernel reduces to the sine kernel of 1D log-gases, revealing crossover between dimensions and highlighting the impact of domain shape on fluctuation statistics (Nagao, 17 Apr 2024).

6. Screening, Engineering, and Applications

Coulomb interaction strength in two-dimensional materials can be engineered via the dielectric environment. For 2D Mott systems, substrate screening reduces the on-site Coulomb repulsion, shifting the position of Hubbard bands by eV-scale energies and inducing insulator–metal transitions. Many-body calculations (Dual Boson or GW+DMFT approaches) capture these effects and match spectroscopic observations; for example, the interaction in momentum space is modified as

$$V(q) = \frac{2\pi e^2}{q} \frac{1}{\epsilon} \frac{1 + x e^{-qh}}{1 - x e^{-qh}}$$

with $$x = (1 - \epsilon_\mathrm{env}/\epsilon)/(1 + \epsilon_\mathrm{env}/\epsilon)$$ (Loon et al., 2020).

For atomically thin semiconductors, both free carrier and exciton-induced screening must be incorporated. The modified Coulomb potential considering free carriers and neutral dipole (exciton) screening is

$$\phi(q) = \frac{e}{q(2\epsilon + q\alpha_\mathrm{2D}^\mathrm{total}) - e\chi_\mathrm{carrier}(q) - 2e \chi_\mathrm{dipole}(q) [1 - J_0(qd)]}$$

This result is derived from linear response theory and enables direct estimates of exciton binding energy shifts and polarizabilities, with predictions matching experiment and simulation (Xiao et al., 2023).

7. Non-equilibrium and Collective Excitations

In disordered 2D Coulomb glasses, the application of an electric field generates nonlinear conductivity, which can be largely mapped to an equilibrium linear response evaluated at an effective temperature $T_\mathrm{eff}$ determined by energy dissipation: $$P = \sigma(T,E) E^2 = c (T_\mathrm{eff}^2 - T^2) T_\mathrm{eff}^3$$ The occupation in the Coulomb gap equilibrates to a Fermi–Dirac distribution with $T_\mathrm{eff}$, and this value is independently corroborated via an extended fluctuation–dissipation theorem (Caravaca et al., 2011).

Classical 2D Coulomb fluids support both longitudinal and transverse collective excitations (plasmons). Molecular dynamics and quasi-crystalline approximations (QLCA/QCA) yield dispersion relations successfully capturing the transition from weakly to strongly coupled regimes: $$\omega_l^2 \simeq \omega_0^2 k a + \mathcal{C} k^2 v_T^2$$ The cutoff wave number below which shear waves cannot propagate scales as $q^* \simeq 15.2\,\Gamma^{-0.9}$ at strong coupling (Khrapak et al., 2018).

In bilayer 2D systems, Coulomb drag generates mode-selective damping: with identical carrier types in each layer, drag damps the acoustic mode; with opposing carrier types, it damps the optical mode. The corresponding plasmon dispersion relation is

$$\omega \left[\omega + \frac{i}{\tau_p} + \frac{i}{\tau_D}(1 \mp \chi_t/\chi_b)\right] = \frac{2\pi n_0 e^2 |q|}{\kappa m}(1 \pm e^{-|q|d})$$

and the selective nature of damping can be probed via resonant linewidths in optical or microwave experiments (Safonov et al., 2023).

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In sum, two-dimensional Coulomb systems exhibit intricate behavior determined by singular potentials, strong correlation effects, boundary conditions, and the capacity for environmental tuning. These features not only underpin rich mathematical structures (determinantal point processes, RG flows, operator expansions) but also drive diverse physical phenomena, including anomalous fluctuation statistics, phase transitions, like-charge attraction, collective mode transformations, and the emergence of new universality classes under confinement or disorder.