---
title: Jellium Problem in Electron Gas Theory
url: https://www.emergentmind.com/topics/jellium-problem
type: topic
---

# Jellium Problem in Electron Gas Theory

The jellium problem is the study of interacting charged particles embedded in a uniform neutralizing background, with the background introduced to enforce global neutrality and to regularize long-range Coulomb energies. In condensed-matter usage, it is the paradigmatic model of the uniform electron gas; in statistical mechanics it is the one-component plasma; and in two dimensions it also appears as the logarithmic Coulomb gas that underlies several exact and asymptotic results. Across these settings, the problem serves as a reference point for simple metals, warm dense matter, density-functional theory, Coulomb-gas asymptotics, and the plasma analogy for fractional quantum Hall states [2103.03720][2203.06952][1705.07495].

## 1. Canonical formulations

In three dimensions, jellium is most commonly written as a gas of electrons of charge \(e\) moving in a compensating positive background. A compact Hamiltonian is
\[
H = \sum_{i=1}^N \frac{p_i^2}{2m} + \frac{1}{2} \sum_{i\neq j} \frac{e^2}{|r_i-r_j|} + V_{\mathrm{bg}},
\]
where \(V_{\mathrm{bg}}\) collects the electron–background and background–background terms needed for neutrality and a finite thermodynamic limit. In the units used for finite-temperature simulations, lengths are scaled by \(a=(4\pi n/3)^{-1/3}\), energies by the Rydberg, and \(r_s=a/a_B\) parametrizes density; small \(r_s\) is the high-density regime and large \(r_s\) the strongly correlated regime [2103.03720].

In two dimensions, the Coulomb kernel is logarithmic,
\[
G(z)=-\log|z|,\qquad -\Delta G = 2\pi \delta_0,
\]
and the classical one-component plasma on a bounded domain \(D\subset \mathbb{R}^2\) with background density \(\rho_b\) has energy
\[
H_{\mathrm{jellium}}(z_1,\dots,z_N)
= -\sum_{1\le i<j\le N}\log|z_i-z_j|
+ \sum_{i=1}^N V_{\mathrm{ext}}(z_i)
-\rho_b \sum_{i=1}^N \int_D \log|z_i-x|\,dx
+ \frac{\rho_b^2}{2}\int_D\int_D \log|x-y|\,dx\,dy.
\]
Neutrality is imposed by \(\rho_b |D|=N\). Hard-wall confinement or a smooth confining potential growing at infinity are standard choices [2203.06952].

A broader formulation replaces the Coulomb law by a Riesz kernel \(g_s(x)=|x|^{-s}\), \(0<s<d\). In dimension \(d\ge 3\), Coulomb corresponds to \(s=d-2\), while the logarithmic case is \(s=0\) in \(d=2\). In this language, the neutralized jellium energy in a domain \(\Omega\subset\mathbb{R}^d\), \(|\Omega|=N\), is
\[
E_{\mathrm{Jel},s}(\Omega,\{x_i\})
=
\sum_{1\le i<j\le N}|x_i-x_j|^{-s}
-
\sum_{i=1}^N \int_{\Omega}|x_i-y|^{-s}\,dy
+
\frac12\iint_{\Omega\times\Omega}|x-y|^{-s}\,dx\,dy.
\]
This Riesz form is the setting for several sharp asymptotic results and for the comparison between jellium and the uniform electron gas in density-functional theory [1707.07664].

## 2. Thermodynamic limit, mean-field theory, and density functionals

At finite temperature, the classical \(2\)D jellium Gibbs measure is
\[
Z_N(\beta)=\frac{1}{N!}\int_{D^N} e^{-\beta H_{\mathrm{jellium}}}\,dz_1\cdots dz_N,
\qquad
\mu_{N,\beta}=\frac{1}{Z_N(\beta)} e^{-\beta H_{\mathrm{jellium}}}\,dz_1\cdots dz_N.
\]
Under the mean-field scaling \(V_{\mathrm{ext}}(x)=N v(N^{-1/2}x)\), with \(v\) smooth and confining, the leading equilibrium is governed by
\[
\mathcal{F}[\sigma]
=
\int_{\mathbb{R}^2} v(x)\sigma(x)\,dx
-
\frac12 \iint_{\mathbb{R}^2\times\mathbb{R}^2}\log|x-y|\,\sigma(x)\sigma(y)\,dx\,dy,
\]
minimized over probability measures \(\sigma\). If \(\sigma_{\mathrm{MF}}\) is the unique minimizer, then the \(k\)-point equilibrium densities converge weakly after the natural rescaling \(x\mapsto N^{-1/2}x\), and the free energy has the local-density expansion
\[
F(N,\beta)
=
-\frac{N^2\log N}{4}
+N^2\inf \mathcal{F}
+
N\int_{\mathbb{R}^2} f_{\mathrm{hom}}\big(\beta,\sigma_{\mathrm{MF}}(x)\big)\,dx
+o(N),
\]
where \(f_{\mathrm{hom}}(\beta,\rho)\) is defined by the homogeneous thermodynamic limit of jellium [2203.06952].

For homogeneous jellium, Lieb–Narnhofer-type thermodynamic limits are central. In \(2\)D, if \(\Omega_L\) is a box of side \(L\), \(N/|\Omega_L|\to \rho\), and \(V_L\) is the background-generated confinement with \(+\infty\) outside \(\Omega_L\), then
\[
f(\beta,\rho)
:=
\lim_{L\to\infty,\;N/|\Omega_L|\to \rho}
\frac{F(L,N,\beta)+E(L,\rho)}{|\Omega_L|}
\]
exists and is independent of the shape of \(\Omega\) in \(2\)D and, more generally, in \(d\ge 2\). This homogeneous free-energy density is the local input of the local-density approximation [2203.06952].

In density-functional theory, the exact relationship between the inhomogeneous electron gas and inhomogeneous jellium is unusually sharp. If \(F_e(\beta,V|n)\) and \(F_j(\beta,V|n)\) are the Mermin intrinsic free-energy functionals of the electron gas and jellium, then
\[
F_j(\beta,V|n)
=
F_e(\beta,V|n)
+
\int d\mathbf{r}\, v_b(\mathbf{r})\,n(\mathbf{r})
+
\beta E_b,
\]
with \(v_b\) the electron–background potential and \(E_b\) the background self-energy. Since the Hartree sector exactly absorbs the background dependence, the exchange–correlation functionals coincide:
\[
F_{\mathrm{xc}}^{j}(\beta,V|n)\equiv F_{\mathrm{xc}}^{e}(\beta,V|n).
\]
This identity justifies the use of jellium-based exchange–correlation information in local and gradient approximations for real inhomogeneous electronic systems [1705.07495].

## 3. Next-order asymptotics, renormalized energy, and equivalent definitions

A major modern formulation of the jellium problem concerns the next-order term beyond the mean-field \(N^2\) energy. For Riesz interactions \(0<s<d\), the minimal confined energy has the asymptotic form
\[
E_{N,s}(V)
=
N^2 I_{s,V}(\mu_V)
+
N^{1+s/d}\, C_{\mathrm{Jel}}(s,d)\,
\int_{\mathbb{R}^d}\rho_V(x)^{1+s/d}\,dx
+
o(N^{1+s/d}),
\]
where \(C_{\mathrm{Jel}}(s,d)\) is defined via a renormalized infinite-volume energy \(W\). In the Caffarelli–Silvestre extended formulation,
\[
-\mathrm{div}(|y|^\gamma E)=C_{d,s}\big(\nu-\mathbf{1}\big),
\]
and
\[
C_{\mathrm{Jel}}(s,d)=\min_{\nu\in \mathrm{Config}} W(\nu).
\]
For the many-marginals optimal-transport formulation of the uniform electron gas, one similarly gets
\[
F_{N,s}(\mu)
=
N^2 I_s(\mu)
+
N^{1+s/d}\, C_{\mathrm{UEG}}(s,d)\,
\int_{\mathbb{R}^d}\rho(x)^{1+s/d}\,dx
+
o(N^{1+s/d}).
\]
The main theorem is
\[
C_{\mathrm{Jel}}(s,d)=C_{\mathrm{UEG}}(s,d)
\]
for \(d\ge 3\) and \(s\in[d-2,d)\), and for \(d=2\) and \(s\in(d-2,d)\). In the Coulomb case \(d=3,s=1\), this verifies the long-standing conjecture that the second-order constants of jellium and the uniform electron gas coincide [1707.07664].

At the microscopic scale, the jellium energy equidistributes. After blow-up at the interparticle spacing \(n^{-1/d}\), the local renormalized energy density in a microscopic cube converges to the minimum jellium energy corresponding to the local macroscopic density. In the Coulomb case this leads to surface-order discrepancy bounds for point counts,
\[
\left|
\nu_n'(K_\ell(a_n))
-
\int_{K_\ell(a_n)} m_V'(x)\,dx
\right|
\lesssim \ell^{d-1},
\]
in the regime described in the theorem. This is a rigorous form of microscopic rigidity that does not amount to a crystallization proof but does identify the local energy density with the jellium minimum [1609.03849].

A complementary question is whether apparently different definitions of the jellium ground-state energy give the same thermodynamic constant. In \(d\ge 2\), one has
\[
e_{\mathrm{Jel}} = e_{\mathrm{per}} = e_{\mathrm{UEG}},
\]
where \(e_{\mathrm{Jel}}\) is the classical background-neutralized energy density, \(e_{\mathrm{per}}\) is the periodic Coulomb or Ewald-type definition, and \(e_{\mathrm{UEG}}\) is the indirect-energy definition with prescribed constant one-body density. In \(2\)D, a modified floating crystal trial state with exact characteristic-function density extends this equivalence and connects \(e_{\mathrm{Jel}}\) to Serfaty’s renormalized energy and to the order-\(n\) term of logarithmic energy on the sphere [2103.07975]. In \(3\)D Coulomb jellium, a related floating Wigner crystal construction suppresses the boundary charge fluctuations that otherwise lead to a macroscopic energy increase, and proves the coincidence of the background, constant-density, and periodic definitions [1905.09138].

## 4. Plasma analogy, screening, and incompressibility in two dimensions

In fractional quantum Hall theory, Laughlin’s wavefunction at filling \(\nu=1/m\),
\[
\psi_m(z_1,\dots,z_N)
=
C\,
\prod_{1\le i<j\le N}(z_i-z_j)^m
\exp\!\Big(-\frac{1}{4\ell_B^2}\sum_{i=1}^N |z_i|^2\Big),
\]
has squared modulus
\[
|\psi_m|^2
\propto
\exp\!\Big(
2m\sum_{i<j}\log|z_i-z_j|
-
\frac{1}{2\ell_B^2}\sum_i |z_i|^2
\Big)
=
\exp\big(-\beta H_{\mathrm{plasma}}\big),
\]
with \(\beta=2m\). Thus Laughlin’s state is the Gibbs measure of a \(2\)D one-component plasma with quadratic confinement. Analytic quasi-hole factors \(F=\prod_j f(z_j)\) insert additional positive charges, because the terms \(-2\log|f(z_i)|\) are superharmonic in each variable [2203.06952].

The central structural result is an incompressibility estimate. In physical units,
\[
\rho(z)\le \frac{1}{2\pi\,m\,\ell_B^2}
\qquad \text{in the bulk, up to edge corrections.}
\]
In the paper’s rescaled units, for minimizing configurations of
\[
H(x_1,\dots,x_N)
=
\frac{\pi}{2}\sum_{j=1}^N |x_j|^2
-
\sum_{i<j}\log|x_i-x_j|
+
W(x_1,\dots,x_N),
\]
with \(W\) superharmonic in each variable, every disk \(D(a,R)\) satisfies
\[
N(a,R)\le \pi R^2\,(1+g(R)),
\qquad g(R)\to 0 \text{ as }R\to\infty.
\]
A mesoscopic version for Laughlin-like states states that for disks of radius \(N^\alpha\), \(\alpha>(\sqrt{5}-1)/4\),
\[
\int_D \rho_{\Psi_F}
\le
\frac{B}{2\pi\ell}\,|D|\,(1+o(1)),
\]
and the probability of violating this bound has an exponential large-deviation estimate.

The mechanism is a screening construction. For any finite set of point charges \(\{z_1,\dots,z_K\}\), one builds a screening region \(S\) of constant negative charge density such that
\[
U(z)
=
-\sum_{i=1}^K \log|z-z_i|
+
\rho_0 \int_S \log|z-x|\,dx
\]
satisfies
\[
U(z)=0 \text{ for } z\in \mathbb{R}^2\setminus S,
\qquad
U(z)>0 \text{ for } z\in S.
\]
This is equivalent to a Thomas–Fermi-like obstacle problem with constraint \(0\le \sigma\le 1\), \(\int \sigma=K\). The resulting exclusion rule yields the density upper bound, and it feeds directly into a stability theorem for the Laughlin phase: for smooth external potential \(v\) and weak smooth interaction \(w\), there exists \(\lambda_0>0\) such that
\[
\frac{E(N,\lambda)}{e(N,\lambda)}\longrightarrow 1
\qquad (N\to\infty,\ |\lambda|\le \lambda_0),
\]
meaning that among all holomorphic perturbations of Laughlin’s state, independent quasi-hole states of product form are asymptotically optimal [2203.06952].

## 5. Broken symmetries, Hartree–Fock pathologies, and screened approximations

Within Hartree–Fock, the uniform Fermi gas is not the full story. In three-dimensional jellium, broken-symmetry states are energetically favored at any density against the homogeneous Fermi gas with isotropic Fermi surface. At high density, Hartree–Fock yields metallic spin-unpolarized incommensurate crystals in which the charge and spin densities form an incommensurate crystal with more crystal sites than electrons; as \(r_s\to 0\), these states approach pure spin-density waves with modulation wavevector \(Q\to 2k_F\). At lower density, the commensurate Wigner crystal is favored for \(r_s>3.4\), the system undergoes several structural phase transitions, and the polarization transition occurs around \(r_s\approx 8.5\) [1307.3081].

A distinct Hartree–Fock issue is the unscreened single-particle dispersion. In \(3\)D jellium, the Hartree–Fock spectrum
\[
\varepsilon(k)=\frac{k^2}{2}+\Sigma_x(k),
\]
with
\[
\Sigma_x(k)
=
-
\frac{k_F}{\pi}
\left(
1+\frac{k_F^2-k^2}{2kk_F}\ln\frac{k_F+k}{k_F-k}
\right),
\]
has a logarithmically divergent derivative at \(k=k_F\). Consequently, the density of states
\[
g(\varepsilon)=\frac{V k^2}{\pi^2}\,\frac{1}{d\varepsilon/dk}
\]
vanishes at the Fermi energy. Slater’s hyper-Hartree–Fock equations do not remove this anomaly; instead they shift it from \(k_F\) to a tunable boundary \(k_R=\Lambda^{1/3}k_F\) between optimized and unoptimized orbitals. This supports the interpretation that the pathology is an artifact of the nonlocal exchange operator rather than exclusively a failure of screening [1502.01479].

Self-consistent \(GW\) addresses different aspects of the same problem. In \(3\)D jellium, the fully self-consistent approximation is built from
\[
G^{-1}=(G^0)^{-1}-\Sigma,
\qquad
W^{-1}=v^{-1}-\Pi,
\qquad
\Sigma=-G\,W,
\]
with dielectric function
\[
\varepsilon(q,\omega)=1-v(q)\Pi(q,\omega),
\qquad
v(q)=\frac{4\pi e^2}{q^2}.
\]
Its most serious deficiency is an incorrect dielectric response, traceable to violation of particle-number conservation in \(\Pi\). Enforcing
\[
\Pi(q,i\omega_n)\to \Pi(q,i\omega_n)-\Pi(0,i\omega_n)+\Pi(0,0)\delta_{n,0}
\]
restores the physically required \(q\to 0\) behavior. At \(r_s=1\), \(q=0.1k_F\), and \(T/E_F=0.02\), the corrected dielectric function exhibits a plasmon zero close to \(\omega_p\), with
\[
\omega_p^{(\mathrm{GW})}\approx 0.89(1)\,\omega_p,
\]
and the same calculations yield benchmark \(GW\) values for \(-E_{XC}\), the quasiparticle residue \(Z\), and \(m^*/m\) for \(r_s=1,2,4,5,10\) [1607.01183].

A recent extension shows how quantum geometry modifies the classical phase logic. In \(2\)D \(\lambda\)-jellium, the lower band has Berry curvature
\[
\Omega(\mathbf{q})=\frac{2\lambda^2}{(1+\lambda^2 q^2)^2},
\qquad
C=\frac{1}{2\pi}\int d^2k\,\Omega(\mathbf{k})=1,
\]
while the dispersion remains quadratic. Self-consistent Hartree–Fock then yields two Fermi liquids, two Wigner crystals, and an anomalous Hall crystal; the anomalous Hall crystal occupies a large region of the \((r_s,\lambda)\) phase diagram, and a continuous transition separates it from a halo Wigner crystal [2503.12704].

## 6. Numerical methods, finite temperature, and open directions

At finite temperature, fermionic path-integral Monte Carlo for jellium is obstructed by the sign problem. A restricted-path, fixed-node formulation combined with a canonical worm algorithm gives a practical route through this obstruction. In the implementation for fully polarized \(N=33\) electrons with periodic boundary conditions, long-range Coulomb interactions are handled through the Fraser correction,
\[
\phi(r)=v(r)-\frac{N}{N-1}D,
\qquad
v(r)=\frac{2\,\mathrm{Ry}}{r_s r},
\qquad
D=\frac{1}{\Omega}\int_{\mathrm{cell}} v(r)\,dr,
\]
which is equivalent in spirit to Ewald summation. The restricted path uses free-fermion trial nodes
\[
\rho_T(R,R';t)=\det[\rho_0(r_i,r_j';t)],
\]
and two canonical variants were studied: Algorithm A, which under-samples exchange at low temperature, and Algorithm B, which restores exchange sampling by a tailored \(G\)-sector. For \(r_s=1,2,4\) and \(\theta=T/T_F=1,0.5,0.25,0.125\), the method reproduces canonical restricted PIMC energies and pair correlations with good agreement at high density, and substantially improves low-\(\theta\) results when permutations are restored [2103.03720].

The same model continues to motivate questions outside the standard metallic-fluid regime. In the dielectric Kirzhnits–Maksimov–Khomskii formalism for the electron–proton jellium model, the screened interaction
\[
W(q,\omega)=\frac{V_C(q)}{\varepsilon(q,\omega)},
\qquad
V_C(q)=\frac{4\pi}{q^2+k_{TF}^2},
\]
leads to a linearized integral equation for \(T_c\). Solving that equation directly gives a dome-shaped \(T_c(r_s)\) with maximum
\[
T_c \approx 0.051\ \mathrm{K} \quad \text{at } r_s=9.0
\]
within Thomas–Fermi–RPA screening, while static Hubbard local-field corrections raise \(T_c\) only several-fold and still keep it below \(1\) K across the explored densities. This supports the conclusion that superconductivity in the jellium model remains weak-coupling over the accessible density range [2606.29306].

Several mathematical and physical questions remain open. In the \(2\)D Laughlin/jellium regime, the universal incompressibility bounds are presently established on disks of radius \(N^\alpha\) with \(\alpha>(\sqrt{5}-1)/4\), and extension to all mesoscopic scales \(L\gg 1\) is identified as a natural open problem; stronger or singular interactions, realistic short-scale disorder, and generalizations beyond independent quasi-holes are likewise unresolved [2203.06952]. In the Riesz/Coulomb asymptotic theory, the logarithmic \(d=2\) case remains exceptional, and the crystallization hypotheses behind the conjectured optimal constants—triangular in \(2\)D logarithmic systems, bcc or fcc in \(3\)D Riesz systems depending on \(s\)—remain open [1707.07664]. These unresolved points help explain why the jellium problem remains a live interface between Coulomb-gas analysis, many-body theory, and computational condensed matter.

Source: https://www.emergentmind.com/topics/jellium-problem