Hartree-Fock Phase Diagrams in Quantum Systems

• Hartree-Fock phase diagrams are mathematical representations of electronic and magnetic phases based on a mean-field approximation, highlighting symmetry breaking through methods like the Overhauser instability.
• They detail transitions between uniform Fermi gases, Wigner crystals, and incommensurate modulated phases as functions of parameters such as electronic density and interaction strength.
• These diagrams serve as benchmarks for advanced many-body techniques by providing clear analytic expressions, despite neglecting correlation effects beyond exchange.

Hartree-Fock phase diagrams are mathematical and graphical representations of the competing electronic and magnetic ground states in quantum many-body systems, as predicted by the Hartree-Fock (HF) mean-field approximation. These diagrams organize the stability domains of phases such as Fermi liquids, Wigner crystals, spin-density waves, charge-density waves, antiferromagnetism, and various forms of superconductivity, as functions of system parameters: notably electronic density, interaction strengths (e.g., on-site repulsion, Hund's exchange), magnetic field, temperature, and others. Unlike correlated methods, the HF approach retains only exchange effects, so the phase boundaries and transition orders reflect symmetry breaking at the mean-field level, and the diagrams often reveal fundamental instabilities and reference points for more advanced treatments.

1. Hartree-Fock Phase Diagrams in Electron Gases

The canonical realizations are the phase diagrams of the homogeneous electron gas ("jellium") in both two and three dimensions. In these systems, the competition between uniform Fermi liquid states and spatially modulated phases is captured via the dimensionless Wigner-Seitz parameter $r_s = a / a_B$, where $a$ is the mean interparticle distance and $a_B$ is the Bohr radius. The fundamental classes of phases in the HF phase diagram are as follows (Bernu et al., 2011, Baguet et al., 2013, Baguet et al., 2014, 0810.3559):

• Uniform Fermi Gas (FG): All electrons fill plane-wave states up to the Fermi momentum $k_F$, with constant charge and spin densities.
• Commensurate Wigner Crystal (WC): Electrons localize into a periodic lattice, usually triangular in 2D or bcc/fcc in 3D, with charge density maxima commensurate with the number of particles.
• Incommensurate Crystal (IC): The lattice of maxima contains more sites than electrons, resulting in metallic ("overoccupied") states; these include spin-density waves (SDW) and charge-density waves (CDW).
• Spin Polarized (Ferromagnetic) Phases: At lower densities (larger $r_s$), the HF solution can favor complete spin alignment.

For 2D unpolarized electrons, the HF zero-temperature phase diagram as a function of $r_s$ exhibits a sequence (Bernu et al., 2011):

$r_s$ interval	HF ground state
$r_s < 1.22$	Unpolarized incommensurate triangular (U-IΔ)
$1.22 < r_s < 1.62$	Unpolarized triangular Wigner crystal (U-WCΔ)
$1.62 < r_s < 2.6$	Unpolarized square Wigner crystal (U-WC□)
$r_s > 2.6$	Fully polarized triangular Wigner crystal (P-WCΔ)

In 3D, the sequence involves unpolarized and polarized incommensurate and commensurate crystals with varying lattice symmetries (bcc, fcc, sc, hexagonal) and a polarization transition at $r_s \approx 8.5$ (Baguet et al., 2013, Baguet et al., 2014).

Notably, the uniform Fermi gas is never the HF ground state at zero temperature; it is always unstable to symmetry breaking via the Overhauser effect (Gontier et al., 2018).

2. Symmetry Breaking and the Overhauser Instability

A central and ubiquitous property of Hartree-Fock phase diagrams for electron gases is the instability of the uniform state—originally predicted by Overhauser—to infinitesimal density/spin modulations: for any $r_s$, a weak CDW or SDW perturbation generates a state of lower HF energy (Bernu et al., 2011, 0810.3559, Gontier et al., 2018). Analytically, the energy gain is exponentially small at high density (small $r_s$), justifying the use of the uniform Fermi gas as a reference for large-$k_F$ expansions, but the broken-symmetry phase strictly pre-empts the homogeneous solution.

The phase transition from Fermi gas to metallic SDW/CDW (incommensurate crystal) and subsequently to a Wigner crystal as $r_s$ increases is continuous in the amplitude of modulation, but the symmetry and commensurability of the ground state undergo sharp changes at critical values of $r_s$ (Bernu et al., 2011, 0810.3559, Baguet et al., 2014).

3. Quantitative Description: Energies, Order Parameters, and Phase Boundaries

The HF energy per particle is constructed as: $E[ρ_1]=T[ρ_1]+V_x[ρ_1]$ with kinetic and exchange contributions explicitly evaluated for each competing phase. For example, in 2D (Bernu et al., 2011):

• Unpolarized Fermi gas:

$$E^{U}_{FG}(r_s) = \frac{1}{2r_s^2} - \frac{8}{3\pi\sqrt{2}r_s}$$

• Fully polarized Fermi gas:

$$E^{P}_{FG}(r_s) = \frac{1}{r_s^2} - \frac{8}{3\pi r_s}$$

• Wigner crystal: Asymptotes to the Madelung form

$$E_{Madelung}(r_s) = -1.1061 / r_s + \mathcal{O}(r_s^{-2})$$

For incommensurate phases, the ground-state is found by variational minimization over the modulation wavevector $Q$ for each crystalline symmetry, resulting in a nontrivial manifold of metallic, symmetry-broken states. The phase boundaries correspond to the crossing points of the HF energies of different modulated phases, determined either numerically or—at high density—by analytic asymptotic expansions (Bernu et al., 2011).

4. Multiband and Multiorbital Hartree-Fock Phase Diagrams

HF phase diagrams are also prominent in lattice fermion models with multiple orbitals or bands, including the Hubbard model (and extensions thereof) and multiorbital systems relevant for transition metal oxides and iron-based superconductors. In these cases, real-space or momentum-space HF solutions describe a rich hierarchy of magnetic, superconducting, and coexistent phases as functions of control parameters such as electron filling, on-site repulsion $U$, Hund's coupling $J$, and inter-orbital hybridization (Luo et al., 2013, Zegrodnik et al., 2012, Matsuyama et al., 2022, Charlier et al., 23 Dec 2025).

For example, the five-orbital Hubbard model for iron pnictides exhibits, in the real-space HF solution, a sequence as a function of filling $n$ and $U$: G-type antiferromagnet, mixed G-C, C-type (stripe) antiferromagnet, E-type (bicollinear), block-antiferromagnet, flux, and double-C phases, interspersed with phase separation regions (Luo et al., 2013). The existence and arrangement of such phases is generic for multiorbital models.

In the Hartree-Fock-Bogoliubov approach to extended schematic models (such as the Agassi model), the interplay between Hartree (density) and pairing channels produces analytic phase boundaries, quantum multicritical points with phase coexistence, and regions of first- and second-order transitions (García-Ramos et al., 2018).

5. Hartree-Fock Phase Diagrams in Correlated Lattice Systems

For the 2D Hubbard model, the HF phase diagram constructed via analytic asymptotics fully delineates regions of paramagnetism, ferromagnetism, antiferromagnetism, and narrow coexisting ("mixed") phases in the space of ($U$, doping) (Charlier et al., 23 Dec 2025). The method classifies solutions into uniform, ferromagnetic, and commensurate antiferromagnetic mean fields, deduces explicit critical curves, and rigorously proves the existence of mixed-phase domains, not captured by naïve mean-field numerics.

In the context of cuprate high-$T_c$ superconductors, symmetry-constrained HF solutions of a realistic effective three-parameter Hamiltonian predict—using only experimentally calibrated interaction strengths—quantum phase transitions at $p_1\approx0.05$ (SDW to d-density wave pseudogap) and $p_2\approx0.16$ (pseudogap to pure $d$-wave superconductor), quantitatively accounting for order parameter evolution, $T_c$, London penetration depth, and spin-wave velocity (Laughlin, 2013).

6. Hartree-Fock–BCS Coexistence and Extended Many-Body Models

Combining Hartree-Fock and BCS-type mean fields in orbitally degenerate models yields phase diagrams exhibiting coexistent spin-triplet superconductivity and magnetic — antiferromagnetic or ferromagnetic — orderings. The precise stability regions depend sensitively on Hund's exchange, band filling, inter-orbital hybridization, and temperature, with analytic and numerical solutions mapping out transitions among normal, pure superconducting, pure magnetic, and various coexisting (SC+AF, SC+FM) phases (Zegrodnik et al., 2012). These paradigms are extensible to higher-dimensional, multi-sublattice, or mixed symmetry models.

7. Implications and Methodological Considerations

Hartree-Fock phase diagrams serve as archetypal benchmarks for many-body symmetry breaking and as limiting cases for more comprehensive treatments (e.g., RPA, correlated quantum Monte Carlo, dynamical mean-field theory). They systematically clarify the nature of mean-field instabilities (e.g., Overhauser-type CDW/SDW collapse for the electron gas), the order of the transitions (1st or 2nd order depending on model details), and the analytic structure of phase boundaries. In certain cases (e.g., the Agassi model), they realize maximal multicriticality—points or lines where up to five phases coexist—providing stringent tests for novel algorithms in quantum many-body theory (García-Ramos et al., 2018).

However, all HF-phase diagrams neglect correlation effects beyond exchange, which can shift quantitative values, alter transition orders, and stabilize new phases not accessible via single-determinant ansätze. Nonetheless, the qualitative sequence of symmetry-broken states, and the explicit analytic expressions, underlie and inform the analysis of strongly correlated electron matter across condensed matter and quantum chemistry.