Correlated Lattice Fermions and Mott Physics

Correlated lattice fermions can become insulating or ordered through mechanisms that a band picture misses. This chapter connects effective Hamiltonians, strong-coupling reductions, spectra, and numerical descriptions of Mott physics.

Enter this chapter

Begin with From Bands and Orbitals to Effective Lattice Hamiltonians to choose the orbitals and interactions of an effective lattice model.

Helpful background. From Bands and Orbitals to Effective Lattice Hamiltonians and Hubbard Models: Symmetries and Controlled Limits develop the definitions used by the chapter’s opening pages.

Topics

1. From Bands and Orbitals to Effective Lattice Hamiltonians — Constructs hopping and interaction tensors while exposing basis and screening ambiguity.
2. Hubbard Models: Symmetries and Controlled Limits — Fixes the Hamiltonian, particle–hole convention, atomic limit, and strong-coupling scale.
3. Superexchange and the t–J Projection — Performs the Schrieffer–Wolff elimination and tracks transformed observables and three-site terms.
4. Hubbard Bands and Spectral-Weight Transfer — Derives atomic poles, moments, coherent weight, and doping transfer.
5. Mott, Slater, Band, and Charge-Transfer Insulators — Provides the mechanism-specific diagnostic matrix, with Anderson localization as a distinct comparison.
6. DMFT Impurity Mapping and Self-Consistency — Derives the local impurity mapping and its infinite-coordination control.
7. Cluster DMFT and Nonlocal-Correlation Validity — Compares cellular and dynamical-cluster constructions and periodization choices.
8. Sign-Free Quantum Monte Carlo and Hubbard-Phase Inference — Proves two standard sign-free domains and builds a bounded finite-size claim.
9. Electron–Phonon Fields and Retarded Interactions — Integrates a harmonic displacement to obtain a retarded attraction and competing channels.
10. Multiorbital Correlations and Hund’s Metals — Connects Kanamori multiplets to orbital-dependent coherence.
11. Competing Orders and Electronic Nematicity — Constructs coupled-order functionals and distinguishes primary from vestigial nematicity.
12. Doped Mott Matter and the Pseudogap Evidence Problem — Triangulates a pseudogap while retaining alternative mechanisms.

How the ideas fit together

At large repulsion near one fermion per site, virtual charge fluctuations generate spin exchange even when real charge motion is suppressed. This connects Hubbard spectra to low-energy magnetism without identifying every insulating gap as a Mott gap. Band filling, magnetic order, and charge-transfer processes provide alternatives Imada, Fujimori, and Tokura 1998, §§ II–V, pp. 1047–1155.

DMFT captures local dynamical correlations through a self-consistent impurity. Cluster and other numerical methods recover additional spatial information under different limitations. Agreement on one spectral feature therefore requires checks against solver, size, and model assumptions before it establishes a mechanism.

The comparison below separates the assumptions and tests needed for these different conclusions.

Correlated-electron validity table

Stage or decision	Required check	PASS permits	FAIL requires
Hamiltonian reduction	Reproduce target bands and symmetries while varying window, basis, range, screening, and double counting	A declared effective lattice Hamiltonian with reduction uncertainty	Revise the retained subspace or interaction model
Low-energy projection	Verify $t/U\ll1$, match the low-energy spectrum and transformed observables, and retain same-order terms	A spin or projected model for low-energy observables	Return to the full Hubbard sector; do not infer its charge spectrum from the projection
Single-site DMFT	Converge both branches, impurity and bath, residuals, causality, and spectral moments	A local, solver-stable observable in the declared regime	No DMFT observation or phase inference
Cluster method	Test size, shape, boundary condition, solver error, and justified reconstructions	A cluster-stable nonlocal observation	No thermodynamic or nonlocal conclusion from that cluster sequence
Sign-free QMC	Prove nonnegative weight and control autocorrelation, time step or projection, covariance, temperature, and size	A finite-model observable with a bounded extrapolation	No asymptotic phase conclusion
Spectral, orbital, or retarded analysis	Control continuation, resolution, matrix elements, basis, interaction tensor, double counting, and vertex regime as applicable	An operational observable with stated method dependence	Report only an unresolved tendency
Insulator mechanism comparison	Test symmetry restoration, adiabatic continuity, orbital gap edges, spectral transfer, compressibility, and disorder scaling	A bounded Mott, Slater, band, charge-transfer, or Anderson attribution	Keep the insulating observation but list viable mechanisms
Pseudogap observation	Show stable low-energy suppression relative to a declared temperature or doping baseline in converged spectra and at least one corroborating observable	An operational pseudogap in the declared regime	Report a method-dependent suppression, not an established pseudogap
Pseudogap mechanism attribution	Measure discriminants for short-range magnetism, precursor pairing, intertwined order, zeros, disorder, and probe effects	A mechanism claim bounded to the tested model and regime	The pseudogap observation may stand, but its mechanism remains open

Where to go next

Continue to Pairing, Superfluidity, and Superconductivity to derive why an attractive channel destabilizes a Fermi surface.

References

• Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, “Metal–Insulator Transitions,” Reviews of Modern Physics 70 (1998) 1039–1263, doi:10.1103/RevModPhys.70.1039.