Correlated Lattice Fermions and Mott Physics

A correlated-electron claim is supported only when four layers agree: a justified low-energy Hamiltonian, a mechanism-specific insulating or ordering diagnosis, a solver inside its validity domain, and observables that survive uncertainty and competing explanations. Large $U/t$, a two-peak spectrum, or one successful fit is never sufficient by itself Imada, Fujimori, and Tokura 1998, §§ II–V, pp. 1047–1155.

This chapter covers downfolded lattice Hamiltonians, Hubbard and strong-coupling limits, spectral-weight transfer, insulator distinctions, DMFT and cluster interpretation, sign-free Hubbard applications, retarded phonons, Hund matter, competing orders, and bounded pseudogap inference. Generic lattice algorithms, sign-problem theory, and numerical certification remain with Volume VIII.

Helpful background. Effective lattice Hamiltonians supplies the orbital, screening, and downfolding choices; Hubbard symmetries and controlled limits supplies the interaction, filling, and strong-coupling conventions.

Enter this chapter

Start with effective lattice Hamiltonians if orbital choice, screening, or double counting is unfamiliar. Then use Hubbard symmetries and limits before any solver or phase label. A reader prepared for the solver branch should also be able to identify a Green-function sum rule and propagate a correlated numerical error.

Reader goal	Suggested route	Capability at the end
Strong-coupling core	Effective Hamiltonian → Hubbard limits → superexchange → spectral transfer	Derive $J=4t^2/U$ and distinguish projected spin physics from charge localization
Insulator diagnosis	Hubbard limits → spectral transfer → insulator distinctions	Classify Mott, Slater, band, charge-transfer, and Anderson mechanisms using observables
DMFT route	Hubbard limits → single-site DMFT → cluster DMFT	State the infinite-coordination control and measure cluster, bath, and periodization drift
Numerical evidence	Hubbard limits → sign-free QMC → competing orders	Make a finite-size phase statement with covariance and sign-free domain explicit
Doped and multiorbital matter	Effective Hamiltonian → DMFT → Hund metals or pseudogap	Preserve basis, solver, continuation, and mechanism alternatives

From reduction to inference

The first map follows a microscopic band problem through Wannier projection, screened interactions, the Hubbard model, superexchange, spectral observables, and the local DMFT mapping. Each arrow has a small parameter or a model-dependence test. The local self-consistency branch is controlled in the large-coordination limit, not merely at large interaction Georges et al. 1996, §§ II–III, pp. 19–50.

Reduction and solution layers for correlated lattice fermions. The figure is schematic: downfolding is not unique, $t/U$ controls only the projected strong-coupling branch, and DMFT is controlled by locality in large coordination rather than by large $U$ alone.

The second map begins at solver output and asks which errors or alternative mechanisms can still change the conclusion. Inspect the separate branches for bath/cluster drift, sign-free but finite-size QMC, analytic continuation, multiorbital basis dependence, and pseudogap nonuniqueness. Cluster-size, boundary, and periodization dependence are independent controls Maier et al. 2005, §§ II–IV, pp. 1030–1059.

Validity and evidence ceiling for solver-based claims. The map is schematic; passing one branch never substitutes for the other error and mechanism tests.

Correlated-electron validity table

Claim or method	Defining input	Required observable or invariant	Solver or scale control	Competing alternative or ambiguity	Decisive failure mode
Downfolded Hamiltonian	Chosen band window and localized orbitals	Reproduced target bands, symmetries, and screened matrix elements	Window, basis, range, and screening convergence	Different admissible orbital gauges or excluded low-energy states	Physical predictions change strongly under admissible reductions
Strong-coupling spin model	Hubbard sector near integer filling	Matched low-energy spectrum and transformed observables	$t/U\ll1$ with the charge gap retained	Projected hopping, three-site terms, and operator corrections	Double occupancy or omitted $O(t^2/U)$ terms are significant
Mott insulator	Interaction-driven localization without required broken symmetry	Paramagnetic charge gap, incompressibility, and spectral transfer	Thermodynamic and zero-temperature limits	Slater order, band structure, charge transfer, or disorder	Gap disappears when order is removed or lacks interaction-driven transfer
Slater insulator	Translation-breaking magnetic order in an itinerant band	Gap onset tracks the order parameter and reconstructed zone	Ordered and symmetry-restored calculations at matched parameters	Pre-existing paramagnetic Mott gap	Gap persists with the same scale after magnetic order is removed
Band insulator	Filled isolated one-particle bands	Adiabatic band gap at integer band filling	Band convergence and symmetry-preserving interaction path	Correlation-driven spectral transfer or symmetry order	Gap cannot be continued to the weakly interacting band limit
Charge-transfer insulator	Ligand and correlated-orbital levels with charge-transfer scale $\Delta$	Orbital-resolved gap edges and ligand-to-metal spectral transfer	Multiband window, matrix elements, and interaction matching	Single-band Mott description with misassigned orbitals	Gap-edge character or transfer disagrees across probes and reductions
Single-site DMFT	Local self-energy and self-consistent impurity	Causal $G$, spectral moments, and branch convergence	Infinite coordination plus solver and bath convergence	Short-range nonlocal correlations	Cluster corrections qualitatively change the phase or moments fail
Cluster DMFT	Cluster self-energy or cumulant	Persistence across size, shape, boundary, and periodization	Systematic cluster sequence plus solver and bath control	Periodization and cluster-geometry artifacts	Qualitative reordering across admissible clusters persists
Sign-free QMC inference	Nonnegative determinant in a stated domain	Size-scaled estimator with covariance	$\Delta\tau$, autocorrelation, temperature or projection, and size	Crossover mistaken for order; departure from the sign-free domain	Scaling is incompatible with the claimed thermodynamic phase
Hund metal	Rotationally consistent multiorbital interaction	Multiplets, local moments, and orbital-resolved coherence	Basis, filling, crystal field, spin–orbit, and impurity solver	Bad metallicity from another interaction or double counting	Trend survives removal of Hund coupling or is basis-choice dependent
Pseudogap	Operational low-energy suppression	Spectral plus thermodynamic or response triangulation	Cluster/method, continuation, resolution, and temperature	Competing order, scattering crossover, disorder, or matrix elements	Onset shifts without convergence or mechanisms remain indistinguishable

Together with the preceding relationship-centered explanations and alt text, this table supplies the nonvisual account of both figures. It keeps model definition, solver validity, observable evidence, and causal mechanism in separate columns.

Guide to the pages

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.

Convention bridge

We write the one-band kinetic term as $-\sum_{ij\sigma}t_{ij}c_{i\sigma}^\dagger c_{j\sigma}$ and the onsite repulsion as $U n_{i\uparrow}n_{i\downarrow}$. On a bipartite nearest-neighbor lattice, particle–hole symmetry occurs at $\mu=U/2$ in this unshifted convention. As in the correlator dictionary, $A=-2\operatorname{Im}G^R$ and $\int_{-\infty}^{\infty}\mathrm d\omega\,A_\sigma(\mathbf k,\omega)/(2\pi)=1$. Every cluster page states whether it periodizes the self-energy, cumulant, or Green function.

Review the chapter

Mechanism diagnosis. A sample becomes insulating at the onset of antiferromagnetic order and its gap closes when that order is suppressed at fixed interaction. This supports a Slater component; it does not exclude mixed Mott physics without spectral-transfer and paramagnetic-gap tests.

Solver assessment. A four-site cluster shows an antinodal pseudogap. A successful conclusion must survive at least one different cluster geometry, periodization choice, bath/solver refinement, and analytic-continuation protocol, while retaining competing order and finite-temperature explanations.

Strong-coupling translation. Derive the bond singlet–triplet splitting $4t^2/U$, then explain why the electron creation operator also needs a Schrieffer–Wolff transformation. Matching the Hamiltonian alone is insufficient for spectral weight.

References

• Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg, “Dynamical Mean-Field Theory of Strongly Correlated Fermion Systems and the Limit of Infinite Dimensions,” Reviews of Modern Physics 68 (1996) 13–125, doi:10.1103/RevModPhys.68.13.
• Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, “Metal–Insulator Transitions,” Reviews of Modern Physics 70 (1998) 1039–1263, doi:10.1103/RevModPhys.70.1039.
• Thomas Maier, Mark Jarrell, Thomas Pruschke, and Matthias H. Hettler, “Quantum Cluster Theories,” Reviews of Modern Physics 77 (2005) 1027–1080, doi:10.1103/RevModPhys.77.1027.