Hubbard Physics in Materials

To apply Hubbard physics to a solid, first choose the orbitals that carry the relevant low-energy charge, then determine their hopping and interactions. The interaction $U$ is a matrix element in that chosen basis, not a universal constant attached to the chemical formula. A useful model explains selected spectra, spin dynamics or transport within a stated energy and temperature range.

The Hubbard Model develops the operator algebra and controlled limits. Here we ask how a real crystal supplies its orbitals, screened interactions and filling, and how additional terms change the answer. Wannier Functions explains the localized basis; Mott Insulators examines the insulating phase itself.

Required background. Hubbard Model supplies the operator Hamiltonian and controlled limits specialized here to a material. What Are Strong Correlations? supplies the diagnostic hierarchy, and Mott Insulators supplies charge-incompressibility and competing-mechanism tests. Tight-Binding Models supplies the localized-orbital basis, hopping matrices, and orbital embedding used to define the active subspace.

Why the Hubbard Description Is Useful

The minimal Hubbard idea isolates a recurring competition:

• hopping spreads charge through a lattice;
• local repulsion penalizes selected charge configurations;
• virtual charge fluctuations generate lower-energy spin or orbital dynamics;
• changing filling releases mobile carriers into a constrained background.

That economy is its strength. The model supplies controlled limits, common diagnostics, and comparable numerical benchmarks. It can reveal why a partially filled band develops local moments, a narrow coherent scale, spectral-weight transfer, or charge incompressibility.

The same economy is also a warning. A real low-energy Hamiltonian can contain several orbitals per site, ligand states, Hund exchange, pair hopping, intersite repulsion, spin–orbit coupling, phonons, disorder, and retarded interactions. The useful question is not whether those terms exist microscopically; they do. The question is whether their effects on the target observables can be absorbed into a smaller parameter set with a controlled residual.

From Crystal Bands to an Active Subspace

Begin with a crystal structure and a reference one-electron problem. The reference may be a density-functional calculation, a quasiparticle calculation, or a fitted band model, but its method, exchange-correlation functional, magnetic constraint, spin–orbit treatment, and structural coordinates must be recorded.

Choose an active subspace $\mathcal T$ containing the bands and orbitals intended for the many-body calculation. In localized orbitals $\lvert w_{ia}\rangle$, the projected one-body matrix is

$$h_{ij}^{ab} = \langle w_{ia}\rvert H_{\mathrm{ref}} \lvert w_{jb}\rangle.$$

Here $i,j$ label cells or correlated sites and $a,b$ label active orbitals. This compact object contains onsite crystal fields, hopping, hybridization inside $\mathcal T$, and spin–orbit matrix elements if spin is included in the basis.

The interaction is a four-index tensor. For orbitals centered on one correlated site, a representative matrix element is

$$U_{abcd}(\omega) ={} \int d\mathbf r\,d\mathbf r'\, w_a^*(\mathbf r) w_b^*(\mathbf r') \times W_{\mathrm r}(\mathbf r,\mathbf r';\omega) w_c(\mathbf r') w_d(\mathbf r).$$

$W_{\mathrm r}$ is not normally the bare Coulomb interaction. It is screened by degrees of freedom excluded from the active low-energy polarization, and it can depend on frequency. Density–density $U$ and Hund coupling $J_{\mathrm H}$ are reduced summaries of this tensor under specified symmetry and interaction conventions.

Isolated and entangled bands

An isolated group of bands can be transformed among smooth Bloch gauges without changing its exact projector. Entangled bands require an outer energy window, an inner frozen window when used, trial orbitals, and a disentanglement criterion. Different legitimate choices can produce different orbital spreads, hopping ranges, and interaction tensors while interpolating similar band energies. Unitary gauge changes within a fixed selected subspace are exact basis changes; changing the selected window or dropping generated hopping and interaction terms changes the approximation.

Wannierization Workflows explains how to construct and test the one-body basis. DMFT for Quantum Materials follows the subsequent impurity and lattice calculation. The physical question here is whether the retained orbitals and interactions are sufficient. Compare plausible orbital windows and recompute their parameters; changing the orbitals while keeping an unrelated scalar $U$ is not a consistent comparison.

Screening and the Meaning of U

Using a consistent retarded-frequency convention, separate the total independent-particle polarization into contributions internal and external to the target subspace,

$$P(\omega) = P_{\mathcal T}(\omega) + P_{\mathrm r}(\omega).$$

In the constrained random-phase approximation (cRPA; Aryasetiawan and collaborators, 2004), screening by $P_{\mathcal T}$ is withheld because the low-energy solver is intended to treat those processes explicitly. The partially screened interaction is

$$W_{\mathrm r}(\omega) = \left[ 1-vP_{\mathrm r}(\omega) \right]^{-1}v,$$

where $v$ is the bare Coulomb kernel and products denote operator composition. Restoring target-space screening gives the fully screened interaction,

$$W(\omega) = \left[ 1-W_{\mathrm r}(\omega)P_{\mathcal T}(\omega) \right]^{-1} W_{\mathrm r}(\omega).$$

This partition explains why a low-energy Hubbard $U$ is neither the bare atomic integral nor the fully screened static interaction measured after all low-energy carriers respond. It is partially screened according to the transitions left for the low-energy calculation. The algebra is exact within the chosen polarization partition, but the RPA polarization itself is an approximation; cRPA does not include all environmental correlations. For entangled bands, defining the target transitions also requires a consistent projector prescription.

Static versus dynamic interactions

A frequency-dependent $U(\omega)$ is a retarded interaction: eliminating screening degrees of freedom leaves a memory of earlier charge fluctuations. It is represented by an interaction kernel in an action, or by an enlarged Hamiltonian with explicit screening modes, rather than by a single instantaneous Hubbard term.

Weak frequency variation inside the low-energy window can support a static approximation, but replacing $U(\omega)$ by $U(0)$ while leaving the hopping unchanged is not automatically sufficient. High-frequency screening can still renormalize coherent motion and produce satellites. A useful static reduction may need adjusted one-body terms and observables as well as $U(0)$ (Aryasetiawan and collaborators, 2004).

The numerical value also depends on orbital extent. Enlarging the target window often makes the active orbitals more localized and can increase direct interaction matrix elements, while explicitly retaining ligand states changes which screening processes are excluded. A smaller $U$ in one basis does not imply weaker physical correlation than a larger $U$ in another basis.

Beyond onsite U

Material models may require:

• interorbital repulsion $U'$ and Hund exchange $J_{\mathrm H}$;
• spin-flip and pair-hopping terms in a rotationally invariant multiorbital atomic interaction;
• nonlocal density interactions $V_{ij}$;
• correlated hopping and longer-range exchange;
• coupling to phonons or other screening modes;
• site-dependent terms at surfaces, interfaces, or disordered environments.

Dropping them is a modeling decision. It should be justified by scale estimates and observable convergence, not by the name “Hubbard.”

Double Counting in DFT-Based Workflows

Density-functional reference bands already contain an approximate average interaction through their Hartree and exchange-correlation potentials. Adding an explicit local interaction without a subtraction would count part of that physics twice. A schematic working Hamiltonian is

$$H_{\mathrm{work}} = H_{\mathrm{ref}}^{\mathcal T} - H_{\mathrm{dc}} + H_{\mathrm{int}}.$$

$H_{\mathrm{dc}}$ shifts and sometimes reshapes the correlated subspace. Unlike a diagrammatic subtraction between two exactly matched approximations, the usual DFT+$U$ or DFT+DMFT double-counting term has no universally exact form because semilocal density functionals do not expose a unique local diagram subset.

This ambiguity is especially consequential in charge-transfer systems. Moving correlated $d$ levels relative to ligand $p$ levels changes occupancy, covalency, gap character, and magnetic exchange. A credible calculation therefore reports the double-counting functional or potential, the resulting orbital occupancies and $p$–$d$ separation, and sensitivity over a physically motivated range. Charge self-consistency can reduce some inconsistencies, but it does not make the model-window or double-counting choices disappear.

For example, take $H_{\mathrm{dc}}=V_{\mathrm{dc}}N_d$ in the displayed subtraction convention. Then $\varepsilon_d^{\mathrm{eff}}=\varepsilon_d^{\mathrm{ref}}-V_{\mathrm{dc}}$: increasing $V_{\mathrm{dc}}$ lowers the $d$ level. Opposite sign conventions occur, so state the effective level shift before interpreting occupancy changes (Karolak and collaborators, 2010).

Filling Is a Subspace Statement

“Half filling” must refer to specified active modes. With $M$ spatial orbitals and two spin states per orbital, the local capacity is $2M$ electrons. Half filling corresponds to

$$N_{\mathrm{loc}} = M,$$

not to every integer formal valence. A $d^1$ ion represented by three $t_{2g}$ orbitals has one electron in six spin-orbital modes; it is at an integer commensurate filling but not at multiorbital half filling. It can still display Mott physics.

Several counts should remain distinct:

Count	Meaning and limitation
Formal oxidation state	Ionic charge assignment; covalency and ligand holes are not resolved.
Active-subspace population	Electrons represented in the chosen orbital window.
Projected orbital occupancy	A basis-dependent expectation value, possibly noninteger in an insulator.
Fermi-volume count	A metallic momentum-space count with symmetry and interaction assumptions.
Dopant concentration	A control parameter that need not equal mobile-carrier density.

Hybridization can give noninteger projected $d$ occupancy even when the many-body state is incompressible at an integer total filling. Conversely, an integer formal $d$ count does not guarantee that a $d$-only model has isolated the correct low-energy charge states. Valence histograms, ligand occupancy, and the character of addition and removal spectra are more informative than a single nominal ion label.

Orbital-selective regimes

Crystal fields, bandwidth differences, Hund coupling, and hybridization can make orbitals lose coherence at different scales. An orbital-selective Mott description requires orbital-resolved spectra and charge fluctuations in a declared basis. Because orbital labels can mix under local rotations, use symmetry-defined subspaces or basis-covariant spectral and self-energy information rather than one arbitrary diagonal entry. Density-matrix eigenvalues alone do not establish a Mott gap. Hybridization between putatively localized and itinerant sectors must be retained when assessing a sharp orbital-selective transition.

Strong Coupling and Magnetism

Near one electron per site in a single isolated orbital, repulsive strong coupling suppresses double occupancy and produces antiferromagnetic kinetic exchange of order $4\lvert t_{ij}\rvert^2/U$. The controlled derivation and its operator conventions live in Effective Hamiltonians in Many-Body Systems.

For a material, that estimate is a baseline rather than a universal exchange formula. Real denominators contain crystal-field and multiplet energies; Hund coupling can favor different intermediate states; ligand-mediated hopping introduces charge-transfer denominators; direct exchange, nonlocal repulsion, and ring exchange may matter; and spin–orbit coupling generates anisotropic exchange and Dzyaloshinskii–Moriya terms. Exchange Interactions explains those mechanisms.

Identify the low-energy local multiplet and the accessible virtual charge states before assigning exchange denominators. Derive the spin-orbital operator in the same basis as the hopping, then compare several exchange paths; one ordering temperature rarely fixes the microscopic couplings.

Magnetic order is also not guaranteed by a successful local-moment model. Dimensionality, frustration, competing exchange, and quantum fluctuations determine the ordered state or its absence.

Metal–Insulator Transitions in a Material Model

The one-band ratio $U/W$ is a useful coordinate only after $U$, $W$, filling, and the orbital window are fixed. Pressure can alter hopping, crystal fields, hybridization, and screening simultaneously. A structural transition can change the active subspace itself. Chemical substitution changes disorder and electrostatics in addition to bandwidth or filling.

Follow quasiparticle weight and linewidth, charge response, orbital populations and local moments along the same structural and filling path. A fit obtained independently at each point can hide a changing Hamiltonian. When coexistence is proposed, compare the competing solutions and their stability.

Mott Insulators gives the phase-level evidence and the distinction from Slater, band, Anderson, and charge-transfer limits.

Doping Changes More Than the Chemical Potential

A fixed-parameter Hubbard model can be solved away from integer filling by changing its chemical potential. Chemical doping of a solid is less surgical. Substitution can change bond angles, local potentials, impurity scattering, ligand energies, screening, and even the number of active orbitals. Electrostatic gating and intercalation have different electrochemical side effects.

The first validation question is therefore whether one parameter set still interpolates spectra and response across doping. Nonrigid spectral-weight transfer, Fermi-surface reconstruction, pseudogaps, and changing local moments may be intrinsic consequences of the interacting model, but they can also expose a changing Hamiltonian.

At strong coupling, projecting out high-energy double occupancy leads toward a t–J Model. The projection is controlled by a scale hierarchy, not by cuprate vocabulary. A material reduction must also retain longer-range hopping, charge-transfer character, and generated interactions at the accuracy required by the target observable.

Material Case Studies

SrVO₃: correlated metal, not Mott insulator

Cubic $\mathrm{SrVO_3}$ has a nominal $d^1$ vanadium configuration and three low-energy $t_{2g}$ orbitals. It is a correlated metal with renormalized quasiparticle bands and incoherent spectral weight, making it a benchmark for Wannier-based DFT+DMFT. A one-band half-filled interpretation is wrong: the orbital degeneracy and filling are different. Quantitative spectra also test surface sensitivity, oxygen vacancies, analytic continuation, nonlocal self-energy, and electron–phonon scattering.

V₂O₃: orbitals and structure at the transition

In $\mathrm{V_2O_3}$, trigonal crystal fields split the $t_{2g}$ manifold into $a_{1g}$ and $e_g^\pi$ sectors. Their occupancies, hybridization, and structural evolution participate in the pressure- and composition-dependent transition. Multiorbital DFT+DMFT captures important correlated metallic and insulating trends, but the result cannot be reduced to changing one scalar $U/W$ in a fixed one-band model.

Cuprates: one-band utility after a charge-transfer problem

The copper-oxide planes contain Cu $3d_{x^2-y^2}$ and O $2p_\sigma$ states. A multiband $p$–$d$ description is natural for charge-transfer spectroscopy, while a downfolded one-band model can organize lower-energy motion when its Zhang–Rice-like reduction and parameter range are justified. Material-dependent longer-range hoppings, apical oxygen, ligand covalency, and screening affect the effective model. Success of one-band numerics at low energy does not make oxygen irrelevant to the parent material.

Molecular salts: the dimer is a modeling choice

In $\kappa$-(BEDT–TTF)$_2X$, molecules form anisotropic layers with strong intradimer hopping. Treating each dimer as one effective site gives a frustrated half-filled model, while a molecule-resolved model retains intradimer charge and additional interactions. The correct choice depends on the energy scale and probe; dielectric or charge-order observables can invalidate a dimer-only reduction that remains useful for lower-energy spin physics.

Sr₂IrO₄: comparable interaction and spin–orbit scales

In $\mathrm{Sr_2IrO_4}$, crystal fields and strong spin–orbit coupling reorganize the $t_{2g}$ manifold into spin-orbital states before moderate repulsion opens an insulating gap. The effective $J_{\mathrm eff}=1/2$ description is useful but approximate: tetragonal distortion, covalency, intersite hopping, and mixing with $J_{\mathrm eff}=3/2$ states must be tested. This case demonstrates that “Hubbard” need not mean a bare spin-degenerate $3d$ orbital.

Predicting an observable

An accurate band interpolation does not guarantee an accurate optical intensity or magnetic response. Project and, when necessary, renormalize the measurement operator along with the Hamiltonian. Compare spectra with the relevant matrix elements, occupations and instrumental resolution. Effective Hamiltonians in Quantum Matter explains operator corrections, while Analytic Continuation treats uncertainty in reconstructing real-frequency response.

Use one set of orbital and interaction parameters to predict several observables that were not all used for fitting. If the spectrum agrees but exchange or orbital populations do not, test the omitted ligand channels, dynamic screening and spatial correlations. A calculation described as “first principles” still contains choices of functional, subspace, screening, subtraction and solver.

Exercises

1. Basis rotation and interaction truncation

Let two active orbitals transform as

$$\lvert w'_\alpha\rangle = \sum_a \lvert w_a\rangle R_{a\alpha},$$

where $R$ is unitary. Derive the transformation laws for the one-body matrix $h$ and interaction tensor $U_{abcd}$. Explain why rotating the basis and then keeping only two intra-orbital Hubbard parameters generally changes the model.

Solution

Creation operators transform as

$$c_\alpha^{\prime\dagger} = \sum_a R_{a\alpha} c_a^\dagger.$$

Substitution into the one-body term gives

$$h' = R^\dagger hR.$$

For an interaction written with matrix elements multiplying $c_a^\dagger c_b^\dagger c_c c_d$, the tensor transforms as

$$U'_{\alpha\beta\gamma\delta} = \sum_{abcd} R_{a\alpha}^* R_{b\beta}^* U_{abcd} R_{c\gamma} R_{d\delta}.$$

Even if the original basis was approximated by intra-orbital density interactions only, a generic rotation produces interorbital density terms, exchange-like terms, and pair hopping. Transforming the full tensor leaves physics invariant; transforming the orbitals and then discarding the generated components does not. A quoted scalar $U$ therefore belongs to a basis and a truncation convention.

2. What is half filling?

A transition-metal site is represented by three $t_{2g}$ spatial orbitals with spin. Find the local capacity and the electron number at half filling. Classify nominal $d^1$, $d^2$, and $d^3$ configurations. Can the non-half-filled integer cases be Mott insulating?

Solution

Three spatial orbitals with two spin states each give six local modes, so the capacity is six electrons. Half filling is

$$N_{\mathrm{loc}}=3.$$

Thus $d^3$ is half filled within the three-orbital model. The $d^1$ and $d^2$ cases are integer commensurate fillings but are not half filled.

Yes. Multiorbital Mott phases can occur at integer fillings other than half filling when interactions make charge addition and removal costly. Their critical scales, local multiplets, and Hund-coupling dependence differ from the half-filled case. Integer filling is a possible condition for incompressibility; “half filled” is not a synonym for “Mott capable.”

3. Restoring target-space screening

Given

$$W_{\mathrm r} = v+vP_{\mathrm r}W_{\mathrm r},$$

show that adding polarization $P_{\mathcal T}$ from the active subspace yields

$$W = W_{\mathrm r} + W_{\mathrm r}P_{\mathcal T}W,$$

and solve for $W$. Why should a low-energy solver not start from the fully screened $W$ if it will explicitly generate $P_{\mathcal T}$?

Solution

After external screening has been resummed into $W_{\mathrm r}$, repeated target-space polarization insertions form

$$W ={} W_{\mathrm r} + W_{\mathrm r}P_{\mathcal T}W_{\mathrm r} + W_{\mathrm r}P_{\mathcal T} W_{\mathrm r}P_{\mathcal T}W_{\mathrm r} +\cdots.$$

This geometric series obeys the stated Dyson equation. Rearranging gives

$$\left( 1-W_{\mathrm r}P_{\mathcal T} \right)W = W_{\mathrm r},$$

and hence

$$W = \left( 1-W_{\mathrm r}P_{\mathcal T} \right)^{-1} W_{\mathrm r}.$$

If the solver begins with fully screened $W$ and then explicitly produces target-space particle–hole screening, those processes are included twice. cRPA supplies an interaction screened by the other transitions while withholding the target transitions included in this approximation.

4. Exchange estimate as a diagnostic

A one-band fit gives $\lvert t\rvert=0.25\ \mathrm{eV}$ and $U_{\mathrm{eff}}=4.0\ \mathrm{eV}$. Estimate the leading antiferromagnetic exchange in millielectronvolts. Neutron scattering instead suggests $J=130\ \mathrm{meV}$. Does the mismatch falsify all Hubbard physics? List at least four checks.

Solution

The leading single-band estimate is

$$\begin{aligned} J_{\mathrm{est}} &= \frac{4t^2}{U_{\mathrm{eff}}}\\ &= \frac{4(0.25\ \mathrm{eV})^2}{4.0\ \mathrm{eV}} = 0.0625\ \mathrm{eV}\\ &= 62.5\ \mathrm{meV}. \end{aligned}$$

The factor-of-two mismatch falsifies the claim that this lowest-order, one-band parameter set quantitatively explains the measured exchange. It does not by itself falsify every Hubbard-family description.

Checks include: uncertainty and window dependence of $t$ and $U$; longer-range and ligand-mediated hopping; charge-transfer rather than $U$ denominators; Hund and multiplet structure; direct exchange; ring exchange and higher orders in $t/U$; spin–orbit anisotropy; and whether the experimental fit used a Hamiltonian with the same exchange convention. Several exchange paths and observables should be fit together.

5. Double counting in a charge-transfer material

A calculation retains $d$ and ligand $p$ orbitals, with $H_{\mathrm{dc}}=V_{\mathrm{dc}}N_d$ subtracted from the reference Hamiltonian. Increase $V_{\mathrm{dc}}$ while holding all other parameters and the total $p+d$ electron count fixed. How do the effective $d$ level, $d$ occupation and ligand-hole count tend to change? Does this fix the change in the measured gap?

Solution

The effective level is $\varepsilon_d^{\mathrm{ref}}-V_{\mathrm{dc}}$, so increasing the subtracted potential lowers it. This favors greater $d$ occupancy. At fixed total $p+d$ population, a larger $d$ population means fewer ligand electrons and therefore more ligand holes. The bare relative level difference $\varepsilon_d^{\mathrm{eff}}-\varepsilon_p$ decreases; the many-body charge-transfer cost also includes interactions and hybridization.

For a simple check, take real hybridization $t_{pd}>0$. The one-electron Hamiltonian in the $(d,p)$ basis is

$$h=\begin{pmatrix}\varepsilon_d^{\mathrm{eff}}&t_{pd}\\t_{pd}&\varepsilon_p\end{pmatrix}.$$

In its lower eigenstate, with $\Delta=\varepsilon_d^{\mathrm{eff}}-\varepsilon_p$,

$$n_d=\frac12\left(1-\frac{\Delta}{\sqrt{\Delta^2+4t_{pd}^2}}\right).$$

Lowering $\Delta$ increases $n_d$, confirming the sign. This elementary check is not a model of an interacting insulating gap. That gap depends on the many-body addition and removal states and need not change monotonically. Compare orbital populations, ligand-hole weight, both spectral edges and sensitivity to the subtraction choice. In a charge-self-consistent calculation, other potentials can also change.

For finite instances and quantum-processor encodings, continue to Simulation of Quantum Materials. Correlated Insulators in Moiré Systems applies these model choices to narrow moiré bands.

References

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