From Bands and Orbitals to Effective Lattice Hamiltonians

An effective lattice Hamiltonian is not read directly from a band plot. It is constructed by choosing a low-energy Hilbert space, selecting localized coordinates within it, integrating out excluded degrees of freedom, matching interactions and observables, and attaching an uncertainty budget. The result is therefore a family of Hamiltonian–observable pairs with a declared energy window and accuracy—not a unique list of hopping and Hubbard parameters.

Required background. Use the microscopic-to-field matching logic to distinguish projection from integrating out and to match observables. Helpful background. Lattice regulators and continuum targets supplies the regulator distinction.

Choose the low-energy Hilbert space

It helps to separate seven operations that are often all called “downfolding”:

1. choose the physical low-energy subspace;
2. choose an orbital basis inside that subspace;
3. integrate out coupled high-energy states;
4. partition low- and high-energy screening channels;
5. decide whether a retarded interaction may be compressed to a static one;
6. transform every measured operator along with the Hamiltonian; and
7. validate the resulting model family on observables not used to construct it.

These operations need not commute. In particular, changing the target subspace changes both the hopping matrices and which polarization processes must be excluded from the screened interaction.

Consider a periodic reference Hamiltonian $h_{\mathrm{ref}}(\mathbf k)$ on a Born–von Kármán crystal with $N_c$ unit cells and $N_c$ sampled momenta. Use full-crystal normalization,

$$\langle\psi_{n\mathbf k}|\psi_{m\mathbf k'}\rangle =\delta_{nm}\delta_{\mathbf k\mathbf k'}.$$

For an isolated group of $N$ bands, the subspace—not a particular basis in it—is encoded by

$$P(\mathbf k)=\sum_{n=1}^{N} |\psi_{n\mathbf k}\rangle\langle\psi_{n\mathbf k}|.$$

When one exists, choose a smooth, Brillouin-zone-periodic frame

$$|u_{a\mathbf k}\rangle =\sum_{n}|\psi_{n\mathbf k}\rangle C_{na}(\mathbf k), \qquad C^\dagger(\mathbf k)C(\mathbf k)=\mathbf 1_N,$$

where $a=1,\ldots,N$ labels retained orbitals. For an isolated $N$-band group, $C(\mathbf k)$ is square and unitary. The corresponding orthonormal Wannier orbitals are

$$|w_{\mathbf R a}\rangle= \frac{1}{\sqrt{N_c}}\sum_{\mathbf k} e^{-i\mathbf k\cdot\mathbf R}|u_{a\mathbf k}\rangle, \qquad \langle w_{\mathbf R a}|w_{\mathbf R'b}\rangle =\delta_{\mathbf R\mathbf R'}\delta_{ab}.$$

When the desired bands are entangled with other bands, an outer energy window first defines an $M_{\mathbf k}$-dimensional space $\mathcal F(\mathbf k)$, and a rectangular semiunitary $M_{\mathbf k}\times N$ matrix $C(\mathbf k)$ selects a smooth subspace $\mathcal S(\mathbf k)\subset\mathcal F(\mathbf k)$. A frozen inner window can require selected eigenstates to be reproduced exactly; outside it, smoothness and localization compete with exact band reproduction. The outer and frozen windows, trial orbitals, localization or spillage functional, momentum mesh, and symmetry constraints are therefore scientific inputs, not software details Souza, Marzari, and Vanderbilt 2001, § III.A and § III.G. For cRPA, the same selected subspace must also define the excluded low-energy polarization Miyake, Aryasetiawan, and Imada 2009, § II.

Here “orbital gauge” means a basis choice inside the retained subspace; it is unrelated to electromagnetic gauge. The projector $P(\mathbf k)$ is invariant under a smooth periodic rotation of the frame, but individual orbital shapes, onsite energies, hoppings, and interaction entries are not.

Symmetry and topology are constraints, not decorations

Let a space-group operation $g$ act on the retained frame by

$$\widehat g|u_{a\mathbf k}\rangle =\sum_b|u_{b,g\mathbf k}\rangle[D_g(\mathbf k)]_{ba}.$$

If the reference Hamiltonian respects $g$, its retained matrix must obey

$$D_g^\dagger(\mathbf k)h(g\mathbf k)D_g(\mathbf k)=h(\mathbf k).$$

A real-space truncation should retain complete symmetry orbits of bonds and enforce the corresponding tensor relations. A band fit that violates this covariance is not a symmetry-respecting reduction even if its eigenvalues look close along a few high-symmetry lines.

There may also be no allowed localized basis for the chosen subspace. A nonzero first Chern class of the entire target projector bundle on a Brillouin-zone two-cycle obstructs exponentially localized orthonormal Wannier functions for that fixed composite subspace; Chern numbers of constituent bands can cancel in a larger composite space Brouder et al. 2007, abstract. A symmetry-protected obstruction asks for localized orbitals that also realize specified crystal symmetries. A fragile obstruction can disappear after suitable trivial bands are added, while a stable obstruction cannot Po, Watanabe, and Vishwanath 2018, abstract. The canonical Bloch and Wannier treatment develops those distinctions. Here the practical rule is simple: enlarge the target space or retain a momentum-space description rather than inventing a short-range Wannier model that the bundle forbids.

Project the one-body Hamiltonian

The retained Bloch matrix and its real-space Fourier coefficients are

$$\begin{aligned} h_{ab}(\mathbf k) &=\langle u_{a\mathbf k}|h_{\mathrm{ref}}(\mathbf k)|u_{b\mathbf k}\rangle,\\ h_{ab}(\mathbf R) &=\frac{1}{N_c}\sum_{\mathbf k} e^{-i\mathbf k\cdot\mathbf R}h_{ab}(\mathbf k),\\ h_{ab}(\mathbf k) &=\sum_{\mathbf R}e^{i\mathbf k\cdot\mathbf R}h_{ab}(\mathbf R). \end{aligned}$$

For the moment assume spin-rotation invariance and no spin-dependent one-body term, so spin $\sigma$ is a spectator. Then

$$H_0^{\mathrm{lat}}= \sum_{\mathbf R\mathbf R'}\sum_{ab\sigma} h_{ab}(\mathbf R'-\mathbf R) c_{\mathbf R a\sigma}^{\dagger}c_{\mathbf R'b\sigma},$$

with

$$h_{ab}(\mathbf R)=h_{ba}^{*}(-\mathbf R).$$

One may instead write $h=-t$ for off-site entries, but the sign convention must then be used consistently. With spin–orbit coupling or noncollinear fields, spin becomes part of the orbital spinor and the general matrix is $h_{as,bs'}(\mathbf R)$; replacing it by $h_{ab}\delta_{ss'}$ would discard physical terms.

Changing the frame sends $h(\mathbf k)$ to $V^\dagger(\mathbf k)h(\mathbf k)V(\mathbf k)$. With the full Fourier series, this leaves the retained spectrum and projector physics unchanged. Truncating long-range matrix elements breaks that exact equivalence, so drift under admissible frames and hopping ranges is a model error to measure. The minimum one-body checks are Hermiticity, symmetry covariance, target-band energies, velocities, Fermi surface, and projector overlap throughout the declared window—not only at a plotted path Marzari et al. 2012, §§ II.A.1–2, II.I, and VI.A–B.

Project the screened interaction

Let $\alpha=(\mathbf R,a)$ be a composite site–orbital index. For a spin-independent scalar kernel, projecting the partially screened interaction $W_{\mathrm r}$ gives

$$\mathcal U_{\alpha\beta;\gamma\delta}(i\nu_n) =\int\mathrm d^3r\,\mathrm d^3r'\, w_{\alpha}^{*}(\mathbf r)w_{\beta}^{*}(\mathbf r') W_{\mathrm r}(\mathbf r,\mathbf r';i\nu_n) w_{\gamma}(\mathbf r)w_{\delta}(\mathbf r'),$$

where $\nu_n=2\pi n/\beta$ is a bosonic Matsubara frequency. “Partially screened” means that screening internal to the retained low-energy space has been excluded. For a reciprocal scalar kernel,

$$\begin{aligned} \mathcal U_{\alpha\beta;\gamma\delta} &=\mathcal U_{\beta\alpha;\delta\gamma},\\ \mathcal U_{\alpha\beta;\gamma\delta}(i\nu_n)^* &=\mathcal U_{\gamma\delta;\alpha\beta}(-i\nu_n). \end{aligned}$$

If the interaction is compressed to an instantaneous tensor $\mathcal U^{\mathrm{stat}}$, its Hamiltonian is

$$H_{\mathrm{int}}=\frac12 \sum_{\alpha\beta\gamma\delta} \sum_{\sigma\sigma'} \mathcal U_{\alpha\beta;\gamma\delta}^{\mathrm{stat}} c_{\alpha\sigma}^{\dagger}c_{\beta\sigma'}^{\dagger} c_{\delta\sigma'}c_{\gamma\sigma}.$$

Hermiticity requires

$$\mathcal U_{\alpha\beta;\gamma\delta}^{\mathrm{stat}} =\left(\mathcal U_{\gamma\delta;\alpha\beta}^{\mathrm{stat}}\right)^*.$$

For a reciprocal scalar matching prescription, one should also preserve $\mathcal U_{\alpha\beta;\gamma\delta}^{\mathrm{stat}} =\mathcal U_{\beta\alpha;\delta\gamma}^{\mathrm{stat}}$.

Fermionic antisymmetry is supplied by the operators and must not be inserted again by accident. For one orbital on one site, $\mathcal U_{ii;ii}^{\mathrm{stat}}=U$ gives

$$\frac{U}{2}\sum_{\sigma\sigma'} c_{i\sigma}^{\dagger}c_{i\sigma'}^{\dagger} c_{i\sigma'}c_{i\sigma} =U n_{i\uparrow}n_{i\downarrow};$$

the equal-spin terms vanish and the two opposite-spin terms cancel the factor $1/2$. Nonlocal density interactions, exchange, pair hopping, and Hund couplings are different entries or combinations of the same tensor. With spinor Wannier functions, use the full spinor matrix elements rather than an automatically conserved external spin label.

The tensor is basis covariant. Let $V$ be unitary on the full composite one-particle basis, with transformed creation operators $c_p'{}^\dagger=\sum_\alpha V_{\alpha p}c_\alpha^\dagger$. Then

$$\mathcal U'_{pq;rs} =V^*_{\alpha p}V^*_{\beta q} \mathcal U_{\alpha\beta;\gamma\delta} V_{\gamma r}V_{\delta s},$$

with repeated composite labels summed. A momentum-dependent rotation of Bloch frames generally mixes different cells in real space. Truncating that range or dropping exchange and pair-hopping entries can therefore destroy covariance even when the untruncated theories are equivalent.

Partition screening without counting it twice

Fix the polarization sign by

$$W^{-1}=v^{-1}-\Pi,$$

where the static density polarization is nonpositive in a stable channel under this convention. Decompose

$$\Pi=\Pi_{\mathrm{low}}+\Pi_{\mathrm{rest}}, \qquad W_{\mathrm r}=(v^{-1}-\Pi_{\mathrm{rest}})^{-1}.$$

Here $v$ is the bare Coulomb kernel. The “rest” polarization is $\Pi-\Pi_{\mathrm{low}}$; it includes transitions between retained and excluded spaces, not only transitions wholly inside the excluded space. Within the same RPA decomposition, restoring the retained polarization gives

$$W=(W_{\mathrm r}^{-1}-\Pi_{\mathrm{low}})^{-1} =(1-W_{\mathrm r}\Pi_{\mathrm{low}})^{-1}W_{\mathrm r}.$$

This is a matrix identity, so factor order matters. It explains why the target-space RPA bubbles must not already be hidden in $W_{\mathrm r}$. It does not say that an interacting low-energy solver must reproduce the Kohn–Sham/RPA polarization or full RPA $W$; vertex corrections and a different irreducible polarization are separate physics. cRPA is a concrete, reproducible prescription, not an expansion protected by a universal small parameter. Entangled subspaces and omitted vertex structure can produce quantitative failure Aryasetiawan et al. 2004, §§ II–III and Shinaoka, Troyer, and Werner 2015, abstract.

A static interaction is another matching step

The frequency dependence of $\mathcal U(i\nu_n)$ is physical. Choosing $\mathcal U^{\mathrm{stat}}=\mathcal U(0)$ is controlled only when the target frequencies are small compared with the relevant screening modes and the induced changes to bandwidths and observables are retained or bounded. Even a high-frequency screening mode can reduce hopping and low-energy photoemission weight when it is integrated out Casula et al. 2012, Eqs. (1)–(6). Other static prescriptions may instead match a chosen low-energy observable and need not equal the zero-frequency value. If no static compression is stable, retain a retarded interaction, introduce explicit screening bosons, or enlarge the electronic subspace.

A two-level elimination you can check

The smallest example shows why projection alone misses virtual effects. Retain $|d\rangle$, place $|h\rangle$ higher by $\Delta=\epsilon_h-\epsilon_d>0$, and write an eigenvector as $a|d\rangle+b|h\rangle$ for

$$H= \begin{pmatrix} \epsilon_d & V\\ V^{*} & \epsilon_h \end{pmatrix}.$$

The coupled equations are

$$(\epsilon_d-E)a+Vb=0, \qquad V^*a+(\epsilon_h-E)b=0.$$

For $E\ne\epsilon_h$, the second gives $b=-V^*a/(\epsilon_h-E)$. Substitution into the first produces the exact nonlinear eigenvalue equation

$$E=H_{\mathrm{eff}}(E), \qquad H_{\mathrm{eff}}(E)=\epsilon_d+\frac{|V|^2}{E-\epsilon_h}.$$

This energy-dependent object is not yet an ordinary static Hamiltonian. The exact lower eigenvalue and its controlled expansion are

$$\begin{aligned} E_-&=\frac{\epsilon_d+\epsilon_h- \sqrt{\Delta^2+4|V|^2}}{2}\\ &=\epsilon_d-\frac{|V|^2}{\Delta} +\frac{|V|^4}{\Delta^3} +O\!\left(\frac{|V|^6}{\Delta^5}\right). \end{aligned}$$

The discarded-state amplitude begins at $b/a=-V^*/\Delta$. Equivalently, with $\Sigma(E)=|V|^2/(E-\epsilon_h)$, the retained-state weight is

$$Z_d=\left[1-\partial_E\Sigma(E_-)\right]^{-1} =\left[1+\frac{|V|^2}{(E_--\epsilon_h)^2}\right]^{-1}.$$

Thus the same mixing that shifts the energy also moves spectral weight into the excluded state. If $|V|/\Delta$ or the target energy range divided by $\Delta$ is not small, no regular static expansion is justified.

For projectors $P$ and $Q=1-P$, the many-state version is

$$H_{\mathrm{eff}}(E)=PHP+PHQ\frac{1}{E-QHQ}QHP,$$

defined when $E$ lies in the resolvent set of $QHQ$. A unitary block diagonalization $e^S$, with $S^\dagger=-S$ and $S=O(\lVert PHQ\rVert/\Delta)$ when a gap $\Delta$ controls the mixing, also sends each observable to

$$\mathcal O_{\mathrm{eff}}=Pe^S\mathcal Oe^{-S}P.$$

Different higher-order effective Hamiltonians can be related by further unitary transformations, but only if their observables are transformed consistently Chernyshev et al. 2004, §§ III.C–D and IV.

Worked construction: a square-lattice Hubbard family

Take one topologically trivial, isolated, spin-degenerate band on a two-dimensional square lattice of spacing $a_{\mathrm{lat}}$. Suppose its retained dispersion is

$$\epsilon(\mathbf k)=\epsilon_0 -2t\bigl[\cos(k_xa_{\mathrm{lat}})+\cos(k_ya_{\mathrm{lat}})\bigr] -4t'\cos(k_xa_{\mathrm{lat}})\cos(k_ya_{\mathrm{lat}}).$$

Fourier inversion gives

$$\begin{aligned} h(\mathbf 0)&=\epsilon_0,\\ h(\pm a_{\mathrm{lat}}\widehat{\mathbf x}) =h(\pm a_{\mathrm{lat}}\widehat{\mathbf y})=-t,\\ h(\pm a_{\mathrm{lat}}\widehat{\mathbf x} \pm a_{\mathrm{lat}}\widehat{\mathbf y})&=-t' \quad\text{for all four diagonal bonds}. \end{aligned}$$

Assume, for illustration, that a static low-frequency compression is justified. The onsite and nearest-neighbor density entries are

$$U=\mathcal U_{ii;ii}(0), \qquad V_1=\mathcal U_{ij;ij}(0) \quad (i,j\ \text{nearest neighbors}).$$

Keeping those terms produces

$$\begin{aligned} H_{\mathrm{model}}={}& -t\sum_{\langle ij\rangle,\sigma} (c_{i\sigma}^\dagger c_{j\sigma}+\mathrm{h.c.}) -t'\sum_{\langle\!\langle ij\rangle\!\rangle,\sigma} (c_{i\sigma}^\dagger c_{j\sigma}+\mathrm{h.c.})\\ &+\epsilon_0\sum_{i\sigma}n_{i\sigma} +U\sum_i n_{i\uparrow}n_{i\downarrow} +V_1\sum_{\langle ij\rangle}n_i n_j +\delta H. \end{aligned}$$

The remainder $\delta H$ is part of the result: it contains every discarded longer-range hopping, interaction, exchange, correlated-hopping, multibody, spin–orbit, or retarded term allowed by the retained symmetries. For real $t,t',U,V_1$ and no flux or spin-dependent term, the displayed model respects translations, the square-lattice point group, time reversal, charge conservation, and spin $SU(2)$. Its dispersion reconstructs the stated target exactly because that target was chosen with only the displayed Fourier harmonics; a real material generally requires a range-convergence test.

At fixed particle number, the filling is part of the model specification. In a grand-canonical calculation one instead studies $H_{\mathrm{model}}-\mu N$; only the combination $\epsilon_0-\mu$ enters this one-orbital example, so an onsite energy shift must not be mistaken for a separately determined chemical potential.

This construction becomes a controlled result only after three further checks. First, vary the outer or frozen window and Wannier frame, then refit the full correlated parameter set $(t,t',U,V_1,\ldots)$. Second, verify a held-out quantity such as a velocity, Fermi-surface contour, optical matrix element, or screening trend. Third, transform the corresponding current, density, or dipole operator; fitting the energy bands does not validate response intensities. If the screening scale is not well separated, $\delta H$ must retain the dynamical kernel and its induced bandwidth or spectral-weight corrections.

Keep the two double-counting problems distinct

Problem	What may be counted twice	Required response
Screening-channel double counting	Retained-space polarization appears in both $W_{\mathrm r}$ and the low-energy treatment	Define $\Pi_{\mathrm{low}}$ with the same target projector and exclude it from $W_{\mathrm r}$
Reference-functional double counting	An average interaction effect is already present in a density-functional or mean-field $h_{\mathrm{ref}}$ and is added again through $H_{\mathrm{int}}$	State the reference functional, correlated projector, subtraction, occupancy convention, and sensitivity to alternatives

There is no method-independent universal reference-functional subtraction across arbitrary density functionals, correlated subspaces, and interaction tensors. An exact overlap can be derived within a specified continuum DFT+DMFT functional Haule 2015, abstract; that result does not turn one formula into a universal correction. Adjusting the subtraction until one occupancy agrees with experiment is calibration, not independent validation.

Validation and uncertainty budget

The chapter’s reduction map places Hamiltonian construction before solver and mechanism claims, while its correlated-electron validity table gives the shared stop conditions. For this page, record the following model-specific checks.

Layer	Minimum reproducible record	Independent check	Stop condition
Target subspace	Outer and frozen windows, projectors, trial orbitals, frame criterion, mesh, filling, and energy zero	Projector overlap, orbital character, and held-out bands across admissible windows	The selected subspace changes discontinuously or omits states carrying a target response
One-body sector	Full $h_{ab}(\mathbf R)$ before truncation, spinor scope, symmetry representation, and retained range	Hermiticity, symmetry covariance, energies, velocities, and Fermi surface	A target observable depends on discarded bonds or the orbital frame
Interaction sector	Full retained tensor, range, frequency dependence, polarization partition, and static-matching rule	Tensor symmetries, causality, basis covariance, and drift under the screening partition	$\mathcal U(\omega)$ varies on the target scale or omitted tensor entries change the claim
High-energy elimination	Retained and excluded spaces, gap, mixing norm, order, and generated operators	Exact small system, resolvent, or second transformation	A discarded pole or soft mode enters the target window
Reference subtraction	Functional, projector, occupancy, subtraction, and defensible alternatives	Filling, gap, and level alignment under the alternative set	The conclusion is fixed mainly by the subtraction choice
Observable matching	Effective density, current, dipole, spin, and other measured operators	Sum rules and at least one held-out response	Energies fit while intensities, charges, or sum rules fail
Uncertainty	Joint parameter ensemble or covariance matrix, solver error, and held-out data	Coverage of withheld observables and stability under the full ensemble	Only fitted observables agree, or the uncertainty band comes from uncorrelated one-at-a-time variations rather than admissible joint model cards

Parameters derived from one orbital and screening construction are correlated. For a smooth observable $\mathcal A(\boldsymbol\theta)$ and a parameter covariance matrix $C_{\theta}$,

$$\operatorname{Var}\mathcal A\simeq (\boldsymbol\nabla_{\theta}\mathcal A)^{\mathsf T} C_{\theta} (\boldsymbol\nabla_{\theta}\mathcal A).$$

Varying $t$, $U$, and a double-counting shift independently can explore combinations that no downfolding ever produced. For nonlinear or multimodal dependence, propagate a joint ensemble of complete model cards instead of relying on the linear formula.

Passing band reconstruction validates only the projected one-body sector. Passing a many-body solver check validates a conclusion conditional on the model. A material claim requires both layers, matched observables, and an uncertainty band that includes window, basis, screening, truncation, subtraction, and solver choices.

Common pitfalls

Treating hopping entries as observables. They are coordinates in an orbital basis. Quote the frame and truncation, and compare spectra, symmetry data, or matched observables that survive the basis change.

Calling one static Hubbard parameter first-principles. A static number suppresses frequency, range, and tensor structure. State which screening channels and target frequencies it represents and where that compression fails.

Confusing projection with elimination. $PHP$ deletes virtual excursions through $Q$; the resolvent or a controlled unitary transformation generates their low-energy effects.

Validating on the fitting set. Reconstructing the bands used to choose the model is necessary but not independent evidence. Reserve at least one response, dispersion feature, or parameter trend as a held-out test.

Exercises

Rotate the orbital frame

Let $V(\mathbf k)$ be smooth, periodic, and unitary. With no real-space truncation, show that $h'(\mathbf k)=V^\dagger hV$ leaves the one-particle spectrum invariant but changes individual real-space hopping entries.

Solution

Unitary similarity preserves the characteristic polynomial and therefore every eigenvalue at each $\mathbf k$. Fourier transforming a momentum-dependent $V(\mathbf k)$ redistributes matrix elements among orbitals and lattice separations, so individual $h_{ab}(\mathbf R)$ change. If the complete Fourier series and the transformed observables are retained, the two descriptions are equivalent. A range truncation breaks that exact equivalence, which is why frame drift is a truncation diagnostic.

Check the eliminated-state weight

For the two-level model, derive the low-energy shift through order $|V|^2/\Delta$ and the leading probability in the discarded state.

Solution

Expanding the exact lower eigenvalue gives

$$E_-=\epsilon_d-\frac{|V|^2}{\Delta} +O\!\left(\frac{|V|^4}{\Delta^3}\right).$$

The second component equation gives $b/a=-V^*/(\epsilon_h-E_-)=-V^*/\Delta+O(|V|^3/\Delta^3)$. After normalization, the discarded-state probability is $|b|^2=|V|^2/\Delta^2+O(|V|^4/\Delta^4)$. This is the leading spectral weight that naive projection misses.

Restore the retained RPA polarization

Treat $v$, $W_{\mathrm r}$, and $\Pi_{\mathrm{low}}$ as noncommuting matrix kernels. Starting from $W_{\mathrm r}^{-1}=v^{-1}-\Pi_{\mathrm{rest}}$, show that restoring $\Pi_{\mathrm{low}}$ reconstructs the full interaction within the same RPA decomposition.

Solution

The full RPA kernel satisfies

$$W^{-1}=W_{\mathrm r}^{-1}-\Pi_{\mathrm{low}} =W_{\mathrm r}^{-1}(1-W_{\mathrm r}\Pi_{\mathrm{low}}).$$

Therefore

$$W=(1-W_{\mathrm r}\Pi_{\mathrm{low}})^{-1}W_{\mathrm r} =W_{\mathrm r}(1-\Pi_{\mathrm{low}}W_{\mathrm r})^{-1}.$$

Both ordered forms are equal, but moving factors arbitrarily is not allowed. The result is an RPA identity; a correlated solver may have a different irreducible polarization and vertex corrections.

Assemble and stress-test a model card

For the square-lattice dispersion in the worked construction, verify the listed Fourier coefficients. Then suppose a second admissible Wannier window changes $(t,t',U,V_1)$ coherently from $(1,0.20,6,1.0)$ to $(0.92,0.15,5.4,0.8)$ in common energy units. Which statement survives without solving either interacting model?

Solution

Summing the four nearest-neighbor coefficients gives $-2t(\cos k_xa_{\mathrm{lat}}+\cos k_ya_{\mathrm{lat}})$; summing the four diagonal coefficients gives $-4t'\cos k_xa_{\mathrm{lat}}\cos k_ya_{\mathrm{lat}}$. Both cards therefore realize the same symmetry-allowed operator basis but different correlated parameter sets. One may conclude that both reductions are interacting, extended Hubbard models with $U/t>5$ and appreciable nonlocal repulsion. One may not claim a Mott phase, magnetic order, or quantitative gap from those ratios alone. Such claims require solving both cards, propagating the correlated window change, and checking observables and alternative mechanisms.

What you can now do

You can now turn a declared band and orbital subspace into a symmetry-respecting lattice Hamiltonian, identify the screening and static-matching assumptions behind its interaction tensor, transform its observables, and state when window, basis, or covariance drift invalidates the reduction. The strongest warning is also the simplest: a fitted Hamiltonian is one representative of a controlled model family, not a unique microscopic truth.

Continue with Hubbard symmetries and controlled limits to test the one-band model before assigning a Mott regime. Use electron–phonon fields and retarded interactions when the static compression fails, and multiorbital correlations and Hund’s metals when the retained tensor cannot be reduced to one orbital.

References

• Ferdi Aryasetiawan, Masatoshi Imada, Antoine Georges, Gabriel Kotliar, Silke Biermann, and Alexander I. Lichtenstein, “Frequency-Dependent Local Interactions and Low-Energy Effective Models from Electronic Structure Calculations,” Physical Review B 70 (2004) 195104, doi:10.1103/PhysRevB.70.195104.
• Christian Brouder, Gianluca Panati, Matteo Calandra, Christophe Mourougane, and Nicola Marzari, “Exponential Localization of Wannier Functions in Insulators,” Physical Review Letters 98 (2007) 046402, doi:10.1103/PhysRevLett.98.046402, Open PDF.
• Michele Casula, Philipp Werner, Laurent Vaugier, Ferdi Aryasetiawan, Takashi Miyake, Andrew J. Millis, and Silke Biermann, “Low-Energy Models for Correlated Materials: Bandwidth Renormalization from Coulombic Screening,” Physical Review Letters 109 (2012) 126408, doi:10.1103/PhysRevLett.109.126408, Open PDF.
• A. L. Chernyshev, D. Galanakis, P. Phillips, A. V. Rozhkov, and A.-M. S. Tremblay, “Higher Order Corrections to Effective Low-Energy Theories for Strongly Correlated Electron Systems,” Physical Review B 70 (2004) 235111, doi:10.1103/PhysRevB.70.235111.
• Kristjan Haule, “Exact Double Counting in Combining the Dynamical Mean Field Theory and the Density Functional Theory,” Physical Review Letters 115 (2015) 196403, doi:10.1103/PhysRevLett.115.196403, Open PDF.
• Nicola Marzari, Arash A. Mostofi, Jonathan R. Yates, Ivo Souza, and David Vanderbilt, “Maximally Localized Wannier Functions: Theory and Applications,” Reviews of Modern Physics 84 (2012) 1419–1475, doi:10.1103/RevModPhys.84.1419, Open PDF.
• Takashi Miyake, Ferdi Aryasetiawan, and Masatoshi Imada, “Ab Initio Procedure for Constructing Effective Models of Correlated Materials with Entangled Band Structure,” Physical Review B 80 (2009) 155134, doi:10.1103/PhysRevB.80.155134, Open PDF.
• Hoi Chun Po, Haruki Watanabe, and Ashvin Vishwanath, “Fragile Topology and Wannier Obstructions,” Physical Review Letters 121 (2018) 126402, doi:10.1103/PhysRevLett.121.126402.
• Hiroshi Shinaoka, Matthias Troyer, and Philipp Werner, “Accuracy of Downfolding Based on the Constrained Random-Phase Approximation,” Physical Review B 91 (2015) 245156, doi:10.1103/PhysRevB.91.245156.
• Ivo Souza, Nicola Marzari, and David Vanderbilt, “Maximally Localized Wannier Functions for Entangled Energy Bands,” Physical Review B 65 (2001) 035109, doi:10.1103/PhysRevB.65.035109, Open PDF.