DMFT for Quantum Materials

Dynamical mean-field theory turns a declared lattice or electronic-structure problem into a self-consistent quantum impurity problem. For quantum materials, the difficult part is not merely iterating that mapping. It is preserving the meaning of the correlated subspace, interaction tensor, double-counting operator, thermodynamic state, solver statistics, continuation, and observable map from the source calculation to the final claim.

This page owns that joined execution and evidence workflow for single-site material DMFT. It starts from a frozen source handoff and ends with a reproducible, bounded claim about a local Green function, occupancy, low-frequency branch, spectral candidate, or thermodynamic solution. It does not choose the material model, rederive impurity physics, teach a general solver, establish a phase from one converged fixed point, or identify a computed spectrum with measured intensity.

Required background. Hubbard Physics in Materials supplies a justified active space, filling convention, interaction tensor, and double-counting model. Anderson Impurity Model Preview supplies the auxiliary bath, impurity Green function and self-energy, and the conceptual single-site mapping.

Helpful background. Thermal Green Functions develops the general imaginary-time and contact-term theory. Analytic Continuation and Spectral Functions are useful when the requested claim requires a real-frequency object. The conventions actually used below are nevertheless declared locally.

Declare the Materials DMFT Claim

Begin with one sentence:

For this frozen material or model source, correlated subspace, state, and interaction convention, use single-site DMFT to estimate this object to this tolerance, test it against these alternatives, and stop at this bounded physical claim.

The object must be explicit. These are different requests:

• converge a local Matsubara Green function and density matrix;
• test whether a finite-temperature branch is compatible with a Fermi liquid;
• resolve a local one-particle gap above a declared continuation resolution;
• compare paramagnetic, magnetic, orbital, or structural fixed points;
• supply an intrinsic spectral object to a measurement forward model;
• decide whether a local self-energy approximation remains adequate.

A converged calculation answers only the represented problem. It does not retroactively validate the active space, screened interaction, double counting, exchange-correlation functional, locality approximation, analytic continuation, or experimental inference.

Before selecting software, require four readiness conditions:

1. the source Hamiltonian or electronic-structure artifact is immutable and independently validated;
2. the correlated modes, projectors, basis metric, interactions, filling, and double-counting sign are declared;
3. the requested estimator and numerical tolerance are meaningful at the chosen temperature and frequency resolution;
4. at least one exact limit, competing branch, and stopping test is available.

If the first two fail, return to Wannierization Workflows or the material-model owner. If the target object is unclear, return to Computational Quantum Matter before running a solver.

Write the Ten-Field DMFT Record

Use the same ten labels for a quick test, production run, archive, and paper-level claim. Fill every field, or write unresolved and stop.

1. Physical problem. State the material or synthetic system, crystal and correlated sites, retained orbitals, dimension, geometry, boundaries, and preparation.
2. State and limit. Give the ensemble, total filling or chemical potential, temperature, field, drive, symmetry sector, thermodynamic target, and frequency and response limits actually taken.
3. Claim and accuracy. Name the target Green function, occupancy, conditional quasiparticle parameter, gap candidate, thermodynamic branch, units, normalization, requested resolution, and physical comparison.
4. Representation and provenance. Record the source artifact, $H_0$, basis metric, projectors, outer and frozen windows, local axes, interaction tensor, double-counting operator and sign, source revision, and omitted states.
5. Method and controlled domain. State the single-site locality approximation, solver family, one-shot or charge-self-consistent branch, allowed order, continuation branch, and why it addresses the requested object.
6. Finite numerical problem. Give normalized $\mathbf k$ weights, Matsubara grid and cutoff, analytic tails, Monte Carlo or bath controls, seeds, mixing, charge cell, and initial self-energies or ordered states.
7. Estimator and forward model. Define $G_{\rm imp}$, $G_{\rm loc}$, $\Sigma_{\rm imp}$, occupancy, conditional $Z$, continued spectrum, thermodynamic estimator, and the declared map to a downstream observable.
8. Convergence and uncertainty. Separate mesh, frequency and tail, stochastic, autocorrelation, loop, solver, interaction, window, projector, double-counting, continuation, structural, nonlocal, and probe uncertainties, with covariance where relevant.
9. Verification, validation, and provenance. List exact, noninteracting, and atomic limits; matrix causality; high-frequency moments; spectral sum rules; lattice–impurity and charge residuals; symmetry; seeds; competing branches; raw artifacts; versions; and hashes.
10. Licensed claim and stopping rule. State the strongest supported conclusion, credible alternatives, missing physics, a falsifier, and the next canonical owner when the local-material claim fails.

The first five fields define the represented physical problem. Fields six through nine define the finite evidence. Field ten prevents a small residual from becoming an oversized material claim.

Freeze the Correlated Subspace and Source Handoff

DMFT must consume a versioned source rather than a band plot or an informal orbital label. The handoff should include:

• the crystal structure, charge and magnetic state, functional, core treatment, basis, and source-code revision;
• the full $H^B_{\mathbf k}$ or model $H_0(\mathbf k)$ on a declared mesh;
• the correlated-site and local-axis convention;
• projector matrices, band ranges, outer and frozen windows, and any overlap matrix;
• the interaction tensor and its screening or fitting provenance;
• the double-counting operator, occupancy convention, and sign;
• hashes for the source, projectors, interaction data, and conversion scripts.

Band Structure Workflows owns the converged independent-particle source. Wannierization Workflows owns numerical subspace construction and interpolation. Hubbard Physics in Materials owns the physical adequacy of the active space and screened interactions. This page audits their frozen outputs; it does not silently revise them during the DMFT loop.

For an orthonormal full Bloch source, use

$$P_{m\nu}(\mathbf k) := \langle\chi_{m\mathbf k}|\psi_{\nu\mathbf k}\rangle.$$

Here $m$ labels the $N_{\rm corr}$ correlated spin-orbital modes and $\nu$ labels source Bloch modes. In this page $P_{\mathbf k}$ denotes the orthonormalized retained analysis map, so its rows obey

$$P_{\mathbf k}P_{\mathbf k}^\dagger = I_{N_{\rm corr}}.$$

The correlated frame, orbital order, spin order, local axes, and reciprocal-space phase convention belong to the archive. If a raw projector is full row rank but does not satisfy this isometry, either orthonormalize it with the declared metric or form the source-space projector as

$$\Pi_{\mathbf k} = P_{\mathbf k}^\dagger \left(P_{\mathbf k}P_{\mathbf k}^\dagger\right)^{-1} P_{\mathbf k}.$$

For a nonorthogonal full basis, record $S_{\mathbf k}$ and metric-aware primal and dual projectors. An implementation that uses duals, orthonormalized local orbitals, or a rectangular embedding must state which matrix maps source coefficients into the correlated space and which adjoint maps a correlated self-energy back. Copying an orthonormal formula into a nonorthogonal basis is not a convention change; it is a different calculation.

Two retained frames can be compared only after placing them in one source space. For equal-rank subspaces define

$$\Pi_{\mathbf k} := P_{\mathbf k}^\dagger P_{\mathbf k},$$

and

$$d_P^{AB}(\mathbf k) := \left\lVert \Pi^{(A)}_{\mathbf k}-\Pi^{(B)}_{\mathbf k} \right\rVert_2 = \sin\theta_{\max}(\mathbf k).$$

This distance is invariant under unitary gauge rotations within either retained frame. Report its full-zone maximum and a distribution summary such as the 95th percentile. Raw orbital coefficients are gauge dependent and are not a subspace metric.

Freezing the source does not mean believing it unconditionally. It means that each window, projector, structure, or interaction change creates a named model branch whose consequences can be compared without confusing them with solver noise.

Declare Interactions, Double Counting, Filling, and Chemical Potential

The local interaction is an operator, not merely the pair of numbers $U$ and $J$. Record the full four-index tensor or the symmetry reduction used by the solver, its orbital and spin order, and every term that has been discarded. For a rotationally invariant degenerate Kanamori interaction,

$$U' = U-2J.$$

This identity need not hold for a general crystal-field interaction tensor. Spin-flip and pair-hopping terms also distinguish a general matrix problem from a density-density segment problem.

A screened interaction may depend on frequency. If the calculation replaces $U(\omega)$ by a static tensor, state the matching prescription and the frequency range over which that reduction is intended. A static interaction and an unexplained retarded one are not interchangeable inputs.

Let $V_{\rm DC}$ denote the static Hermitian double-counting operator and fix the sign through

$$\Sigma_{\rm eff}(i\varepsilon_n) := \Sigma_{\rm imp}(i\varepsilon_n)-V_{\rm DC}.$$

The same sign must be used in the impurity levels, lattice embedding, occupancy record, and archived postprocessing. Fully localized limit, around-mean-field, and nominal formulas are prescriptions. Formalism-specific exact-overlap constructions also exist, so the defensible statement is not that exact double counting is universally impossible. The defensible statement is that the adopted construction and its domain must be named.

Filling has at least three levels:

• the total electron count in the full source cell;
• the occupancy of the correlated subspace under a declared projector;
• orbital- and spin-resolved occupancies inside that subspace.

These quantities need not be integers and need not agree across different windows. If total particle number is fixed, the chemical-potential loop must be closed within each DMFT iteration or outer cycle. At fixed chemical potential, changing $V_{\rm DC}$ generally changes the grand-canonical state. In an isolated all-correlated model at fixed total filling, a scalar double-counting shift can instead be redundant with a shift of $\mu$. That gauge lesson does not extend to a full $p$–$d$ embedding in which relative levels matter.

Build the Local Self-Energy Embedding and Impurity Problem

Set $\hbar=1$ and

$$\beta := \frac{1}{k_BT}.$$

The fermionic Matsubara energy is

$$\varepsilon_n := \frac{(2n+1)\pi}{\beta}.$$

Do not call $\varepsilon_n$ an angular frequency or insert an undeclared factor of $\hbar$. The imaginary-time Green function and transform used here are

$$G_{ab}(\tau) := -\left\langle T_\tau c_a(\tau)c_b^\dagger(0) \right\rangle,$$ $$G(i\varepsilon_n) := \int_0^\beta d\tau\, e^{i\varepsilon_n\tau}G(\tau).$$

For the orthonormal full-source convention of the previous section,

$$G^B_{\mathbf k}(i\varepsilon_n) = \left[ (i\varepsilon_n+\mu)I-H^B_{\mathbf k} -P_{\mathbf k}^\dagger \Sigma_{\rm eff}(i\varepsilon_n) P_{\mathbf k} \right]^{-1}.$$

With normalized Brillouin-zone weights,

$$G_{\rm loc}(i\varepsilon_n) = \sum_{\mathbf k}w_{\mathbf k} P_{\mathbf k}G^B_{\mathbf k}(i\varepsilon_n) P_{\mathbf k}^\dagger, \qquad \sum_{\mathbf k}w_{\mathbf k}=1.$$

For a nonorthogonal source the inverse instead contains

$$(i\varepsilon_n+\mu)S_{\mathbf k}-H_{\mathbf k} -\bar P_{\mathbf k}^\dagger \Sigma_{\rm eff}(i\varepsilon_n) \bar P_{\mathbf k},$$

where $\bar P_{\mathbf k}$ denotes the declared metric-aware embedding. Its downfolding partner must be defined consistently rather than inferred from the orthonormal equation.

The current impurity Dyson relation is

$$\mathcal G_0^{-1}(i\varepsilon_n) = G_{\rm imp}^{-1}(i\varepsilon_n) +\Sigma_{\rm imp}(i\varepsilon_n).$$

If $E_{\rm imp}$ is the declared local one-body matrix, the corresponding hybridization function is

$$\begin{aligned} \Delta(i\varepsilon_n) ={}& (i\varepsilon_n+\mu)I-E_{\rm imp}\\ &-\Sigma_{\rm imp}(i\varepsilon_n) -G_{\rm imp}^{-1}(i\varepsilon_n). \end{aligned}$$

The auxiliary impurity problem uses that bath and the same local interaction tensor as the declared correlated site. After solving it, embed $\Sigma_{\rm imp}$ into the lattice, recompute $G_{\rm loc}$, and update the next bath. At a fixed point,

$$G_{\rm imp}(i\varepsilon_n) = G_{\rm loc}(i\varepsilon_n)$$

in the same correlated basis within the declared covariance.

Single-site locality means that the self-energy is local in the chosen correlated representation. It can still be a matrix in orbital, spin, sublattice, or inequivalent-site labels. Projecting that matrix into Bloch states can create momentum-dependent spectral effects without creating a nonlocal self-energy.

Single-site DMFT becomes exact in the appropriately scaled infinite-coordination or infinite-dimensional limit. For a finite-dimensional material, locality remains an approximation in the declared correlated representation and must be tested against credible nonlocal alternatives.

Choose the Solver, Matsubara Grid, Tails, and Statistics

Solver choice follows the represented impurity problem and requested observable:

• general matrix CT-HYB naturally treats finite-temperature multiorbital hybridization expansions, but its sign, autocorrelation, and matrix interaction cost must be audited;
• a segment CT-HYB solver is restricted to a compatible density-density interaction and cannot represent general spin-flip and pair-hopping terms;
• exact diagonalization exposes real-frequency poles directly but introduces bath discretization and many-body truncation;
• numerical renormalization group is powerful for low-temperature, real-frequency impurity scales in suitable few-channel problems, but is not a universal realistic multiorbital solver;
• perturbative solvers are controlled only in their stated weak-coupling, strong-coupling, or crossing-selected regimes.

This page records why a solver fits the material claim. It leaves algorithm derivation and software implementation to their canonical methods owners.

The Matsubara grid must state temperature, number of positive and negative energies retained or reconstructed by symmetry, transform convention, cutoff, and tail treatment. For the $N_{\rm corr}$ orthonormal correlated modes,

$$G(i\varepsilon_n) = \frac{I_{N_{\rm corr}}}{i\varepsilon_n} +\frac{M_1}{(i\varepsilon_n)^2} +\cdots,$$ $$\Sigma(i\varepsilon_n) = \Sigma_\infty +\frac{\Sigma_1}{i\varepsilon_n} +\cdots.$$

Archive the measured moments and compare them with equal-time identities. Analytic tails belong in occupancies, energies, transforms, and sum rules. A finite truncated Matsubara sum without a tail is not the declared estimator.

For Monte Carlo, report:

• warm-up, measurement count, update types, and acceptance diagnostics;
• integrated autocorrelation estimates and the resulting blocking choice;
• independent seeds and replica identities;
• covariance across frequencies, orbitals, and transformed observables;
• mean sign together with its time history and rare excursions;
• failed, metastable, or symmetry-breaking runs rather than only the selected fixed point.

A large update count is not an effective sample size. A mean sign near one is not by itself an uncertainty analysis. Convergence criteria should be compared with combined statistical uncertainty, not imposed below the noise floor.

Close the Impurity, Lattice, and Charge Loops

A reproducible iteration separates the nested loops:

1. freeze the source, correlated frame, interaction, and double counting;
2. solve the current impurity problem for $G_{\rm imp}$ and $\Sigma_{\rm imp}$;
3. embed the self-energy and integrate the lattice Green function;
4. adjust $\mu$ if the total filling rather than chemical potential is fixed;
5. update and mix the Weiss field or hybridization;
6. repeat until matrix and occupancy residuals meet their covariance-aware criteria;
7. only for charge self-consistency, rebuild the correlated density and one-body source in a separate outer loop.

One-shot DMFT closes the impurity and lattice loops while holding the source charge density and $H_0$ fixed. Charge-self-consistent DFT+DMFT additionally updates the density and source Hamiltonian. It is a different workflow, not a certificate that subspace, interaction, double counting, or functional error has disappeared.

Evaluate correlated occupancy with the all-frequency sum

$$n_{\rm corr} = k_BT \sum_{n=-\infty}^{\infty} e^{i\varepsilon_n0^+} \operatorname{Tr}G_{\rm loc}(i\varepsilon_n),$$

including analytic tails and the declared contact and trace conventions.

For the first $N_\varepsilon=100$ positive Matsubara energies, define

$$\begin{aligned} R_G :={}& \left[ \frac{1}{N_\varepsilon} \sum_{n=0}^{N_\varepsilon-1} \left\lVert G_{\rm imp}(i\varepsilon_n) -G_{\rm loc}(i\varepsilon_n) \right\rVert_F^2 \right]^{1/2}\\ &\bigg/ \max\!\left\{ \left[ \frac{1}{N_\varepsilon} \sum_{n=0}^{N_\varepsilon-1} \left\lVert G_{\rm loc}(i\varepsilon_n) \right\rVert_F^2 \right]^{1/2}, \sqrt{N_{\rm corr}}\,G_{\rm floor} \right\}, \end{aligned}$$

with

$$G_{\rm floor} = 10^{-3}\ \text{eV}^{-1}.$$

This scalar residual does not replace componentwise statistics. Require every matrix component of $G_{\rm imp}-G_{\rm loc}$ to agree within two combined standard errors. Also report

$$R_n := \left|n_{\rm imp}-n_{\rm loc}\right|,$$

and, for charge self-consistency,

$$R_\rho := \frac{ \int_{\Omega_{\rm CSC}}d^3r\, \left| \rho_{\rm out}(\mathbf r)-\rho_{\rm in}(\mathbf r) \right| }{N_{e,\Omega_{\rm CSC}}}.$$

Here $\Omega_{\rm CSC}$ is the repeated charge-self-consistency cell and $N_{e,\Omega_{\rm CSC}}$ is its electron count.

Only the semicircular Bethe benchmark with half-bandwidth $D=2t_*$ permits the special update

$$\Delta(i\varepsilon_n) = t_*^2G_{\rm imp}(i\varepsilon_n).$$

It is an exact workflow check for that benchmark, not a self-consistency formula for a general material band structure.

Validate Causality, Moments, Metastability, and Uncertainty

Define the Hermitian real and imaginary parts by

$$\operatorname{Re}_{\rm H}X := \frac{X+X^\dagger}{2}, \qquad \operatorname{Im}_{\rm H}X := \frac{X-X^\dagger}{2i}.$$

For positive Matsubara energy, and for retarded boundary values in the same sign convention, require

$$-\operatorname{Im}_{\rm H}G\succeq0, \qquad -\operatorname{Im}_{\rm H}\Delta\succeq0, \qquad -\operatorname{Im}_{\rm H}\Sigma\succeq0.$$

Test eigenvalues of these Hermitian matrices. Causal diagonal entries do not guarantee a causal matrix when off-diagonal components are present.

For the $N_{\rm corr}$ orthonormal correlated modes,

$$A(\omega) := -\frac{1}{\pi} \operatorname{Im}_{\rm H}G^R(\omega),$$

and the one-particle sum rule is

$$\int_{-\infty}^{\infty}d\omega\, \operatorname{Tr}A(\omega) = N_{\rm corr}.$$

Check this rule before interpreting redistributed spectral weight. Also check the noninteracting and atomic limits, high-frequency moments, Hermiticity, symmetry relations, total electron count, and agreement among independent seeds and initial self-energies.

Use

$$Z = \left[ I- \left. \frac{\partial \operatorname{Re}_{\rm H}\Sigma^R(\omega)} {\partial\omega} \right|_{\omega=0} \right]^{-1}$$

only on a differentiable Fermi-liquid branch. A finite-temperature Matsubara-axis fit is provisional and basis aware. An orbital diagonal element is not invariant under an arbitrary rotation of the correlated frame.

Numerical convergence and model validity need separate ledgers:

• numerical controls include $\mathbf k$ mesh, Matsubara cutoff and tails, solver statistics, autocorrelation, mixing, residuals, and continuation;
• representation controls include window, projector, basis metric, local axes, and omitted bands;
• physical-model controls include $U,J$, retardation, double counting, structure, charge feedback, locality, and allowed broken symmetry;
• inference controls include forward-model parameters, experimental resolution, backgrounds, and covariance.

Seed symmetry-related, magnetic, orbital, and structural alternatives. A smooth residual history establishes a fixed point, not a thermodynamic phase. Hysteresis or coexistence does not locate a transition without matched thermodynamic potentials and a controlled metastability analysis.

To test locality, compare aligned self-energies or observables from a cluster or another controlled nonlocal calculation. Momentum dependence of $A(\mathbf k,\omega)$ alone is insufficient because a local orbital self-energy can become momentum dependent after projection into Bloch states. If the aligned nonlocal diagnostic or the licensed observable changes beyond tolerance, stop the single-site material claim.

Continue to Real Frequency and Build Observable Forward Models

Analytic continuation is required only when the claim genuinely depends on a real-frequency object. Preserve the raw imaginary-time or Matsubara data, covariance, tails, transforms, and exact constraints before fitting. Then:

1. name the object being continued, such as a diagonal Green function, a positive matrix spectral density, or a compatible self-energy;
2. calibrate attainable resolution on synthetic data with the same grid and covariance;
3. use at least two defensible methods or targets when a feature matters;
4. back-transform the result and inspect covariance-whitened residuals;
5. vary priors, regularization, support, moments, and covariance treatment;
6. report only features stable within that admissible family.

Analytic Continuation owns the inverse problem, priors, covariance, and identifiability standards. This page owns carrying those uncertainties into the final DMFT record.

A continued local one-particle gap is not automatically the exact fundamental charge gap, neutral optical gap, mobility gap, or transport activation scale. Likewise, a coherent spectral ridge is not proof of an asymptotic zero-temperature Fermi liquid.

Spectral Functions owns the interpretation of poles, continua, linewidths, and spectral weight. Angle-Resolved Photoemission Spectroscopy adds matrix elements, surface sensitivity, photon-energy dependence, backgrounds, occupation, and resolution. Transport, Response, and Optics owns response selection, vertices, contacts, and order of limits. A single-particle bubble without the required vertex analysis is not automatically a measured conductivity.

The result of this section is therefore an intrinsic, uncertainty-bearing input to a probe owner, not simulated detector data unless the complete forward model has also been supplied.

Worked Audit: A Three-Orbital Correlated Metal

This complete record is synthetic. It tests whether numerical convergence, source variation, interaction variation, and continuation support a bounded local-metal claim.

1. Physical problem. Use an infinite periodic cubic three-$t_{2g}$-orbital metal with bandwidth $W=2.4\ \text{eV}$ and six retained spin-orbital modes. It is a teaching system rather than a measured material.

2. State and limit. Fix total correlated filling $n=1$, $T=290\ \text K$, no field, drive, or SOC, and the paramagnetic single-site branch. The lowest positive Matsubara energy is $\varepsilon_0=0.07851\ \text{eV}$.

3. Claim and accuracy. Test local occupancy and a finite-temperature, Fermi-liquid-compatible low-frequency renormalization. Target numerical stability in $Z$ to $0.02$ and a continued low-energy maximum only to $30\ \text{meV}$.

4. Representation and provenance. Consume immutable artifact T2G-CUBIC-PBE-018B-001E-v1, which contains eighteen stored spinor bands and an accepted rank-six target. Relative to source $E_F=0$, the central outer and frozen windows are $[-2.0,1.5]\ \text{eV}$ and $[-0.80,0.60]\ \text{eV}$. An independently accepted ligand-tailed rank-six branch uses $[-3.0,2.5]\ \text{eV}$ with the same frozen window, has full-zone maximum $d_P=0.07$, and gives $W=2.55\ \text{eV}$.

5. Method and controlled domain. Use one-shot single-site DMFT with a general matrix CT-HYB solver and rotational Kanamori $U=4.0\ \text{eV}$, $J=0.60\ \text{eV}$, and $U'=2.8\ \text{eV}$. The central scalar $V_{\rm DC}=2.0\ \text{eV}$ uses the sign declared above.

6. Finite numerical problem. Use every point of the half-shifted $12^3$, $16^3$, and $20^3$ meshes,

   $$\mathbf k_{\mathbf n} = \sum_{i=1}^{3} \frac{n_i+1/2}{N_k}\mathbf b_i, \qquad w_{\mathbf k}=N_k^{-3}.$$

   Retain 1,024 positive Matsubara energies and exact analytic tails. For each mesh run eight independent replicas, each with $2\times10^6$ warm-up and $4\times10^7$ measurement updates stored in 200 batches. Use three distinct initial self-energies and require mean sign at least $0.97$.

7. Estimator and forward model. The mesh sequence gives $n=1.002,1.000,1.000$ and provisional $Z=0.42,0.40,0.40$. Fits to the first four, six, and eight positive Matsubara points give $0.397,0.404,0.416$. Maximum-entropy and stochastic continuations place the same coarse low-energy maximum at $-0.01\ \text{eV}$ and $+0.02\ \text{eV}$. No probe forward model is included, so these are intrinsic local spectral estimates.

8. Convergence and uncertainty. Report $Z=0.40\pm0.02$ only as the combined solver and fit-window result. After retuning $\mu$ to the same filling and keeping central $U,J$, the central and ligand-tailed projectors give $Z=0.40$ and $0.37$. Recomputing screened interactions for the alternate basis as $U=4.30$, $J=0.65\ \text{eV}$ gives $Z=0.35$ and defines a new model branch. Recompute $U'=U-2J$ on every rotational-Kanamori branch. At fixed $J=0.60\ \text{eV}$, $U=3.6,4.0,4.4\ \text{eV}$ gives $Z=0.47,0.40,0.33$. At fixed $U=4.0\ \text{eV}$, $J=0.50,0.60,0.70\ \text{eV}$ gives $Z=0.43,0.40,0.36$. At fixed $\mu$, changing $V_{\rm DC}=2.0\pm0.2\ \text{eV}$ changes occupancy by $\pm0.05$ and spans $Z=0.36$–$0.45$; this is a changed grand-canonical state, not a same-filling error bar. At fixed filling in this isolated all-correlated model, the same scalar shift is redundant with $\mu$.

9. Verification, validation, and provenance. Require $R_G<5\times10^{-4}$, every matrix component within two combined standard errors, $R_n<2\times10^{-4}$, causal $G,\Delta,\Sigma$, correct high-frequency moments and spectral weight, and noninteracting and atomic checks. Archive all replicas, covariance, tails, seeds, projectors, interactions, continuation inputs, and failed branches.

10. Licensed claim and stopping rule. License a numerically converged, source- and model-conditional local finite-temperature Fermi-liquid-compatible branch. The representation and interaction spread dominates the $\pm0.02$ solver/fit uncertainty. Do not claim an asymptotic zero-temperature Fermi liquid, a unique material phase, absence of nonlocality, or agreement with a probe. Escalate when any of those claims is requested.

The audit succeeds because it weakens the conclusion when the model branch changes more than the numerical refinement. Reporting only the central $Z=0.40$ would conceal the dominant uncertainty.

Worked Audit: A Charge-Transfer Transition Candidate

This second synthetic record is designed to stop. It asks whether an explicit-ligand, charge-self-consistent single-site calculation licenses a charge-transfer gap and then exposes representation, double-counting, structural, ordered, continuation, and nonlocal alternatives.

1. Physical problem. Use a synthetic cubic perovskite-like $DX_3$ source in $Pm\bar{3}m$ No. 221 with lattice constant 3.90 Å, $D$ at the origin, and $X$ on the three face centers. It is not a real compound.

2. State and limit. Fix filling $N_e=8$ electrons per primitive cell, with $\mu$ adjusted accordingly, at $T=300\ \text K$, zero external field, no drive or SOC, and a paramagnetic charge-self-consistent branch. Then $\varepsilon_0=0.08122\ \text{eV}$. Treat the infinite periodic cell and seed competing two-sublattice order separately.

3. Claim and accuracy. Test whether the declared local model supports a charge-transfer gap candidate. Resolve continued gaps only above $0.15\ \text{eV}$. Do not identify the local one-particle object with an exact charge, neutral optical, mobility, or transport gap.

4. Representation and provenance. Consume DX3-PM3M-PBE-010B-008E-v1, an accepted PBE-density artifact with a $90\ \text{Ry}$ teaching cutoff, ten spin-orbital modes, and eight electrons: two correlated $e_g$ spatial orbitals plus three explicit ligand $p_\sigma$ orbitals. Use $U=6.0\ \text{eV}$, $J=0.80\ \text{eV}$, $U'=4.4\ \text{eV}$. Use the full rotational two-orbital Kanamori interaction, including spin-flip and pair-hopping terms, and central $V_{\rm DC}=3.50\ \text{eV}$ with $H_{\rm loc}\mapsto H_{\rm loc}-V_{\rm DC}$.

5. Method and controlled domain. Use charge-self-consistent single-site DMFT with a general matrix CT-HYB solver. Compare the full explicit-ligand branch with separately defined $d$-only and one-shot branches. Those are different represented problems, not numerical refinements.

6. Finite numerical problem. Use every point of the half-shifted $10^3$, $14^3$, and $18^3$ meshes with the formula and equal weights from the first audit. Retain 1,024 positive Matsubara energies plus exact tails. Run eight independent replicas with $5\times10^7$ measurement updates per replica, require mean sign at least $0.94$, and use impurity mixing $0.5$ and charge mixing $0.3$.

7. Estimator and forward model. The mesh sequence gives $n_d=2.225,2.212,2.210$ and continued local one-particle gaps $0.34,0.31,0.31\ \text{eV}$. The $d$-only one-shot branch gives $(n_d,E_g)=(2.00,0.62\ \text{eV})$; the explicit-ligand one-shot branch gives $(2.34,0.18\ \text{eV})$; and the explicit-ligand charge-self-consistent branch gives $(2.21,0.31\ \text{eV})$. No detector-level forward model is part of these estimates.

8. Convergence and uncertainty. At fixed filling $N_e=8$ per primitive cell, $V_{\rm DC}=3.30,3.50,3.70\ \text{eV}$ gives $n_d=2.12,2.21,2.30$ and gaps $0.48,0.31,0.08\ \text{eV}$; the last is unresolved. At central $J$, recompute $U'=U-2J$ as $U=5.5,6.0,6.5\ \text{eV}$ gives gaps $0.18,0.31,0.47\ \text{eV}$. Maximum entropy gives $0.31\pm0.08\ \text{eV}$ and stochastic continuation gives $0.27\pm0.10\ \text{eV}$. An alternate structural artifact shifts the $p$–$d$ alignment by $+0.15\ \text{eV}$ and leaves only a $0.14\ \text{eV}$ gap candidate.

9. Verification, validation, and provenance. Require $R_G<8\times10^{-4}$, every component within two combined standard errors, $R_n<3\times10^{-4}$, $R_\rho<5\times10^{-6}$, and correlated spectral weight $3.997\pm0.006$ against exact $N_{\rm corr}=4$. A separately converged two-sublattice AFM branch has correlated-spin projected moment $m=0.62\mu_B$ per $D$ site and gap $0.55\ \text{eV}$ but lacks a matched thermodynamic-potential comparison. In aligned local gauges define

   $$r_{\rm nl} := \frac{ \left[ 20^{-1} \sum_{n=0}^{19} \left\lVert \Sigma_{01}(i\varepsilon_n) \right\rVert_F^2 \right]^{1/2} }{ \left[ 20^{-1} \sum_{n=0}^{19} \left\lVert \Sigma_{00}(i\varepsilon_n) \right\rVert_F^2 \right]^{1/2} }.$$

   An external two-site diagnostic gives $r_{\rm nl}=0.12\pm0.03$ and cluster gap $0.22\pm0.10\ \text{eV}$. The nonlocal component is resolved, but the central gap displacement is only $|0.31-0.22|=0.09\ \text{eV}$, smaller than the declared $0.15\ \text{eV}$ feature resolution. Because the two calculations share model and continuation systematics and no cross-covariance is supplied, no combined-significance claim is licensed. The gap consequence remains unresolved.

10. Licensed claim and stopping rule. License only the existence of converged single-site candidate branches and strong subspace, double-counting, continuation, structural, and nonlocal sensitivity. Do not select a unique phase or transition and do not claim a measured gap. The resolved nonlocal ratio and unresolved shifted gap require a nonlocal follow-up before any stronger claim, but they do not establish a changed gap or phase by themselves.

Here a small $R_G$ coexists with a failed phase claim. The calculation has converged; the interpretation has not.

Exit Checkpoint, Failure Modes, and Canonical Handoffs

You are ready to leave this page when you can:

• reproduce the source, projector, interaction, double-counting, and state handoff from immutable artifacts;
• distinguish total filling, correlated occupancy, and chemical potential;
• state whether the calculation is one-shot or charge self-consistent;
• close impurity, lattice, chemical-potential, and charge loops separately;
• defend the solver from the interaction, temperature, sign, and requested observable;
• test matrix causality, moments, spectral weight, and componentwise lattice–impurity agreement;
• separate solver uncertainty from window, interaction, double-counting, structure, locality, continuation, and probe uncertainty;
• enumerate credible magnetic, orbital, structural, and nonlocal branches;
• archive raw Matsubara data, covariance, tails, seeds, failed branches, versions, environments, and hashes;
• write a bounded claim, falsifier, and next owner without calling one fixed point a material phase.

Use these canonical handoffs:

• Computational Quantum Matter owns routing, cross-method readiness, and the general ten-field claim ledger.
• Hubbard Physics in Materials owns active-space adequacy, screened interactions, double counting as a material-model input, and model-to-material inference.
• Anderson Impurity Model Preview owns impurity charge, bath physics, the conceptual mapping, and the generic solver survey.
• Wannierization Workflows owns numerical windows, projections, disentanglement, localization, and source-operator validation.
• Thermal Green Functions owns imaginary-time ordering, contact terms, and general thermal Green-function identities.
• Analytic Continuation owns the inverse problem, covariance, priors, regularization, and feature identifiability.
• Spectral Functions owns intrinsic spectral semantics, weights, linewidths, and sum-rule interpretation.
• Angle-Resolved Photoemission Spectroscopy owns photoemission kinematics, matrix elements, surfaces, calibration, resolution, and detector inference.

For physical interpretation, What Are Strong Correlations? owns the evidence hierarchy and Mott Insulators owns interaction-driven charge incompressibility and material phase diagnosis. For a failed locality test, route to Computational Many-Body QM and the applicable cluster, diagrammatic, or lattice-model owner.

Common failures have precise repairs:

Treating single-site as scalar. Retain every orbital, spin, sublattice, and inequivalent-shell component permitted by the problem.

Changing the source inside the solver error bar. A new window, projector, interaction, double-counting prescription, or structure is a new represented problem. Compare it as model sensitivity.

Using a segment solver for rotational interactions. Either use a general matrix solver or declare and justify a density-density truncation.

Reading a finite Matsubara sum as an occupancy. Restore analytic tails and the contact convention before comparing charge.

Testing causality element by element. Diagonalize the Hermitian imaginary part; off-diagonal entries can create a negative eigenvalue.

Calling a fitted $Z$ universal. Restrict it to a stable, Fermi-liquid-compatible window and declared correlated frame.

Calling a fixed point a phase. Seed competing states and compare matched thermodynamic quantities. Residual convergence is not phase selection.

Calling a local gap a measured gap. Carry the intrinsic object to the appropriate spectral, response, and probe owners with continuation and instrument uncertainty intact.

The final archive should include source matrices, overlaps and projectors, interaction and double-counting files, all loop histories, covariance and tail records, raw and continued spectra, back-transforms, initial states, failed runs, environment manifests, and scripts that regenerate every quoted number. A code manual may explain a version-dependent switch; it does not replace this physics and validation record.

Exercises

Exercise 1: Matsubara energies and the Bethe loop

At $T=290\ \text K$, compute $\varepsilon_0$ and $\varepsilon_3$. Then consider a semicircular Bethe benchmark with $D=2t_*=2\ \text{eV}$ and

$$G_{\rm imp}(i\varepsilon_0) = (-0.10-0.62i)\ \text{eV}^{-1}.$$

Find $\Delta(i\varepsilon_0)$ and check its causal sign.

Solution

Using $k_B=8.617333262\times10^{-5}\ \text{eV/K}$,

$$\pi k_BT = 0.07851\ \text{eV}.$$

Therefore

$$\varepsilon_0 = 0.07851\ \text{eV}, \qquad \varepsilon_3 = 7\pi k_BT = 0.54956\ \text{eV}.$$

The half-bandwidth convention gives $t_*=1\ \text{eV}$, so

$$\Delta(i\varepsilon_0) = t_*^2G_{\rm imp}(i\varepsilon_0) = (-0.10-0.62i)\ \text{eV}.$$

For positive Matsubara energy, $-\operatorname{Im}\Delta=0.62\ \text{eV}>0$, so the scalar hybridization passes the causal sign check.

Exercise 2: High-frequency moments

For one correlated orbital, take $\epsilon_d=-0.50\ \text{eV}$, $U=2.0\ \text{eV}$, opposite-spin occupancy $\bar n=0.40$, and zero chemical and double-counting shifts. Using

$$\Sigma_\infty=U\bar n, \qquad \Sigma_1=U^2\bar n(1-\bar n),$$

find $\Sigma_\infty$, $\Sigma_1$, and the first Green-function moment $M_1=\epsilon_d+\Sigma_\infty$.

Solution

Direct substitution gives

$$\Sigma_\infty = (2.0)(0.40) = 0.80\ \text{eV},$$

and

$$\Sigma_1 = (2.0)^2(0.40)(0.60) = 0.96\ \text{eV}^2.$$

The first Green-function moment is

$$M_1 = -0.50+0.80 = 0.30\ \text{eV}.$$

A fitted tail with a different constant or first moment violates the declared equal-time data before any continuation is attempted.

Exercise 3: Matrix causality

Two candidate matrices for $-\operatorname{Im}_{\rm H}\Sigma^R$ have diagonal entries $0.20$ and $0.10\ \text{eV}$. The real symmetric off-diagonal entry is first $0.06\ \text{eV}$ and then $0.16\ \text{eV}$. Compute the eigenvalues and decide which matrix is causal.

Solution

For

$$K(b) = \begin{pmatrix} 0.20 & b\\ b & 0.10 \end{pmatrix} \text{eV},$$

the eigenvalues are

$$\lambda_\pm = \frac{0.30\pm\sqrt{0.10^2+4b^2}}{2} \text{eV}.$$

At $b=0.06\ \text{eV}$ they are $0.228$ and $0.072\ \text{eV}$, so $K\succeq0$. At $b=0.16\ \text{eV}$ they are $0.318$ and $-0.018\ \text{eV}$, so the second matrix fails. Positive diagonal entries did not guarantee matrix causality.

Exercise 4: Double-counting sensitivity

In the charge-transfer audit, $V_{\rm DC}=3.30,3.50,3.70\ \text{eV}$ gives $n_d=2.12,2.21,2.30$ and gaps $0.48,0.31,0.08\ \text{eV}$. Estimate $dn_d/dV_{\rm DC}$ across the interval. Classify the variation and decide whether the final gap is resolved at $0.15\ \text{eV}$ resolution.

Solution

The secant slope is

$$\frac{\Delta n_d}{\Delta V_{\rm DC}} = \frac{2.30-2.12}{3.70-3.30} = 0.45\ \text{electrons/eV}.$$

Every point is separately converged at fixed filling $N_e=8$ per primitive cell, but changing $V_{\rm DC}$ changes relative $p$–$d$ levels and hence the represented material model. This is model and state sensitivity, not mesh or Monte Carlo error. The $0.08\ \text{eV}$ candidate lies below the declared $0.15\ \text{eV}$ resolution and must be reported as unresolved.

Exercise 5: Choose an impurity solver regime

A three-orbital impurity has rotational Kanamori interactions including spin-flip and pair-hopping terms at finite temperature. Choose among a segment CT-HYB solver, a general matrix CT-HYB solver, finite-bath exact diagonalization, and NRG. State the appropriate central choice and one important limitation of each alternative.

Solution

The central choice is a general matrix CT-HYB solver because it can represent the full rotational interaction at finite temperature. Its sign, autocorrelation, and matrix-cost scaling still require measurement.

A segment solver assumes a compatible density-density form and is invalid without an explicit interaction truncation. Exact diagonalization replaces the continuum bath by finitely many levels and must extrapolate bath discretization and many-body truncation. NRG is powerful for very low-energy real-frequency scales in suitable few-channel problems, but the realistic multiorbital matrix structure can be prohibitive. Solver names do not remove their control parameters.

Exercise 6: Continuation nonuniqueness

Maximum entropy gives a local gap $0.31\pm0.08\ \text{eV}$, while stochastic continuation gives $0.27\pm0.10\ \text{eV}$. The calibrated feature resolution is $0.15\ \text{eV}$. What is licensed?

Solution

The one-standard-deviation intervals are $[0.23,0.39]\ \text{eV}$ and $[0.17,0.37]\ \text{eV}$, with a substantial overlap. Both support a coarse central gap candidate larger than the calibrated resolution. They do not resolve a sharp edge, fine substructure, or a difference between the two central estimates. The report must preserve the method-dependent intervals and back-transform checks rather than average the curves into false precision.

Exercise 7: Apply the nonlocal stopping rule

An aligned two-site comparison gives $r_{\rm nl}=0.12\pm0.03$ and a cluster gap $0.22\pm0.10\ \text{eV}$, compared with the single-site central gap $0.31\pm0.08\ \text{eV}$. Decide whether to defend a definitive single-site material phase.

Solution

The nonlocal self-energy ratio is resolved away from zero at roughly four standard deviations. The central gap displacement is

$$|0.31-0.22| = 0.09\ \text{eV},$$

which is smaller than the $0.15\ \text{eV}$ feature resolution. The two gap errors share source, model, and continuation systematics, and no cross-covariance was supplied, so they cannot be combined as independent errors. The comparison detects nonlocal self-energy structure but does not resolve its effect on the gap or establish a different phase. A definitive material-phase claim is already unlicensed; archive this unresolved systematic and escalate before making any stronger locality-dependent claim.

Exercise 8: Complete the charge-transfer record

Using the second worked audit, fill every ten-field label. Identify which comparisons change the represented problem and state the strongest licensed claim and next owners.

Solution

1. Physical problem. The system is synthetic cubic $DX_3$ in $Pm\bar{3}m$, with one $D$ site, three face-centered $X$ sites, and an infinite periodic geometry.
2. State and limit. It has fixed filling $N_e=8$ electrons per primitive cell with $\mu$ adjusted, $T=300\ \text K$, zero external field, no SOC or drive, and a paramagnetic charge-self-consistent single-site branch; two-sublattice order is a separate seed.
3. Claim and accuracy. The target is a local one-particle charge-transfer gap candidate resolvable only above $0.15\ \text{eV}$, not an exact charge, optical, mobility, or transport gap.
4. Representation and provenance. The source is DX3-PM3M-PBE-010B-008E-v1, with two correlated $e_g$ orbitals, three explicit ligand $p_\sigma$ orbitals, $U=6.0\ \text{eV}$, $J=0.80\ \text{eV}$, $U'=4.4\ \text{eV}$, the full rotational two-orbital Kanamori interaction including spin-flip and pair-hopping, and $V_{\rm DC}=3.50\ \text{eV}$ under the declared subtraction sign.
5. Method and controlled domain. The central method is charge-self-consistent single-site DMFT with general matrix CT-HYB. The $d$-only, one-shot, AFM, structural, and two-site calculations are distinct comparison branches.
6. Finite numerical problem. It uses half-shifted $10^3,14^3,18^3$ meshes, 1,024 positive Matsubara energies and exact tails, eight replicas with $5\times10^7$ measurements each, mean sign at least $0.94$, impurity mixing $0.5$, and charge mixing $0.3$.
7. Estimator and forward model. The record contains $G_{\rm imp}$, $G_{\rm loc}$, $\Sigma_{\rm imp}$, $n_d$, charge density, and continuation-dependent local gaps. A detector forward model is not included.
8. Convergence and uncertainty. Mesh convergence is small, while the subspace, charge feedback, $U$, double counting, continuation, structure, ordered branch, and nonlocal comparison generate much larger changes. Those branches are model uncertainties, not solver error bars.
9. Verification, validation, and provenance. The central branch requires $R_G<8\times10^{-4}$, componentwise two-standard-error agreement, $R_n<3\times10^{-4}$, $R_\rho<5\times10^{-6}$, spectral weight $3.997\pm0.006$ against $N_{\rm corr}=4$, matrix causality, moments, seeds, and archived artifacts. The nonlocal diagnostic is $0.12\pm0.03$, while its effect on the gap remains unresolved.
10. Licensed claim and stopping rule. The evidence licenses converged single-site candidate branches with strong model sensitivity. It does not license a unique phase, transition, or measured gap. Subspace and interaction questions return to Hubbard Physics in Materials, real-frequency resolution to Analytic Continuation, nonlocality to a cluster owner, phase classification to the relevant Quantum Matter page, and measured spectra to the probe owner.

The $d$-only, explicit-ligand, one-shot, charge-self-consistent, structural, AFM, and cluster records are not successive error bars on one calculation. Each changes the represented problem or approximation and must remain separately named.

References

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• Kristjan Haule, Chuck-Hou Yee, and Kyoo Kim, “Dynamical Mean-Field Theory within the Full-Potential Methods: Electronic Structure of CeIrIn₅, CeCoIn₅, and CeRhIn₅,” Physical Review B 81, 195107 (2010).
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• Frank Lechermann, Antoine Georges, Alexander Poteryaev, Silke Biermann, Michel Posternak, Akihisa Yamasaki, and Ole K. Andersen, “Dynamical Mean-Field Theory Using Wannier Functions: A Flexible Route to Electronic Structure Calculations of Strongly Correlated Materials,” Physical Review B 74, 125120 (2006).
• Walter Metzner and Dieter Vollhardt, “Correlated Lattice Fermions in $d=\infty$ Dimensions,” Physical Review Letters 62, 324–327 (1989); erratum 62, 1066 (1989).
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• Philipp Werner, Armin Comanac, Luca de’ Medici, Matthias Troyer, and Andrew J. Millis, “Continuous-Time Solver for Quantum Impurity Models,” Physical Review Letters 97, 076405 (2006).