Computational Quantum Matter

Computation is an inference chain, not a machine that turns a Hamiltonian or a crystal file into a material fact. This gateway starts with the physical claim, identifies its canonical theory and measurement owners, chooses the shortest numerical route that can test it, and requires a record of approximation, convergence, uncertainty, and provenance before the result is promoted.

Required background. Choosing a Model supplies the one universal capability: declaring the physical system, target, and approximation boundary before selecting a computational method. All other preparation is branch-specific.

Helpful background. Quantum Matter Conventions fixes shared signs and normalizations. Math Needed for Computational QM and Numerical Mathematics supply the numerical language. Interacting calculations should consult Computational Many-Body QM, while comparison with data should begin at How Quantum Matter Is Measured.

Enter Computational Quantum Matter

Begin by writing one sentence of the form:

For the declared system, state, model, geometry, and data provenance, compute this object to this accuracy in order to test this bounded physical claim.

The sentence must name the object. A Kohn–Sham eigenvalue, a quasiparticle pole, a finite-cluster level, a Monte Carlo estimator, a retarded spectral density, a terminal conductance, and a probe-weighted intensity are not interchangeable. Neither are a model prediction, a fitted parameter, and a measured observable.

This chapter gateway owns four decisions:

whether the question is sufficiently declared to compute;
which theory, method, and measurement owners are needed;
which error and validation tests license the requested conclusion;
where to stop when a production workflow or canonical algorithm owner is still planned.

It does not derive eigensolvers, density-functional theory, tensor networks, Monte Carlo, topological algorithms, quasiclassical solvers, or instrument inversion.

Write the Computational Claim Ledger

Use the same ten fields for a quick route choice, a notebook, a production run, and a paper-level claim. Fill every field or write unresolved.

Physical problem. State the material or model, degrees of freedom, dimension, geometry, boundaries, specimen or unit cell, and preparation.
State and limit. Give ensemble, filling or chemical potential, temperature, field, disorder, drive, equilibrium status, finite-size or thermodynamic target, and the required order of limits.
Claim and accuracy. Name the target observable or decision, units, normalization, spatial, frequency, or time resolution, tolerance, and the physical comparison that motivates it.
Representation and provenance. Record the Hamiltonian, functional, action, active subspace, basis, interactions and material parameters, their sources, and omitted degrees of freedom.
Method and controlled domain. State the approximation, solver family, asymptotic or variational control, sign or locality structure, and why the method can estimate the requested observable.
Finite numerical problem. Declare the cell and boundaries, meshes, cutoffs, system size, symmetry sector, bond or bath truncation, time step, regulator, stochastic ensemble, and random seeds.
Estimator and forward model. Give the operator, estimator, continuation or response kernel, matrix elements, contacts or instrument model, and the map from computed object to reported observable.
Convergence and uncertainty. Separate discretization, truncation, solver, self-consistency, statistical, finite-size, continuation, fit, model, and experimental-comparison uncertainties, including covariance when relevant.
Verification, validation, and provenance. List residuals, identities, conservation and symmetry checks, exact limits, independent methods or codes, registered benchmarks, raw artifacts, environment, and versioning.
Licensed claim and stopping rule. State the strongest supported conclusion, credible alternatives, known missing physics, a falsifier, and the next canonical owner if the tolerance or evidence requirement fails.

A plot without this ledger is an exploratory output, not a computational claim.

Pass the Readiness Gate

Before choosing software or an algorithm, ask:

Is the requested output defined by the declared theory?
Are units, charge signs, Fourier conventions, basis order, and normalizations explicit?
Does the chosen finite or periodic geometry represent the physical question?
Is the target an equilibrium object, a real-time object, or an order-of-limits response?
Are the intended filling, temperature, and symmetry sector representable?
Is there at least one analytic limit, benchmark, or cross-method check?
Is the desired accuracy meaningful compared with model-form and experimental uncertainty?

If the physical object is unclear, return to the Quantum Matter Map and the relevant chapter gateway. If the numerical limit is unclear, use Convergence Tests and Conditioning and Stability. If the observable-to-data map is unclear, route through the measurement gateway before computing.

Follow the Nonlinear Dependency Graph

The catalog order is not a universal prerequisite chain. Use the branch that matches the declared claim:

physical claim
  -> canonical theory owner
  -> model and parameter provenance
  -> numerical object and controlled regime
  -> algorithm owner
  -> verification and convergence
  -> observable forward model
  -> validation against benchmark or data
  -> bounded conclusion, falsifier, and archived artifacts

band/material input
  -> band theory and electronic-structure semantics
  -> converged source calculation
  -> optional localized representation
  -> response, topology, or correlated-method branch

interacting lattice claim
  -> many-body model and phase owner
  -> finite-size/statistical/method selection
  -> controlled estimator and scaling

response or transport claim
  -> response owner and order of limits
  -> bulk or terminal geometry
  -> contacts, disorder, continuation, and probe forward model

Verification asks whether the declared equations were solved correctly. Validation asks whether those equations and their forward model are adequate for the physical claim. A calculation may pass one and fail the other.

Route Band and Localized-Representation Questions

For a declared lattice Hamiltonian, begin at Lattices, Reciprocal Space, and Bloch Electrons and Band Theory and Electronic Structure. Those owners define Bloch Hamiltonians, fillings, gaps, effective masses, and the distinction among model, Hartree–Fock, Kohn–Sham, quasiparticle, spectral, and measured band objects.

Route a crystalline calculation in this order:

verify structure, units, electron count, symmetry, and basis provenance;
converge the self-consistent or model input before drawing a band path;
validate the full Brillouin-zone subspace, not only eigenvalues on a high-symmetry line;
treat degeneracies through projectors or subspaces rather than arbitrary eigenvector labels;
compare with data only through the relevant matrix-element and resolution model.

Wannier Functions owns the theory of localized frames, gauge freedom, localization, and obstructions. A numerical interpolation that reproduces one band path does not therefore validate the subspace, real-space truncation, symmetry, Berry geometry, or other operators.

Band Structure Workflows owns the crystal-input, self-consistency, convergence, band-output, probe-validation, and artifact pipeline. Wannierization Workflows owns numerical window and projection choice, disentanglement, localization, full-zone interpolation, operator validation, and the resulting reproducibility record.

Route Interacting and Correlated Questions

Computational Many-Body QM is the live method-selection gateway for exact diagonalization, Monte Carlo, tensor networks, finite-size scaling, and dynamical correlations. What Are Strong Correlations? owns the material and model claim; the computational route is chosen only after the active degrees of freedom, dimensionality, sign structure, and target observable are declared.

Use the following distinctions:

exact diagonalization is exact only for the represented finite Hamiltonian;
tensor-network control is expressed through bond dimension, discarded weight, geometry, entanglement structure, and competing initial states;
Monte Carlo uncertainty requires equilibration, autocorrelation, estimator bias, and sign or phase diagnostics;
a self-consistent mean-field or embedding solution is not automatically the global thermodynamic state;
a local self-energy approximation does not certify the absence of nonlocal correlations;
analytic continuation is an ill-conditioned inference problem, not a change of variable.

DMFT for Quantum Materials owns the executable materials workflow: recording and propagating a declared correlated subspace, interaction, and double-counting choice; auditing impurity-solver controls, lattice and charge self-consistency, continuation sensitivity, and cross-probe validation. Hubbard Physics in Materials owns the physical active-space, interaction, and double-counting choices. The conceptual impurity mapping remains with the live many-body theory owners, and material claims remain with the relevant Quantum Matter pages.

Route Response, Pairing, and Disorder Questions

Use Transport, Response, and Optics before selecting a Kubo, Boltzmann, or Landauer calculation. The response owner fixes the current, geometry, contacts, and order of limits. A bulk conductivity and a terminal conductance are different computational objects.

Use Topological Quantum Matter before computing an invariant or boundary spectrum. An integer returned on one mesh does not establish the occupied subspace, the bulk or mobility gap, the protecting symmetry, or the material phase. Projector, Wilson-loop, and finite-boundary checks are complementary rather than interchangeable.

Use Superfluidity and Superconductivity before a Ginzburg–Landau, BCS, Bogoliubov–de Gennes, quasiclassical, or vortex calculation. Declare gauge, Nambu redundancy, self-consistency, boundary conditions, and whether the target is a pair field, spectral edge, stiffness, vortex observable, or terminal response.

Use Disorder in Quantum Matter before averaging random Hamiltonians. One disorder realization, one inverse participation ratio, or one finite transfer matrix is not a localization transition. Record the ensemble, correlations, rare-event sensitivity, boundary conditions, size scaling, and uncertainty of the disorder average.

Production algorithms for these branches belong to the dedicated Computational QM volume. Those routes are still planned; this gateway names the coverage gap without linking scaffold pages.

Validate, Quantify Uncertainty, and Preserve Provenance

For a reported value $\widehat Q$ , distinguish the physical target $Q_{\mathrm{phys}}$ , the declared model prediction $Q_{\mathcal M}$ , the discretized or truncated result $Q_h$ , and the computed estimator:

Q_{\mathrm{phys}}-\widehat Q = \bigl(Q_{\mathrm{phys}}-Q_{\mathcal M}\bigr) + \bigl(Q_{\mathcal M}-Q_h\bigr) + \bigl(Q_h-\widehat Q\bigr).

The three terms represent model-form, discretization or truncation, and solver or sampling discrepancies. They are not automatically independent random variables, so do not combine them in quadrature without a covariance model. Probe calibration and forward-model uncertainty form an additional layer.

A convergence statement must name a limit:

Q(h,L,N_b,N_s,\varepsilon) \longrightarrow Q_\star

as mesh $h$ , size $L$ , retained basis or bond dimension $N_b$ , sample count $N_s$ , and solver tolerance $\varepsilon$ are taken through a controlled sequence. Varying one axis while freezing an uncontrolled one does not establish convergence.

For a stationary Markov chain, define $\tau_{\mathrm{int}}=\tfrac12+\sum_{t>0}\rho(t)$ after equilibration. With $N_s$ retained samples, an autocorrelation-aware standard error is approximately

\operatorname{SE}(\overline O) \simeq \sqrt{ \frac{2\tau_{\mathrm{int}}\operatorname{Var}(O)} {N_s} }.

The formula assumes equilibration, a defined estimator, and an integrated autocorrelation time $\tau_{\mathrm{int}}$ that is itself resolved. A small average sign can make the effective information exponentially poorer.

For an approximate eigenpair, report a dimensionless residual such as

r_n := \frac{ \lVert H\widetilde\psi_n-\widetilde E_n\widetilde\psi_n\rVert }{ (\lVert H\rVert+|\widetilde E_n|)\lVert\widetilde\psi_n\rVert }.

A small residual verifies only the represented eigenproblem. It does not validate the basis, effective Hamiltonian, physical model, or observable. Interpret it with spectral gaps and conditioning: near a degeneracy, a well-converged invariant subspace can be meaningful even when individual eigenvectors and band labels are not.

For iterative self-consistency, a small fixed-point residual $\lVert F(x)-x\rVert$ proves neither uniqueness nor stability. Compare free energies or the correct variational functional where available, scan initial conditions and symmetry sectors, and test hysteresis and metastability.

Archive at minimum:

the exact input and preprocessing path;
source revision, dependencies, compiler or interpreter, and hardware-relevant numerical settings;
random seeds and sampling schedule;
raw estimator data, not only plotted curves;
convergence sweeps and failed or excluded runs;
scripts that regenerate every reported number and figure;
a machine-readable statement of the licensed claim and known stopping conditions.

Worked Audit: A Small-Gap SOC Crystal

The live Band Structure Workflows page now owns the complete synthetic small-gap SOC record: declared crystal, SCF and full-zone meshes, path-versus-global gap, method spread, force and density checks, archived artifacts, and the bounded Kohn–Sham claim. This gateway retains only the routing lesson: a path separation cannot license a global material-gap claim, and model dependence can dominate numerical refinement.

Routing Lesson: A Three-Orbital Correlated Metal

A three-orbital DFT+DMFT calculation can converge while its active subspace, interaction tensor, double-counting convention, continuation, or probe map remains unvalidated. Route the complete executable record through DMFT for Quantum Materials, which owns the synthetic audit, lattice and impurity residuals, solver statistics, sensitivity branches, archived artifacts, and stopping tests.

This gateway retains only the routing rule: a stable local self-energy licenses no stronger material, phase, or measured-intensity claim until the material-model, continuation, spectral, and probe owners accept their parts of the evidence chain.

Exit Checkpoint

You are ready to leave this gateway when you can:

identify the canonical owner of the physical claim;
distinguish the model object from the computed estimator and measured observable;
justify the method from scales, geometry, sign structure, and target output;
name every approximation and control parameter that could change the claim;
separate model-form, discretization, solver, sampling, continuation, and experimental uncertainties;
provide at least one analytic or cross-method benchmark and one falsifier;
reproduce the result from archived inputs, code, data, and environment;
stop before converting numerical agreement into an unlicensed phase, mechanism, topology, or material claim.

If one item fails, route back to the corresponding theory, numerical-analysis, many-body, or measurement owner.

Canonical Boundaries and Coverage Gaps

This gateway is the Quantum Matter entry point and validation compiler. It is not the canonical home of general numerical algorithms. The Computational QM volume owns reusable numerical assembly and implementation of declared lattice Hamiltonians, band diagonalization, Berry and Chern discretizations, Wilson loops, transport numerics, mean-field self-consistency, Bogoliubov–de Gennes solvers, disorder algorithms, exact diagonalization, tensor networks, Monte Carlo, reproducibility infrastructure, and benchmark implementations. Those production pages remain planned and are not linked here as required destinations.

Three narrowly material-facing workflows are live in this chapter:

Band Structure Workflows owns the crystal-input, self-consistency, convergence, band-output, probe-validation, and artifact pipeline—not the semantics of electronic-structure objects or the Bloch eigensolver.
Wannierization Workflows owns numerical windows, projections, disentanglement, localization, interpolation and operator validation, and stored artifacts—not the Wannier theory or topology classification.
DMFT for Quantum Materials owns the executable DFT+DMFT materials workflow—not the impurity mapping, Hubbard-material model choice, or a universal phase interpretation.

The former local Overview, Tight-Binding Numerics, Topological Invariant Computation, Berry Curvature Numerics, Transport Simulation, Mean-Field Models, Bogoliubov–de Gennes Numerics, Exact Diagonalization for Lattice Models, DMRG and Tensor Networks, Quantum Monte Carlo, Disorder Numerics, Reproducible Notebooks, and Benchmark Problems routes were zero-inbound scaffolds. Their contracts are absorbed by this gateway, the live Many-Body and theory owners, and the canonical Computational QM plan. No substantive content or deployed inbound route was removed.

Exercises

1. Route six computational claims

Route each request to its physical owner and then to the smallest suitable computational capability: (a) bands of a periodic four-orbital Hamiltonian, (b) a real-frequency electron-removal spectrum, (c) a Hall response, (d) correlations on an interacting cylinder, (e) localization in a disordered wire, and (f) a material-specific DMFT claim.

Solution

(a) begins with the band-theory owner, then uses deterministic diagonalization and full-zone subspace checks. (b) begins with Spectral Functions; a real-frequency solver or analytic continuation is downstream, and a measured photoemission intensity needs its probe owner. (c) begins with the Transport, Response, and Optics gateway, which selects Kubo, Landauer, or another regime before any numerical kernel. (d) begins with Computational Many-Body Quantum Mechanics and may select a tensor-network cylinder calculation with bond, length, and width scaling. (e) begins with Disorder in Quantum Matter and the appropriate transport or localization owner, followed by ensemble and size numerics. (f) begins with the material Hamiltonian and strong-correlation owners; the live DFT+DMFT execution workflow does not substitute for the physical claim. None of these routes makes the method name the canonical owner of the result.

2. Separate seven error sources

A finite-cluster Monte Carlo calculation is solver-converged and agrees with a broad experimental peak after analytic continuation. Classify at least one possible model, representation, finite-size, solver, stochastic, continuation, and inference error. Which can be combined in quadrature without additional evidence?

Solution

Model error includes omitted orbitals or interactions. Representation error includes a truncated basis, constrained symmetry sector, or discretized bath. Finite-size error includes cluster shape and boundary dependence. Solver error includes a residual or incomplete optimization at fixed represented problem. Stochastic error includes autocorrelation, equilibration, and sign reweighting. Continuation error includes prior, kernel, and regularization sensitivity. Inference error includes the instrument forward model, backgrounds, parameter non-identifiability, and the possibility that the broad peak has another origin. None of these terms may be combined in quadrature merely because each has been assigned a number; independence or a covariance model is required.

3. Interpret a machine-precision residual

A normalized eigenpair residual is $r_n=10^{-14}$ for a Hamiltonian projected onto eight orbitals in a finite cell. What has been verified, and what has not?

Solution

The result verifies that the reported vector and eigenvalue solve the projected finite-dimensional eigenproblem to high numerical precision, subject to the conditioning and norm convention. It does not verify that eight orbitals form an adequate active space, that the effective Hamiltonian represents the material, that the finite cell is large enough, or that the eigenvalue maps to the requested observable. Those require basis enlargement, model comparison, size checks, physical benchmarks, and a forward model. Near degeneracy, test the invariant subspace rather than attach physical meaning to one numerically selected eigenvector.

4. Repair a high-symmetry band-gap claim

A plotted high-symmetry path has a minimum direct separation of $50\ \mathrm{meV}$ , and the caption calls the material a $50\ \mathrm{meV}$ insulator. Repair the computation and the claim.

Solution

First name the spectrum object: Kohn–Sham, quasiparticle, addition/removal, or another gap. Search the full Brillouin zone on a converged mesh, refine around candidate extrema, track the intended occupied and empty subspaces through crossings, and distinguish the minimum direct separation from the global indirect gap at the declared filling. Vary basis or cutoff, sampling, structure, SOC and magnetic state, smearing, and the physical approximation. Until those checks are complete, only the path-local direct separation is licensed. Even a positive converged Kohn–Sham global gap remains an auxiliary model result, not automatically a fundamental, optical, or mobility gap.

5. Compute an effective sample count

A stationary record contains $N_s=200{,}000$ samples and uses the convention $\tau_{\mathrm{int}}=\tfrac12+\sum_{t>0}\rho(t)$ . If $\tau_{\mathrm{int}}=40$ , estimate the effective independent sample count. When is that estimate itself unreliable?

Solution

The variance formula implies

N_{\mathrm{eff}} \simeq \frac{N_s}{2\tau_{\mathrm{int}}} = 2500.

The estimate is unreliable if the chain is not equilibrated, the record is not stationary, the window is too short to resolve a long autocorrelation tail, different chains disagree, rare transitions are absent, or the estimator is a sign-reweighted ratio whose covariance has not been included. Report the windowing rule and stability of $\tau_{\mathrm{int}}$ , not just $N_{\mathrm{eff}}$ .

6. Diagnose a nonconverged Chern calculation

A code reports $0.93$ , $1.07$ , and $-0.98$ on successively refined meshes, then rounds every value to “ $C=1$ .” The minimum direct gap is also shrinking. Give a four-part diagnostic that tests projector isolation, orientation, gauge treatment, and mesh admissibility.

Solution

First verify a fixed-rank occupied projector remains isolated everywhere; a closing gap invalidates the insulating Chern claim. Second declare the Brillouin-zone orientation and link ordering, and confirm that reversing the orientation flips the sign exactly. Third use projectors or gauge-covariant link variables and test invariance under random occupied-frame rotations; energy-sorted eigenvector phases are not a gauge prescription. Fourth inspect plaquette phases and refine adaptively until the discretization is admissible and stable. Rounding hides failure rather than repairing it. The licensed interim statement is only that the implementation produced unstable near-integer diagnostics under the tested conventions.

7. Compare noncommuting response limits

A finite periodic calculation with level broadening $\eta$ reports a large zero-frequency current response and calls it both dc conductivity and superfluid stiffness. Explain why the thermodynamic, dc, broadening, and finite-size limits must be registered separately.

Solution

A finite spectrum contains discrete levels, so $\eta$ can merely smooth level spacing. For bulk dissipative dc transport, take a controlled thermodynamic and reservoir or relaxation limit, then examine $\omega\to0$ with the physical collision and contact model; sending $\eta\to0$ first can recover finite-size delta functions instead. Equilibrium stiffness is a static free- energy or twist curvature with its own equilibrium and thermodynamic limits, not the same dynamical limit as conductivity. The $\mathbf q\to0$ and $\omega\to0$ orders also distinguish longitudinal, transverse, static, and transport responses. Report several limit paths and their physical meanings; do not name a common finite- $L$ , finite- $\eta$ number as both observables.

8. Build a complete new-case dossier

A calculation reports a small band gap, a nonzero Berry diagnostic, a surface state in a slab, and agreement with an ARPES-like ridge. Fill the ten-field ledger and identify four separate escalation routes.

Solution

The dossier must state: (1) the crystal, model, geometry, and physical owner; (2) state, preparation, ensemble, and requested limits; (3) the precise band, topology, surface, and data claims with accuracy targets; (4) Hamiltonian or functional, active space, parameters, and provenance; (5) why the selected methods are controlled for those objects; (6) basis, meshes, symmetry, gauge, slab termination, boundaries, tolerances, and seeds; (7) Kohn–Sham or spectral estimators and the ARPES forward map; (8) convergence along every relevant axis with separated and covariance-aware uncertainties; (9) residuals, symmetries, sum rules, null models, independent checks, artifacts, and exact environment; and (10) the bounded conclusion, credible alternatives, falsifier, stopping rule, and next owner.

The four escalation routes are: band/electronic-structure validation for the gap, topological owners for the invariant and protecting conditions, bulk–boundary and surface owners for the slab state, and the ARPES owner for matrix elements, resolution, and inversion. Agreement among all four can support a declared-model evidence package; it does not automatically prove a material topological phase.

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