Quantum Matter Reference and Data

A compact reference answer is useful only when another reader can tell what object it describes, which conventions make it true, where it came from, and where it stops. A bare number, symbol, formula, Hamiltonian, or probe label can look authoritative while concealing a change of basis, tensor index, sign, normalization, specimen state, or data version. This gateway turns a lookup request into a transferable record rather than another isolated fact.

The page does not replace the site’s glossary, formula cards, model cards, tables, bibliographies, or canonical explanations. It teaches how to locate the right handle, recover the complete context, perform a convention-safe translation, and cite the result with enough provenance to reuse or revise it.

Helpful background. Quantum Matter Map locates the stable concept, model, material, and probe branches. Conventions for Quantum Matter owns the full lattice, basis, Fourier, Berry, electromagnetic, response, normalization, and sign system used here. Neither is a universal prerequisite: the record structure and the conventions used in the worked audits are stated locally.

Enter Quantum Matter Reference and Data

Use this gateway when the immediate task sounds simple but the answer must be portable:

• What does this symbol mean in this paper or page?
• Which formula applies, and under which assumptions?
• Is this quantity per cell, per spin, per mole, per area, or per volume?
• Which Hamiltonian defines the model being compared?
• What observable does this probe actually report?
• Can two numerical values be compared after translating their conventions?
• Which source, dataset version, correction, and canonical explanation belong with the result?

The transferable object is not the displayed value alone. It is a record that binds the value to its physical owner, system and state, convention block, validity regime, provenance, uncertainty, failure modes, and reuse conditions. The same discipline applies to a one-line definition and to a multi-gigabyte dataset.

By the end of the page, you should be able to classify a lookup request, find its live canonical owner and compact Reference handle, translate units and normalizations without losing physics, test whether the source supports the requested reuse, and stop before a reference entry is inflated into a material, mechanism, topological, lifetime, sum-rule, or universality claim.

Decide What Kind of Record You Need

Choose the record class before searching. The class determines what must be preserved and which owner should answer the question.

Term. A compact definition needs distinguishing features, nearby terms, scope, and a link to the page that explains the concept in context. A glossary entry is not a derivation or a phase diagnosis.

Symbol. A symbol record needs mathematical type, indices, units, basis, sign, normalization, and collision notes. The same $\rho$ may denote a density operator, charge density, mass density, or resistivity.

Formula. A formula record needs every variable, assumption, domain, limiting procedure, convention, unit check, canonical derivation, and a small consistency test. A familiar shape does not make two convention variants interchangeable.

Hamiltonian or operator. Record the Hilbert space, degrees of freedom, basis, boundary conditions, parameters and their units, operator ordering, symmetries, and omitted terms. A displayed sum without this information is not a complete model.

Model. A model record adds the physical question, solvability or numerical regime, observables, controlled limits, and failure boundary. It must separate an abstract model from a material realization.

Material-scale datum. A gap, velocity, lattice constant, transition temperature, moment, carrier density, or interaction scale needs composition, specimen, prepared state, method, normalization, uncertainty, and source version. Rounded pedagogical values and fitted material parameters are not fundamental constants.

Probe or observable map. Preserve the chain from detector record through calibration, background, resolution, matrix elements, estimator, and forward model to the claimed response or correlation function. The probe name alone does not identify what was measured.

Computational benchmark. Freeze the code and version, environment, input artifact, precision, metric, reference result, tolerance, and pass/fail semantics. Agreement among implementations sharing one uncontrolled approximation is not independent physical validation.

Bibliography. A source record needs complete metadata, stable identifier, purpose, evidence class, exact locator, and any correction or retraction. A reading list is not evidence until the claim and source location are aligned.

Correction or version record. Preserve the previous statement, changed artifact, reason, date, responsible source, downstream impact, and replacement. Silent overwriting destroys the evidence needed to audit reuse.

Some tasks cross classes. A Hall-density lookup, for example, combines a probe map, formula, material datum, and model assumption. Name the primary task and attach the other records rather than forcing the whole question into one card.

Write the Ten-Field Reference Record

Every worked record on this page uses the same ten fields. Write not applicable with a reason when a field genuinely does not apply; never omit a field silently.

1. Lookup task and record class. State the exact question and whether the primary object is a term, symbol, formula, Hamiltonian/operator, model, material-scale datum, probe/observable map, computational benchmark, bibliography, or correction/version record.

2. Canonical owner and reference handle. Name the live page that owns the explanation, derivation, or measurement and the compact Reference handle that supports fast lookup. Record an explicit coverage gap when a handle does not yet exist.

3. System, state, geometry, and scope. Freeze composition or model, specimen or device, preparation, boundaries, dimension, temperature, pressure, field, filling, strain, disorder, drive, history, and the region or interval to which the record applies.

4. Symbols, units, basis, and sign conventions. Define every symbol and index, dimensional unit, coordinate or reciprocal basis, tensor ordering, Fourier phase, charge sign, orientation, normalization, and sheet, cell, mole, particle, spin, area, or volume convention.

5. Definition, formula, model, or probe map. Give the compact statement actually being reused and the map from inputs or raw records to the output. Link the canonical derivation or forward model instead of reproducing it.

6. Assumptions, validity regime, and order of limits. State approximations, symmetry and isolation assumptions, response regime, finite window, and the order of zero-frequency, zero-temperature, infinite-size, long-time, or weak-field limits.

7. Source, provenance, version, and lifecycle. Preserve authors or institution, DOI or stable identifier, exact table, figure, equation, or dataset locator, version or hash, acquisition and processing history, correction state, access date, and review date.

8. Transformation, normalization, and uncertainty. Show every unit, basis, tensor, degeneracy, cell, geometry, and normalization conversion. Propagate covariance and distinguish standard uncertainty, confidence interval, fit error, resolution, systematic effect, and model spread.

9. Cross-checks, collisions, and failure modes. State dimensional and limiting checks, alternate convention translations, independent records, ambiguous symbols, false matches, nuisance models, and conditions under which the reuse fails.

10. Licensed answer, citation, reuse conditions, and review. Give the strongest answer supported, what is not licensed, how to cite the exact record, the conditions under which it may be reused, its live owner, and the review or correction trigger.

This ledger is longer than a search result because it records the information that search usually strips away. Once it is complete, the final answer may be short. Its brevity is earned by preserving the audit trail elsewhere in the same record.

Freeze Conventions Before Reuse

Convention translation begins by naming distinct quantities. Energy $E$, ordinary frequency $f$, angular frequency $\omega$, temperature-equivalent energy $k_{\rm B}T_E$, and spectroscopic wavenumber $\widetilde\nu$ obey

$$E := h f := \hbar\omega := k_{\rm B}T_E, \qquad \omega=2\pi f, \qquad \widetilde\nu=\frac{E}{hc}.$$

Do not label an energy axis “frequency” and then compare its numerical values with $f$ or $\omega$ without the corresponding factor of $h$ or $\hbar$. Preserve the uncertainty during the conversion, and state whether a quoted linewidth is an energy width, ordinary-frequency width, or angular-frequency width.

For the crystallographic reciprocal basis used throughout Quantum Matter,

$$\mathbf b_i\cdot\mathbf a_j := 2\pi\delta_{ij}, \qquad \mathbf Q := \sum_i H_i\mathbf b_i := \boldsymbol\tau+\mathbf q.$$

A tuple in reciprocal-lattice units is a set of coordinates in this declared basis, not an inverse-length magnitude. State explicitly when a source uses reciprocal vectors without $2\pi$, changes the primitive basis, or reports $\mathbf q$ relative to a different reciprocal vector $\boldsymbol\tau$.

Hall records require tensor indices rather than a generic transverse label. For the in-plane convention

$$\boldsymbol\sigma := \begin{pmatrix} \sigma_{xx} & \sigma_{xy}\\ -\sigma_{xy} & \sigma_{xx} \end{pmatrix},$$

matrix inversion gives

$$\rho_{yx} := \frac{\sigma_{xy}} {\sigma_{xx}^2+\sigma_{xy}^2}, \qquad \rho_{xy}=-\rho_{yx}.$$

Thus $\rho_{yx}$ and $\sigma_{xy}$ have the same sign in this convention, while $\rho_{xy}$ has the opposite sign. State lead polarity, field direction, current direction, and tensor order before assigning a carrier sign. Under the site’s complete Berry convention—electron charge $q_{\rm e}=-e$, $\mathcal A=i\langle u|\nabla_{\mathbf k}u\rangle$, $dk_x\wedge dk_y$, and $\sigma_{xy}=J_x/E_y$—a filled band has $\sigma_{xy}=-C_{\rm occ}e^2/h$. Copy that whole convention block, not only the integer.

Sheet and bulk quantities are also different records. For a uniform film with justified current-carrying thickness $t$,

$$\rho_{ij}=tR_{ij}^{\square}, \qquad \sigma_{ij}=\frac{G_{ij}^{\square}}{t}.$$

Surface states, parallel layers, depletion, and nonuniform current can make that conversion invalid. When the same $t$ enters the numerator and denominator of a derived ratio, its uncertainty is correlated and may cancel; counting it as independent twice overstates the uncertainty.

For a derived scalar $y=f(\mathbf x)$, covariance-aware first-order propagation is

$$u_y^2 \simeq \boldsymbol\nabla f^{\mathsf T} \boldsymbol\Sigma \boldsymbol\nabla f.$$

The covariance matrix must correspond to the actual shared calibrations and processing. Extra printed digits do not create accuracy, and a model range is not interchangeable with a standard uncertainty. Both worked records below use one-standard uncertainties for primitive inputs. Each record declares the primitive covariance or independence assumptions and preserves induced covariance among quantities derived from the same primitive.

If observed and instrumental widths are independent Gaussians, one may define

$$w_{\rm int} := \sqrt{w_{\rm obs}^2-w_{\rm res}^2}.$$

This expression is conditional. A negative radicand, non-Gaussian line shape, correlated width estimate, dispersive resolution volume, or unresolved multicomponent peak is a stopping test. Even a valid intrinsic Gaussian width is not automatically the inverse lifetime of an exponentially decaying mode.

Finally, keep normalization axes orthogonal. If $D_{\rm cell}$ is a density of states per cell and the cell volume is $\Omega_c$, then

$$D_{\rm vol} := \frac{D_{\rm cell}}{\Omega_c}.$$

Converting per cell to per volume says nothing about whether spin, valley, orbital, layer, or formula-unit multiplicities are included. Perform each conversion explicitly.

Follow the Nonlinear Lookup Graph

A reliable lookup is a short graph, not a requirement to read the site in sidebar order.

1. Classify the request. Decide what record you possess and what answer is being requested. Separate a formula lookup from a material inference.
2. Find the stable owner. Use the Quantum Matter Map for the explanation, derivation, model, material, or probe page that owns the physical meaning.
3. Find the compact handle. Use the Reference for a term, symbol, formula card, model card, source guide, table, benchmark, or correction record.
4. Freeze fields 3–7. Record the system, state, geometry, convention block, definition or forward map, validity regime, and exact source before doing a conversion.
5. Transform and test. Carry units, signs, bases, multiplicities, covariance, and significant figures. Run dimensional, limit, symmetry, and independent-record checks.
6. Write the licensed answer. State what may be reused, what cannot be inferred, how to cite it, and what change triggers review.

Several branches may be needed. A reported Chern number can require a compact formula handle, the band or many-body topology owner, a basis and orientation record, a numerical convergence audit, and a separate experimental owner if a Hall response is claimed. No one lookup card should absorb all of those roles.

Route Terms, Symbols, Formulas, and Models

For a term, start with the Glossary and follow its canonical link to the Quantum Matter page that defines the object in context. A one-sentence quasiparticle definition cannot establish that a measured ridge is coherent or that a continuum contains a particular excitation.

For a symbol, use the Symbol Index and then check Conventions for Quantum Matter. Record collisions before substituting. Common examples include $k$ as wave number or crystal momentum, $K$ as a Brillouin-zone corner or coupling, $A$ as a vector potential or spectral function, $D$ as a density of states, stiffness, diffusion constant, or Drude weight, and $\rho$ as either a density operator or resistivity tensor.

For a formula, begin with Quantum Matter Formulas. Each compact card should expose assumptions, symbols, normalization, and the canonical derivation. Use the card to recall Bloch form, density of states, effective mass, Berry curvature, or Chern normalization; follow the linked Quantum Matter owner before extending the formula beyond its stated domain.

For an operator or Hamiltonian, use the Operator and Hamiltonian Library for the compact record. A useful Hamiltonian entry includes the Hilbert space, degrees of freedom, basis, parameters, boundaries, symmetries, and omitted terms. The long derivation and material mapping remain with the model owner.

For a model, Condensed-Matter Models provides short cards, while Choosing a Model for Quantum Matter owns the problem-to-minimal-model decision. Many-Body and Quantum Statistical Mechanics Reference retains many-body formula, Hamiltonian, quasiparticle, and model-index handoffs. Do not infer material realism from a model’s solvability or pedagogical familiarity.

For a topological lookup, attach the invariant’s domain, occupied subspace or many-body state, protecting symmetry when relevant, orientation, gauge or projector convention, and convergence. Topological Quantum Matter owns phase and evidence criteria. A formula card supplies a handle, not a material diagnosis.

Route Materials, Scales, Probes, and Computation

A material scale should lead first to the page that owns the system and the method that produced the number. Record composition, specimen, phase or prepared state, geometry, temperature, field, pressure, filling, strain, frequency or time window, normalization, uncertainty, and source locator. There is not yet a single sourced and versioned sitewide material-scale table; that is an explicit Reference coverage gap, not permission to create an unsourced local list.

A crystallographic or reciprocal-space datum routes to Crystals and Lattices, Reciprocal Lattice, and Brillouin Zones as appropriate. Preserve setting, primitive versus conventional cell, vector ordering, orientation, $2\pi$ convention, and source version. A high-symmetry point label without its lattice and setting is not portable.

A probe or observable map begins with How Quantum Matter Is Measured and continues to the specialist probe page. The record should name detector output, calibration, background, resolution, matrix elements, response or correlation function, estimator, nuisance model, and claimed layer. A probe table is a router; it cannot replace the forward model.

A computational result routes through Computational Quantum Matter for method selection and evidence discipline, then to the exact method owner. The durable benchmark record belongs in Software, Notebooks, and Benchmarks. Freeze environment, input, pseudopotential or basis, random seed when relevant, convergence path, reference result, tolerance, and code version.

A source request starts with Condensed Matter References and ends at the exact paper, book edition, dataset, equation, table, or figure. A correction request belongs in Errata and Version History and the page that owns the affected claim. Preserve the superseded value and downstream consequences rather than silently replacing it.

Historical context and current frontier status have their own owners. This gateway is neither a timeline nor a news feed. Use Quantum Matter Frontiers and Open Problems when a lookup has become a dated scientific-status audit.

Check Provenance, Precision, and Version

A reusable source record should let a reader recover the exact artifact, not merely the work’s title. Record the DOI or stable identifier, edition or dataset version, table, figure, equation, panel, row, column, or file locator, access date for mutable resources, and correction or retraction state. For a derived value, preserve the raw inputs and processing script or notebook with its environment and hash.

Distinguish the following before combining uncertainties:

• standard uncertainty, associated with an estimated standard deviation;
• confidence or credible interval, which requires its construction and level;
• fit covariance, conditional on the likelihood and forward model;
• instrument resolution, which is a response width rather than automatically an uncertainty on a fitted center;
• systematic effect, which may be bounded, corrected, or modeled rather than sampled;
• specimen or batch variation, which is not repeat-readout noise; and
• model spread, which records alternative admissible representations and should not be hidden inside a precision-only error bar.

Apply significant figures after the uncertainty and provenance audit. Exact defined constants, measured constants, fitted parameters, rounded classroom values, and model inputs are different record classes. Printing every digit returned by software does not make a source more accurate.

Versioning is part of the physics when a value depends on a corrected geometry, new calibration, changed pseudopotential, revised structure, or reprocessed background. A good correction record says what changed, why it changed, which derived quantities move, and whether the licensed conclusion changes. If the convention, geometry, covariance, provenance, or validity regime cannot be reconstructed, stop and route to the canonical owner rather than repairing the record by assumption.

Worked Audit: A Signed Hall Record

This synthetic audit starts from signed terminal data and asks what can be reused after geometry and model conversion. The purpose is not to reteach Hall theory; Hall Measurements owns acquisition and reduction, while Hall Effect owns the physical models.

1. Lookup task and record class. Translate a signed Hall-bar resistance record into a bulk Hall coefficient, one-carrier Hall density, bulk resistivity, and Hall mobility while preserving signs and shared uncertainties. The linked record classes are formula, material-scale datum, and probe/observable map.

2. Canonical owner and reference handle. Hall Effect owns the transport meaning; Conventions for Quantum Matter owns tensor and sign choices; Hall Measurements owns the raw-to-resistance workflow. Reference units and formula handles supply constants and compact conversions.

3. System, state, geometry, and scope. The synthetic record contains three nominally identical nonmagnetic Hall bars from two wafers. Use right-handed $(x,y,z)$ axes, $I_x>0$, $B_z>0$, and a $+V_y$ lead at $+y$. The state is $T=20.0\pm0.1\ \text K$, reversible, and in weak-current response. The uniform electrically active, current-carrying thickness is $t=50.0\pm0.5\ \text{nm}$.

4. Symbols, units, basis, and sign conventions. Define $R_{yx}=V_y/I_x$, $\rho_{yx}=tR_{yx}$, and $\rho_{xx}=tR_\square$. With

   $$\boldsymbol\sigma = \begin{pmatrix} \sigma_{xx} & \sigma_{xy}\\ -\sigma_{xy} & \sigma_{xx} \end{pmatrix},$$

   matrix inversion gives $\rho_{yx}=\sigma_{xy}/(\sigma_{xx}^2+\sigma_{xy}^2)$ and $\rho_{xy}=-\rho_{yx}$.

5. Definition, formula, model, or probe map. The antisymmetrized $\pm2\ \text T$ trace is linear with $s=dR_{yx}/dB=-3.200\pm0.040\ \Omega/\text T$, and the sheet resistance is $R_\square=16.00\pm0.20\ \Omega$. The requested one-carrier map is

   $$R_H=ts, \qquad n_H=\frac{1}{e|R_H|}, \qquad \rho_{xx}=tR_\square, \qquad \mu_H=\frac{|R_H|}{\rho_{xx}} =\frac{|s|}{R_\square}.$$

6. Assumptions, validity regime, and order of limits. The odd-in-$B$ slope is the reversible weak-field branch after current reversal and matched field antisymmetrization. The film is treated as a uniform three-dimensional conductor over the voltage region. The density $n_H$ is licensed only in a one-carrier Drude interpretation. Multiband, parallel-channel, nonuniform, magnetic, or nonlinear transport stops that density claim. Bulk $R_H$ and $\rho_{xx}$ also stop if parallel surface or edge conduction, a dead layer, or thickness inhomogeneity is not excluded or modeled.

7. Source, provenance, version, and lifecycle. Archive raw traces for all four current and field sign states, both sweep directions, reciprocity checks, bar geometry, thickness method, wafer and specimen identifiers, field, current, voltage, and thermometer calibrations, processing script and version hash, publication or dataset locator, correction state, access date, and review date.

8. Transformation, normalization, and uncertainty. Every quoted primitive $\pm$ value is a one-standard uncertainty. In this synthetic record, $T$, $t$, $s$, and $R_\square$ are mutually independent, so $\operatorname{Cov}(s,R_\square)=0$; the derived $R_H$ and $\rho_{xx}$ share the same $t$ and are correlated. Geometry conversion gives

   $$R_H = (-1.600\pm0.026)\times10^{-7}\ \text{m}^3/\text C, \qquad \rho_{xx} = 80.0\pm1.3\ \mu\Omega\,\text{cm}.$$

   Under the one-carrier branch,

   $$n_H = (3.90\pm0.06)\times10^{25}\ \text{m}^{-3},$$

   and

   $$\begin{aligned} \mu_H &= 0.2000\pm0.0035\ \text{m}^2/(\text{V}\,\text s) \\ &= (2000\pm35)\ \text{cm}^2/(\text{V}\,\text s). \end{aligned}$$

   The common thickness cancels exactly from $\mu_H=|s|/R_\square$. Its uncertainty must not be propagated independently through both $R_H$ and $\rho_{xx}$. The quoted mobility uncertainty comes from the slope and sheet-resistance uncertainties under their declared covariance.

9. Cross-checks, collisions, and failure modes. The negative slope means negative $\rho_{yx}$ and negative $\sigma_{xy}$ in the declared tensor convention; $\rho_{xy}$ would be positive. Calling that sign electron-like is licensed only within the one-carrier branch, not as a microscopic carrier inventory in a multiband system. Check current independence, linearity, both sweep directions, reciprocity, multiple contact pairs, multiple bars, active thickness, and simultaneous longitudinal and transverse fits. Compare the one-carrier density with capacitance, stoichiometry, spectroscopy, or quantum oscillations. A sign reversal or curvature under an expanded field window exposes the one-carrier failure.

10. Licensed answer, citation, reuse conditions, and review. The signed $R_H$ and $\rho_{xx}$ may be reused with the full state, geometry, lead, field, tensor, covariance, and source record. The $n_H$ and $\mu_H$ values may be reused only as conditional one-carrier Hall quantities. Cite the exact dataset and processing version together with Hall Measurements and Hall Effect. Do not infer a microscopic carrier inventory, Fermi-surface topology, scattering mechanism, or universality. Review after a geometry correction, new channel model, expanded field range, or source revision.

This record demonstrates why “the Hall density” is not a self-contained table entry. The reusable result includes a signed tensor component and a model qualifier, while the common geometry uncertainty has a known covariance structure.

Worked Audit: A Neutron-Scattering Record

The second synthetic audit begins with coordinates, energy, linewidth, and spectral weight. It tests whether a compact record can survive reciprocal-space, unit, resolution, and multiplicity translations without being promoted into a lifetime or moment sum rule.

1. Lookup task and record class. Translate a reported neutron peak’s reciprocal-space coordinate, energy, linewidth, and integrated weight. The linked record classes are formula, material-scale datum, and probe/observable map. A lifetime and magnetic-moment sum rule are explicitly outside the initial request.

2. Canonical owner and reference handle. Reciprocal Lattice and Conventions for Quantum Matter own the momentum coordinates. Neutron Scattering owns the detector-to-structure-factor map, resolution, polarization, form factor, detailed balance, and evidence semantics. Reference handles supply constants, unit conversions, and source records.

3. System, state, geometry, and scope. Use a synthetic tetragonal sample with $a=3.900\pm0.004\ \text{\AA}$, $c=12.40\pm0.02\ \text{\AA}$, and two equivalent magnetic ions per formula unit. The specimen is measured at $20\ \text K$ in the declared orientation matrix and instrument configuration.

4. Symbols, units, basis, and sign conventions. Define $\mathbf b_i\cdot\mathbf a_j=2\pi\delta_{ij}$ and use neutron energy loss $E=E_i-E_f>0$. In this synthetic record the peak coordinate is exact by construction, not a fitted coordinate with an omitted uncertainty:

   $$\mathbf Q = (1.25,0,0)\ \text{rlu} = \boldsymbol\tau_{(1,0,0)} + \mathbf q_{(0.25,0,0)}.$$

   Ordinary frequency is $f$, angular frequency is $\omega$, and spectroscopic wavenumber is $\widetilde\nu$.

5. Definition, formula, model, or probe map. The peak energy is $E=18.00\pm0.12\ \text{meV}$. A one-dimensional energy cut gives observed Gaussian FWHM $w_{\rm obs}=1.10\pm0.10\ \text{meV}$; a calibration at the same kinematic point gives Gaussian resolution $w_{\rm res}=0.80\pm0.05\ \text{meV}$. Vanadium normalization yields an integrated weight $0.420\pm0.040\ \mu_{\rm B}^2$ per formula unit. Magnetic form-factor and polarization corrections have not been applied.

6. Assumptions, validity regime, and order of limits. Coordinate conversion assumes the stated tetragonal reciprocal basis and origin. Width deconvolution assumes independent Gaussian instrument and intrinsic profiles with negligible unresolved dispersion through the resolution volume. A lifetime requires a justified dynamical line shape. A moment sum rule additionally requires form factor, polarization, Bose or detailed- balance, momentum and energy window, multiplicity, and absolute normalization controls.

7. Source, provenance, version, and lifecycle. Archive specimen and state, orientation matrix, incident energy, chopper or analyzer configuration, detector mask and efficiency, monitor and vanadium normalization, background model, resolution file, raw event and processed data, integration region, code and environment, script version and hash, source locator, correction state, access date, and review date.

8. Transformation, normalization, and uncertainty. Every quoted primitive $\pm$ value is a one-standard uncertainty. Treat $a$, $c$, $E$, $w_{\rm obs}$, $w_{\rm res}$, and the integrated weight as mutually independent in this synthetic record, explicitly including $\operatorname{Cov}(w_{\rm obs},w_{\rm res})=0$. The reduced and total momentum magnitudes are

   $$|\mathbf q| = 0.4028\pm0.0004\ \text{\AA}^{-1}, \qquad |\mathbf Q| = 2.0138\pm0.0021\ \text{\AA}^{-1}.$$

   The energy converts to

   $$f = \frac{E}{h} = 4.352\pm0.029\ \text{THz},$$ $$\omega = \frac{E}{\hbar} = 27.35\pm0.18\ \text{ps}^{-1}, \qquad \widetilde\nu = 145.18\pm0.97\ \text{cm}^{-1}.$$

   Conditional Gaussian deconvolution gives

   $$w_{\rm int} = \sqrt{1.10^2-0.80^2} = 0.75\pm0.16\ \text{meV}.$$

   Dividing by the declared multiplicity gives a raw normalized weight $0.210\pm0.020\ \mu_{\rm B}^2$ per magnetic ion. This arithmetic does not supply the missing form-factor or polarization corrections. Because $|\mathbf q|$ and $|\mathbf Q|$ share only the same uncertain $a$ here, their derived uncertainties are perfectly positively correlated. Likewise $f$, $\omega$, and $\widetilde\nu$ are perfectly positively correlated because all three are deterministic transforms of the same $E$. Division by the exact multiplicity of two ions is deterministic, so the per-ion weight is perfectly correlated with the per-formula-unit weight and its standard uncertainty is exactly halved.

9. Cross-checks, collisions, and failure modes. Check the reciprocal basis, orientation matrix, reciprocal origin, $2\pi$ convention, $\mathbf Q$ versus reduced $\mathbf q$, energy-transfer sign, $f$ versus $\omega$, equivalent zones, detailed balance, resolution-model sensitivity, background, magnetic form factor, polarization factor, finite integration window, and per-formula-unit versus per-ion normalization. If $w_{\rm obs}\le w_{\rm res}$ within uncertainty, the deconvolved width is an upper limit or unresolved result rather than a negative physical width.

10. Licensed answer, citation, reuse conditions, and review. The converted coordinates, energy scales, conditional Gaussian intrinsic width, and raw per-ion normalized weight may be reused with the complete basis, instrument, resolution, normalization, multiplicity, uncertainty, and source record. Cite the exact dataset, cut, resolution file, and processing version together with Neutron Scattering. Do not infer a lifetime, fluctuating moment, spin sum rule, quasiparticle identity, or phase. Review after reprocessing, revised resolution, polarization or form-factor correction, extended integration, or source correction.

The kinematic conversions are durable. The lifetime and moment interpretations are not yet licensed because their missing assumptions belong to the probe forward model, not to the unit conversion.

Exit Checkpoint

You are ready to use this gateway when you can take an unfamiliar reference request and perform all of the following:

• classify the record before choosing a destination;
• name one live canonical owner and one compact reference handle or explicit coverage gap;
• freeze system, state, geometry, symbols, units, basis, signs, normalization, and order of limits;
• reconstruct the formula, model, or probe map without duplicating its canonical derivation;
• preserve DOI or stable identifier, exact locator, version, processing history, correction state, and review date;
• propagate shared covariance and keep precision, resolution, specimen variation, systematic effects, and model spread distinct;
• run dimensional, sign, limit, collision, and independent-record checks; and
• state the licensed answer, citation, reuse conditions, stopping point, and review trigger.

A completed form is not sufficient if its fields contain vague labels such as “standard convention,” “typical sample,” “the usual normalization,” or “from the literature.” The record must be specific enough that a reader can detect a changed basis, sample, processing version, or claim.

Stop when a convention, provenance chain, active geometry, covariance, normalization, or validity condition cannot be reconstructed. The correct next step is a source or owner query, not an undocumented assumption.

Canonical Boundaries and Coverage Gaps

This gateway owns task classification, the ten-field transferable record, domain-specific convention and provenance audits, nonlinear routing, and stopping rules. Its boundaries are deliberate.

• The Reference owns durable sitewide glossary and symbol handles, compact formulas, Hamiltonian and model cards, sourced tables, bibliographies, benchmarks, data and figure indices, and errata and version records. This gateway routes into those collections rather than building a second local library.
• Quantum Matter concept, model, phase, material, method, and probe pages own definitions in context, derivations, validity arguments, calculations, forward models, interpretation, and scientific claims.
• Quantum Matter Map owns branch location. Conventions for Quantum Matter owns the complete lattice, Fourier, Berry, electromagnetic, response, normalization, and sign system. This page applies compact subsets without becoming a competing convention owner.
• Choosing a Model owns problem-to-minimal-model selection. Many-Body Reference and its cross-link indexes retain many-body formulas, Hamiltonians, quasiparticle taxonomy, and global model routing.
• How Quantum Matter Is Measured and each specialist probe page own detector records, calibration, resolution, matrix elements, forward models, and inference. A probe lookup cannot establish an experimental conclusion.
• Computational Quantum Matter and the computational volumes own algorithms, workflow, convergence, environments, and reproducibility. Reference benchmark pages own durable benchmark records.
• Historical and frontier-status owners retain timelines, changing status, corrections, and living reviews. This gateway is neither a news feed nor a material leaderboard.

Two useful sitewide gaps remain: a sourced and versioned material-scale lookup and a convention-aware comparison of invariants. If built, both belong in the Reference with provenance, units, assumptions, canonical links, and review metadata. They do not justify restoring local glossary, symbol, Hamiltonian, lattice, Brillouin-zone, band-notation, crystallography, material-scale, response-formula, invariant, quasiparticle, model, experiment, probe, benchmark, bibliography, timeline, or errata leaves under this chapter.

The consolidation is pedagogical as well as architectural. A single gateway forces every lookup to name its owner and validity boundary; a parallel set of thin local tables would drift away from the formulas, models, probes, and sources they summarize.

Exercises

1. Classify and route ten lookup requests

Assign each request to one primary record class, one live canonical owner, and one compact Reference handle: a definition of quasiparticle; the meaning of $\rho$ in a transport equation; the density-of-states formula; the Hubbard Hamiltonian; the purpose of the SSH model; a material’s measured band gap; the observable behind an ARPES intensity map; an SSH numerical benchmark; a source for condensed-matter field theory; and a corrected lattice constant.

Solution

1. Quasiparticle is a term. Quasiparticles owns the physical meaning; the Glossary is the compact handle.
2. $\rho$ in transport is a symbol. Conventions for Quantum Matter owns the tensor convention; the Symbol Index handles symbol lookup. The record must say whether $\rho$ means $\rho_{xy}$ or $\rho_{yx}$.
3. Density of states is a formula. Density of States owns interpretation and derivation; the formula card is the compact handle.
4. Hubbard Hamiltonian is a Hamiltonian/operator record. Hubbard Model owns the model; the Hubbard Hamiltonian card is the handle.
5. SSH purpose is a model request. Tight-Binding Models owns the localized-orbital and hopping construction; the SSH Model card is the compact handle. The record must also route a topological claim to its topology owner and state the symmetry and boundary assumptions.
6. Measured band gap is a material-scale datum. Band Theory Overview owns the stable gap semantics; the exact material and probe pages own the reported value. A sourced sitewide material-scale handle is a current coverage gap, so cite the experiment or calculation rather than inventing a local table entry.
7. ARPES intensity is a probe/observable map. Angle-Resolved Photoemission Spectroscopy owns the intensity-to-spectral-function relation; the Reference has no replacement for its forward model.
8. SSH benchmark is a computational benchmark. Computational Quantum Matter owns the material-facing workflow and evidence standard; Software, Notebooks, and Benchmarks is the collection handle. A substantive, versioned SSH benchmark record is still a coverage gap, so do not link its planned scaffold.
9. Condensed-matter field-theory source is bibliography. Condensed Matter References supplies the annotated handle; the exact book edition or paper remains the source.
10. Corrected lattice constant is a correction/version record attached to a material-scale datum. Crystals and Lattices owns the definition and the material page owns interpretation, while Errata and Version History owns the correction record.

Several answers require a second class, but the primary classification tells the reader where to begin. None licenses a scientific claim merely by locating the compact handle.

2. Translate Hall tensor indices without losing the sign

In the convention $\boldsymbol\sigma=\bigl(\begin{smallmatrix}\sigma_{xx}&\sigma_{xy}\\ -\sigma_{xy}&\sigma_{xx}\end{smallmatrix}\bigr)$, a record gives $\rho_{xx}=100\ \mu\Omega\,\text{cm}$ and $\rho_{yx}=-5.0\ \mu\Omega\,\text{cm}$. Find $\rho_{xy}$, $\sigma_{xx}$, and $\sigma_{xy}$. Which transverse conductivity has the same sign as the supplied transverse resistivity?

Solution

Antisymmetry gives $\rho_{xy}=-\rho_{yx}=+5.0\ \mu\Omega\,\text{cm}$. Converting to SI,

$$\rho_{xx}=1.00\times10^{-6}\ \Omega\,\text m, \qquad \rho_{yx}=-5.0\times10^{-8}\ \Omega\,\text m.$$

For the declared tensor order,

$$\sigma_{xx} = \frac{\rho_{xx}} {\rho_{xx}^2+\rho_{yx}^2}, \qquad \sigma_{xy} = \frac{\rho_{yx}} {\rho_{xx}^2+\rho_{yx}^2}.$$

Therefore

$$\sigma_{xx} = 9.98\times10^5\ \text{S/m}, \qquad \sigma_{xy} = -4.99\times10^4\ \text{S/m}.$$

The components $\rho_{yx}$ and $\sigma_{xy}$ have the same sign. The component $\rho_{xy}$ has the opposite sign. A source that calls its transverse entry “$\rho_{xy}$” must be translated by its index definition, not by the plot label alone.

3. Recompute the signed Hall audit

Use a uniform electrically active, current-carrying thickness $t=50.0\pm0.5\ \text{nm}$, $s=-3.200\pm0.040\ \Omega/\text T$, and $R_\square=16.00\pm0.20\ \Omega$. Treat the three direct measurements as independent one-standard uncertainties. Compute $R_H$, $\rho_{xx}$, the conditional one-carrier $n_H$, and $\mu_H$. Identify the induced covariance between $R_H$ and $\rho_{xx}$, explain why the thickness uncertainty cancels from the mobility, and state the stopping rule.

Solution

The geometry conversions give

$$R_H=ts=-1.600\times10^{-7}\ \text{m}^3/\text C$$

and

$$\rho_{xx}=tR_\square =8.00\times10^{-7}\ \Omega\,\text m =80.0\ \mu\Omega\,\text{cm}.$$

Independent propagation gives relative standard uncertainties

$$\frac{u(R_H)}{|R_H|} = \sqrt{ \left(\frac{0.5}{50.0}\right)^2 + \left(\frac{0.040}{3.200}\right)^2} =0.0160$$

and the same value for $\rho_{xx}$ because $0.20/16.00=0.040/3.200$. Thus

$$R_H=(-1.600\pm0.026)\times10^{-7}\ \text{m}^3/\text C, \qquad \rho_{xx}=80.0\pm1.3\ \mu\Omega\,\text{cm}.$$

Using the exact elementary charge in the one-carrier branch,

$$n_H = \frac{1}{e|R_H|} = (3.90\pm0.06)\times10^{25}\ \text{m}^{-3}.$$

For mobility, substitute the original observables before propagating:

$$\mu_H = \frac{|ts|}{tR_\square} = \frac{|s|}{R_\square} = 0.2000\pm0.0035\ \text{m}^2/(\text{V}\,\text s) = (2000\pm35)\ \text{cm}^2/(\text{V}\,\text s).$$

The derived $R_H$ and $\rho_{xx}$ are correlated because they share the same measured $t$. In their ratio, that same thickness occurs in numerator and denominator and cancels; propagating two independent thickness errors would invent uncertainty. Stop the bulk $R_H$ and $\rho_{xx}$ conversion if parallel surface or edge conduction, a dead layer, or thickness inhomogeneity is not excluded or modeled. Stop the carrier-density inference if the Hall trace is nonlinear, multiband or parallel-channel transport is credible, the active thickness is unknown, or an external density constraint disagrees. The signed $R_H$ remains a useful conditional response record when its geometry is valid, even when $n_H$ does not.

4. Translate reciprocal coordinates and energy scales

For the tetragonal neutron record, use $a=3.900\pm0.004\ \text{\AA}$, $\mathbf Q=(1.25,0,0)$ rlu exactly by construction, $\boldsymbol\tau=(1,0,0)$, and $E=18.00\pm0.12\ \text{meV}$. Compute $|\mathbf q|$, $|\mathbf Q|$, $f$, $\omega$, and $\widetilde\nu$. Treat the lattice and energy uncertainties as independent one-standard uncertainties, and identify the covariance induced within each set of quantities derived from the same primitive.

Solution

With $\mathbf b_1=(2\pi/a)\hat{\mathbf x}$,

$$|\mathbf q| = 0.25\frac{2\pi}{a} = 0.4028\ \text{\AA}^{-1}, \qquad |\mathbf Q| = 1.25\frac{2\pi}{a} = 2.0138\ \text{\AA}^{-1}.$$

Both magnitudes have relative uncertainty $u(a)/a=0.004/3.900$, giving

$$|\mathbf q| = 0.4028\pm0.0004\ \text{\AA}^{-1}, \qquad |\mathbf Q| = 2.0138\pm0.0021\ \text{\AA}^{-1}.$$

Using CODATA constants,

$$f=\frac{E}{h}=4.352\pm0.029\ \text{THz},$$ $$\omega = \frac{E}{\hbar} = 27.35\pm0.18\ \text{ps}^{-1}, \qquad \widetilde\nu = \frac{E}{hc} = 145.18\pm0.97\ \text{cm}^{-1}.$$

The $c$ lattice constant does not enter this particular $(H,0,0)$ magnitude. The tuple in rlu and the inverse-length magnitude remain distinct records. Because both magnitudes depend on the same $a$, their derived uncertainties are perfectly positively correlated. The three energy conversions depend on the same $E$ and are likewise perfectly positively correlated; $f$ and $\omega$ remain distinct quantities despite that shared uncertainty.

5. Deconvolve a width and audit spectral weight

Use $w_{\rm obs}=1.10\pm0.10\ \text{meV}$, $w_{\rm res}=0.80\pm0.05\ \text{meV}$, and integrated weight $0.420\pm0.040\ \mu_{\rm B}^2$ per formula unit with two equivalent magnetic ions. Treat all quoted $\pm$ values as one-standard uncertainties, set $\operatorname{Cov}(w_{\rm obs},w_{\rm res})=0$, and assume independent Gaussian widths. Compute $w_{\rm int}$ and its first-order uncertainty, and convert the weight per ion. Why do neither result license a lifetime or moment sum rule?

Solution

Conditional Gaussian deconvolution gives

$$w_{\rm int} = \sqrt{w_{\rm obs}^2-w_{\rm res}^2} = \sqrt{1.10^2-0.80^2} = 0.755\ \text{meV}.$$

For independent inputs,

$$u^2(w_{\rm int}) = \left( \frac{w_{\rm obs}}{w_{\rm int}}u_{\rm obs} \right)^2 + \left( \frac{w_{\rm res}}{w_{\rm int}}u_{\rm res} \right)^2,$$

which gives $u(w_{\rm int})=0.16\ \text{meV}$. Report $w_{\rm int}=0.75\pm0.16\ \text{meV}$ rather than unsupported extra digits. The declared multiplicity gives

$$I_{\rm ion} = \frac{0.420\pm0.040}{2} = 0.210\pm0.020\ \mu_{\rm B}^2/\text{ion}.$$

The ion multiplicity is exact in this record, so this division is deterministic: the per-ion and per-formula-unit weights are perfectly correlated, and the standard uncertainty is exactly halved.

A Gaussian energy width is not the Lorentzian width produced by a single exponential decay, and dispersion through the multidimensional resolution volume may contribute. A lifetime therefore needs a dynamical line-shape and resolution model. The weight lacks magnetic form-factor, polarization, detailed-balance, integration-window, and sum-rule normalization controls, so it remains a raw normalized per-ion weight rather than a fluctuating moment.

6. Keep volume and spin normalizations separate

A source quotes $D_{\rm cell}=1.80$ states/(eV cell) for a cell of volume $\Omega_c=160\ \text{\AA}^3$. Convert to states/(eV $\text m^3$). If the source explicitly says the value includes two degenerate spin sectors, also give the per-spin value per cell. Explain why these are independent transformations.

Solution

Because $1\ \text{\AA}^3=10^{-30}\ \text m^3$,

$$\Omega_c = 160\times10^{-30}\ \text m^3 = 1.60\times10^{-28}\ \text m^3.$$

Therefore

$$D_{\rm vol} = \frac{D_{\rm cell}}{\Omega_c} = 1.125\times10^{28} \ \frac{\text{states}}{\text{eV}\,\text m^3}.$$

Only because the source explicitly includes two degenerate spin sectors may one write

$$D_{\rm cell,spin} = \frac{1.80}{2} = 0.900 \ \frac{\text{states}}{\text{eV cell spin}}.$$

The first operation changes spatial normalization; the second changes an internal degeneracy convention. Applying one does not imply the other. Valley, orbital, layer, and formula-unit multiplicities would require their own source statements and conversions.

7. Repair a stale copied data record

A slide states “the gap is $42.137\ \text{meV}$” and cites only an article’s title. The paper’s supplementary table was corrected after publication, and a repository contains both the original and revised processed files. Write the minimum repaired record before the value is reused.

Solution

The repaired record should contain:

• the material, specimen or model, prepared state, geometry, and quantity definition;
• units, normalization, sign, basis, fit or extraction method, and validity interval;
• authors, full source metadata, DOI, and the exact supplementary table, row, column, figure, or equation locator;
• the repository identifier, corrected dataset version, file name and hash, correction date, and access date;
• the raw inputs and processing script or notebook version used to reproduce the corrected value;
• the uncertainty type and budget, significant figures justified by that uncertainty, and any model spread;
• the superseded value and the reason for correction, rather than silent deletion;
• the live canonical owner, compact reference handle or coverage gap, licensed reuse, downstream consequences, and next review trigger.

The number $42.137\ \text{meV}$ should not survive merely because it has many digits. Use the corrected value and precision supported by the source’s uncertainty. If the corrected artifact or state cannot be identified, the licensed answer is that the slide value is not reusable.

8. Complete a ten-field ARPES velocity record

A synthetic ARPES dataset reports $dE/dk=2.40\pm0.08\ \text{eV}\,\text{\AA}$ along $k_x$. The measurement uses a cleaved (001) surface of layered metal A, $T=15\ \text K$, $B=0$, photon energy $70\ \text{eV}$, linear polarization, $12\ \text{meV}$ energy resolution, and $0.15^\circ$ angular resolution. The fit spans $0.06$–$0.12\ \text{\AA}^{-1}$ and binding energies $25$–$120\ \text{meV}$ below a gold-referenced Fermi level. Fill all ten fields, compute the group velocity, and state the stopping rule.

Solution

1. Lookup task and record class. Convert a fitted ARPES dispersion slope into a reusable group-velocity datum. The primary class is material-scale datum, with formula and probe/observable-map records attached.

2. Canonical owner and reference handle. Angle-Resolved Photoemission Spectroscopy owns the intensity, spectral-function, matrix-element, resolution, surface, and fit semantics. Conventions for Quantum Matter owns energy, momentum, and sign choices. The Reference supplies units and constants; no compact material-velocity table is presently the canonical handle.

3. System, state, geometry, and scope. Record layered metal A, specimen and batch identifier, freshly cleaved (001) surface, surface age and vacuum, $T=15\ \text K$, $B=0$, composition and filling, photon energy $70\ \text{eV}$, linear polarization, analyzer geometry, and the stated $k_x$ cut. The velocity applies only to the fitted momentum and binding- energy window.

4. Symbols, units, basis, and sign conventions. Define binding energy, gold-referenced $E_F$, positive $k_x$, surface Brillouin-zone basis, whether $E(k)$ increases toward or away from $E_F$, and $v_x=\hbar^{-1}\partial E/\partial k_x$. The quoted slope is $+2.40\pm0.08\ \text{eV}\,\text{\AA}$ in that basis.

5. Definition, formula, model, or probe map. The raw count map is divided by detector normalization and mapped through energy, angle, work-function, and momentum calibration. A declared MDC or EDC fitting procedure extracts the slope. The compact conversion is

   $$v_x = \frac{1}{\hbar} \frac{dE}{dk_x}.$$

6. Assumptions, validity regime, and order of limits. The fitted feature is treated as a differentiable quasiparticle ridge over $0.06$–$0.12\ \text{\AA}^{-1}$ and $25$–$120\ \text{meV}$ binding energy. The result is surface and photon-matrix-element selected. It is a finite- temperature, finite-resolution, renormalized dispersion slope, not automatically a bare-band or zero-energy Fermi velocity.

7. Source, provenance, version, and lifecycle. Preserve DOI and exact figure or dataset locator, specimen and cleave records, raw counts, calibration files, gold-reference trace, analyzer settings, processing and fit code with version and hash, fit covariance, correction state, access date, and review date.

8. Transformation, normalization, and uncertainty. Using $1\ \text{eV}\,\text{\AA}/\hbar =1.51927\times10^5\ \text{m/s}$ gives

   $$v_x = (3.65\pm0.12)\times10^5\ \text{m/s}.$$

   The conversion constant is negligible compared with the stated slope uncertainty. Calibration covariance, fit-window variation, resolution, and model dependence must be reported separately if not included in $0.08\ \text{eV}\,\text{\AA}$.

9. Cross-checks, collisions, and failure modes. Compare MDC and EDC fits, vary the fit window, photon energy and polarization, inspect energy and momentum residuals, test surface aging, and compare symmetry-related cuts. Check against quantum oscillations or another bulk-sensitive record when a bulk velocity is requested. Matrix-element extinction, unresolved bands, a kink, self-energy curvature, surface reconstruction, or an energy-zero shift can invalidate the single-slope reuse.

10. Licensed answer, citation, reuse conditions, and review. License the positive renormalized group velocity $(3.65\pm0.12)\times10^5\ \text{m/s}$ for the declared surface, cut, photon condition, temperature, and fit window. Cite the exact dataset, calibration, fit version, and ARPES owner. Do not call it a universal material velocity, bare-band velocity, transport velocity, or proof of a coherent bulk quasiparticle outside the fitted regime. Review after a calibration, fit, surface assignment, dataset, or source correction.

References

• M. I. Aroyo, ed., International Tables for Crystallography, Volume A: Space-Group Symmetry, 6th print edition / 2nd online edition, International Union of Crystallography (2016), doi:10.1107/97809553602060000114.
• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston (1976).
• Bureau International des Poids et Mesures, The International System of Units (SI Brochure), 9th ed. (2019), version 4.01 (June 2026), doi:10.59161/AUEZ1291.
• H. B. G. Casimir, “On Onsager’s Principle of Microscopic Reversibility,” Reviews of Modern Physics 17, 343–350 (1945), doi:10.1103/RevModPhys.17.343.
• E. H. Hall, “On a New Action of the Magnet on Electric Currents,” American Journal of Mathematics 2, 287–292 (1879), doi:10.2307/2369245.
• Joint Committee for Guides in Metrology, Evaluation of Measurement Data—Guide to the Expression of Uncertainty in Measurement, JCGM 100:2008, doi:10.59161/JCGM100-2008E.
• S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter, Clarendon Press (1984).
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• G. L. Squires, Introduction to the Theory of Thermal Neutron Scattering, 3rd ed., Cambridge University Press (2012), doi:10.1017/CBO9781139107808.
• E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Taylor, “CODATA Recommended Values of the Fundamental Physical Constants: 2018,” Reviews of Modern Physics 93, 025010 (2021), doi:10.1103/RevModPhys.93.025010.
• M. D. Wilkinson et al., “The FAIR Guiding Principles for Scientific Data Management and Stewardship,” Scientific Data 3, 160018 (2016), doi:10.1038/sdata.2016.18.