Conventions for Quantum Matter

This page defines the default notation contract for lattices, Bloch states, band Hamiltonians, Berry geometry, electromagnetic coupling, response functions, and material observables. It supplements the sitewide Conventions Overview, Fourier Transform Conventions, and Many-Body Symbols and Conventions.

For magnetic claims, Magnetism and Spin Systems supplies the moment, normalization, exchange, ordering-wavevector, field-history, and probe ledger; the convention definitions themselves remain canonical here and in their specialist owners.

Choosing a Model for Quantum Matter owns the preceding choice of retained description. This page begins after that choice and fixes the signs, bases, normalizations, units, and limits needed to compare calculations. Superfluidity and Superconductivity owns the chapter-level choice among pairing, stiffness, electromagnetic response, defects, weak links, and proximity; this page supplies the charge, gauge, Fourier, and response conventions used along those routes. It does not choose the local crystalline learning branch; use the Lattices, Reciprocal Space, and Bloch Electrons gateway for that dependency audit, then return here before comparing formulas.

When the calculation produces band data, pair this notation contract with the Band Theory and Electronic Structure gateway. Its band ledger records the description and parameter provenance, basis and symmetry content, energy reference, filling and state, direct or indirect gap claim, temperature and broadening, target observable, and validity window. This page owns the conventions used to state those fields; the gateway owns the downstream interpretation route.

When the calculation instead produces a dynamical matrix, mode spectrum, dielectric pole, or dressed material excitation, pair this notation contract with the Lattice Vibrations and Collective Modes gateway. This page fixes cell phases, angular-frequency and energy variables, normalizations, response signs, and orders of limits; the gateway owns the status-aware material route.

A specialized article may use another standard convention when that choice clarifies the subject. It must declare the override near the first affected formula and translate back when comparing results. A silent change of sign, normalization, basis embedding, or tensor order is an error.

Minimum Declaration Block

A reusable quantum-matter calculation must make the following information recoverable:

Item	Required data
lattice	primitive vectors, basis positions, dimension, orientation
finite geometry	number of cells, boundary conditions, sample shape
reciprocal space	reciprocal vectors, Brillouin-zone representative, momentum grid
basis	orbital, sublattice, layer, spin, band, and gauge conventions
Fourier transform	phases, normalization, and whether basis positions enter
Hamiltonian	hopping orientation, interaction factors, charge signs, units
state	filling, chemical potential, temperature, preparation
response	perturbation sign, current or density convention, Fourier variable
geometry	field, current, polarization, surface, and tensor-axis directions
limits	thermodynamic, clean, dc, uniform, zero-temperature, and their order

The compact rule is:

A formula is defined only together with its basis, normalization, orientation, charge, units, and limits.

Typography and Index Register

Symbol	Default meaning
$\mathbf r$	continuum position
$\mathbf R$	Bravais-lattice vector
$\boldsymbol\tau_\alpha$	position of basis object $\alpha$ within the chosen cell
$\mathbf r_{\mathbf R\alpha}$	embedded position $\mathbf R+\boldsymbol\tau_\alpha$
$\mathbf a_i$	primitive real-space vector
$\mathbf b_i$	primitive reciprocal-lattice vector
$\mathbf k$	crystal wavevector or Bloch label
$\mathbf q$	transferred wavevector
$\mathbf G$	reciprocal-lattice vector
$i,j$	sites or Cartesian components, declared locally
$\alpha,\beta$	orbital, sublattice, layer, or basis indices
$n,m$	band, eigenstate, integer, or Matsubara indices
$\sigma,\sigma'$	physical spin unless explicitly called pseudospin
$N_c$	number of primitive cells
$N_{\mathrm{orb}}$	number of retained orbitals per primitive cell
$\Omega_c$	primitive-cell volume or area
$V$	physical sample volume or area in the modeled dimension

Repeated Cartesian indices are not summed unless an article declares an Einstein convention. Lattice, orbital, band, and spin sums are normally explicit.

Boldface denotes geometric vectors. It does not denote a generic column vector in orbital space; write $u_{n\alpha}(\mathbf k)$ or declare a bold orbital-space vector locally.

Real-Space Lattice

Primitive vectors

In $d$ dimensions, a Bravais vector is

$$\mathbf R = \sum_{i=1}^{d} n_i\mathbf a_i, \qquad n_i\in\mathbb Z.$$

The primitive vectors are ordered. In three dimensions the default orientation is right handed:

$$\Omega_c = \mathbf a_1\cdot \left( \mathbf a_2\times\mathbf a_3 \right) > 0.$$

In two dimensions embedded in the $xy$ plane,

$$\Omega_c = \left( \mathbf a_1\times\mathbf a_2 \right)\cdot\hat{\mathbf z} > 0.$$

If a source uses a left-handed ordering, reverse the orientation before comparing Berry-curvature or Hall signs.

Basis positions

A crystal site or localized orbital is embedded at

$$\mathbf r_{\mathbf R\alpha} = \mathbf R + \boldsymbol\tau_\alpha.$$

$\boldsymbol\tau_\alpha$ is defined modulo a Bravais vector. Moving a basis representative into a neighboring cell changes Fourier phases and matrix representations but not physical positions.

An orbital is not always centered exactly at an atomic coordinate. Wannier centers, bond orbitals, and effective degrees of freedom may have different embeddings. Record the position used by the position operator and by electromagnetic phases.

Cell count and physical size

For a periodic finite lattice with $N_i$ cells along $\mathbf a_i$,

$$N_c = \prod_{i=1}^{d}N_i.$$

When the modeled sample is exactly tiled by primitive cells,

$$V = N_c\Omega_c.$$

Use $N_c$ for the number of primitive cells rather than the ambiguous $N_s$. The number of localized modes is $N_cN_{\mathrm{orb}}$ before spin or other internal multiplicities.

Reciprocal Lattice and Brillouin Zone

Reciprocal vectors

The reciprocal basis is defined by

$$\mathbf a_i\cdot\mathbf b_j = 2\pi\delta_{ij}.$$

A reciprocal vector is

$$\mathbf G = \sum_{i=1}^{d} m_i\mathbf b_i, \qquad m_i\in\mathbb Z.$$

With the right-handed three-dimensional convention,

$$\begin{aligned} \mathbf b_1 &= 2\pi \frac{\mathbf a_2\times\mathbf a_3}{\Omega_c}, \\ \mathbf b_2 &= 2\pi \frac{\mathbf a_3\times\mathbf a_1}{\Omega_c}, \\ \mathbf b_3 &= 2\pi \frac{\mathbf a_1\times\mathbf a_2}{\Omega_c}. \end{aligned}$$

The reciprocal-cell volume is

$$\Omega_{\mathrm{BZ}} = \frac{(2\pi)^d}{\Omega_c}.$$

Here $\Omega_{\mathrm{BZ}}$ means the measure of any primitive reciprocal cell, including the first Brillouin zone.

First Brillouin zone

The default first Brillouin zone is the Wigner–Seitz cell of the reciprocal lattice around $\mathbf k=\mathbf0$. Any primitive reciprocal cell contains one representative of each equivalence class

$$\mathbf k \sim \mathbf k+\mathbf G.$$

Boundary points have multiple equivalent representatives. A numerical mesh must assign each boundary state once. Integrals are insensitive to that measure-zero choice, while discrete sums are not.

Finite periodic grid

For periodic boundary conditions over $N_i\mathbf a_i$, use

$$\mathbf k_{\mathbf m} = \sum_{i=1}^{d} \frac{m_i}{N_i} \mathbf b_i, \qquad m_i=0,\ldots,N_i-1.$$

The listed grid is a parallelepiped representative. Points may be folded into the first Brillouin zone for plotting.

With a boundary twist $\theta_i$,

$$\mathbf k_{\mathbf m}(\boldsymbol\theta) = \sum_{i=1}^{d} \frac{m_i+\theta_i/(2\pi)}{N_i} \mathbf b_i.$$

State whether $\boldsymbol\theta$ is a boundary phase, a physical flux, or an auxiliary parameter used for many-body geometry.

Crystal Momentum Is a Wavevector Label

$\mathbf k$ has units of inverse length. In a continuum free-particle problem,

$$\mathbf p = \hbar\mathbf k$$

is physical momentum. In a crystal, $\hbar\mathbf k$ is commonly called crystal momentum, but it is defined modulo $\hbar\mathbf G$ and need not equal mechanical momentum or $m\mathbf v$.

The band velocity of an isolated differentiable band is

$$\mathbf v_n(\mathbf k) = \frac{1}{\hbar} \boldsymbol\nabla_{\mathbf k} \varepsilon_n(\mathbf k).$$

The effective inverse-mass tensor is

$$\left[ \left( m_n^\ast \right)^{-1} \right]_{ij} = \frac{1}{\hbar^2} \frac{\partial^2\varepsilon_n} {\partial k_i\partial k_j}.$$

Neither equation defines a global particle mass. Both describe local band response under the assumptions of semiclassical wave-packet dynamics.

Lattice Fourier Transform

Default cell convention

For an annihilation operator $c_{\mathbf R\alpha\sigma}$, the default lattice transform is

$$\begin{aligned} c_{\mathbf k\alpha\sigma} &= \frac{1}{\sqrt{N_c}} \sum_{\mathbf R} e^{-i\mathbf k\cdot\mathbf R} c_{\mathbf R\alpha\sigma}, \\ c_{\mathbf R\alpha\sigma} &= \frac{1}{\sqrt{N_c}} \sum_{\mathbf k} e^{i\mathbf k\cdot\mathbf R} c_{\mathbf k\alpha\sigma}. \end{aligned}$$

The basis position $\boldsymbol\tau_\alpha$ is not included in the phase. This is called the cell convention.

Discrete orthogonality is

$$\begin{aligned} \frac{1}{N_c} \sum_{\mathbf R} e^{i(\mathbf k-\mathbf k')\cdot\mathbf R} &= \delta_{\mathbf k,\mathbf k'}, \\ \frac{1}{N_c} \sum_{\mathbf k} e^{i\mathbf k\cdot(\mathbf R-\mathbf R')} &= \delta_{\mathbf R,\mathbf R'}. \end{aligned}$$

The symmetric normalization preserves canonical anticommutators or commutators without extra cell factors.

Orbital-position convention

Some sources define

$$\bar c_{\mathbf k\alpha\sigma} = \frac{1}{\sqrt{N_c}} \sum_{\mathbf R} e^{-i\mathbf k\cdot (\mathbf R+\boldsymbol\tau_\alpha)} c_{\mathbf R\alpha\sigma}.$$

The two conventions are related by

$$\bar c_{\mathbf k\alpha\sigma} = e^{-i\mathbf k\cdot\boldsymbol\tau_\alpha} c_{\mathbf k\alpha\sigma}.$$

Define the diagonal embedding matrix

$$D_{\alpha\beta}(\mathbf k) = \delta_{\alpha\beta} e^{-i\mathbf k\cdot\boldsymbol\tau_\alpha}.$$

Then

$$\bar{\mathbf c}_{\mathbf k} = D(\mathbf k)\mathbf c_{\mathbf k}.$$

The two bases give identical observables when Hamiltonians, eigenvectors, position operators, and derivatives are transformed together.

Sum-to-integral rule

In the thermodynamic limit,

$$\frac{1}{N_c} \sum_{\mathbf k} \longrightarrow \frac{\Omega_c}{(2\pi)^d} \int_{\mathrm{BZ}} d^dk.$$

Equivalently,

$$\sum_{\mathbf k} \longrightarrow \frac{V}{(2\pi)^d} \int_{\mathrm{BZ}} d^dk.$$

The second form assumes $V=N_c\Omega_c$. A formula per cell uses the first normalization; a formula per physical volume uses the second.

Hopping and Bloch Hamiltonian

Hopping orientation

Define

$$H_0 = \sum_{\mathbf R,\boldsymbol\Delta} \sum_{\alpha,\beta} t_{\alpha\beta}(\boldsymbol\Delta) c_{\mathbf R\alpha}^\dagger c_{\mathbf R+\boldsymbol\Delta,\beta}.$$

$t_{\alpha\beta}(\boldsymbol\Delta)$ is the amplitude multiplying a hop from $(\mathbf R+\boldsymbol\Delta,\beta)$ to $(\mathbf R,\alpha)$ in this operator order. Hermiticity requires

$$t_{\alpha\beta}(\boldsymbol\Delta) = t_{\beta\alpha}^\ast(-\boldsymbol\Delta).$$

With the default cell transform,

$$H_0 = \sum_{\mathbf k} \mathbf c_{\mathbf k}^\dagger h(\mathbf k) \mathbf c_{\mathbf k},$$

where

$$h_{\alpha\beta}(\mathbf k) = \sum_{\boldsymbol\Delta} t_{\alpha\beta}(\boldsymbol\Delta) e^{i\mathbf k\cdot\boldsymbol\Delta}.$$

For a one-dimensional chain written as

$$H = -t \sum_j \left( c_j^\dagger c_{j+1} +c_{j+1}^\dagger c_j \right),$$

the dispersion is

$$\varepsilon(k) = -2t\cos(ka).$$

The minus sign belongs to the Hamiltonian definition. Some sources absorb it into a signed hopping parameter.

Periodicity and embedding

In the cell convention,

$$h(\mathbf k+\mathbf G) = h(\mathbf k).$$

In the orbital-position convention,

$$\bar h(\mathbf k) = D(\mathbf k) h(\mathbf k) D^\dagger(\mathbf k).$$

Since

$$D(\mathbf k+\mathbf G) = D_{\mathbf G}D(\mathbf k), \qquad \left(D_{\mathbf G}\right)_{\alpha\beta} = \delta_{\alpha\beta} e^{-i\mathbf G\cdot\boldsymbol\tau_\alpha},$$

the embedded Hamiltonian is generally quasi-periodic:

$$\bar h(\mathbf k+\mathbf G) = D_{\mathbf G} \bar h(\mathbf k) D_{\mathbf G}^\dagger.$$

Its eigenvalues remain periodic. Confusing matrix periodicity with spectral periodicity is a common source of apparent disagreement.

Bloch States and Normalization

Translation convention

The active translation operator acts as

$$\langle\mathbf r| T_{\mathbf R} |\psi\rangle = \psi(\mathbf r-\mathbf R).$$

A Bloch eigenstate satisfies

$$T_{\mathbf R} |\psi_{n\mathbf k}\rangle = e^{-i\mathbf k\cdot\mathbf R} |\psi_{n\mathbf k}\rangle.$$

Equivalently,

$$\psi_{n\mathbf k}(\mathbf r+\mathbf R) = e^{i\mathbf k\cdot\mathbf R} \psi_{n\mathbf k}(\mathbf r).$$

Write

$$\psi_{n\mathbf k}(\mathbf r) = e^{i\mathbf k\cdot\mathbf r} u_{n\mathbf k}(\mathbf r),$$

where

$$u_{n\mathbf k}(\mathbf r+\mathbf R) = u_{n\mathbf k}(\mathbf r).$$

This structural form suppresses any overall finite-volume normalization. Bloch’s Theorem gives the proof and the distinction between finite-volume eigenvectors and infinite-crystal generalized eigenfunctions.

Finite-crystal normalization

For a finite periodic crystal, use

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

For cell-periodic functions, define

$$\langle u|v\rangle_c = \frac{1}{\Omega_c} \int_{\mathrm{cell}} d^dr\, u^\ast(\mathbf r)v(\mathbf r),$$

and normalize

$$\langle u_{n\mathbf k} | u_{m\mathbf k} \rangle_c = \delta_{nm}.$$

With this cell-average convention and $V=N_c\Omega_c$, the corresponding globally normalized wavefunction is

$$\psi_{n\mathbf k}^{\mathrm{norm}}(\mathbf r) = \frac{1}{\sqrt{N_c\Omega_c}} e^{i\mathbf k\cdot\mathbf r} u_{n\mathbf k}(\mathbf r).$$

The factor $1/\sqrt{N_c\Omega_c}$ is omitted only when the structural Bloch form or a generalized infinite-volume state is intended.

An article using $\int_{\mathrm{cell}}|u|^2=1$ has moved a factor of $\Omega_c$ into the cell inner product. Translate all matrix elements consistently.

Band and basis indices

$n,m$ label energy bands after diagonalization. $\alpha,\beta$ label orbitals or sublattices before diagonalization:

$$\sum_\beta h_{\alpha\beta}(\mathbf k) u_{n\beta}(\mathbf k) = \varepsilon_n(\mathbf k) u_{n\alpha}(\mathbf k).$$

At degeneracies, individual band eigenvectors are not unique. Use a projector onto the isolated band subspace when a gauge-invariant statement does not require choosing a basis inside it.

Bloch Gauge and Basis Gauge

Eigenvector phase gauge

For an isolated band,

$$|u_{n\mathbf k}\rangle \longrightarrow e^{i\chi_n(\mathbf k)} |u_{n\mathbf k}\rangle$$

does not change the state ray or energy. This is the Bloch eigenvector gauge.

For a degenerate isolated subspace,

$$|u_{a\mathbf k}\rangle \longrightarrow \sum_b |u_{b\mathbf k}\rangle U_{ba}(\mathbf k)$$

with unitary $U(\mathbf k)$ is the corresponding nonabelian gauge freedom.

Orbital-basis gauge

A $\mathbf k$-dependent unitary change of orbital basis also transforms $h(\mathbf k)$. This is distinct from multiplying a solved eigenvector by a phase. Derivatives with respect to $\mathbf k$ must include the basis connection when the orbital basis itself depends on $\mathbf k$.

In a compatible $\mathbf k$-independent cell basis, the velocity matrix is

$$v_i(\mathbf k) = \frac{1}{\hbar} \frac{\partial h(\mathbf k)}{\partial k_i}.$$

After a $\mathbf k$-dependent basis transformation, differentiating only the transformed matrix can omit connection terms. State the basis before using derivative formulas for velocity, Berry curvature, or orbital response.

Berry Connection, Curvature, and Orientation

Default sign

For a normalized cell-periodic eigenvector, use

$$\mathcal A_{n,i}(\mathbf k) = i \langle u_{n\mathbf k} | \partial_{k_i} u_{n\mathbf k} \rangle_c.$$

Under

$$|u_{n\mathbf k}\rangle \longrightarrow e^{i\chi_n(\mathbf k)} |u_{n\mathbf k}\rangle,$$

the connection transforms as

$$\mathcal A_{n,i} \longrightarrow \mathcal A_{n,i} - \partial_{k_i}\chi_n.$$

The curvature is

$$\Omega_{n,ij} = \partial_{k_i}\mathcal A_{n,j} - \partial_{k_j}\mathcal A_{n,i}.$$

In three-vector notation,

$$\boldsymbol\Omega_n = \boldsymbol\nabla_{\mathbf k} \times \boldsymbol{\mathcal A}_n.$$

This sign agrees with Berry Connection and the Berry Curvature formula card.

Chern-number orientation

For an oriented two-dimensional Brillouin zone with coordinates $(k_x,k_y)$,

$$C_n = \frac{1}{2\pi} \int_{\mathrm{BZ}} dk_x\,dk_y\, \Omega_{n,xy}(\mathbf k).$$

Reversing the coordinate orientation changes the sign of both the area form and reported Chern number. Always state the real-space and reciprocal-space axis orientation before comparing signs.

Periodic gauge and topology

A periodic gauge would satisfy

$$|u_{n,\mathbf k+\mathbf G}\rangle = |u_{n\mathbf k}\rangle$$

in a periodic orbital basis. Such a gauge may not be globally smooth for a topologically nontrivial isolated band. Numerical calculations should use gauge-covariant overlaps, projectors, or patching rather than enforce an impossible global phase choice.

Charge, Electromagnetic Coupling, and Peierls Phase

Charge sign

$e>0$ denotes the elementary charge magnitude. The electron charge is

$$q_{\mathrm e} = -e.$$

For a particle of charge $q$, minimal coupling is

$$\mathbf p \longrightarrow \mathbf p-q\mathbf A.$$

For an electron,

$$\mathbf p-q_{\mathrm e}\mathbf A = \mathbf p+e\mathbf A.$$

The electromagnetic potentials obey

$$\begin{aligned} \mathbf B &= \boldsymbol\nabla\times\mathbf A, \\ \mathbf E &= - \boldsymbol\nabla\phi - \frac{\partial\mathbf A}{\partial t}. \end{aligned}$$

Gauge transformation

Use

$$\begin{aligned} \mathbf A' &= \mathbf A + \boldsymbol\nabla\Lambda, \\ \phi' &= \phi - \frac{\partial\Lambda}{\partial t}, \\ \psi' &= e^{iq\Lambda/\hbar} \psi. \end{aligned}$$

For an electron, the wavefunction phase is $e^{-ie\Lambda/\hbar}$.

Peierls substitution

For a hopping term $t_{ij}c_i^\dagger c_j$, which annihilates at $j$ and creates at $i$, use

$$t_{ij} \longrightarrow t_{ij} \exp \left[ \frac{iq}{\hbar} \int_{\mathbf r_j}^{\mathbf r_i} \mathbf A\cdot d\boldsymbol\ell \right].$$

For an electron,

$$t_{ij} \longrightarrow t_{ij} \exp \left[ - \frac{ie}{\hbar} \int_{\mathbf r_j}^{\mathbf r_i} \mathbf A\cdot d\boldsymbol\ell \right].$$

Reversing the hopping orientation complex conjugates the phase. Products around a closed oriented loop encode the magnetic flux with the same charge and orientation convention.

The Peierls substitution is an approximation tied to localized orbitals and slowly varying fields. Intra-cell dipoles, orbital magnetic moments, and field-induced changes of the basis may require additional terms.

Energy, Frequency, Temperature, and Units

Explicit constants

The default is to keep $\hbar$ and $k_B$ explicit. SI units are used for dimensionful electromagnetic formulas unless an article declares Gaussian, atomic, natural, or lattice units.

Use:

• $E$ or $\varepsilon$ for energy;
• $\omega$ for angular frequency;
• $\nu$ for ordinary frequency only when needed;
• $\beta=1/(k_BT)$ for inverse energy.

The conversions are

$$E = \hbar\omega = h\nu, \qquad \omega = 2\pi\nu.$$

A spectrum labeled in electronvolts is energy resolved even if the horizontal variable is traditionally called $\omega$ in the literature. State once whether $\omega$ is angular frequency or an energy-valued variable used with $\hbar=1$.

Chemical potential and Fermi energy

$\mu$ denotes chemical potential. $E_F$ denotes the zero-temperature chemical potential when that limit is well defined:

$$E_F = \lim_{T\to0}\mu(T).$$

At finite temperature, prefer $\mu$. A band energy crossing $\mu$ need not imply a sharp quasiparticle if interactions or disorder broaden the spectrum.

Occupation functions

For fermions,

$$f(E) = \frac{1} {e^{\beta(E-\mu)}+1}.$$

For bosons with an allowed chemical potential,

$$n_B(E) = \frac{1} {e^{\beta(E-\mu)}-1}.$$

The energy argument and chemical-potential convention must match the Hamiltonian. If time evolution uses the grand Hamiltonian $K=H-\mu\hat N$, say so explicitly.

Filling, Density, and Degeneracy

The default filling is electrons per primitive cell:

$$\nu = \frac{N_{\mathrm e}}{N_c}.$$

This differs from:

$$n = \frac{N_{\mathrm e}}{V},$$

the physical number density, and from occupation per orbital,

$$\bar n = \frac{N_{\mathrm e}} {N_cN_{\mathrm{orb}}}.$$

Spin, valley, layer, and orbital degeneracies are not silently included. A phrase such as “half filling” must state the denominator. One electron per site can mean half filling for one spinful orbital, full filling for one spinless orbital, or another fraction in a multiorbital basis.

Use doping with an explicit reference:

$$\delta\nu = \nu-\nu_{\mathrm{ref}}.$$

State whether positive $\delta\nu$ means electron doping. Hole doping is often denoted $p>0$ with $\nu=\nu_{\mathrm{ref}}-p$; never assume the sign from the letter alone.

Real-Time Fourier and Retarded Response

Time transform

For an ordinary time-dependent quantity,

$$\begin{aligned} A(\omega) &= \int_{-\infty}^{\infty} dt\, e^{i\omega t} A(t), \\ A(t) &= \int_{-\infty}^{\infty} \frac{d\omega}{2\pi} e^{-i\omega t} A(\omega). \end{aligned}$$

The Heisenberg operator is

$$A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}.$$

With a perturbation

$$\delta H(t) = -f(t)B,$$

the retarded response is

$$\chi_{AB}^{\mathrm R}(t) = - \frac{i}{\hbar} \theta(t) \langle [A(t),B(0)] \rangle.$$

Then

$$\delta\langle A\rangle(\omega) = \chi_{AB}^{\mathrm R}(\omega) f(\omega).$$

If a source uses $\delta H=+fB$, its susceptibility or response equation carries the compensating sign.

Energy-valued Green function

For fermions, define

$$G_{\alpha\beta}^{\mathrm R}(\mathbf k,t) = -i\theta(t) \left\langle \left\{ c_{\mathbf k\alpha}(t), c_{\mathbf k\beta}^\dagger(0) \right\} \right\rangle.$$

When the spectral variable $E$ has energy units, use

$$G^{\mathrm R}(\mathbf k,E) = \frac{1}{\hbar} \int_{-\infty}^{\infty} dt\, e^{iEt/\hbar} G^{\mathrm R}(\mathbf k,t).$$

The spectral function is

$$A(\mathbf k,E) = - \frac{1}{\pi} \operatorname{Im} G^{\mathrm R}(\mathbf k,E).$$

With canonical normalization,

$$\int_{-\infty}^{\infty} dE\, A_{\alpha\alpha}(\mathbf k,E) = 1.$$

A source using energy-valued $\omega$ has renamed $E$ and usually set $\hbar=1$. Check dimensions before comparing self-energies or linewidths.

Structure Factors

For a local operator $O_{\mathbf R}$, define

$$O_{\mathbf q} = \frac{1}{\sqrt{N_c}} \sum_{\mathbf R} e^{-i\mathbf q\cdot\mathbf R} O_{\mathbf R}.$$

The default dynamical structure factor is

$$S_{OO}(\mathbf q,\omega) = \frac{1}{2\pi} \int_{-\infty}^{\infty} dt\, e^{i\omega t} \left\langle O_{\mathbf q}(t) O_{-\mathbf q}(0) \right\rangle.$$

It satisfies the equal-time sum rule

$$\int_{-\infty}^{\infty} d\omega\, S_{OO}(\mathbf q,\omega) = \left\langle O_{\mathbf q} O_{-\mathbf q} \right\rangle.$$

Experimental cross sections can include $2\pi$, $\hbar$, form-factor, polarization, unit-cell, and detailed-balance factors. Quote the measured normalization separately from the theoretical correlator.

When the operator has basis components, include embedding phases or form factors explicitly:

$$O_{\mathbf q} = \frac{1}{\sqrt{N_c}} \sum_{\mathbf R,\alpha} e^{-i\mathbf q\cdot (\mathbf R+\boldsymbol\tau_\alpha)} F_\alpha(\mathbf q) O_{\mathbf R\alpha}.$$

This physical probe phase is not removed merely because the Hamiltonian uses the cell Fourier convention.

Conductivity and Hall Signs

Transport, Response, and Optics routes these shared conventions to the appropriate material or terminal framework.

Drude Theory and Boltzmann Transport apply the Fourier, signed-charge, tensor-order, electrochemical-field, and field-orientation conventions below to dc, thermal, optical, and Hall response.

Tensor order

Define conductivity by

$$J_i(\omega) = \sum_j \sigma_{ij}(\omega) E_j(\omega),$$

where $\mathbf J$ is current density. Thus $\sigma_{xy}$ means current along $x$ in response to electric field along $y$.

Resistivity is the matrix inverse:

$$\rho = \sigma^{-1},$$

not the componentwise reciprocal. In an isotropic two-dimensional system,

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

Then

$$\rho_{xy} = - \frac{\sigma_{xy}} {\sigma_{xx}^2+\sigma_{xy}^2}.$$

This minus sign is a frequent source of apparent Hall-sign contradictions.

Coordinate and field orientation

Use a right-handed laboratory frame:

$$\hat{\mathbf x} \times \hat{\mathbf y} = \hat{\mathbf z}.$$

State the direction of magnetic field, current, voltage leads, magnetization, and sample normal. A reported sign without these directions is not portable.

Chern-band convention

With:

• electron charge $q_{\mathrm e}=-e$;
• $\mathcal A=i\langle u|\boldsymbol\nabla_{\mathbf k}u\rangle$;
• $C=(2\pi)^{-1}\int dk_xdk_y\,\Omega_{xy}$;
• $\sigma_{xy}=J_x/E_y$;

the filled-band intrinsic contribution is

$$\sigma_{xy} = - C \frac{e^2}{h}.$$

Equivalently, before a complete occupied band has been integrated,

$$\sigma_{xy}^{\mathrm{int}} = -\frac{e^2}{\hbar} \sum_n \int_{\mathrm{BZ}} \frac{d^2k}{(2\pi)^2} f_n(\mathbf k)\, \Omega_{n,xy}(\mathbf k).$$

The minus sign belongs to this complete convention block: it combines the electron current with the stated Berry-connection sign and tensor order. A source using $\mathcal A=-i\langle u|\boldsymbol\nabla_{\mathbf k}u\rangle$ reverses both $C$ and the written Chern-to-Hall bridge, leaving the physical response unchanged.

Equivalently,

$$\sigma_{yx} = C \frac{e^2}{h}.$$

Sources can report the opposite sign by reversing the Berry-connection sign, coordinate orientation, or tensor-index order. Compare the complete convention block, including the fixed physical charge, rather than treating carrier charge as an independent notational toggle.

Polarization, Position, and Origin

Bulk electric polarization in a periodic crystal is defined modulo a polarization quantum. Shifting an orbital center by a lattice vector changes a cell dipole while leaving the infinite charge distribution equivalent.

In one dimension, moving charge $q$ by one lattice vector $a$ changes polarization by

$$\Delta P = q$$

when polarization is expressed as charge per cell boundary in the one-dimensional convention. In $d$ dimensions, a lattice-vector transport changes polarization by

$$\Delta\mathbf P = \frac{q\mathbf R}{\Omega_c}.$$

State the origin, ionic contribution, electron-charge sign, occupied-band subspace, and branch. A Berry phase is not an absolute dipole without these data.

The canonical geometric treatment is Zak Phase Preview and the mathematical Berry pages. This conventions page does not choose a material topology claim: enter Topological Quantum Matter to select the appropriate live phase, response, boundary, or computation owner. Berry-Phase Polarization and Charge Pumping is the material owner for polarization branches, ionic-plus-electronic charge, and integrated current over an adiabatic cycle.

Disorder, Averages, and Broadening

Use:

• $\langle\cdots\rangle$ for a quantum or thermal expectation;
• $\overline{(\cdots)}$ or $[\cdots]_{\mathrm{dis}}$ for disorder averaging;
• $\langle\cdots\rangle_{\mathrm{cell}}$ only after defining a unit-cell average;
• $\eta>0$ for a causal infinitesimal;
• $\Gamma>0$ for a physical linewidth or damping scale.

The operations

$$\overline{ \langle A\rangle_H }$$

and

$$\left\langle A \right\rangle_{\overline H}$$

are generally different. The first solves each disorder realization and then averages; the second replaces the Hamiltonian by an average before solving.

Do not use $\eta$ as a fitted lifetime. A numerical broadening should be reported with convergence checks and distinguished from physical self-energy.

Boundary Conditions and Limits

State boundary conditions separately for each direction:

• periodic or twisted for bulk calculations;
• open for edges and finite samples;
• lead-coupled for transport;
• slab or semi-infinite for surfaces.

The following limits can fail to commute:

$$\begin{gathered} N_c\to\infty, \qquad \omega\to0, \qquad \mathbf q\to0, \\ T\to0, \qquad \Gamma\to0, \qquad t\to\infty. \end{gathered}$$

For conductivity, specify whether

$$\lim_{\omega\to0} \lim_{\mathbf q\to0}$$

or

$$\lim_{\mathbf q\to0} \lim_{\omega\to0}$$

is intended, and whether the thermodynamic and clean limits occur first. The first often represents a spatially uniform time-dependent drive; the second can represent static long-wavelength equilibrium response.

For spontaneous order, a source field $h$ may require

$$\lim_{h\to0} \lim_{N_c\to\infty} \langle M\rangle_h.$$

Reversing the limits restores the symmetry in each finite system.

Numerical Mesh and Plotting Conventions

Computational Quantum Matter owns the calculation-specific record that applies these conventions to model, solver, convergence, benchmark, and probe validation; this page retains the shared sign, gauge, mesh, unit, and limit contracts.

Reciprocal coordinates

Fractional reciprocal coordinates mean

$$\mathbf k = \sum_i \kappa_i\mathbf b_i.$$

Cartesian coordinates use inverse-length units. Label axes explicitly; a number such as $0.5$ is meaningless without the basis and units.

High-symmetry paths

A band plot samples a one-dimensional path through a higher-dimensional Brillouin zone. It is not the full band structure or density of states. State:

• reciprocal coordinates of every labeled point;
• path order and interpolation;
• whether horizontal distance is cumulative Euclidean distance;
• energy zero and chemical potential;
• spin, orbital, or symmetry projections;
• whether degeneracies are exact or plotting-tolerance coincidences.

Mesh integration

For a uniform mesh with weights $w_{\mathbf k}$,

$$\sum_{\mathbf k} w_{\mathbf k} = 1$$

for an average per cell. A Brillouin-zone integral is then approximated by

$$\frac{\Omega_c}{(2\pi)^d} \int_{\mathrm{BZ}} d^dk\, F(\mathbf k) \simeq \sum_{\mathbf k} w_{\mathbf k} F(\mathbf k).$$

Symmetry-reduced meshes require weights for the full star. Berry curvature near avoided crossings, Fermi-surface integrals, and singular densities of states need convergence beyond a visually smooth plot.

Translation Between Common Conventions

Difference	Translation
cell versus orbital Fourier phase	multiply orbital component by $e^{-i\mathbf k\cdot\boldsymbol\tau_\alpha}$
$e^{+i\omega t}$ versus $e^{-i\omega t}$ forward transform	replace $\omega\to-\omega$ and translate causal prescriptions
angular frequency versus energy	$E=\hbar\omega$ and transform measures accordingly
electron charge $-e$ versus signed $e$	replace every Lorentz, minimal-coupling, and current factor consistently
$\mathcal A=+i\langle u	\nabla u\rangle$versus$-i\langle u
$\sigma_{xy}=J_x/E_y$ versus $J_y/E_x$	swap tensor indices and account for antisymmetry
per-cell versus per-volume response	multiply or divide by $\Omega_c$ with operator normalization
$S=\hbar\sigma/2$ versus dimensionless $S=\sigma/2$	restore $\hbar$ in operators and coupling units
$h(\mathbf k)$ periodic versus quasi-periodic	identify whether basis embedding enters the Fourier phase

A reliable translation checks an invariant:

• canonical commutator or anticommutator;
• total state count;
• Hermiticity;
• gauge covariance;
• an equal-time or spectral sum rule;
• dimensions of the final observable.

Common Mistakes

1. Including $\boldsymbol\tau_\alpha$ in one Fourier transform but not its inverse.
2. Calling $\mathbf k$ momentum while using $m\mathbf v=\hbar\mathbf k$ in a nonparabolic band.
3. Comparing Berry signs without comparing reciprocal orientation and tensor order.
4. Differentiating a $\mathbf k$-dependent basis Hamiltonian without connection terms.
5. Using $N_s$ for sites, cells, and orbitals in the same calculation.
6. Calling a finite-temperature chemical potential the Fermi energy without qualification.
7. Treating resistivity as $1/\sigma_{ij}$ component by component.
8. Using energy-valued $\omega$ in one equation and angular frequency in another.
9. Applying a Peierls phase with the electron charge sign but the reverse link orientation.
10. Taking a dc, clean, or thermodynamic limit silently.
11. Plotting a high-symmetry line and inferring the absence of off-path crossings.
12. Calling numerical $\eta$ a physical lifetime without convergence evidence.

Convention Audit Checklist

Before accepting a calculation, verify:

• primitive and reciprocal vectors obey $\mathbf a_i\cdot\mathbf b_j=2\pi\delta_{ij}$;
• real-space and reciprocal orientations are stated;
• every basis position $\boldsymbol\tau_\alpha$ is declared;
• lattice Fourier phases and normalization form an inverse pair;
• the Bloch Hamiltonian follows from the declared hopping orientation;
• state counting gives $N_cN_{\mathrm{orb}}$ modes before internal multiplicities;
• cell and orbital Fourier conventions are not mixed;
• Bloch eigenvectors use a stated cell inner product;
• Berry and Hall signs include charge, orientation, and tensor order;
• energy and angular frequency are dimensionally distinct;
• response specifies the perturbation sign and current normalization;
• broadening and disorder averages are defined;
• every noncommuting limit is written in order;
• numerical meshes reproduce state counts and sum rules.

Exercises

Exercise 1: lattice state counting

Use the default Fourier transform to show that one orbital on each of $N_c$ cells produces exactly $N_c$ independent crystal-momentum modes.

Solution

The transform matrix is

$$U_{\mathbf k\mathbf R} = \frac{1}{\sqrt{N_c}} e^{-i\mathbf k\cdot\mathbf R}.$$

Discrete orthogonality gives

$$\sum_{\mathbf R} U_{\mathbf k\mathbf R} U^\ast_{\mathbf k'\mathbf R} = \delta_{\mathbf k,\mathbf k'}.$$

The inverse relation gives

$$\sum_{\mathbf k} U^\ast_{\mathbf k\mathbf R} U_{\mathbf k\mathbf R'} = \delta_{\mathbf R,\mathbf R'}.$$

Thus $U$ is an $N_c\times N_c$ unitary matrix. It maps $N_c$ real-space modes to $N_c$ momentum modes without changing the Hilbert-space dimension.

Exercise 2: cell and orbital conventions

Starting from $\bar{\mathbf c}_{\mathbf k}=D(\mathbf k)\mathbf c_{\mathbf k}$, derive the relation between $h(\mathbf k)$ and $\bar h(\mathbf k)$.

Solution

The Hamiltonian in the cell basis is

$$H = \sum_{\mathbf k} \mathbf c_{\mathbf k}^\dagger h(\mathbf k) \mathbf c_{\mathbf k}.$$

Since

$$\mathbf c_{\mathbf k} = D^\dagger(\mathbf k) \bar{\mathbf c}_{\mathbf k},$$

substitution gives

$$H = \sum_{\mathbf k} \bar{\mathbf c}_{\mathbf k}^\dagger D(\mathbf k) h(\mathbf k) D^\dagger(\mathbf k) \bar{\mathbf c}_{\mathbf k}.$$

Therefore

$$\bar h(\mathbf k) = D(\mathbf k) h(\mathbf k) D^\dagger(\mathbf k).$$

The spectra agree because the transformation is unitary at each $\mathbf k$.

Exercise 3: hopping sign

For

$$H = +t \sum_j \left( c_j^\dagger c_{j+1} +c_{j+1}^\dagger c_j \right),$$

find the dispersion and compare its band minimum with the convention using $-t$.

Solution

The Fourier transform gives

$$\varepsilon(k) = +2t\cos(ka).$$

For $t>0$, its minimum occurs at $k=\pi/a$, whereas the Hamiltonian with coefficient $-t$ gives

$$\varepsilon(k) = -2t\cos(ka)$$

and has its minimum at $k=0$. On a bipartite nearest-neighbor chain the two signs can be related by the gauge transformation $c_j\mapsto(-1)^jc_j$, which shifts momentum by $\pi/a$. Additional hoppings, boundaries, or flux can make hopping signs physically meaningful.

Exercise 4: Berry gauge transformation

Derive the transformation of

$$\mathcal A_i = i\langle u|\partial_i u\rangle$$

under $|u'\rangle=e^{i\chi}|u\rangle$.

Solution

Differentiate:

$$\partial_i|u'\rangle = e^{i\chi} \left( i(\partial_i\chi)|u\rangle + \partial_i|u\rangle \right).$$

Using $\langle u'|=\langle u|e^{-i\chi}$ and $\langle u|u\rangle=1$,

$$\begin{aligned} \mathcal A_i' &= i \langle u| \left[ i(\partial_i\chi)|u\rangle + \partial_i|u\rangle \right] \\ &= -\partial_i\chi + \mathcal A_i. \end{aligned}$$

The curl is unchanged because mixed derivatives commute on a smooth patch:

$$\Omega_{ij}' = \Omega_{ij}.$$

Exercise 5: Hall tensor inversion

Invert

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

and verify the sign of $\rho_{xy}$.

Solution

The determinant is

$$\det\sigma = \sigma_{xx}^2+\sigma_{xy}^2.$$

Therefore

$$\rho = \sigma^{-1} = \frac{1} {\sigma_{xx}^2+\sigma_{xy}^2} \begin{pmatrix} \sigma_{xx} & -\sigma_{xy}\\ \sigma_{xy} & \sigma_{xx} \end{pmatrix}.$$

Hence

$$\rho_{xy} = - \frac{\sigma_{xy}} {\sigma_{xx}^2+\sigma_{xy}^2}.$$

The sign follows from matrix inversion, not from a new carrier convention.

Exercise 6: Peierls gauge covariance

Show that the Peierls-dressed term

$$t_{ij} e^{(iq/\hbar)\int_j^i\mathbf A\cdot d\boldsymbol\ell} c_i^\dagger c_j$$

is invariant under the electromagnetic gauge transformation defined above.

Solution

The link phase changes as

$$\exp \left[ \frac{iq}{\hbar} \int_j^i (\mathbf A+\boldsymbol\nabla\Lambda) \cdot d\boldsymbol\ell \right] = e^{iq(\Lambda_i-\Lambda_j)/\hbar} \exp \left[ \frac{iq}{\hbar} \int_j^i \mathbf A\cdot d\boldsymbol\ell \right].$$

The annihilation operator transforms as

$$c_j \longrightarrow e^{iq\Lambda_j/\hbar}c_j,$$

so

$$c_i^\dagger c_j \longrightarrow e^{-iq\Lambda_i/\hbar} e^{iq\Lambda_j/\hbar} c_i^\dagger c_j.$$

The operator phase cancels the added link phase. Reversing the link orientation reverses the line integral and complex conjugates the phase.

Connections

• Quantum Matter Overview routes readers to the definition, conceptual map, convention ledger, and next technical chapter.
• Choosing a Model for Quantum Matter selects the smallest adequate retained description before this page fixes its calculation contract.
• How to Use This Volume provides goal-based routes and identifies when this convention ledger is needed as a lookup.
• Quantum Matter Reference and Data applies this page’s convention system to transferable lookup records; this page remains the canonical owner of the lattice, basis, Fourier, Berry, electromagnetic, normalization, and response conventions themselves.
• Conventions Overview owns sitewide state, operator, unit, and Fourier choices.
• Many-Body Symbols and Conventions owns generic Fock-space, ensemble, correlator, and limit notation.
• Phonons applies the cell-phase, angular-frequency, energy, Brillouin-zone, and per-cell normalization contracts to lattice dynamics.
• Drude Theory applies the signed-charge, Fourier, conductivity-tensor, Hall, and optical conventions.
• Boltzmann Transport applies the phase-space, degeneracy, collision-integral, electrochemical-field, and thermoelectric conventions.
• Hall Effect fixes the transverse tensor order, voltage polarity, carrier sign, dimensionality, and resistivity-to-conductivity inversion.
• Notation Collisions catalogs overloaded symbols.
• Bloch’s Theorem owns the theorem, proof, finite-size state counting, and Bloch–Floquet qualification.
• Wannier Functions applies the Bloch-frame, Fourier-phase, orbital-embedding, and reciprocal-sewing conventions fixed here to the construction and localization of band-derived orbitals.
• Symmetry of Bloch States applies the momentum, reciprocal-sewing, basis, and band-index conventions fixed here to little-group representations and symmetry-constrained degeneracies.
• Bloch Theorem formula card gives the compact translation result.
• Berry Curvature formula card gives gauge and projector formulas.
• Chern Number formula card records normalization and isolation assumptions.
• Retarded and Advanced Response owns causal analyticity and Fourier structure.
• Peierls Phase Preview develops lattice gauge covariance.
• Quantum Matter Map shows where each convention enters model and experiment.

References

1. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976. Standard lattice, reciprocal-space, Bloch, band, and transport notation.
2. C. Kittel, Introduction to Solid State Physics, 8th ed., Wiley, 2004. A compact reference for crystal and reciprocal-lattice conventions.
3. G. D. Mahan, Many-Particle Physics, 3rd ed., Springer, 2000. Green-function, response, spectral, and transport conventions.
4. A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003. Operator normalization, propagators, and response functions.
5. D. Vanderbilt, Berry Phases in Electronic Structure Theory, Cambridge University Press, 2018. Modern Bloch gauge, polarization, Wannier, and Berry conventions.
6. D. Xiao, M.-C. Chang, and Q. Niu, “Berry Phase Effects on Electronic Properties,” Reviews of Modern Physics 82, 1959–2007 (2010), doi:10.1103/RevModPhys.82.1959. Berry-curvature and semiclassical sign conventions with explicit electron charge.
7. J. M. Blount, “Formalisms of Band Theory,” Solid State Physics 13, 305–373 (1962), doi:10.1016/S0081-1947(08)60459-2. A foundational treatment of position, derivatives, and gauge in Bloch representations.
8. R. Resta, “Macroscopic Polarization in Crystalline Dielectrics: The Geometric Phase Approach,” Reviews of Modern Physics 66, 899–915 (1994), doi:10.1103/RevModPhys.66.899. Establishes branch and polarization-quantum conventions.
9. R. Peierls, Quantum Theory of Solids, Oxford University Press, 1955. Classical source for lattice models and magnetic phases on hopping amplitudes.