Conventions for Quantum Matter

Signs and normalization factors are part of a physical formula. This reference gives the lattice, Bloch-state, electromagnetic and response conventions used in this volume. It explains how to translate between common alternatives, with particular care over orbital positions, spectral matrices and Hall signs. A page using another convention states it beside the affected formula.

Required background. Fourier transforms explain transform signs and integration measures. General conventions and many-body notation introduce state normalization, operators and thermal averages.

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. The velocity depends on how the physical position operator is represented, not just on whether the orbital coefficient labels depend on $\mathbf k$.

In an orthonormal representation where the retained position operator is $r_i=i\partial_{k_i}$, the velocity matrix is

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

The cell Fourier convention used above instead gives $r_i=i\partial_{k_i}+\tau_i$ for orthonormal point orbitals, where $\tau_i$ is the diagonal matrix of their intracell coordinates. In that model,

$$v_i(\mathbf k) =\frac{1}{\hbar} \left( \frac{\partial h(\mathbf k)}{\partial k_i} -i[\tau_i,h(\mathbf k)] \right).$$

For example, hopping between two different positions within one cell can produce a nonzero optical velocity even when $h$ has no $\mathbf k$ dependence. More general retained bases require their full position connection, including intra-atomic dipole matrix elements when those are kept. A basis transformation must transform that connection along with the Hamiltonian.

The extra term describes motion between different positions within a cell. It is therefore relevant to interband optical matrix elements even when the band energies are unchanged by an embedding convention. Esteve-Paredes and Palacios give the general local-orbital treatment. See tight-binding models for the construction and Kubo response for the field coupling.

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 chosen orientation of the Brillouin-zone surface reverses the reported Chern number. A passive change of coordinates on the same oriented surface does not: the curvature components and integration measure transform together. Comparing Hall signs also requires the laboratory axes and tensor-index order below.

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:

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

For a metal this convention identifies the Fermi-level energy. In a gapped system, the zero-temperature chemical potential can depend on how the limit and particle number are specified; a finite interval of chemical potentials can select the same ground-state particle number. At finite temperature, use $\mu$. Neither a chemical potential nor a crossing of it guarantees a sharp quasiparticle.

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 bosonic expression applies where $E-\mu>0$; a macroscopically occupied ground mode is treated separately. Equilibrium phonons and photons ordinarily have $\mu=0$ because their number is not conserved. The energy argument must match the Hamiltonian: if time evolution uses $K=H-\mu\hat N$ rather than $H$, the corresponding single-particle energy is measured relative to $\mu$.

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

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

For a scalar or a diagonal element, the spectral function is $A=-\operatorname{Im}G^{\mathrm R}/\pi$. For an orbital matrix the corresponding definition is

$$\begin{aligned} A(\mathbf k,E) &=-\frac{G^{\mathrm R}(\mathbf k,E)-G^{\mathrm R}(\mathbf k,E)^\dagger}{2\pi i},\\ A_{\alpha\beta} &=-\frac{G^{\mathrm R}_{\alpha\beta}-(G^{\mathrm R}_{\beta\alpha})^*}{2\pi i}. \end{aligned}$$

The imaginary part here is the Hermitian imaginary part of a matrix, not the imaginary part of each entry. Off-diagonal spectral entries can be complex. The fermionic spectral matrix is Hermitian and positive semidefinite for a stationary equilibrium state in a canonical orthonormal orbital basis.

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 Hermitian 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

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

Then $O_{-\mathbf q}=O_{\mathbf q}^\dagger$, the spectral weights are nonnegative, and the equal-time sum rule is

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

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

Conductivity converts electric field into current density. Both the order of its indices and its matrix inverse matter for the Hall effect.

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 ionic and electronic contributions must be combined consistently. Berry-phase polarization and charge pumping explains how polarization differences are obtained from integrated current and how the branch ambiguity enters.

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

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

Taking the source to zero first in a finite, symmetry-invariant equilibrium state gives zero symmetry-breaking expectation. At strictly zero temperature, exact finite-system degeneracies require separate care: a selected pure ground state need not share the symmetry of the equal-weight ground-state ensemble.

Numerical Mesh and Plotting Conventions

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
$\mathcal A=+i\langle u\vert\nabla u\rangle$ versus $-i\langle u\vert\nabla u\rangle$	reverse connection, curvature, and Chern signs
$\sigma_{xy}=J_x/E_y$ 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.

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.

Exercise 7: a complex spectral matrix

In units where the energy scale is one, let

$$h=\begin{pmatrix}0&i\\-i&0\end{pmatrix}, \qquad G^{\mathrm R}(E)=[(E+i\eta)I-h]^{-1},\quad \eta>0.$$

Find the positive-energy projector and explain why taking the imaginary part of every entry of $G^{\mathrm R}$ is not the correct matrix spectral function.

Solution

Since $h^2=I$, the energies are $\pm1$ and $P_+=(I+h)/2$. Thus

$$A(E)=\frac{\eta}{\pi[(E-1)^2+\eta^2]}P_+ +\frac{\eta}{\pi[(E+1)^2+\eta^2]}P_-.$$

Each coefficient is nonnegative and each projector is Hermitian positive semidefinite. In particular, $P_+$ has imaginary off-diagonal entries $+i/2$ and $-i/2$. Entrywise imaginary parts followed by a real factor would give a real matrix and lose these entries. The Hermitian difference of the retarded and advanced matrices reproduces the projector expansion and the matrix sum rule $\int A(E)\,dE=I$.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976. Standard lattice, reciprocal-space, Bloch, band, and transport notation.
• J. J. Esteve-Paredes and J. J. Palacios, “A comprehensive study of the velocity, momentum and position matrix elements for Bloch states: Application to a local orbital basis,” SciPost Physics Core 6, 002 (2023), doi:10.21468/SciPostPhysCore.6.1.002, arXiv:2201.12290. Position-connection terms and the conditions for Hamiltonian-derivative velocity formulas.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer, 2000. Green-function, response, spectral, and transport conventions.
• 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.
• 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.