Conditional band velocity and anomalous term

Assume Axiom of Choice and Countable Choice. Let ${\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3\in {\mathbb{R}}^3$ be a basis of direct-lattice vectors, each measured in metres, and set $\mathrm{\Lambda }={\sum }_{j=1}^3\mathbb{Z}{\mathbf{a}}_j$ and ${\mathrm{\Lambda }}^{\ast }:=\{\mathbf{G}\in {\mathbb{R}}^3:\mathbf{G}\cdot \mathbf{R}\in 2\pi \mathbb{Z}\text{ for all }\mathbf{R}\in \mathrm{\Lambda }\}$, whose vectors have units of inverse metres. The reciprocal-space torus is ${\mathbb{R}}^3\mathrm{/}{\mathrm{\Lambda }}^{\ast }$; let $U\subset {\mathbb{R}}^3$ be an open set on which the quotient map is injective, giving local wave-vector coordinates. For $\delta \in {\mathbb{Z}}^3$ put ${\mathbf{R}}_{\delta }={\sum }_j{\delta }_j{\mathbf{a}}_j$ and ${\theta }_j(\mathbf{k})=\mathbf{k}\cdot {\mathbf{a}}_j(\mathrm{m}\mathrm{o}\mathrm{d}2\pi )$. Fix finite $S\subset {\mathbb{Z}}^3$ with $S=-S$ and hopping matrices $T_{\delta }\in M_r(\mathbb{C})$ satisfying $T_{-\delta }=T_{\delta }^{\ast }$; assign the entries of $T_{\delta }$ units of joules. The finite-range physical fiber is

$h(\mathbf{k})={\sum }_{\delta \in S}{T}_{\delta }{e}^{-i\mathbf{k}\cdot {\mathbf{R}}_{\delta }}={\sum }_{\delta \in S}{T}_{\delta }{e}^{-i\theta (\mathbf{k})\cdot \delta },$

which is the fiber in Lattice Bloch transform and band projectors written in physical reciprocal coordinates. Its entries and its eigenvalues have energy units (joules). Let $\epsilon :U\to \mathbb{R}$ be a $C^2$ simple eigenvalue of $h(\mathbf{k})$ and, after shrinking $U$ if needed, assume its distance from the other eigenvalues is at least some $\gamma >0$ throughout $U$. Choose a $C^2$ local normalized eigenvector $u:U\to {\mathbb{C}}^r$ with $h(\mathbf{k})u(\mathbf{k})=\epsilon (\mathbf{k})u(\mathbf{k})$ and $u(\mathbf{k}{)}^{\ast }u(\mathbf{k})=1$.

Let $\mathbf{x}\in {\mathbb{R}}^3$ denote a position coordinate measured in metres, $\mathbf{k}\in U$ the physical crystal wave vector measured in inverse metres (radians are dimensionless), $q$ a constant signed charge in coulombs, ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\in {\mathbb{R}}^3$ a constant electric field in volts per metre, and $\mathrm{\hslash }>0$ the reduced Planck constant in joule seconds. Under the adopted single-band effective description, take

$L=\mathrm{\hslash }\mathbf{k}\cdot \dot{\mathbf{x}}+i\mathrm{\hslash }u(\mathbf{k}{)}^{\ast }{\mathrm{\nabla }}_{\mathbf{k}}u(\mathbf{k})\cdot \dot{\mathbf{k}}-\epsilon (\mathbf{k})+q{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\cdot \mathbf{x}\mathrm{.}$

Here each term of $L$ has units of joules. Define the real Berry connection ${\mathcal{A}}_j(\mathbf{k}):=iu(\mathbf{k}{)}^{\ast }{\mathrm{\partial }}_{k_j}u(\mathbf{k})$, whose components have units of metres, and the Berry-curvature vector $\mathbf{\Omega }(\mathbf{k}):={\mathrm{\nabla }}_{\mathbf{k}}\times \mathcal{A}(\mathbf{k})$, whose components have units of square metres. For time $t$ measured in seconds, a compact interval $I=[t_0,t_1]$ with $t_0<t_1$, and curves $\mathbf{x},\mathbf{k}\in C^2(I;{\mathbb{R}}^3)$ with $\mathbf{k}(t)\in U$, define the action $\mathcal{S}[\mathbf{x},\mathbf{k}]={\int }_{t_0}^{t_1}L(\mathbf{x}(t),\mathbf{k}(t),\dot{\mathbf{x}}(t),\dot{\mathbf{k}}(t))dt$. A curve is stationary if ${\frac{d}{ds}\mid }_{s=0}\mathcal{S}[{\mathbf{x}}_s,{\mathbf{k}}_s]=0$ for every jointly $C^2$ variation $(s,t)\mapsto ({\mathbf{x}}_s(t),{\mathbf{k}}_s(t))$, $\mathrm{\mid }s\mathrm{\mid }<\eta$, with $({\mathbf{x}}_0,{\mathbf{k}}_0)=(\mathbf{x},\mathbf{k})$, fixed endpoints ${\mathbf{x}}_s(t_i)=\mathbf{x}(t_i)$ and ${\mathbf{k}}_s(t_i)=\mathbf{k}(t_i)$ for $i=0,1$, and ${\mathbf{k}}_s(t)\in U$. Every such $C^2$ stationary curve of this adopted effective action satisfies

$\mathrm{\hslash }\dot{\mathbf{k}}=q{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}},\dot{\mathbf{x}}=\frac{1}{\mathrm{\hslash }}{\mathrm{\nabla }}_{\mathbf{k}}\epsilon -\dot{\mathbf{k}}\times \mathbf{\Omega }\mathrm{.}$

The action has units of joule seconds. The equations are invariant under a local phase change of $u$. The words “wave packet” and “slow-force” name only the physical motivation and regime label of this adopted effective model; no wave-packet object is used in the deduction, and no approximation-validity or error-bound claim about underlying quantum dynamics is made. For the two-band specialization, take $r=2$. Let ${\mathbf{b}}_1,{\mathbf{b}}_2,{\mathbf{b}}_3$ be the reciprocal basis, defined by ${\mathbf{b}}_j\cdot {\mathbf{a}}_{\mathrm{\ell }}=2\pi {\delta }_{j\mathrm{\ell }}$, so ${\mathrm{\Lambda }}^{\ast }={\sum }_j\mathbb{Z}{\mathbf{b}}_j$. Fix $\tau \in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ and the closed reciprocal two-torus

${\mathrm{\Sigma }}_{\tau }=\{[\mathbf{k}]\in {\mathbb{R}}^3\mathrm{/}{\mathrm{\Lambda }}^{\ast }:{\theta }_3(\mathbf{k})=\tau \},{\kappa }_{\tau }({\theta }_1,{\theta }_2)=\left[\frac{{\theta }_1{\mathbf{b}}_1+{\theta }_2{\mathbf{b}}_2+\tau {\mathbf{b}}_3}{2\pi }\right]\mathrm{.}$

Here ${\kappa }_{\tau }:{\mathbb{T}}_{2\pi }^2\to {\mathrm{\Sigma }}_{\tau }$ is a diffeomorphism and fixes the orientation by $d{\theta }_1\wedge d{\theta }_2$. Suppose the restriction has the two-band form $h\circ {\kappa }_{\tau }=d_{0}I+d\cdot \sigma$ of Two-band Chern number and degree formula, with $d$ nowhere zero on this entire torus. Put $n=d\mathrm{/}\mathrm{\mid }d\mathrm{\mid }$ and $P_s=(I+sn\cdot \sigma )\mathrm{/}2$, $s\in \{+1,-1\}$; the chosen local eigenvector belongs to that band wherever its chart meets the slice. Interpret $P_s$ also as the corresponding globally defined projector on ${\mathrm{\Sigma }}_{\tau }$. Define the global real two-form

${\mathcal{B}}_s:=i\mathrm{tr}(P_s(d{P}_s\wedge d{P}_s)P_s)\mathrm{.}$

On every local normalized eigenvector chart this is $d\mathcal{A}$ for $\mathcal{A}=i{u}^{\ast }du$; it does not require a global eigenvector. In physical reciprocal coordinates, the coordinate conversion on every such chart is

${\kappa }_{\tau }^{\ast }{\mathcal{B}}_s=\frac{\mathbf{\Omega }({\kappa }_{\tau }(\theta ))\cdot ({\mathbf{b}}_1\times {\mathbf{b}}_2)}{(2\pi {)}^2}d{\theta }_1\wedge d{\theta }_2\mathrm{.}$

The coefficient is dimensionless, since $\mathbf{\Omega }$ has units of square metres and each reciprocal basis vector has inverse-metre units. With this orientation and the same band sign,

$\frac{1}{2\pi }{\int }_{{\mathrm{\Sigma }}_{\tau }}{\mathcal{B}}_s=\frac{i}{2\pi }{\int }_{{\mathbb{T}}_{2\pi }^2}\mathrm{tr}(P_s(d{P}_s\wedge d{P}_s)P_s)=C(P_s)=-s\mathrm{deg}(n)\mathrm{.}$

The integrals are over the whole closed torus, and the exterior derivatives in the second integral are in the dimensionless $\theta$ coordinates. This is the Chern convention of Two-band Chern number and degree formula.

Lattice Bloch transform and band projectors

Exact Bloch transform and regularity of gapped band projectors

Two-band Chern number and degree formula