Sites, one-step shift, and total power on a periodic $q$-dimensional lattice

Sites. For natural numbers $q$ and $L$, a site is a function $x : \{0,\dots,q-1\} \to \mathbb{Z}/L\mathbb{Z}$, i.e. an element of $(\mathbb{Z}/L\mathbb{Z})^q$, the discrete $q$-dimensional torus with side $L$. Coordinates take values in $\{0,\dots,L-1\}$, and all addition and negation of coordinates is modulo $L$. No constraint on $q$ or $L$ is made at this stage. If $q = 0$ there is exactly one site (the empty function). If $L = 0$ and $q \ge 1$ there are no sites.

Shift. For a site $x$, a direction $a \in \{0,\dots,q-1\}$ and an amount $s \in \mathbb{Z}/L\mathbb{Z}$, $\mathrm{shift}(x,a,s)$ is the site that agrees with $x$ in every coordinate $b \ne a$ and has $a$-th coordinate $x_a + s \pmod L$:

Write $x + e_a := \mathrm{shift}(x,a,1)$ and $x - e_a := \mathrm{shift}(x,a,-1)$. Here $1$ is the residue of $1$ modulo $L$ and $-1$ is its additive inverse modulo $L$, i.e. the residue $L-1$. The shift wraps around: if $x_a = L-1$ then $(x+e_a)_a = 0$, and if $x_a = 0$ then $(x-e_a)_a = L-1$. In the degenerate case $L = 1$ we have $1 = -1 = 0$, so $x \pm e_a = x$. When $L = 2$ we have $1 = -1$, so $x + e_a = x - e_a$.

Power. The function $\mathrm{power}$ takes explicit natural numbers $q, L, d$, the assumption $L \neq 0$ (which is what gives the residue $1$ above its meaning), and two further arguments:

• $W$, which assigns to every site $x$ and direction $a \in \{0,\dots,q-1\}$ an arbitrary real $d \times d$ matrix $W(x,a)$. Nothing is assumed about these matrices: no symmetry, definiteness, invertibility, or relation between different $(x,a)$.
• $v$, which assigns to every site $x$ an arbitrary real vector $v(x) \in \mathbb{R}^d$.

It returns the real number

Here $\cdot$ is the plain real dot product $u\cdot w = \sum_{i=0}^{d-1} u_i w_i$, with no conjugation or transpose, and $M u$ is the ordinary matrix-vector product $(Mu)_i = \sum_{j} M_{ij} u_j$. The first term uses the matrix at the site $x$ itself, and the second uses the matrix at the neighbouring site $x - e_a$, both in the same direction $a$. Both sums are finite and range over all $L^q$ sites and all $q$ directions.

Degenerate cases. If $q = 0$ the inner sum is empty, so $\mathrm{power} = 0$. If $d = 0$ every dot product is empty, so $\mathrm{power} = 0$. If $L = 1$ then $x \pm e_a = x$, each summand is $v(x)\cdot(W(x,a)-W(x,a))v(x) = 0$, and so $\mathrm{power} = 0$. If $L = 2$ the two neighbours $x+e_a$ and $x-e_a$ are the same site. The file defines only these objects and asserts no properties about them.