Local charge current and finite-volume correlation conventions

Fix either a finite open chain with sites $\{1,\dots ,L\}$, $L\ge 2$, or a periodic chain with sites $\mathbb{Z}\mathrm{/}L\mathbb{Z}$, $L\ge 3$, and denote its site set by ${\mathrm{\Lambda }}_L$. Let ${\mathcal{H}}_L$ be a nonzero finite-dimensional complex Hilbert space. For each site $j$, let $q_j=q_j^{\ast }$ be a bounded charge observable, with physical unit $\mathsf{Q}$. For each oriented nearest-neighbor bond $(j,j+1)$, let $h_{j,j+1}=h_{j,j+1}^{\ast }$ be an energy operator in joules. Require $[h_{k,k+1},q_j]=0$ whenever $j\mathrm{\notin }\{k,k+1\}$, and require local charge conservation $[h_{j,j+1},q_j+q_{j+1}]=0$. On an open chain set $H_L={\sum }_{j=1}^{L-1}{h}_{j,j+1}$; on a ring set $H_L={\sum }_{j\in \mathbb{Z}\mathrm{/}L\mathbb{Z}}{h}_{j,j+1}$ with periodic indices. For an open chain define the absent boundary currents $j_{0,1}=j_{L,L+1}=0$. Use $\mathrm{\hslash }>0$ in joule-seconds. These finite local data may be realized on a tensor-product spin space or on a finite fixed-particle site space, provided the stated commutation conditions hold.

The oriented bond current from $j$ to $j+1$ is $j_{j,j+1}:=\frac{i}{\mathrm{\hslash }}[h_{j,j+1},q_{j+1}]\mathrm{.}$ It has unit $\mathsf{Q}{\mathrm{s}}^{-1}$. The total current is the sum over existing bonds, $J_L:={\sum }_{(j,j+1)}{j}_{j,j+1}$, and its Heisenberg evolution for $t\in \mathbb{R}$ is $J_L(t):=e^{i{H}_{L}t\mathrm{/}\mathrm{\hslash }}{J}_L{e}^{-i{H}_{L}t\mathrm{/}\mathrm{\hslash }}$. For finite inverse temperature $\beta >0$ in joules${}^{-1}$, define ${\rho }_{\beta ,L}:=e^{-\beta H_L}\mathrm{/}\mathrm{Tr}(e^{-\beta H_L})$ and ${\omega }_{\beta ,L}(A):=\mathrm{Tr}({\rho }_{\beta ,L}A)$. Write $J_L^{\circ }:=J_L-{\omega }_{\beta ,L}(J_L)I$ and $J_L^{\circ }(t):=e^{i{H}_{L}t\mathrm{/}\mathrm{\hslash }}{J}_L^{\circ }{e}^{-i{H}_{L}t\mathrm{/}\mathrm{\hslash }}$. Define the real, connected, symmetrized correlation per site by $C_L(t):=\frac{1}{L}\mathrm{Re}{\omega }_{\beta ,L}\text{\hspace{0.17em}⁣}(J_L^{\circ }(t)J_L^{\circ })=\frac{1}{2L}{\omega }_{\beta ,L}\text{\hspace{0.17em}⁣}(J_L^{\circ }(t)J_L^{\circ }+J_L^{\circ }{J}_L^{\circ }(t))\mathrm{.}$ The correlation has unit ${\mathsf{Q}}^2{\mathrm{s}}^{-2}$ per site.

This definition assigns no transport coefficient by itself. Any proposed finite-volume or thermodynamic transport quantity must additionally state its time cutoff, frequency regulator (for example $e^{-\eta t}$ with $\eta >0$ in seconds${}^{-1}$), spatial normalization, and the exact order in which $L\to \mathrm{\infty }$, the time cutoff tends to infinity, and any regulator is removed. Existence of those limits is not part of this definition.

Finite-time Kubo formula with norm remainder

Sirker, §2

MIT OpenCourseWare 22.103, Lecture 3