Energy current in a quantum chain

Question

I have seen in (e.g. this paper) the definition of the energy current in a chain with $H = \sum_{j=1}^L h_j$ where $H_j$ has support on the $k$ sites $j,j+1,j+2,...,j+(k-1)$ as $$J_j - J_{j+1} = i[H, h_j].$$

This definition is motivated as a discrete version of the continuity equation, namely $J_j - J_{j+1} = \frac{dh_j}{dt}.$

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However, I have only found explicit forms for $J_j$ in terms of the set of $\{h_i\}$ in the case of $k=2$, for which

$$J_j - J_{j+1} = i([h_{j-1}, h_j]+[h_{j+1}, h_j]) = i[h_{j-1}, h_j]-i[h_{j}, h_{j+1}]$$

which has the immediate solution $J_j = i[h_{j-1}, h_j]$.

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What is the explicit solution of $J_j$ for $k>2$?

Answer 1

After a bit of staring at the problem and adding and subtracting quantities in pairs, I've come up with the correct general form for $H_j$ with support on $j,j+1, ..., j+k-1$.

$$J_j = i\sum_{q=1}^{k-1} \sum_{r=0}^{q-1} [h_{j-k+q}, h_{j+r}]$$

To see this, notice that separating out the $r=0$ terms in $J_j$ gets

$$\begin{align} J_j &= i\sum_{q=1}^{k-1} \sum_{r=0}^{q-1} [h_{j-k+q}, h_{j+r}]\\ &= i\sum_{q=1}^{k-1}[h_{j-k+q}, h_j] + i\sum_{q=2}^{k-1} \sum_{r=1}^{q-1} [h_{j-k+q}, h_{j+r}] \\& = i\sum_{q=-(k-1)}^{-1}[h_{j+q}, h_j] + i\sum_{q=2}^{k-1} \sum_{r=1}^{q-1} [h_{j-k+q}, h_{j+r}] \end{align} $$

while separating out the $q=k-1$ terms in $J_{j+1}$ gives

$$\begin{align} J_{j+1} &= i\sum_{q=1}^{k-1} \sum_{r=0}^{q-1} [h_{j+1-k+q}, h_{j+1+r}]\\ &= i\sum_{r=0}^{k-2}[h_j, h_{j+1+r}] + i\sum_{q=1}^{k-2} \sum_{r=0}^{q-1} [h_{j+1-k+q}, h_{j+1+r}] \\&= -i\sum_{r=1}^{k-1}[h_{j+r}, h_j] + i\sum_{q=1}^{k-2} \sum_{r=1}^{(q+1)-1} [h_{j-k+(q+1)}, h_{j+r}] \\&= -i\sum_{r=1}^{k-1}[h_{j+r}, h_j] + i\sum_{q=2}^{k-1} \sum_{r=1}^{q-1} [h_{j-k+q}, h_{j+r}] \end{align}$$

Thus, after all this index manipulation, we find at last

$$J_j - J_{j+1} = i\sum_{q=-(k-1)}^{k-1} [h_{j+q}, h_j] = i[H,h_j]$$

which confirms the correctness of the form of $J_j$.