Exercise Sheet 9

Take the oscillator’s Lax matrix $L=\begin{pmatrix}p&q\\q&-p\end{pmatrix}$ of Sheet 8, Exercise 1. (i) Compute $\pb{\leg L1}{\leg L2}=\sum_{a,b,c,d}\pb{L_{ab}}{L_{cd}}\,E_{ab}\otimes E_{cd}$ as a sum of terms $\pm E_{ab}\otimes E_{cd}$. (ii) Write the same object as a $4\times4$ matrix, and check its entry in row $(1,2)$ and column $(2,2)$ against the sum of (i). (iii) Write $\perm$ as a $4\times4$ matrix. Check $(E_{ab}\otimes E_{cd})\perm=E_{ad}\otimes E_{cb}$ in two cases, and show that $\perm(A\otimes B)\perm=B\otimes A$.

For one site, $L(\lambda)=\begin{pmatrix}\lambda-p&e^{-q}\\-e^{q}&0\end{pmatrix}$. (i) Find the four non-zero brackets $\pb{L_{ab}(\lambda)}{L_{cd}(\mu)}$. (ii) Assemble them into the $4\times4$ matrix $\pb{\leg L1(\lambda)}{\leg L2(\mu)}$. (iii) Compute $L(\lambda)\otimes L(\mu)=\leg L1(\lambda)\leg L2(\mu)$ as a $4\times4$ matrix, then $\comm{\rmat(\lambda-\mu)}{L(\lambda)\otimes L(\mu)}$ with $\rmat(u)=-\perm/u$, and check the quadratic Sklyanin bracket of Lecture 9 entry by entry.

(i) For $N=2$ compute $\transfer(\lambda)=\tr\big(L_2(\lambda)L_1(\lambda)\big)=\lambda^{2}+c_1\lambda+c_2$, and check that $H=\tfrac12c_1^{2}-c_2$ is the Hamiltonian $\tfrac12(p_1^{2}+p_2^{2})+e^{q_1-q_2}+e^{q_2-q_1}$ of the two-particle ring. (ii) Lecture 8 found the $N=2$ ring degenerate: its Lax matrix is $L=\begin{pmatrix}p_1&a_1+a_2\\a_1+a_2&p_2\end{pmatrix}$ with $a_1=e^{(q_1-q_2)/2}$, $a_2=e^{(q_2-q_1)/2}$, and $\tr L^{2}=2H+4$. Show that nevertheless $\det(\lambda-L)=\transfer(\lambda)-2$. (iii) For $N=3$ find $\transfer(\lambda)$, and check $\det(\lambda-L)=\transfer(\lambda)-2$ for the ring’s $3\times3$ Lax matrix of Lecture 8, $L=\begin{pmatrix}p_1&a_1&a_3\\a_1&p_2&a_2\\a_3&a_2&p_3\end{pmatrix}$, $a_n=e^{(q_n-q_{n+1})/2}$, $q_4=q_1$. Where does the constant $-2$ come from?

For the two-site ring, with $\transfer(\lambda)=\lambda^{2}+c_1\lambda+c_2$ as in Exercise 3: (i) Show that $\pb{\transfer(\lambda)}{\transfer(\mu)}=(\lambda-\mu)\pb{c_1}{c_2}$. (ii) Compute $\pb{c_1}{c_2}$ directly, and conclude. (iii) What does the flow of $c_1$ do, and why does that make (ii) work?

Carry out the separation Lecture 9 promises for $N=2$: put $Q=\tfrac12(q_1+q_2)$, $P_Q=p_1+p_2=2\bar p$, $x=q_1-q_2$ and $p_x=\tfrac12(p_1-p_2)$. (i) Show that $\pb Q{P_Q}=\pb x{p_x}=1$ and that the other four brackets among $Q,x,P_Q,p_x$ vanish. (ii) Write $H$ in the new variables. Show that the centre of mass moves uniformly, that $(x,p_x)$ moves under $\mathcal E=p_x^{2}+2\cosh x$, and that for $\mathcal E>2$ the level set of $\mathcal E$ is a closed curve, so Lecture 6’s action–angle variables apply to it. (iii) Show that $\transfer(\lambda)=u^{2}-\mathcal E$ with $u=\lambda-\bar p$. Find the bands $\abs{\transfer}\le2$ and the gap, and show that the gap closes exactly at the equilibrium.

(i) For $N=3$, extend the eigenvalue recursion $a_{n-1}\psi_{n-1}+p_n\psi_n+a_n\psi_{n+1}=\lambda\psi_n$ periodically to all $n\in\Z$ and impose the Bloch condition $\psi_{n+3}=z\psi_n$, $z\ne0$ (Lecture 9’s reading). Show that it becomes $L(z)\psi=\lambda\psi$ for $\psi=(\psi_1,\psi_2,\psi_3)$, where $L(z)$ is $L$ of Exercise 3 (iii) with $(L(z))_{13}=a_3/z$ and $(L(z))_{31}=a_3z$. Then show that $\det(\lambda-L(z))=\transfer(\lambda)-z-1/z$. (ii) Show that $\tr L(z)$ and $\tr L(z)^{2}$ do not depend on $z$, and that $\tr L(z)^{3}=\tr L^{3}+3(z+1/z-2)$. In what sense is this independent of $z$ as a function on phase space? (iii) (Computational.) For $N=6$ take the state $q=0.35\,(0,0.9,0.5,-0.4,0.3,-0.2)$, $p=0.35\,(0.9,-0.5,0.6,0.2,-0.8,-0.4)$ with its mean subtracted. Compute $\transfer(\lambda)$ by multiplying the site matrices, plot it with the lines $\transfer=\pm2$, and find the band edges as the eigenvalues of $L(1)$ and $L(-1)$. Integrate the motion to $t=20$ and check that the band edges do not move while $p_1$ does.

The open chain’s linear bracket. For the open two-particle Toda chain, $L=\begin{pmatrix}b_1&a\\a&b_2\end{pmatrix}$ with $a=e^{(q_1-q_2)/2}$ and $b_n=p_n$ (Lecture 8), Lecture 9’s reading asks for a constant $\rmat$ with $\pb{\leg L1}{\leg L2}=\comm{\rmat}{\leg L1+\leg L2}$. (i) Compute the sixteen brackets $\pb{L_{ab}}{L_{cd}}$, show that $\pb{\leg L1}{\leg L2}=\tfrac a2\big[(E_{12}+E_{21})\otimes(E_{11}-E_{22})-(E_{11}-E_{22})\otimes(E_{12}+E_{21})\big]$, and write it as a $4\times4$ matrix. (ii) Compute $\comm{\rmat}{\leg L1+\leg L2}$ for $\rmat=\tfrac12\big(E_{21}\otimes E_{12}-E_{12}\otimes E_{21}\big)$ as a $4\times4$ matrix, and compare entry by entry. (iii) Prove that any $L$ with $\pb{\leg L1}{\leg L2}=\comm{\rmat}{\leg L1+\leg L2}$, for a constant $\rmat$, has $\pb{\tr L^{j}}{\tr L^{k}}=0$ for all $j,k\ge1$.