Exercise Sheet 14

(i) On $\C^{2}_a\otimes\C^{2}_n$ show that $P_{an}=\tfrac12(\Id+\boldsymbol\sigma_a\cdot\boldsymbol\sigma_n)$. (ii) Deduce Lecture 14’s matrix form of $\Lop{an}(\lambda)$ in the auxiliary space, with entries $\lambda+\tfrac{\ii}2\pm\ii S^{z}_n$ on the diagonal and $\ii S^{-}_n$, $\ii S^{+}_n$ off it. (iii) Write $\Lop{an}(\lambda)$ as a $4\times4$ matrix in the basis $\ket{\uparrow\uparrow}$, $\ket{\uparrow\downarrow}$, $\ket{\downarrow\uparrow}$, $\ket{\downarrow\downarrow}$ (auxiliary state first). Check that exactly six entries are not identically zero (as functions of $\lambda$), and explain, with the vertex picture of Lecture 14, why two of them are $\lambda+\ii$ and not $\lambda$. (iv) For which $\lambda$ is $\Lop{an}(\lambda)$ not invertible? Show that there it is a multiple of a projector, and that $\Lop{an}(0)=\ii P_{an}$.

Let $\Rmat_{ab}=\alpha+\beta P_{ab}$ with numbers $\alpha$, $\beta$ (which may depend on $\lambda$, $\mu$). (i) Following the proof of Lecture 14’s $RLL$ theorem, show that $$\Rmat_{ab}\Lop{bn}(\mu)\Lop{an}(\lambda)-\Lop{an}(\lambda)\Lop{bn}(\mu)\Rmat_{ab}=\big[\alpha-\ii\beta(\lambda-\mu)\big]\big(C-C^{-1}\big).$$ (ii) Deduce that the non-zero $\Rmat_{ab}$ of this form satisfying $RLL$ are exactly the non-zero multiples of $\Rmat_{ab}(\lambda-\mu)=\lambda-\mu-\ii P_{ab}$. (iii) At $\lambda=\mu$ this $\Rmat$ is a multiple of $P_{ab}$. Show directly that $P_{ab}\Lop{bn}(\lambda)\Lop{an}(\lambda)=\Lop{an}(\lambda)\Lop{bn}(\lambda)P_{ab}$.

Let $\mono$ satisfy $RTT$ with $\Rmat(u)=u-\ii P$. (i) Using $\Rmat(u)\Rmat(-u)=-(u^{2}+1)\Id$ (Lecture 14), show that for $\lambda-\mu\ne\pm\ii$ the $RTT$ relation is equivalent to $\tilde\Rmat_{ab}(\lambda-\mu)\mono_a(\lambda)\mono_b(\mu)=\mono_b(\mu)\mono_a(\lambda)\tilde\Rmat_{ab}(\lambda-\mu)$, with $\tilde\Rmat(u)=u+\ii P$: the other order of the monodromies, with the other sign in $\Rmat$. (ii) Show that $\mono'(\lambda)=\mono(-\lambda)$ satisfies the other order with the same $\Rmat$: $\Rmat_{ab}(\lambda-\mu)\mono'_a(\lambda)\mono'_b(\mu)=\mono'_b(\mu)\mono'_a(\lambda)\Rmat_{ab}(\lambda-\mu)$ for all $\lambda$, $\mu$ (where $\mono$ is a polynomial in $\lambda$, as for the chain). For the Heisenberg chain, what is the Lax operator of $\mono'$? (iii) Classically, let $\mono$ satisfy Lecture 9’s $\pb{\leg{\mono}{1}(\lambda)}{\leg{\mono}{2}(\mu)}=\comm{\rmat(\lambda-\mu)}{\leg{\mono}{1}(\lambda)\leg{\mono}{2}(\mu)}$ with $\rmat(\lambda)=-P/\lambda$. Show that $\mono'(\lambda)=\mono(-\lambda)$ satisfies the same bracket with $-\rmat$ in place of $\rmat$. Then, restoring $\hbar$ ($\Rmat=\lambda-\ii\hbar P=\lambda(\Id+\ii\hbar\rmat)$), expand the relation of (ii) to order $\hbar$ as in Lecture 14’s classical limit, taking the symbol of $\mono'_a\mono'_b-\mono'_b\mono'_a$ to be $\ii\hbar\pb{\leg{\mono'}{1}}{\leg{\mono'}{2}}+O(\hbar^{2})$, and check that it gives the same.

Take the Heisenberg chain with $N=3$. (i) Show that $\tra P_{an}=\Id$, $\tra(P_{an}P_{am})=P_{nm}$ for $n\ne m$, and $\tra(P_{a3}P_{a2}P_{a1})=P_{32}P_{31}$, and deduce $$\transfer(\lambda)=2\lambda^{3}+3\ii\lambda^{2}-\lambda\,(P_{12}+P_{13}+P_{23})-\ii\,P_{32}P_{31}.$$ (ii) Show that $U=P_{32}P_{31}$ sends $x_1\otimes x_2\otimes x_3$ to $x_3\otimes x_1\otimes x_2$, so that $\transfer(0)=-\ii U$ is $-\ii$ times the shift by one site. Show that $U$ commutes with $P_{12}+P_{13}+P_{23}$, and deduce $\comm{\transfer(\lambda)}{\transfer(\mu)}=0$ directly. (iii) Show that $\ii\,\transfer(0)^{-1}\transfer'(0)=P_{12}+P_{23}+P_{31}$, and so that Lecture 10’s Hamiltonian at $N=3$ is $H=\tfrac J2\big(\ii\,\transfer(0)^{-1}\transfer'(0)-3\big)$. (iv) Find the eigenvalues of $\transfer(\lambda)$ and of $H$, with their multiplicities.

One degree of freedom, $\comm qp=\ii\hbar$, $p=-\ii\hbar\,\partial_q$ on functions of $q$. (i) Show that $p\,g(q)=g\,p-\ii\hbar g'$, and by induction $p^{k}g=\sum_{j=0}^{k}\binom kj(-\ii\hbar)^{j}g^{(j)}p^{k-j}$: the symbol formula of Lecture 14 for $F=p^{k}$, $G=g$. (ii) Compute $\comm{p^{2}}{e^{q}}$ exactly, and compare its symbol with $\ii\hbar\pb{p^{2}}{e^{q}}$. (iii) Show that $\comm{p}{g(q)}=\ii\hbar\pb pg$ exactly, for every $g$. Deduce that the commutators of the entries of the quantum Toda site matrix $\begin{pmatrix}\lambda-p&e^{-q}\\ -e^{q}&0\end{pmatrix}$ are exactly $\ii\hbar$ times the Poisson brackets of the corresponding functions. Why does (ii) not contradict this?

Quantise Lecture 9’s periodic Toda chain as in Lecture 14: $\comm{q_m}{p_n}=\ii\hbar\,\delta_{mn}$, $\Lop{n}(\lambda)=\begin{pmatrix}\lambda-p_n&e^{-q_n}\\ -e^{q_n}&0\end{pmatrix}$, $\transfer(\lambda)=\tr\big(\Lop{N}(\lambda)\cdots\Lop{1}(\lambda)\big)=\sum_kc_k\lambda^{N-k}$. (i) For $N=2$, find $c_1$ and $c_2$, and show that $H=\tfrac12c_1^{2}-c_2=\tfrac12(p_1^{2}+p_2^{2})+e^{q_1-q_2}+e^{q_2-q_1}$. (ii) Show directly that $\comm{c_1}{c_2}=0$ for $N=2$. (iii) For $N=3$, show that $c_3=-p_1p_2p_3+p_1e^{q_2-q_3}+p_2e^{q_3-q_1}+p_3e^{q_1-q_2}$, and explain why no ordering question arises. Then verify $\comm{H}{c_3}=0$ by hand, with $H=\tfrac12\sum_np_n^{2}+e^{q_1-q_2}+e^{q_2-q_3}+e^{q_3-q_1}$, and check that the terms of order $\hbar^{2}$ cancel too.

Any spin. Let $\mathbf S$ be spin operators on a space $V$, $\comm{S^{j}}{S^{k}}=\ii\epsilon_{jkl}S^{l}$, and on $\C^{2}_a\otimes V$ put $\Lop{an}(\lambda)=\lambda+\tfrac{\ii}2+\ii\,\boldsymbol\sigma_a\cdot\mathbf S$. Lecture 14 cites that it satisfies $RLL$ with the same $\Rmat$; prove it. (i) With $X=\boldsymbol\sigma_a\cdot\mathbf S$ and $Y=\boldsymbol\sigma_b\cdot\mathbf S$, show that $RLL$ for all $\lambda$, $\mu$ is equivalent to $\comm XY=(Y-X)P_{ab}$. (ii) Prove this identity, using only the commutation relations of $\mathbf S$. (iii) For an irreducible spin $s\ge\tfrac12$, find the eigenvalues of $\Lop{an}(\lambda)$ and their multiplicities, and show that for $s=\tfrac12$ you recover $\lambda+\ii P_{an}$.