Exercise Sheet 15

(i) For $v\ne0,-\ii$ show that $B(v)\ket0$ is a plane wave, with amplitudes in the ratio $e^{\ii k}=v/(v+\ii)$ from site to site, and that $v=\ii e^{\ii k}/(1-e^{\ii k})$ when $e^{\ii k}\ne1$. (ii) Find $B(0)\ket0$ and $B(-\ii)\ket0$, and show that $B(v)\ket0/v^{N-1}\to\ii S^{-}\ket0$ as $v\to\infty$, where $S^{-}=\sum_nS^{-}_n$, so $S^{-}\ket0=\sum_n\ket n$. (iii) Using the exchange relations, show that for $\lambda\ne v$ $$\begin{gather*} \transfer(\lambda)B(v)\ket0=\Lambda(\lambda)\,B(v)\ket0+\frac{\ii}{\lambda-v}\big[(v+\ii)^{N}-v^{N}\big]B(\lambda)\ket0,\\ \Lambda(\lambda)=(\lambda+\ii)^{N}f(\lambda-v)+\lambda^{N}f(v-\lambda), \end{gather*}$$ and deduce that for $v\ne0,-\ii$, $B(v)\ket0$ is an eigenvector of every $\transfer(\lambda)$ if and only if $e^{\ii kN}=1$. (iv) For such $v$, find the energy from $\Lambda$.

Take $N=3$. (i) What are the eigenvalue of $\transfer(\lambda)$ and the energy on $\ket0$? (ii) Find the regular solutions $v$ of the Bethe equation for $M=1$, their momenta $k$ and rapidities, and show that $\Lambda(\lambda)=2\lambda^{3}+3\ii\lambda^{2}-\ii e^{-\ii k}$ on each. What are the energy and the eigenvalue of $U$? (iii) Letting $v\to\infty$ in Exercise 1 (iii), show that $\transfer(\lambda)S^{-}\ket0=\big[(\lambda+\ii)^{N}+\lambda^{N}\big]S^{-}\ket0$ for every $N$.

(i) From the $RTT$ relation entry by entry, with $(i,j,k,l)=(2,1,1,2)$ and $(2,2,1,1)$, show that for $\lambda\ne\mu$ $$C(\lambda)B(\mu)-B(\mu)C(\lambda)=g(\lambda-\mu)\big[D(\lambda)A(\mu)-D(\mu)A(\lambda)\big],$$ and that $C(\lambda)C(\mu)=C(\mu)C(\lambda)$ for all $\lambda$, $\mu$. (ii) Let $\bra0$ be the row vector of $\ket0$. Show that for $u\ne v$ $$\bra0C(u)B(v)\ket0=\frac{\ii}{u-v}\big[u^{N}(v+\ii)^{N}-v^{N}(u+\ii)^{N}\big].$$ (iii) Find the limit $u\to v$, and show that for a real rapidity it is $-\norm{B(v)\ket0}^{2}$: the sign $(-1)^{M}$ of Lecture 15’s reading on norms, at $M=1$.

(i) Show that $\{v_1,v_2\}=\{0,-\ii\}$ satisfies Lecture 15’s Bethe equations for $M=2$, for every $N$. (ii) Show that $B(0)=\ii^{N}U\,S^{-}_N$, where $S^{-}_N$ lowers the spin of site $N$. (iii) Deduce that $B(0)B(-\ii)\ket0=0$ for every $N$, although neither factor alone annihilates $\ket0$. What does the algebraic Bethe ansatz give for this pair, and why is Lecture 15’s energy formula stated for regular roots?

(i) Show that $f(a)g(b)+g(a)g(b-a)=g(b)f(a-b)$ for $a$, $b$, $a-b$ non-zero, and that the same identity holds for $F(u)=f(-u)$ and $G(u)=-g(u)$ in place of $f$ and $g$. (ii) For pairwise distinct $\lambda$, $v_1$, $v_2$, derive from the exchange relations $$\begin{split} A(\lambda)B(v_1)B(v_2)={}&f(\lambda-v_1)f(\lambda-v_2)\,B(v_1)B(v_2)A(\lambda)+g(\lambda-v_1)f(v_1-v_2)\,B(\lambda)B(v_2)A(v_1)\\ &+g(\lambda-v_2)f(v_2-v_1)\,B(\lambda)B(v_1)A(v_2), \end{split}$$ and the same for $D$ with $f(u)\to f(-u)$, $g(u)\to-g(u)$: the case $M=2$ of Lecture 15’s lemma. (iii) Acting on $\ket0$, find the unwanted coefficient of $B(\lambda)B(v_2)\ket0$ in $\transfer(\lambda)B(v_1)B(v_2)\ket0$.

Let $Q(\lambda)=\prod_{j=1}^{M}(\lambda-v_j)$ with pairwise distinct $v_j$. (i) Show that Lecture 15’s $\Lambda(\lambda)=(\lambda+\ii)^{N}\prod_j\frac{\lambda-v_j-\ii}{\lambda-v_j}+\lambda^{N}\prod_j\frac{\lambda-v_j+\ii}{\lambda-v_j}$ satisfies $\Lambda(\lambda)Q(\lambda)=(\lambda+\ii)^{N}Q(\lambda-\ii)+\lambda^{N}Q(\lambda+\ii)$. (ii) For $M=1$, show that $\Lambda$ is a polynomial if and only if $(v+\ii)^{N}=v^{N}$. (iii) For $N=4$ the tutorial’s ground state has $Q(\lambda)=\lambda^{2}+\ii\lambda-\tfrac13$. Without using its roots, find $\Lambda$ by dividing the right side of the T–Q relation by $Q$, and read off $K$ and $E$.

The next charge, on magnons. Lecture 15’s reading on the tower of charges states that for $N\ge4$, $\frac{\dd^{2}}{\dd\lambda^{2}}\ln\transfer\big|_0=N+\sum_n\comm{P_{n,n+1}}{P_{n+1,n+2}}$. (i) Let $v_1,\dots,v_M$ be regular roots of the Bethe equations with $\Psi\ne0$, and $N\ge3$. Show that on $\Psi$ the operator $\frac{\dd^{2}}{\dd\lambda^{2}}\ln\transfer\big|_0$ has the eigenvalue $N+\sum_j\big(2\ii\sin2k_j-4\ii\sin k_j\big)$: a sum of one function of each momentum. (ii) Compute $\sum_n\comm{P_{n,n+1}}{P_{n+1,n+2}}$ on $\ket x$, and hence on a magnon $\sum_xe^{\ii kx}\ket x$ with $e^{\ii kN}=1$ and $e^{\ii k}\ne1$, and check the reading’s formula on it for $N\ge4$. (iii) Check (i) on the four-site ground state of Exercise 6.