Exercise Sheet 10

(i) For two spins $\tfrac12$, show that $\mathbf S\cdot\mathbf S'-\tfrac14=\tfrac12(P-1)$ by applying both sides to $\ket{\uparrow\uparrow}$ and $\ket{\uparrow\downarrow}$. (ii) Lecture 10 counts $\binom NM-\binom N{M-1}$ multiplets of spin $\tfrac N2-M$, $0\le M\le\tfrac N2$. List the counts for $N=4$ and $N=6$, and check that they account for all $2^{N}$ states. (iii) Show that each bond term $\tfrac J2(P_{n,n+1}-1)$ has eigenvalues $0$ and $-J$. Deduce that for $J<0$ the state $\ket\Uparrow$ with every spin up is a ground state, and for $J>0$ a highest level.

(i) Show that $H\ket x=\tfrac J2\big(\ket{x-1}+\ket{x+1}-2\ket x\big)$ for one down spin at $x$, and that $\ket k=\sum_xe^{\ii kx}\ket x$ with $e^{\ii kN}=1$ has $E(k)=J(\cos k-1)$. (ii) Show that $e^{\ii k}=(\lambda+\ii/2)/(\lambda-\ii/2)$ exactly when $\lambda=\tfrac12\cot\tfrac k2$, for $0<k<2\pi$, and that then $E=-2J/(4\lambda^{2}+1)$. (iii) Show that $S^{+}\ket k=0$ for $k\ne0$ and not for $k=0$, and identify $\ket{k=0}$.

(i) Lecture 10 shows that Bethe’s two-magnon ansatz $\psi(x_1,x_2)=A_{12}z_1^{x_1}z_2^{x_2}+A_{21}z_2^{x_1}z_1^{x_2}$, $z_j=e^{\ii k_j}$, solves the equations exactly when the contact condition $\psi(x,x)+\psi(x+1,x+1)-2\psi(x,x+1)=0$ holds for all $x$. For finite rapidities, show that it holds exactly when $(\lambda_1-\lambda_2+\ii)A_{21}=(\lambda_1-\lambda_2-\ii)A_{12}$, that is, $A_{21}/A_{12}=S(\lambda_1-\lambda_2)$ where $\lambda_1-\lambda_2\ne-\ii$ and $A_{12}\ne0$, and $A_{12}=0$ at the pole $\lambda_1-\lambda_2=-\ii$. (ii) Show that $\abs{S(\lambda)}=1$ for real $\lambda$, that $S(\lambda)S(-\lambda)=1$, and that $S(0)=-1$. What does $S(0)=-1$ say about two magnons with equal rapidities? (iii) Show that at total momentum $K=k_1+k_2=\pi$ the energy $E(k_1)+E(k_2)$ is $-2J$ however $K$ is split between the two. Why does this make the absence of new momenta after a collision a non-trivial fact?

Take $N=4$ and $M=2$. (i) Solve the logarithmic Bethe equations $N\theta_1(\lambda_j)-\sum_{l\ne j}\theta_2(\lambda_j-\lambda_l)=2\pi I_j$ for real roots with the quantum numbers $\{I_1,I_2\}=\{\tfrac12,-\tfrac12\}$: show that the roots are $\lambda=\pm\frac1{2\sqrt3}$, and find $E$ and $K$. (ii) Write $\sum_n\mathbf S_n\cdot\mathbf S_{n+1}=\tfrac12\big(\mathbf S^{2}-\mathbf S_A^{2}-\mathbf S_B^{2}\big)$ with $\mathbf S_A=\mathbf S_1+\mathbf S_3$ and $\mathbf S_B=\mathbf S_2+\mathbf S_4$, and use it to find the energies of the two spin-$0$ multiplets, the highest-weight states with $M=2$. Which one is the Bethe state of (i)? What does Lecture 10 say about the other?

(i) Show that $(x+\ii n/2)/(x-\ii n/2)=-e^{-\ii\theta_n(x)}$ for real $x$, and take logarithms of the Bethe equations $\big(\frac{\lambda_j+\ii/2}{\lambda_j-\ii/2}\big)^{N}=\prod_{l\ne j}\frac{\lambda_j-\lambda_l+\ii}{\lambda_j-\lambda_l-\ii}$ for real roots to get $N\theta_1(\lambda_j)-\sum_{l\ne j}\theta_2(\lambda_j-\lambda_l)=2\pi I_j$, and show that $I_j\in\Z+\tfrac{N-M+1}2$. (ii) Using $k=\pi-\theta_1(\lambda)$, derive $K=\pi M-\frac{2\pi}N\sum_jI_j$ modulo $2\pi$. (iii) Show that for one magnon the equation is $kN=2\pi n$, and compare it with the Bohr–Sommerfeld rule of Lecture 6 for a particle going once round a ring of circumference $N$.

(Computational: Lecture 10’s table and figure of the $N=6$, $M=2$ check.) (i) At each total momentum $K=2\pi m/6$, find all the regular solutions of the Bethe equations for $M=2$ from the polynomial of Lecture 10’s tutorial, $-(w+1)z^{7}+2wz^{6}+2z^{2}-(w+1)z=0$ with $w=e^{\ii K}$ and $z=e^{\ii k_1}$: discard $z=0$ and $z=1$, and keep the roots whose two rapidities are finite and distinct. Add the singular pair ($E=-J$, $K=\pi$). (ii) Separately, diagonalise $H$ on the $15$ states with two down spins, keep the eigenvectors killed by $S^{+}$, and sort them by momentum. Compare the two lists. (iii) Find the quantum numbers $I_j$ of the real solutions. (iv) Plot the nine levels in the plane of $K/\pi$ and $E/J$: the exact diagonalisation as rings, the Bethe solutions as dots marked by type (real, string, singular). This is Lecture 10’s figure of the check.

Bethe vectors are highest weight, for two magnons. Lecture 10 cites the theorem that a non-zero Bethe vector satisfies $S^{+}\psi=0$. Prove it for $M=2$. Let $\psi=\sum_{1\le x_1<x_2\le N}\psi(x_1,x_2)\ket{x_1,x_2}$ with Bethe’s ansatz, $z_1,z_2\ne1$, $A_{21}/A_{12}=S(\lambda_1-\lambda_2)$, and the periodicity conditions $A_{12}=A_{21}z_1^{N}$, $A_{21}=A_{12}z_2^{N}$ of Lecture 10. (i) Show that $S^{+}\psi=\sum_y\Big(\sum_{x<y}\psi(x,y)+\sum_{x>y}\psi(y,x)\Big)\ket y$. (ii) Sum the geometric series, and show that the coefficient of every $\ket y$ vanishes.