Exercise Sheet 18

(i) On one mode, show that $(c+c^{\dagger})(1-2n)=c^{\dagger}-c$, $(c^{\dagger}-c)(1-2n)=c^{\dagger}+c$ and $(1-2n)(c^{\dagger}-c)=-(c^{\dagger}+c)$. (ii) Show that $\sigma^{y}=-\ii(\sigma^{+}-\sigma^{-})$ and that $\sigma^{y}_n\sigma^{y}_{n+1}=-(c_n^{\dagger}+c_n)(c_{n+1}^{\dagger}-c_{n+1})$ for $1\le n\le N-1$. (iii) Show that $\sigma^{y}_N\sigma^{y}_1=\Pi\,(c_N^{\dagger}+c_N)(c_1^{\dagger}-c_1)$, and that with $c_{N+1}=\mp c_1$ in the two sectors it has the bulk form of (ii).

Lecture 18’s reading states that $$H_{XY}=-J\sum_n\Big(\frac{1+\eta}2\sigma^{x}_n\sigma^{x}_{n+1}+\frac{1-\eta}2\sigma^{y}_n\sigma^{y}_{n+1}\Big)-h\sum_n\sigma^{z}_n$$ has the spectrum of Lecture 18’s theorem with $\varepsilon(k)=2\sqrt{(h-J\cos k)^{2}+\eta^{2}J^{2}\sin^{2}k}$ and $e_0$, $e_\pi$ unchanged, and leaves the derivation as an exercise. (i) Show that in each sector $H_{XY}$ is $-J\sum_n\big[(c_n^{\dagger}c_{n+1}+c_{n+1}^{\dagger}c_n)+\eta\,(c_n^{\dagger}c_{n+1}^{\dagger}+c_{n+1}c_n)\big]+2h\sum_nn_n-hN$, with $c_{N+1}=\mp c_1$. (ii) Derive $\varepsilon(k)$ and the unpaired energies. (iii) (Computational.) Check the whole spectrum against exact diagonalisation at $N=8$, and give the lowest level of each sector at $\eta=\tfrac12$, $h=0.7J$.

(i) For $f(k)=\sum_m\hat f_me^{\ii mk}$ with $\sum_m\abs{\hat f_m}<\infty$, show that $\sum_{k\in\mathrm{NS}}f(k)-\sum_{k\in\mathrm R}f(k)=-2N\sum_{j\ \mathrm{odd}}\hat f_{jN}$. (ii) For $h<J$ the lowest odd level is $E_0^{-}=-\tfrac12\sum_{\mathrm R}\varepsilon(k)$. Compute the splitting $E_0^{-}-E_0^{+}$ of the TFIM at $h=0.4J$ for $N=8$, $10$, $12$, and the ratios from $N$ to $N+2$. (iii) At $N=8$ and $h=0.4J$, how far above the ground state is the first level above the doublet?

(i) From $E_0^{+}=-\tfrac12\sum_{\mathrm{NS}}\varepsilon(k)$, show that $e_0=\lim E_0^{+}/N=-\frac1{2\pi}\int_0^{\pi}\varepsilon(k)\dd k$, and evaluate it at $h=0$ and at $J=0$. (ii) At $h=J$, show that $\varepsilon(k)=4J\abs{\sin\frac k2}$ and $e_0=-4J/\pi$. (iii) Show that $e_0$ is unchanged by $J\leftrightarrow h$. (iv) For the XX chain ($\eta=0$ in Exercise 2) with $J>0$, where does $\varepsilon$ vanish?

Let $Q_-=\sum_n\big(\sigma^{x}_n\sigma^{y}_{n+1}-\sigma^{y}_n\sigma^{x}_{n+1}\big)$ (Lecture 18’s tutorial). (i) Show that $\sigma^{x}_n\sigma^{y}_{n+1}-\sigma^{y}_n\sigma^{x}_{n+1}=2\ii\big(\sigma^{+}_n\sigma^{-}_{n+1}-\sigma^{-}_n\sigma^{+}_{n+1}\big)=2\ii\big(c_{n+1}^{\dagger}c_n-c_n^{\dagger}c_{n+1}\big)$ for $n\le N-1$. (ii) Show that in each sector $Q_-=4\sum_k\sin k\,c_k^{\dagger}c_k$, and deduce that $Q_-$ commutes with $H$, and with $H_{XY}$.

Drop the boundary bond: $H_{\mathrm{open}}=-J\sum_{n=1}^{N-1}\sigma^{x}_n\sigma^{x}_{n+1}-h\sum_{n=1}^{N}\sigma^{z}_n$. Let $\alpha_n=c_n+c_n^{\dagger}$ and $\beta_n=\ii(c_n-c_n^{\dagger})$. (i) Show that $\alpha_n^{2}=\beta_n^{2}=1$, that different Majoranas anticommute, and that $H_{\mathrm{open}}=-\ii J\sum_{n=1}^{N-1}\beta_n\alpha_{n+1}-\ii h\sum_n\alpha_n\beta_n$. (ii) At $h=0$, show that $f=\tfrac12(\alpha_1+\ii\beta_N)$ is a fermion commuting with $H_{\mathrm{open}}$, and deduce that every level is at least doubly degenerate.

The edge mode for $0<h<J$. Keep the open chain of Exercise 6, now with $0<h<J$, and let $x=h/J$. (i) Show that $\Gamma=\sum_{n=1}^{N}x^{n-1}\alpha_n$ satisfies $\comm{H_{\mathrm{open}}}{\Gamma}=2\ii h\,x^{N-1}\beta_N$. (ii) Deduce that on the half-infinite chain $\sum_{n\ge1}x^{n-1}\alpha_n$ commutes with the Hamiltonian, and is normalisable exactly when $h<J$: a zero mode localised at the edge. (iii) (Computational.) At $h=0.4J$ compute the splitting of the two lowest levels of $H_{\mathrm{open}}$ for $N=6$, $7$, $8$, and compare with $2h\,x^{N-1}(1-x^{2})$.