Exercise Sheet 19

(i) Show that $\norm Q\le N\norm q$, and that $\langle Q\rangle=N\langle q\rangle$ in a translation-invariant state. (ii) Show that the local charges of range at most $1$ form a space of dimension $4$, and those of range at most $2$ a space of dimension $13$ when $N\ge3$, below Lecture 19’s bound $4^{2}=16$. What is it when $N=2$?

(i) For $\omega\ne0$ show that $\big\lvert\frac1T\int_0^{T}e^{-\ii\omega t}\dd t\big\rvert\le\frac2{\abs\omega T}$. (ii) Let $H=H_c\otimes\Id$, a chain with Hamiltonian $H_c$ and eigenbasis $\ket n$, beside a spectator spin that feels nothing, and start from $\ket{\varphi}\otimes\ket{{+x}}$. Show that $\langle\sigma^{x}_{\mathrm s}\rangle(t)=1$ for all $t$; that $\sum\abs{c}^{2}\langle\sigma^{x}_{\mathrm s}\rangle$ over the product eigenbasis $\ket n\otimes\ket\uparrow$, $\ket n\otimes\ket\downarrow$ is $0$; and that Lecture 19’s dephased state gives $1$. What went wrong with the second?

(i) Show that $\cos2\chi_k=\frac{a_ka^{0}_k+b_k^{2}}{\varepsilon_k\varepsilon^{0}_k}$ and $\sin2\chi_k=\frac{4J(h_0-h)\sin k}{\varepsilon_k\varepsilon^{0}_k}$. (ii) For $J=1$, $h_0=0.4\to h=1.9$, compute $n_{\pi/2}$, and show that $n_k$ is even in $k$. (iii) Lecture 19 warns that the sum angle $\theta_k+\theta^{0}_k$ gives a plausible and wrong $n_k$. For this quench, how wrong, at worst over $k$?

(Computational.) Take the quench of Exercise 3. (i) Compute $m^{z}(\infty)$. (ii) Compute the energy per site $e=\int\frac{\dd k}{2\pi}\varepsilon_k(n_k-\tfrac12)$ after the quench, the temperature $T$ at which the Gibbs state of $H(h)$ has this energy (thermal occupations $1/(e^{\varepsilon_k/T}+1)$), and the Gibbs value of $m^{z}$. (iii) At $h=J$, show that $\cos2\theta_k=\varepsilon_k/(4J)$, and deduce that $m^{z}=-e/(2J)$ in every state diagonal in the occupations: there the observable cannot tell the two ensembles apart.

(i) Show that $\sin2\theta_k\sin2\chi_k=\frac{8J^{2}(h_0-h)\sin^{2}k}{\varepsilon_k^{2}\,\varepsilon^{0}_k}$. (ii) For $h,h_0\ne J$ and $h_0\ne h$, show that it has a double zero at the stationary points $k=0,\pi$ of $\varepsilon_k$; at $h_0=J$ (with $h\ne J$), only a simple zero in $\abs k$ at $k=0$. What if $h_0=h$? (Lecture 19’s stationary-phase step turns these into decays like $t^{-3/2}$ and $t^{-1}$.) (iii) What does $m^{z}(t)$ do at $h=0$?

Lecture 19’s proof of the relaxation theorem needs the pre-quench ground state in the even sector. Take $J,h_0>0$. (i) In the basis of joint eigenvectors of every $\sigma^{x}_n$, show that $H(h_0)$ has off-diagonal entries $-h_0$ between configurations differing in one spin, and $0$ otherwise. (ii) Perron–Frobenius (an outside fact): a real symmetric matrix whose off-diagonal entries are $\le0$, and which connects every basis vector to every other through non-zero entries, has a non-degenerate lowest eigenvalue with a strictly positive eigenvector. Deduce that the ground state of $H(h_0)$ is unique and, in this basis, positive. (iii) Deduce that it is even.

Which charges are local. For an even $f$ let $Q[f]=\sum_kf(k)(n_k-\tfrac12)$, whose real-space hopping and pairing coefficients, in each sector, are the Fourier coefficients of $A=fa_k/\varepsilon_k$ and $B=fb_k/\varepsilon_k$ (Lecture 19’s sketch). Take $J,h>0$ with $h\ne J$. (i) Show that $(h+J\cos k)a_k-J\sin k\,b_k=2(h^{2}-J^{2})$. (ii) Deduce that $Q[f]$ has finite range if and only if $f=\varepsilon\cdot P$ with $P$ a trigonometric polynomial. (iii) (Computational.) For $f=1$ and $h=0.5J$, compute the ratios $\abs{\hat A_{m+1}/\hat A_m}$ of consecutive Fourier coefficients of $a_k/\varepsilon_k$ at $m=8$, $16$, $24$, $32$. What do they say about $Q[1]$?