Exercise Sheet 16

(i) Show that $\hat a_n(\omega)=e^{-n\abs\omega/2}$. (ii) Transform the ground-state equation $\rho+a_2*\rho=a_1$ to find $\hat\rho(\omega)=1/(2\cosh(\omega/2))$. Lecture 16 states that this is the transform of $1/(2\cosh\pi\lambda)$: prove it. (Parts (i) and (ii) use the residue theorem and the uniqueness of the Fourier transform, neither among Lecture 0’s prerequisites; Lecture 16 uses both.) (iii) Show that $\int_{-\infty}^{\lambda}\rho=\frac1\pi\arctan e^{\pi\lambda}$, and find the fraction of the ground state’s roots with $\abs\lambda<1$ in the thermodynamic limit.

(i) For the gas put $z(k)=k+\frac1L\sum_l\theta(k-k_l)$, so that $Lz(k_j)=2\pi I_j$. Assuming, as Lecture 16 does, that roots and holes fill out densities $\rho$, $\rho_h$, derive $\rho+\rho_h=\frac1{2\pi}+\frac1{2\pi}\int K(k-k')\rho(k')\dd k'$, and Lieb’s equation for the ground state. (ii) Show that $z$ is strictly increasing for every set of real roots. Is $\zeta$? (iii) Write both models as $Lp_0(\lambda_j)+\sum_l\varphi(\lambda_j-\lambda_l)=2\pi I_j$ and find the two kernels $K=\varphi'$.

(i) For $N$ free fermions on the ring, $N$ odd, with momenta $2\pi I/L$, $I=-\tfrac{N-1}2,\dots,\tfrac{N-1}2$, show that $E/N=\frac{\pi^{2}n^{2}}3\big(1-\frac1{N^{2}}\big)$, which tends to Lecture 16’s Tonks–Girardeau value $\pi^{2}n^{2}/3$. (ii) For $\gamma\gg1$, use $K(k-k')\approx2/c$ on the sea to show that $\rho\approx\frac1{2\pi}(1+\frac2\gamma)$, that $q\approx\pi n/(1+\frac2\gamma)$, and that $e(\gamma)=\frac{\pi^{2}}3\big(1-\frac4\gamma+O(\gamma^{-2})\big)$. Compare $q$ with Lecture 11’s ring lengthened by $2N/c$.

(Computational.) Set $c=1$. (i) Solve Lieb’s equation at $q=1$ by the Nyström method of Lecture 16’s tutorial with $40$ Gauss–Legendre points and give $n$, $E/L$, $\gamma$ and $e$. How far is $n$ from this value with $5$, $10$ and $20$ points? (ii) By Hellmann–Feynman the interaction energy is $c\,\partial E/\partial c$ at fixed $N$, $L$. Show that the kinetic and interaction energies are equal exactly where $\dd\ln e/\dd\ln\gamma=\tfrac12$, and find that $\gamma$ and $e/2$ there. (iii) Plot $e(\gamma)$ for $0.02\le\gamma\le300$ on a logarithmic axis, with the two series of Lecture 16’s reading, and give $e(0.1)$ and $e(40)$ beside them.

(Computational.) Take $N=100$, $M=50$ and $I_j=-\tfrac{49}2,\dots,\tfrac{49}2$. (i) Solve the logarithmic Bethe equations by damped Newton, and give $E_0/N$. (ii) Plot the local density $2/[N(\lambda_{j+1}-\lambda_{j-1})]$ against $1/(2\cosh\pi\lambda)$, and give the worst relative deviation five roots in from each end. (iii) Lecture 16’s reading predicts $E_0(N)=Ne_0-\pi c\,v/(6N)+o(N^{-1})$ with $v=\pi J/2$. Compute $-(E_0-Ne_0)\,6N/\pi$ at $N=100$, $200$, $400$, and estimate the central charge $c$.

(Computational.) Lecture 16’s tutorial meets two roots of $N=22$, $M=2$ that share $I_1=I_2=\tfrac{19}2$. (i) Solve the two logarithmic Bethe equations with $I_1=I_2=\tfrac{19}2$ by Newton from $(2.23,2.37)$, to $12$ decimals. (ii) Evaluate $\zeta'$ at both roots and at their midpoint. Which hypothesis of Lecture 16’s vacancy count fails, and which of its statements survive? (iii) Build Lecture 10’s two-magnon wavefunction on them, $\psi(x_1,x_2)=e^{\ii(k_1x_1+k_2x_2)}+A_{21}e^{\ii(k_2x_1+k_1x_2)}$ ($x_1<x_2$), with $e^{\ii k_j}=\frac{\lambda_j+\ii/2}{\lambda_j-\ii/2}$ and $(\lambda_1-\lambda_2+\ii)A_{21}=\lambda_1-\lambda_2-\ii$, and check that it is an eigenvector of $H$ ($J=1$).

Why Lieb’s equation has one positive solution. Fix $c>0$ and $q>0$, and let $(\mathcal K f)(k)=\frac1{2\pi}\int_{-q}^{q}K(k-k')f(k')\dd k'$ on continuous functions on $[-q,q]$ with the sup norm. (i) Show that $\norm{\mathcal K}=\frac2\pi\arctan\frac qc<1$, and deduce that Lieb’s equation has exactly one continuous solution, $\rho=\sum_{m\ge0}\mathcal K^{m}\frac1{2\pi}$. (ii) Show that $\rho\ge\frac1{2\pi}$. (iii) For $q<q'$ show that $\rho_{q'}\ge\rho_q$ on $[-q,q]$, and deduce that $n(q)$ is strictly increasing, so that $\gamma=c/n$ is strictly decreasing in $q$: the step Lecture 16’s proof of Lieb’s theorem uses.