Exercise Sheet 17

(i) Show that $2\pi\rho_t=(p_0')^{\mathrm{dr}}$ for every $\vartheta$. (ii) At $T=0$, with $\varepsilon(\pm q)=0$, show that $\varepsilon'=(\varepsilon_0')^{\mathrm{dr}}$ on $(-q,q)$. (iii) The edge condition matters: for $f(k)=k$, which does not vanish at $\pm q$, compute $(f^{\mathrm{dr}})'(0)$ and $(f')^{\mathrm{dr}}(0)$ at $c=q=1$.

(i) Show that $\ln\binom AB=A\ln A-B\ln B-(A-B)\ln(A-B)+O(\ln A)$ for $0<B<A$. (ii) Deduce Lecture 17’s entropy density $s=\int\big[\rho_t\ln\rho_t-\rho\ln\rho-\rho_h\ln\rho_h\big]$, and show that its integrand is $\rho_t\,\sigma(\vartheta)$ with $\sigma(\vartheta)=-\vartheta\ln\vartheta-(1-\vartheta)\ln(1-\vartheta)$. (iii) Where is $\sigma$ largest, and what is it at $\vartheta=0$ and $1$? (iv) Show that $F(x,y)=y\,\sigma(x/y)$ is concave on $0<x<y$, and deduce that $s$ is concave in $\rho$ (Lecture 17 uses this to show that the Yang–Yang point is a minimum).

(i) With $K\to0$ (impenetrable), show that the Yang–Yang equation gives free fermions: $\varepsilon=k^{2}-\mu$, $\rho_t=\frac1{2\pi}$, and $n=\partial P_{\mathrm F}/\partial\mu$ is the Fermi–Dirac density. (ii) As $\mu/T\to-\infty$ (dilute), show that $P_{\mathrm F}/T^{3/2}\to e^{\mu/T}/(2\sqrt\pi)$, Lecture 17’s Boltzmann asymptote. (iii) With $k=\sqrt T\,\kappa$ and $c=\sqrt T\,\hat c$, show that $\varepsilon/T$ depends on $(\kappa,\mu/T,\hat c)$ only, and $P/T^{3/2}$ on $(\mu/T,\hat c)$ only.

Lecture 17 proves that as $c\to\infty$, $\varepsilon(k)=k^{2}-\mu-2P/c+O(c^{-2})$ uniformly on compact sets. (i) Deduce that $P=P_{\mathrm F}(\mu+2P/c)+O(c^{-2})$, and compare with hard rods of length $a$, $P=P_{\mathrm F}(\mu-aP)$: what is $a$? (ii) (Computational.) At $c=100$, $T=\mu=1$, solve the Yang–Yang equation by the tutorial’s iteration. By how much does $P$ exceed $P_{\mathrm F}(\mu)$, and how well does $P=P_{\mathrm F}(\mu+2P/c)$, solved for $P$, account for it?

(Computational.) Take $\gamma=4$ and $n=1$ (so $c=4$), at $T=0$. (i) Find $q$ (from $n=\int_{-q}^{q}\rho=1$), $\mu$ (from $\varepsilon(\pm q)=0$), and the sound velocity $v_s=\varepsilon'(q)/(2\pi\rho_t(q))$. (ii) With $p(k)=k+\int_{-q}^{q}\theta(k-k')\rho(k')\dd k'$, $\theta(k)=2\arctan(k/c)$, plot Type I, $(p(k)-p(q),\varepsilon(k))$ for $k>q$, and Type II, $(p(q)-p(k),-\varepsilon(k))$ for $\abs k<q$. Show that Type II spans $0\le p\le2\pi n$, vanishing at both ends, and find its top. (iii) Show that both branches leave $p=0$ with slope $v_s$.

(Computational.) Take $c=1$ and the $\mu$ of the $T=0$ sea with $q=1.5$. (i) At $T=0$ give $\mu$, $\gamma$, $e$ and $v=\varepsilon'(q)/(2\pi\rho_t(q))$, and show that $P(0)=-\frac1{2\pi}\int_{-q}^{q}\varepsilon=\mu n-E/L$. (ii) At $T=0.02$, solve the Yang–Yang equation at the same $\mu$ and compare $P(T)-P(0)$ with the Sommerfeld form $\pi T^{2}/(6v)$ of a Luttinger liquid with central charge $1$. (iii) At $T=0.25$ check the thermodynamics: $n=\int\rho$ against $\partial P/\partial\mu$, and Lecture 17’s $s$ against $\partial P/\partial T$.

Free spins in Takahashi’s equations. Lecture 17’s reading gives, for the Heisenberg chain at $T>0$ in a field $h\ge0$, $\ln\eta_n=-\delta_{n1}\frac{\pi J}Ts+s*\ln[(1+\eta_{n-1})(1+\eta_{n+1})]$, $\eta_0\equiv0$, $f=-J\ln2-T\int s\ln(1+\eta_1)$, with $s(\lambda)=1/(2\cosh\pi\lambda)$ and the field entering only through $\lim_n(\ln\eta_n)/n=h/T$. (i) At $J=0$ and $h=0$, look for constant $\eta_n$. Show that they must satisfy $\eta_n^{2}=(1+\eta_{n-1})(1+\eta_{n+1})$, and that $\eta_n=n(n+2)$ does, with $f=-T\ln2$. (ii) Show that $1+\eta_n=\big(\sinh((n+1)x)/\sinh x\big)^{2}$ also does, for any $x>0$, with $(\ln\eta_n)/n\to2x$ and $f=-T\ln(2\cosh x)$. (iii) Compare with a free spin in $H=-h\sum_nS^{z}_n$, $h>0$, and deduce why the boundary condition is $h/T$, not the $2h/T$ of a book whose field term is $-2h\sum_nS^{z}_n$. (iv) Show that, as long as its terms are positive, a solution of the recursion with $\eta_0=0$ is fixed by $\eta_1$, and that $\eta_n>0$ for every $n$ only if $\eta_1\ge3$. Deduce that the positive constant solutions are exactly those of (i) and (ii).