Exact calculation of the magnetocaloric effect in the spin-$\frac{1}{2}$ $XXZ$ chain

We calculate the entropy and cooling rate of the antiferromagnetic spin-$\frac{1}{2}$ $XXZ$ chain under an adiabatic demagnetization process using the quantum transfer-matrix technique and nonlinear integral equations. The limiting case of the Ising chain (corresponding to infinitely large anisotropy) is presented for comparison. Our exact results for the Heisenberg chain are used as a cross-check for the numerical exact diagonalization as well as quantum Monte Carlo simulations and allow us to benchmark the numerical methods. Close to field-induced quantum phase transitions we observe a large magnetocaloric effect. Furthermore, we verify universal low-temperature power laws in the cooling rate and entropy, in particular, linear scaling of entropy with temperature $T$ in the gapless Luttinger-liquid state and scaling as $\sqrt{T}$ at field-induced transitions to gapped phases.

Figure 1

(Color online) Zero-temperature phase diagram of the spin-$\frac{1}{2}$ $XXZ$ chain in a magnetic field.

Figure 2

(Color online) Cooling rate ${\Gamma }_H$ for the Heisenberg chain $\left(\Delta =1\right)$. Shown are ED for $N=20$ sites, QMC for $N=100$ sites, and the exact solution for the infinite system. The different panels are for (a) $T/J=0.3025$, (b) 0.1472, and (c) 0.1037.

Figure 3

(Color online) Cooling rate ${\Gamma }_H$ for different values of the anisotropy. Shown is the exact solution for the infinite system. The different panels are for (a) $T/J=0.05$, (b) 0.5, and (c) 5.

Figure 4

(Color online) Cooling rate ${\Gamma }_H$ of the Ising model ($\Delta \to \infty$ limit of the $XXZ$ model). Shown is the exact solution for the infinite system. The different panels show (a) the cooling rate for a wider range of temperatures and (b) the behavior at low temperatures in the vicinity of the critical point.

Figure 5

(Color online) Entropy per spin $S\left(H,T\right)$ for $\Delta =1$. The isentropes are for $S=0.05,0.1,\dots ,0.6$.

Figure 6

(Color online) Entropy per spin $S\left(\Delta ,T\right)$ for different values of (a) $H=0$, (b) $1.5J$, and (c) $2.5J$. The black lines are the isentropes for $S=0.05,0.1,\dots ,0.6$. Panel (a) contains three additional isentropes with $S=0.005$, 0.01, and 0.02 which are shown by white lines.

Figure 7

(Color online) Entropy per spin $S$ of the $XXZ$ chain at $H=0$ for $\Delta =0.5$, 1, and 1.5. For $\Delta =0.5$ and 1 we also show the Luttinger-liquid expression (27) with the velocity $v$ given by Eq. (30).