Dynamics of a Quantum Phase Transition

We present two approaches to the dynamics of a quench-induced phase transition in the quantum Ising model. One follows the standard treatment of thermodynamic second order phase transitions but applies it to the quantum phase transitions. The other approach is quantum, and uses Landau-Zener formula for transition probabilities in avoided level crossings. We show that predictions of the two approaches of how the density of defects scales with the quench rate are compatible, and discuss the ensuing insights into the dynamics of quantum phase transitions.

Figure 1

Density of kinks ($\nu$) in quantum Ising model after a quench that starts in a ground state at $J=5W$ and ends at $J=0$, plotted versus the dimensionless quench rate ${\tau }_0/{\tau }_Q=\hslash v/4W^2$ for $N=50,60,70,80,90,100$ (solid lines; bottom to top). The simulation data are consistent with the scaling ${\widehat{\nu }}_{\mathrm{KZM}}\sim \sqrt{{\tau }_0/{\tau }_Q}$, Eq. (10), predicted by the Kibble-Zurek mechanism. (See 19 for details on the numerical method). Agreement improves with $N$; for 100 spins, a fit gives $\nu \sim {\tau }_Q^{-0.58}$ (dashed line). As in the classical case [5] Eq. (10) is an overestimate; the best fit is $\nu \simeq 0.16{\widehat{\nu }}_{\mathrm{KZM}}$.

Figure 2

(a) Energies of lowest excitations (see 19) for $N=20$. The energies of the ground state and the first accessible excited state are the lowest (horizontal) solid line and the second lowest solid line. (b) (i) Quench time ${\tau }_Q/{\tau }_0=4W^2/\hslash v$ that yields $f$ of 99%, and (ii) the fidelity for a fixed ${\tau }_Q=200\hslash /W=400{\tau }_0$ as a function of the number of spins $N$ in the quantum Ising chain. A power-law fit to the data corresponding to ${\tau }_{Q_{99\%}}$ gives a power of 1.93 (LZF yields 2, as would KZM). The best fit for the fidelity with Landau-Zener dependence $f=1-\exp \{-aW{\tau }_Q/\hslash N^2\}$ yields $a\simeq 59$, compared to theoretical $a=2{\pi }^3\simeq 62$ based on the steeper slope in the marked energy level above [$a=4{\pi }^3\simeq 124$ when the more shallow slope in Fig. 2a consistent with Eq. (3) is taken]. (c) Upper (dashed lines) and lower (solid lines) bounds on fidelity as a function of ${\tau }_0/{\tau }_Q=\hslash v/4W^2$ for $N=90$ (i), $N=70$ (ii), $N=50$ (iii), $N=30$ (iv). Fits to LZF (dotted lines) lie between these bounds, and for $f>0.6$ give $a\simeq 59$ (i), 59 (ii), 57 (iii), and 54 (iv).

Figure 3

Number of kinks in chains ($N=50,60,70,80,90,100$, spins, bottom to top) after a quench versus the quench rate ${\tau }_0/{\tau }_Q=\hslash v/4W^2$. Both the scaling $\widehat{\nu }\sim 1/\sqrt{\tau }Q$ predicted by KZM (red lines), Eq. (10), and the LZF estimate $1-f$ (blue lines) are valid where expected. The red lines are linear fits in the range (0.025,0.25) yielding slopes between 0.66 and 0.58. The blue lines are the fit results from Fig. 2c. Numerical data (obtained using the method developed in 19) include these used in Fig. 1. However, we now go beyond the expected range of validity of KZM: For quenches slow enough to create “less than a kink” LZF provides reliable predictions, while very fast quenches are “all impulse,” and—according to KZM—the number of kinks should saturate, as is indeed seen.