Time scales at quantum phase transitions in the Lipkin-Meshkov-Glick model

We report on quantum revivals and classical periodicities of wave packets centered around the ground state within the Lipkin-Meshkov-Glick model. Special attention is paid to the behavior at first- and second-order phase transitions. In line with previous studies, we find that away from criticality, characteristic times exhibit smooth, nonsingular behavior, but upon approaching the transition points they diverge as power laws with associated critical exponents. Finite-size effects are studied and the observed phenomenology is discussed in the framework of the time-energy uncertainty relation.

Figure 1

${\gamma }_x\text{−}{\gamma }_y$ phase diagram of the LMG model. The straight lines ${\gamma }_x=-{\gamma }_y$ and ${\gamma }_x=-{\gamma }_y-4$ denote the trajectories discussed in the text. The various phase transition points along these trajectories are represented by thick dots.

Figure 2

Ground-state expectation value of $J_z/j$ as a function of ${\gamma }_x$. Left panel: 3/4 of the particles occupy the ground state at the transition point in the thermodynamic limit. From top to bottom, the lines correspond to $j=5,10,15,20$, and 100. Right panel: All the particles are accommodated in the ground state in the whole range ${\gamma }_x\in [-1,1]$.

Figure 3

Gaussian coefficients $c_k=\langle \psi |u_k\rangle$ for a wave packet centered at the ground state and $\sigma =2$. Left and right panels correspond, respectively, to first- and second-order phase transitions. The $k$ index runs over positive-parity levels only, which depend on the type of transition.

Figure 4

Squared modulus of the autocorrelation function for the early time evolution (top panel) of an initial Gaussian wave packet centered around the ground state. The bottom panel displays the long-time evolution of the same initial wave packet. The vertical, dashed blue lines stand for the classical and revival times as computed through their analytical expressions. (Atomic units.)

Figure 5

Plots of the autocorrelation function for the time evolution of a Gaussian wave packet centered around the ground state. Dark and light regions correspond, respectively, to collapses and regenerations. The top (bottom) panel is for the first-order (second-order) phase transition. (Atomic units.)

Figure 6

Classical time vs. ${\gamma }_x$. Left panel: Within the numerical precision, the maximum is located right at ${\gamma }_x=-2$ for the first-order phase transition. Notice the logarithmic scale on the vertical axis. Right panel: The maxima are located near the predicted points $\pm 1$ for second order phase transitions. In either case, first- or second-order, classical times do not scale with $j$ except at the transition points. This is exemplified here for $j=400$ and $j=800$. (Atomic units.)

Figure 7

The maxima of $T_{\mathrm{cl}}$ scale with $j$ at the transition points, with scaling exponents depending on the type of phase transition: $T_{\mathrm{cl}}\sim j$ for the first-order transition (left panel) and $T_{\mathrm{cl}}\sim j^{0.35\left(3\right)}$ for the second (right panel). The red solid lines are straight-line fits to the data. (Atomic units.)

Figure 8

At fixed $j,T_{\mathrm{cl}}$ scales with the distance to the transition point as $|{\gamma }_x-{\gamma }_c{|}^{-1/2}$ for both first- and second-order transitions alike. (Atomic units.)

Figure 9

$T_{\mathrm{rev}}$ as a function of ${\gamma }_x$ for the first-order phase transition. In the right panel, except at the transition point, the data collapse onto a single curve after normalization by the associated system size $j$. From top to bottom, the lines correspond to $j=1280$, 640, and 320. (Atomic units.)

Figure 10

$T_{\mathrm{rev}}$ as a function of ${\gamma }_x$ for the second-order phase transition. Except at the transition points, the data collapse onto a single curve after normalization by the associated system size $j$ (see inset). The divergences related to the phase transition approach $\pm 1$ as $j\to \infty$. (Atomic units.)

Figure 11

Left panel: Finite maxima of $T_{\mathrm{rev}}$ at ${\gamma }_x=-2$ as functions of $j$ for the first-order phase transition. Right panel: Finite-size corrections to the location of the transition points for the second-order phase transition. (Atomic units.)

Figure 12

For the second-order phase transition $T_{\mathrm{rev}}$ scales with the distance to the value of ${\gamma }_x$ at which it diverges. (Atomic units.)

Figure 13

The wave packet; the standard deviation of the energy drops to almost zero as the transition points are approached. (Atomic units.)

Figure 14

For the first-order phase transition, $\Delta H$ decays as a power law with the distance to the transition point ${\gamma }_c=-2$ with an exponent 1/2. (Atomic units.)

Figure 15

Left panel: Minimum of $\Delta H$ as a function of $j$ for the second-order phase transition. Right panel: Finite-size correction to the location of the minimum of $\Delta H$ as a function of $j$. (Atomic units.)