Symmetry-breaking dynamics of the finite-size Lipkin-Meshkov-Glick model near ground state

We study the dynamics of the Lipkin-Meshkov-Glick (LMG) model with a finite number of spins. In the thermodynamic limit, the ground state of the LMG model with an isotropic Hamiltonian in the broken phase breaks to a mean-field ground state with a certain direction. However, when the spin number $N$ is finite, the exact ground state is always unique and is not given by a classical mean-field ground state. Here, we prove that when $N$ is large but finite, through a tiny external perturbation, a localized state which is close to a mean-field ground state can be prepared, which mimics spontaneous symmetry breaking. Also, we find the localized in-plane spin polarization oscillates with two different frequencies $\sim O(1/N)$, and the lifetime of the localized state is long enough to exhibit this oscillation. We numerically test the analytical results and find that they agree very well with each other. Finally, we link the phenomena to quantum time crystals and time quasicrystals.

Figure 1

Semiclassical picture of infinite degeneracy of the isotropic LMG model in the broken phase ($N\to \infty$).

Figure 2

Two modes of oscillation related to the parity of $N$ and $Nh$. $m_x=2\langle S_x\rangle /N$ is the in-plane polarization along the $x$ axis. ${\omega }_0=0$ specifies the round mode; ${\omega }_0=\nu$ identifies the crescent mode.

Figure 3

Frequency spectra in an $N=100$ and $h=0.714$ system with different magnitudes of the symmetry-breaking potential $V=-g{S}_x$. (a) $g=1\times {10}^{-4}$. The energy difference between this localized state and the exact ground state $\mathrm{\Delta }E=6.60\times {10}^{-4}$. (b) $g=1\times {10}^{-3}$, with $\mathrm{\Delta }E=5.10\times {10}^{-3}$.

Figure 4

(a) An example of $m_x\left(t\right)$ oscillation with $N=100, h=0.716$. The localized state is obtained by bringing into it an instantaneous symmetry-breaking field $V=-g{S}_x$ at $t=0$, where $g=1/N^2$. (b) The frequency spectrum of the oscillation in (a).

Figure 5

The above frequency spectra are computed from an $N=100$ system with different $h$. (a) $h=0.710$; since $Nh=71$ is odd, (a) characterizes a crescent mode. (b) $h=0.713$. (c) $h=0.719$. (d) $h=0.720$; both $N$ and $Nh$ are even, so (d) represents a round mode. See Table 1.

Figure 6

(a) An example of aperiodic oscillation ($S_{x_0}$ component) in an $N=100$ and $h=52\nu +{\omega }_0$ system, where $\nu =1/100$ and ${\omega }_0=(\sqrt{5}-2)\nu$. Two frequency components for this oscillation: $f_1=\nu -{\omega }_0$ and $f_2=\nu +{\omega }_0$. $f_1/f_2=\left(\sqrt{5}-1\right)/2$ is the golden ratio, and thus it is related to the Fibonacci quasicrystal [23, 45]. (b) Generating 1D quasicrystals by the intersection method. An irrational slice, whose slope is equal to the golden ratio $(\sqrt{5}-1)/2$, cuts through a 2D lattice. We label $U$ (up triangles) or $D$ (down triangles) when horizontal or vertical lines are crossed.

Figure 7

Twofold degeneracy when $\gamma =0$.