Quantum memory at an eigenstate phase transition in a weakly chaotic model

We study a fully connected quantum spin model resonantly coupled to a small environment of noninteracting spins, and investigate how initial state properties are remembered at long times. We find memory of initial state properties, in addition to the total energy, that are not conserved by the dynamics. This memory occurs in the middle of the spectrum where an eigenstate quantum phase transition (ESQPT) occurs as a function of energy. The memory effect at that energy in the spectrum is robust to system-environment coupling until the coupling changes the energy of the ESQPT. This work demonstrates the effect of ESQPT memory as independent of integrability and suggests a wider generality of this mechanism for preventing thermalization at ESQPTs.

Figure 1

(a) The classical trajectories of the collective spin variables $({\varphi }_s,z_s)$ for the semiclassical limit of the system Hamiltonian $H_s$. The trajectories with energy at the ESQPT are highlighted in black and form two separatrices that meet at the unstable fixed point at $({\varphi }_s,z_s)=(\pi ,0)=(-\pi ,0)$. The separatrix separates trajectories with sign changing $z_s$ (marked in bright pink) and trajectories that do not change sign (marked in dark blue). (b) The long-time average of $x$ polarization of the collective spin, $S_s^x$ for $V=0$ as a function of the system spin size $S$; it demonstrates the memory of the initial phase ${\varphi }_s(t=0)$ at the ESQPT. The initial states that produce these long-time averages all have energy at the ESQPT $e_n/\left(hS\right)=1$, but with different initial phases ${\varphi }_s(t=0)$ from ${\varphi }_s(t=0)=0$ (darkest blue) to ${\varphi }_s(t=0)=\pi$ (brightest pink). (c) The expectation value of $S_s^x$ in the eigenstate $|n\rangle$ at energy $e_n$. At the energy of the ESQPT, $e_n/\left(hS\right)=1, \langle S_s^x\rangle$ approaches $-1$ nonanalytically.

Figure 2

(a), (b) Chaotic Poincaré maps for $V/h=5$ (a) and $V/h=15$ (b). The plane fixing two of the four phase space variables is defined by the total energy constraint and $z_e=-0.1$. The initial state for both (a) and (b) had ${\varphi }_s(t=0)=0$ and $({\varphi }_e,z)=(0,0)$, but with different initial $z_s$ fixed by $E(z_s,{\varphi }_s)=1.05$ (a) and $E(z_s,{\varphi }_s)=1$ (b). (c) The Lyapunov exponent vs the initial energy of the system, $E$, and the coupling strength, $V$, for an initial state with ${\varphi }_s=0$ and $z_s$ fixed by the energy constraint $E({\varphi }_s,z_s)=E$ ($y$ axis) and $({\varphi }_e,z_e)=(0,0)$. The Lyapunov exponent is the exponential rate at which nearby trajectories diverge [39] and is positive (darker blue) for chaotic dynamics and zero (white) for regular dynamics. The largest Lyapunov exponent found was $\lambda =1.5$ and corresponds to the darkest blue.

Figure 3

Mean-level spacing $\langle r_n\rangle$, for states in the even symmetry sector ($U_{Z2}|\psi \rangle =|\psi \rangle$), as a function of system-environment coupling strength $V$ for $S=40,50$, and 60. When $V=0$, the model is integrable and has mean-level spacing $\langle r_n\rangle \approx 0.38$. Upon increasing $V$, the level spacing increases, indicating integrability breaking. While integrability is broken, the level spacing never increases to $\langle r_n\rangle =0.54$, which would indicate full quantum chaos.

Figure 4

Typical dynamics of $\langle S_s^z\left(t\right)\rangle$ for the initial state in Eq. (6), $\varphi =0$ (solid lines) and $\varphi =\pi$ (dashed line) and coupling, $V$, shown in the plot. For $\varphi =\pi$ the state is $Z_2$ symmetric and $\langle S_s^z\rangle =0$ remains symmetric at all times and for all coupling $V$. For small $V$, the steady state of $\langle S_s^z\rangle$ distinguishes the $\varphi =\pi$ and $\varphi =0$ states, while for large values of $V$, this late-time memory of $\varphi$ is lost in the steady state.

Figure 5

Dependence of the long-time averages $\overline{S_s^x}\left({\varphi }_{s,0}\right)$ and $\overline{S_s^z}\left({\varphi }_{s,0}\right)$ on $S$ for different initial states $\left|\psi \right({\varphi }_{s,0})\rangle$ with ${\varphi }_{s,0}$ varying from ${\varphi }_{s,0}=0$ (dark blue) to ${\varphi }_{s,0}=\pi$ (bright pink). The different columns correspond to different coupling strengths $V/h=2$ (a), $V/h=13$ (b), and $V/h=20$ (c). For $V/h=2$, long-time memory (sensitivity of the long-time observable to the initial system phase ${\varphi }_{s,0}$) is observed even at large values of $S$ similar to the $V/h=0$ case shown in Fig. 1, while for $V/h=20$ sensitivity of initial conditions is quickly loss as the spin size $S$ increases. Exact calculations are shown with solid lines, while semiclassical observables calculated via TWA (with sampling error $\mathrm{\Delta }O/S\approx 0.025$) are shown with dashed lines. We leave out TWA calculations for $V/h=20$ because observable values are smaller than the sampling error.

Figure 6

Memory quantifiers ${\mathrm{\Delta }}_{S_s^z}$ and ${\mathrm{\Delta }}_{S_s^x}$ for the long-time averages $\overline{S_s^x}\left({\varphi }_{s,0}\right)$ and $\overline{S_s^z}\left({\varphi }_{s,0}\right)$ as a function of the system-environment coupling $V$. Exact calculations are shown with solid lines, while TWA calculations are shown with dashed lines. Panel (a) indicates memory of ${\varphi }_{s,0}$ is maintained until $V/h=20$ in the long-time average of $S_s^x$, while panel (b) indicates it is lost earlier for $V/h\sim 16$ in the long-time average of $S_s^z$. The weakening of memory with increasing spin size, $S$, is due to the classicality of the collective spin in the $S\to \infty$ limit and is consistent with the logarithmic scaling of $S$ found in Ref. [23].

Figure 7

The memory quantifier ${\mathrm{\Delta }}_{S_e^z}$ for the long-time average of environment $z$ polarization $S_e^z$ as a function of the system-environment coupling $V$. It indicates that for $0<V/h<16$ a type of memory proximity effect occurs, where the initial state of the system spins is remembered by the long-time average of the environment spins. Exact calculations are shown with solid lines, while TWA calculations are shown with dashed lines.

Figure 8

Scatter plot of the degenerate energies for different values of $V/h$. We consider two energy levels degenerate if $|e_n-e_{n+1}|<{10}^{-7}$, where $e_n$ is the $n\text{th}$ largest eigenenergy: $H|e_n\rangle =e_n|e_n\rangle$. It shows that for $V/h<15$, degeneracy occurs only above $e_n=0$, while for $V/h>20$ the ESQPT responsible for memory at $V=0$ shifts to lower energy. The purple dashed line marks the $V=0$ ESQPT at $e_n/\left(hS\right)=0$.