Surprises in the deep Hilbert space of all-to-all systems: From superexponential scrambling to slow entanglement growth

The quantum dynamics of spin systems with uniform all-to-all interaction are often studied in the totally symmetric space (TSS) of maximal total spin. However, the TSS states are atypical in the full many-body Hilbert space. In this paper, we explore several aspects of the all-to-all quantum dynamics away from the TSS, and reveal surprising features of the “deep Hilbert space” (DHS). We study the out-of-time order correlator (OTOC) in the infinite-temperature ensemble of the full Hilbert space. We derive a phase-space representation of the DHS OTOC and show that the OTOC can have a superexponential initial growth in the large $N$ limit, due to the fast dynamics in an unbounded phase space (in finite systems, we observe numerically that the superexponential growth ends precociously and gives way to a power-law one until saturation). By a similar mechanism, the Krylov complexity grows explosively. We also study the entanglement growth in a quantum quench from a DHS product state, i.e., one of nonaligned spins that resemble the DHS infinite-temperature ensemble with respect to the statistics of the collective spins. Using a field-theoretical method, We exactly calculate the entanglement entropy in the large $N$ limit. We show that, in the DHS, fast OTOC growth does not imply fast entanglement growth, in contrast to the Zurek-Paz relation derived in the TSS.

Figure 1

A cartoon of the many-body Hilbert space of a spin system with all-to-all interaction and uniform coefficients. It is divided into conserved sectors. In the totally symmetric space (TSS), which is invariant under permutation of sites, the collective spin $S={\sum }_j{S}_j$ scales as the system size $S\sim N$. In the large $N$ limit, a semiclassical description leads to a phase space of a two sphere. The vast majority of the states are in the deep Hilbert space (DHS), where the collective spin scales as $S\sim \sqrt{N}$. The phase space of the DHS is the interior the TSS sphere and becomes noncompact in the $N\to \infty$ limit.

Figure 2

Lanzcos coefficients $b_n$ of the operator ${\mathbf{S}}_z=N^{-1/2}{\sum }_j{S}_j^z$ in the LMG model $H={\sum }_j{S}_j^x+N^{-1/2}J{\sum }_{jk}{S}_j^z{S}_k^z/2$ [see Eq. (26)] with $J=1$, and with respect to the infinite-temperature ensemble ${\rho }_{\infty }$. We plot $b_n$ for a few system sizes, as well as in the thermodynamic limit. The latter is fitted by the prediction $b_n=A{n}^{3/2}/ln\left(nB\right)$ [see Eq. (27)]; the fit parameters are $A=0.95$ and $B=27.2$. Inset: Data of different finite-system sizes are collapsed by plotting $b_n/b_N$ against $n/N$. This confirms that the growth of $b_n$ saturates at $n_{\text{sat}}\sim N$ [see Eq. (30)].

Figure 3

Numerical simulation of the DHS OTOC (31) growth in the Euler top model $H={\sum }_{jk}(J_x{S}_j^x{S}_k^x+J_y{S}_j^y{S}_k^y+J_z{S}_j^z{S}_k^z)/\left(2\sqrt{N}\right)$ with $J_x=0,J_y=-1,J_z=1/2$. The evolving operator is $\mathbf{O}={\mathbf{S}}_x=N^{-1/2}{\sum }_j{S}_j^x$. See Appendix pp4 for methods. The time dependence of the OTOC for different system sizes $N$ is compared to the superexponential growth prediction (55) with $c=0.26$ (fit). For each system size, a range for the estimated pre-saturation time scale $t_p$ is indicated by an orange interval; its extremities are the maximum of ${\partial }_{t}lnC\left(t\right)$ and the minimum of ${\partial }_t^{2}lnC\left(t\right)$, respectively. $C\left(t\right)$ becomes significantly $N$ dependent for $t\gtrsim t_p$, invalidating the semiclassical theory. Inset: The presaturation value $C\left(t_p\right)$ (the error bar results from the width of the interval) as a function of $N$, compared to the power law $N^{-0.75}$ (57).

Figure 4

The DHS OTOC (31) with $\mathbf{O}={\mathbf{S}}_z=N^{-1/2}{\sum }_j{S}_j^z$ in the LMG model $H={\sum }_j{S}_j^x+N^{-1/2}J{\sum }_{jk}{S}_j^z{S}_k^z/2$ with $J=1$. The eventual saturation of the OTOC. We plot $\mathbf{C}\left(t\right)/N$ as a function of $t/N^{0.5}$, collapsing the data with different system sizes, except the initial growth regime. This confirms the saturation time scaling $t_S\sim N^{0.5}$ (58). The intermediate-time growth is compared to the power law $t^{1.5}$, predicted by (59) (with $b=0.75$). Inset: The same data plotted without rescaling, and restricted to $t\le 10$. The initial fast growth is consistent with the prediction (54) with a fitted constant $c=0.16$.

Figure 5

Illustration of the path-integral contour with replica number $n=3$. The forward contours ($\sigma =+$, the time $u$ increases from 0 to $t$) start from the right, and are connected to the backward contours ($\sigma =-, u$ decreases from $t$ to 0), which end at the left. In the subsystem $A$ (and not $B$), a cyclic permutation is applied when connecting the forward and backward contours, as indicated by the dashed lines.

Figure 6

Entanglement entropy (Renyi, $n=2$, equal bipartition) in a quantum quench from a DHS product state (61), with $s_1=s_{N/2+1},s_2=s_{N/2+2},\cdots$ equally spaced on the great circle $\{x=0\}$ of the Bloch sphere (example 3 in Sec. 5c2). The two panels correspond to two Hamiltonians. (a) LMG model $H={\sum }_j{S}_j^x+J{\sum }_{jk}{S}_j^z{S}_k^z/\left(2S\right)$, where $S=N/2$ and $J=2$. Note that we need to use the “$1/N$ normalization” (1). (b) The isotropic Euler top $H={\sum }_{jk,a}{S}_j^a{S}_k^a/\left(2S\right)$. the finite-$N$ numerical data (obtained from direct simulation) are compared to semiclassical prediction [obtained from the one-loop determinant (90) discretized in time], as well as to the asymptotic logarithmic growth prediction $S_2\sim clnt$ (115), (119), respectively. For each $N$, we measure the saturation value $S_{2,max}$ as the first local maximum of $S_2$ (indicated by a star). Inset: The $N$ dependence of $S_{2,max}$, compared to $\frac{c}{2}lnN$ [see Eq. (120)]. The simulated time evolution is Trotterized (kicked) with $\delta t=0.5$, see (122) and Appendix pp7. The semiclassical prediction has the same time discretization, and should thus match exactly the Trotterized numerics in the large $N$ limit.