Late-time ensembles of quantum states in quantum chaotic systems

TL;DR

The paper investigates the late-time statistical structure of quantum states evolving under chaotic dynamics in the presence of symmetries. It identifies two universal regimes: (i) typical initial states that effectively mix symmetry sectors yield late-time ensembles indistinguishable from Haar random states for finite $k$-moments in the thermodynamic limit, and (ii) atypical or constrained initial conditions that produce non-universal ensembles captured by constrained random-matrix theory. These results are supported by analytically tractable models with a U(1) constraint and extensive numerical simulations of U(1) random circuits and chaotic Hamiltonians, with entanglement-entropy distributions serving as the primary diagnostic. The findings illuminate how higher statistical moments reveal universality or its breakdown beyond traditional eigenstate statistics, with implications for quantum chaos, information scrambling, and the design of quantum devices under symmetry constraints.

Abstract

Quantum states undergoing quantum chaotic dynamics are expected to evolve into featureless states at late times. While this expectation holds true on an average, coarse-grained level, it is unclear if this expectation applies to higher statistical moments, as symmetries typically present in physical systems constrain the exploration of phase space. Here we study the universal structure of late-time ensembles obtained from unitary dynamics in quantum chaotic systems with symmetries, such as charge or energy conservation. We identify two limiting universal regimes depending on the initial condition. When the initial state is typical -- all the moments of the symmetry operators are equal to those of pure random states -- then the late-time ensemble is indistinguishable from the Haar ensemble in the thermodynamic limit and at the level of higher statistical moments. Otherwise, atypical initial states evolve into non-universal ensembles which can be distinguished from the Haar ensemble from simple measurements or subsystem properties. Interestingly, such atypical initial conditions are not rare, even when considering product state initial conditions, and can be found in the middle of the spectrum of Hamiltonians known to be `maximally' chaotic. In the limiting case of initial states with negligible variance of the symmetry operator (e.g., states with fixed particle number or states with negligible energy variance), the late-time ensemble has universal behavior captured by constrained RMT ensembles. Our work shows that even though midspectrum states do not explore ergodically all of phase space at late times, the late-time ensemble typically -- but not always -- exhibits the same average and sample-to-sample fluctuations as the Haar ensemble.

Figures (5)

• Figure 1: The late-time behavior of an initial state $|\Psi_0\rangle$ can be classified by its distribution of outcome probabilities of the symmetry operator $O = \sum_n O_n|n\rangle\langle n|$ relative to that produced by the ensemble of Haar random states. A typical initial state has the same distribution of outcome probabilities as pure random states. In contrast, an atypical initial state has a distribution which has smaller than typical fluctuations of the symmetry operator. Schematics for typical and atypical states are shown for a magnetic charge $O = \sum_i Z_i$ with quantum number $M$.
• Figure 2: (a) Distribution of half-system entanglement entropy (EE) at late-times for a random quantum circuit with a U(1) conservation law. All the initial conditions considered have zero average magnetization $\langle \hat{S}_z \rangle = 0$ but different magnetization variance $\delta S_z^2$, ranging from 0 (AFM$_z$) to $L/4$ (FM$_x$). Also shown is a generic EE distribution for a spin spiral state, Eq. (\ref{['eq:spiral']}), with $\theta = \pi/4$. The dotted line indicates the exact EE distribution for pure random states (Haar), and the dotted-dashed lines indicates the EE distribution of constrained random states [U(1)] derived in Ref. 2019PRD_BianchiDona. The inset shows the projection $p_M$ of the initial condition within symmetry sectors $M$ for each of the initial conditions. (b) Distribution of EE plotted relative to the maximum EE value ($\delta S_A = L_A\log 2 -S_A$) for spin spiral states, Eq.(\ref{['eq:spiral']}), as a function of the canting angle $\theta$.
• Figure 3: Projection of the (a) FM$_y$ and (b) AFM$_x$ states on the eigenstate basis, $f(E) = \sum_n |\langle n | \Psi_0\rangle|^2\delta(E-\varepsilon_n)$, of the mixed field Ising model (MFIM), Eq. (\ref{['eq:Ham']}). In both panels, datapoints are shown for system sizes $L=12$ (triangle), $L=14$ (squares), and $L=16$ (diamonds). Also shown with solid lines is the average projection of pure random states on the eigenstate basis.
• Figure 4: (a) Distribution of EE at late times for the MFIM and different initial conditions. Also shown is the microcanonical distribution of EE for midspectrum eigenstates (MC). The dotted lines indicate the EE distribution of Haar random states, and the dotted-dashed lines indicate the EE distribution for states constrained to the $M=0$ magnetic sector. (b) Finite-size scaling of the EE distributions plotted relative to the maximum EE value ($\delta S_A = L_A\log 2-S_A$) for different initial conditions. The dots indicated the average EE value, and the bars indicated their standard deviation. The shaded areas indicates the regions limited by $S_A = \mu_{\rm H}\pm\sigma_{\rm H}$ (blue) and $S_A = \mu_M\pm\sigma_M$ (red) for the EE distribution of Haar and constrained random states, respectively.
• Figure 5: (a) Distribution of EE at late-times for the MFIM as a function of the transverse field $g$ and initial conditions (FM$_y$ and AFM$_x$). Also shown is the EE distribution of the microcanonical eigenstate ensemble (MC). All values are plotted relative to the maximum EE ($\delta S_A = L_A\log 2 - S_A$). The symbols indicate the average EE value for each value of $g$, and the bars indicate their standard deviation. The shaded areas indicates the regions limited by $S_A = \mu\pm\sigma$ (blue) and $S_A = \mu_M\pm\sigma_M$ (red) for the EE distribution of Haar and constrained random states, respectively. (b) Zoom of the dotted region in panel (a) for the FM$_y$ initial condition showing EE data for different system sizes $L=12,14,16$. The width $\sigma_L$ of the shaded regions indicate the standard deviation of the EE for the different system sizes.