Grand Canonical-like Thermalization of Quantum Many-body Scars

Abstract

Quantum many-body scar (QMBS) in kinetically constrained quantum systems challenges the conventional eigenstate thermalization hypothesis (ETH). We develop an effective open-system description for constrained dynamics and introduce the definition of quasiparticle number in the system. Based on this, we formulate a revised ETH that accounts for both diagonal and off-diagonal structures of local observables. By introducing the cross coherence purity (CCP), we obtain a unified characterization of off-diagonal matrix elements and show that the relevant density of states (DOS) is determined by the distribution of eigenstates on the energy--quasiparticle-number plane. We numerically verify an inverse relation between the CCP and this generalized DOS. Applied to the quantum many-body scar model, the revised ETH accurately predicts long-time averages and temporal fluctuations of local observables and explains their dependence on initial states. Our framework shows that the anomalous fluctuations and quasi-periodic dynamics of scar states arise naturally from low-DOS regions. These results provide a unified understanding of thermalization and QMBS in kinetically constrained systems.

Figures (6)

• Figure 1: Adjacency graphs in Hilbert space demonstrating the actions of the Hamiltonian. We consider a three-site chain with spin size $j=1$, where nodes represent all product states and lines indicate the mappings between them. Plot (a) shows the basic actions generated by the free Hamiltonian $H_0$, The action of $H_0$ forms a hypercubic structure in the adjacency graph, with each line denoting a bidirectional mapping between the two endpoint states. Plot (b) depicts the constrained Hamiltonian $H$, in which all the states containing patterns as $|1\rangle_k\otimes |-1\rangle_{k+1}(k=1,2,3)$ are blockaded, with light gray lines indicating the prohibited mappings. Plot (c) illustrates the effect of the non-Hermitian Hamiltonian $H_N$. Arrowed lines indicate unidirectional mappings from the prohibited states to the constrained subspace, corresponding to the terms $M_k^{\dagger}$ in Eq.(\ref{['nonHermitianHamiltonian']}). In plots (b) and (c), the green nodes represent the states within the constrained subspace ${\cal H}$, while the red ones denote prohibited states. As seen in plot (c) that, when the system is initiated within ${\cal H}$, the $M_k^{\dagger}$ terms will not affect the constrained dynamics.
• Figure 2: Plot (a) shows the EEVs of a local observable $\hat{O}=s_1^z \otimes s_2^z$, with $s_k^z$ being the $z$-direction spin operator on $k$-th site. The red line represents a single-parameter fit $\mathcal{O}(\mathcal{E})$ as a function of the energy $\mathcal{E}$. Plot (b) displays the same EEVs fitted as a smooth function of both $\mathcal{E}$ and the quasi-particle number $\mathcal{N}$. In plots (a) and (b), scar eigenstates are highlighted by red circles, while the color bars indicate the deviation of each eigenstate from the fitted curve (or surface). These results show that the EEVs cannot be described solely by the energy, with scar states exhibiting particularly large deviations, whereas all states are unified by a smooth function of $\mathcal{E}$ and $\mathcal{N}$. Plot (c) shows the difference between the EEVs and the ensemble averages: $\mathrm{Tr}(\hat{O}\rho)-\langle \hat{O}\rangle_i$, where $\rho$ is taken as either the canonical ensemble $\rho_c(\beta_c)$ or the grand canonical-like ensemble $\rho(\beta,\mu)$. Plot (d) shows the trace-norm distance between the reduced DMs of eigenstates, $\sigma_i^A=\mathrm{Tr}_{\bar{A}}(|E_i\rangle\langle E_i|)$, and that of the corresponding ensembles, $\rho^A=\mathrm{Tr}_{\bar{A}}(\rho)$, with the local region $A$ being a single site. In plots (c) and (d), crosses denote the canonical ensemble results, while dots correspond to the grand canonical ensemble. Results for scar states are marked in red. It can be seen that the grand canonical ensemble provides significantly more accurate predictions for the local properties of eigenstates, particularly for scar states. We employ a spin chain with length $D=10$ and spin size $j=1$, This setting is used throughout the main text.
• Figure 3: Plot (a) shows the DOS $\Omega(\mathcal{E},\mathcal{N})$ defined in Eq. (\ref{['DensityofStates']}) on the energy--quasiparticle-number plane. The kernel widths $h_{1,2}$ are chosen as the standard deviations of the sets $\{E_i\}$ and $\{N_i\}$, respectively. The DOS values associated with all eigenstates are indicated by points on the surface, with scar states highlighted by red circles, revealing a pronounced suppression of DOS in the scarred region. Plot (b) shows the distribution of all off-diagonal terms on the $\omega$--$\nu$ plane, where the color encodes the local density of data points. Concentric ellipses enclosing $10\%, 20\%, \cdots, 90\%$ of the data are constructed from equal-density contours, and the enclosed subsets are used for statistical analysis. Plot (c) uses the innermost $10\%$ of the data points in the $\omega$--$\nu$ plane as samples. The $\mathcal{E}$--$\mathcal{N}$ plane is discretized into bins, within which the averaged quantities $\{E_{\mathrm{grid}}, N_{\mathrm{grid}}, \mathcal{T}_{\mathrm{grid}}\}$ are evaluated. The inverse averaged CCP, $\mathcal{T}_{\mathrm{grid}}^{-1}$, is then fitted by the function $a\,\Omega(E_{\mathrm{bin}},N_{\mathrm{bin}})$, showing excellent agreement with the generalized DOS. Plot (d) shows that, by varying the sample size from $10\%$ to $90\%$, different estimates of the kernel widths $h_{1,2}$ are obtained. The fitted values of $h_{1,2}$ for different sample percentages are shown as blue and red dots, with error bars indicating the corresponding standard deviations. The averaged kernel widths are shown as solid lines, while the coefficient of determination $R^2$, plotted as yellow points, remains high for all fittings.
• Figure 4: Plots (a)--(d) show the time evolution of a local observable $\hat{O}=|0\rangle\langle 0|_{1}+|-1\rangle\langle -1|_{1}$ for different initial states. The blue curves represent the expectation value $\langle \psi(t)|\hat{O}|\psi(t)\rangle$, while the green horizontal lines indicate its long-time average $\overline{O}$. The dark-blue and red dotted lines denote the ensemble averages predicted by the grand canonical and canonical ensembles, respectively. The light-yellow shaded region shows the long-time averaged temporal fluctuations around the mean, $\overline{O}\pm\Delta_t O$. Insets display the short-time dynamics. In the insets of (a)--(c), the gray curves also show the fidelity evolution $|\langle \psi(t)|\psi(0)\rangle|^2$. We can see that the long-time average of the local observable is accurately captured by the grand canonical ensemble, indicating thermalization to a grand-canonical-like local state. In plots (e) and (f), all translation-invariant initial states are considered, and the long-time averaged temporal fluctuations are computed in the same manner as in (a)--(d). These results are compared with the estimates based on the CCP and the DOS, given by $\sqrt{\sum_{i\neq i'} |c_i|^2 |c_{i'}|^2\mathcal{T}_{i,i'}}$ and $\sqrt{\sum_{i\neq i'} |c_i|^2 |c_{i'}|^2\Omega(\mathcal{E},\mathcal{N})^{-1}}$, respectively. The observed temporal fluctuations are found to scale proportionally with these estimates.
• Figure 5: Plot (a) and (b) show the overlaps between eigenstates and two initial states, $|\psi_1(0)\rangle$ and $|\psi_2(0)\rangle$, which give rise to quasi-periodic dynamical oscillations, as defined in Sec. IV. Scar eigenstates are highlighted by red circles. Plot (c) shows the ratio $r_i^{+}=n_i/d_i^{+}$ evaluated for all eigenstates, where the color bar indicates the magnitude of the denominator $d_i^{+}$. Plot (d) displays the denominator $d_i^{+}$ itself, with the color encoding the corresponding values of $r_i^{+}$. These results demonstrate that $r_i^{+}$ is essentially inversely correlated with $d_i^{+}$. Plot (e) shows that, after removing the DOS dependence, the squared off-diagonal matrix elements of the local observable $\hat{R}_k$, together with the factor $1/(\omega+1)^2$, form a single-peaked wave packet along the $\omega$ axis. The blue dots depict the averaged results after resolving the $\omega$ axis into bins. The red dotted line fits these dots with a gaussian function $g(\omega)$. By inserting the fitted $g(\omega)$ into the expression for $d_i^{+}$, we obtain the estimate $d_{\mathrm{fit}}^{+}$ predicted by the modified ETH, as shown in plot (f), where the color of each point represents the corresponding exact value $\ln(r_i^{+})$. In plots (a)-(d) and (f), scar eigenstates are marked by red circles.