Asymptotic Quantum Many-Body Scars

We consider a quantum lattice spin model featuring exact quasiparticle towers of eigenstates with low entanglement at finite size, known as quantum many-body scars (QMBS). We show that the states in the neighboring part of the energy spectrum can be superposed to construct entire families of low-entanglement states whose energy variance decreases asymptotically to zero as the lattice size is increased. As a consequence, they have a relaxation time that diverges in the thermodynamic limit, and therefore exhibit the typical behavior of exact QMBS, although they are not exact eigenstates of the Hamiltonian for any finite size. We refer to such states as asymptotic QMBS. These states are orthogonal to any exact QMBS at any finite size, and their existence shows that the presence of an exact QMBS leaves important signatures of nonthermalness in the rest of the spectrum; therefore, QMBS-like phenomena can hide in what is typically considered the thermal part of the spectrum. We support our study using numerical simulations in the spin-1 XY model, a paradigmatic model for QMBS, and we conclude by presenting a weak perturbation of the model that destroys the exact QMBS while keeping the asymptotic QMBS.

Figure 1

Top: Squared overlap of $|n,k⟩$ for $n=L/2$ and $k=\pi -(2\pi /L)$ with the eigenstates $|E_i⟩$ of Hamiltonian (1) with zero magnetization, $S_z=0$; the parameters of the simulation are $\{J,h,D,J_3\}=\{1,0,0.1,0.1\}$ and $L=10$. The information on $|⟨E_i|n,k⟩{|}^2$ is also encoded in the color code of the marker of all panels using a logarithmic scale, see color bar. Middle and bottom: We plot the data of the top panel in a diagram with the energy $E$ on the abscissa and the bipartition entanglement entropy $S_E$ or the average square magnetisation $S^{z2}(E)$ of the eigenstate on the ordinate, respectively. For the entanglement entropy, we use the natural logarithm and we divide the result by $L/2$ to obtain an intensive quantity. The state $|n,k⟩$ has overlap only with states whose $S_{E_i}$ or $S_{E_i}^{z2}$ lies on the continuous thermal curve. The red circle and the blue square highlight the regions of the plots where the QMBS $|n=L/2,\pi ⟩$ appear: the absence of any gray mark means that the scalar product is compatible with the numerical zero.

Figure 2

The properties of the state $e^{-iHt}|n,k⟩$ for $n=L/2$ and $k=\pi -2\pi /L$ as a function of time for various system sizes $L$. Left: time evolution of the squared magnetisation $S^{z2}(t)$. Right: time evolution of the fidelity with the initial state $F(t)$.

Figure 3

Properties of the eigenstates of Hamiltonian $H^{\prime }$ in the zero magnetization sector $S_z=0$; the parameters of the simulation are $\{J,h,D,J_3,J_z\}=\{1,0,0.1,0.1,1\}$ and $L=10$. Top: Squared overlap of $|n,\pi ⟩$ for $n=L/2$ with the eigenstates $|E_i⟩$ of Hamiltonian $H^{\prime }$ with zero magnetisation, $S_z=0$. The information on $|⟨E_i|n,\pi ⟩{|}^2$ is also encoded in the color code of the marker of all panels using a logarithmic scale, see color bar. Middle and bottom: We plot the data of the top panel in a diagram with the energy $E$ on the abscissa and the bipartition entanglement entropy $S_E$ or the average square magnetisation $S^{z2}(E)$ of the eigenstate on the ordinate, respectively. The state $|n,\pi ⟩$ has overlap only with states whose $S_{E_i}$ or $S^{z2}(E_i)$ lies on a continuous curve. In the Supplemental Material [47] we show the entire spectrum and show that the model does not have any QMBS (here the spectrum is incomplete because we plot only state that have a non-negligible overlap with $|n,\pi ⟩$).

Figure 4

The properties of the state $e^{-i{H}^{\prime }t}|n,\pi ⟩$ for $n=L/2$ as a function of time; the parameters of the Hamiltonian employed in the simulation are the same of Fig. 3. Left: time evolution of the squared magnetization $S^{z2}(t)$; right: time evolution of the fidelity with the initial state $F(t)$. The inset shows the scaling as a function of size of the values of $F(t=3/J)$; we find a scaling to 1 as $1/L\to 0$.