Quantum many-body scars in spin models with multibody interactions

We introduce and study several classes of quantum spin models with multibody interactions that exhibit quantum many-body scars. The models are constructed by two different methods: one exploiting boundary states in integrable spin chains and the other based on a variant of existing methods such as restricted spectrum generating algebras. The first method allows us to construct deformations of the Majumdar-Ghosh and Affleck-Kennedy-Lieb-Tasaki models—prototypes of frustration-free systems. With the second method, we construct a large class of spin-1 models involving scalar spin chirality in both one and two dimensions. Interestingly, in some cases, the models so constructed have towers of scar states of different character. For each example, we show that the scar states behave differently from thermal states by comparing their spectral and dynamical properties with those of other states. We also show that a superposition of the scar states constructed by the second method exhibits perfectly periodic revivals in the dynamics.

Figure 1

The definition of subsystems $A$ and $B$ for one-dimensional systems. Note that the number of sites in $A$ is one less than that in $B$ when $L\mathrm{is}\mathrm{odd}$.

Figure 2

Level-spacing statistics in the middle half of the spectrum of $H\left(t\right)$ in Eq. (7) with $t=8$ and $L=20$. The data are taken in the symmetry sector where $({\mathcal{S}}^z,\mathcal{S},\mathcal{T},\mathcal{F})=(0,0,1,1)$. The curves $P{\left(s\right)}_{\mathrm{GOE}}$ (orange) and $P{\left(s\right)}_{\mathrm{Poisson}}$ (green) are shown for comparison. The distribution follows $P{\left(s\right)}_{\mathrm{GOE}}$.

Figure 3

The mean level-spacing ratio $\langle r\rangle$ as a function of $t$ for the Hamiltonian (7) in the symmetry sector $({\mathcal{S}}^z,\mathcal{S},\mathcal{T},\mathcal{F})=(0,0,1,1)$ for $L=16$, 18, and 20. The green and orange dotted lines indicate $\langle r_{\mathrm{GOE}}\rangle \approx 0.536$ and $\langle r_{\mathrm{Poisson}}\rangle \approx 0.386$, respectively.

Figure 4

Entanglement entropy $S_A$ in all eigenstates of $H\left(t\right)$ in Eq. (7) with $t=8$ in the symmetry sector $({\mathcal{S}}^z,\mathcal{T})=(0,1)$ for $L=14$, 16, and 18. The density of data points is color coded. The red and green circles indicate the dimer state $|\mathrm{dimer}\rangle$ and the ferromagnetic state $|F_{L/2}\rangle$, respectively. The orange dotted line indicates $S_A=2ln2\simeq 1.386$.

Figure 5

Size dependence of the half-chain entanglement entropy of the ferromagnetic state $|F_{L/2}\rangle$. The red line represents the right-hand side of Eq. (13).

Figure 6

The expectation values of $H_{\mathrm{st}}$ with $\epsilon =0.2$ in all eigenstates of $H\left(t\right)$ in Eq. (7) with $t=8$, and $L=18$ in the symmetry sector $({\mathcal{S}}^z,\mathcal{T})=(0,1)$. The density of data points is color coded. The red circle indicates the dimer state $|\mathrm{dimer}\rangle$ with $\langle H_{\mathrm{st}}\rangle \simeq -6.72$, which agrees with the analytical value $\langle H_{\mathrm{st}}\rangle =-1143/170$ obtained from Eq. (15).

Figure 7

Level-spacing statistics in the middle half of the spectrum of $H\left(t\right)$ in (17) with $t=3$ and $L=14$. The data are taken in the symmetry sector where $({\mathcal{S}}^z,\mathcal{S},\mathcal{T},\mathcal{F})=(0,0,1,1)$. The curves $P{\left(s\right)}_{\mathrm{GUE}}$ (magenta) and $P{\left(s\right)}_{\mathrm{Poisson}}$ (green) are shown for comparison. The distribution follows $P{\left(s\right)}_{\mathrm{GUE}}$.

Figure 8

Entanglement entropies in all eigenstates of $H\left(t\right)$ in Eq. (17) for $L=9$, $t=3$ in the ${\mathcal{S}}^z=0$ sector. The density of data points is color coded. The points enclosed by the red and green circles indicate the VBS state $|{\mathrm{\Psi }}_{\mathrm{VBS}}\rangle$ and ferromagnetic state $|F_L\rangle$, respectively. The orange dotted line indicates $S_A=2ln2\simeq 1.386$.

Figure 9

Level-spacing statistics in the middle half of the spectrum of the inhomogeneous model $\tilde{H}\left(t\right)$ in Eq. (30) with $t=1$ and $L=11$. Each $c_j$ is randomly chosen from $[-1,1]$. The data are taken in the symmetry sector where $({\mathcal{S}}^z,\mathcal{S},\mathcal{F})=(0,0,1)$. The curves $P{\left(s\right)}_{\mathrm{GUE}}$ (magenta) and $P{\left(s\right)}_{\mathrm{Poisson}}$ (green) are shown for comparison. The distribution follows $P{\left(s\right)}_{\mathrm{GUE}}$.

Figure 10

Entanglement entropies in all eigenstates of the inhomogeneous model $\tilde{H}\left(t\right)$ in Eq. (30) for $L=9$, $t=1$ in the ${\mathcal{S}}^z=0$ sector. The density of data points is color coded. Each $c_j$ is randomly chosen from $[-1,1]$. The red and green circles indicate the VBS and the ferromagnetic states, respectively.

Figure 11

The expectation values of $H_{\mathrm{AKLT}}$ in all eigenstates of (30) with $t=1$, $L=9$ in the symmetry sector where ${\mathcal{S}}^z=0$. The density of data points is color coded. The red circle indicates the VBS state.

Figure 12

Level-spacing statistics in the middle half of the spectrum of the model (34) with $t=3$ and $L=13$. The data are taken in the symmetry sector where $({\mathcal{S}}^z,\mathcal{S},\mathcal{T},\mathcal{F})=(0,0,1,1)$. The curves $P{\left(s\right)}_{\mathrm{GOE}}$ (orange) and $P{\left(s\right)}_{\mathrm{Poisson}}$ (green) are shown for comparison. The distribution follows $P{\left(s\right)}_{\mathrm{GOE}}$.

Figure 13

Entanglement entropies in all eigenstates of $H\left(t\right)$ in Eq. (34) with $t=3$ for $L=10$ in the ${\mathcal{S}}^z=0$ sector. The density of data points is color coded. The red and green circles indicate the VBS and ferromagnetic states, respectively. The orange dotted line indicates $S_{\mathrm{A}}=2ln2\simeq 1.386$.

Figure 14

Entanglement entropies in all eigenstates of (36) for $L=8$, $h=1$. The density of data points is color coded. The red and blue circles indicate the positions of $|A_{0,n}\rangle$ in Eq. (38) and $|B_{0,n}\rangle$ in Eq. (39), respectively.

Figure 15

Entanglement entropies in all eigenstates of $H_1(h,\left\{D_j\right\})$ in Eq. (45) for $L=8$, $h=1$, $D_j\in [-1,1]$. The density of data points is color coded. The red circle corresponds to $|{\overline{B}}_n\rangle (n=0,1,...,L)$.

Figure 16

Entanglement entropies in all eigenstates of $H_1(h,\left\{D_j\right\})$ in Eq. (45) for $L=10$, $h=1$ in the symmetry sector $({\mathcal{S}}^z,\mathcal{F})=(0,1)$. Each $D_j$ is randomly chosen from $[-5,5]$. The density of data points is color coded. The entanglement entropy of each state obeys a volume-law except for $|{\overline{B}}_{L/2}\rangle$, the symmetric state $|{\psi }_s\rangle =|+-+-\cdots \rangle +|-+-+\cdots \rangle$, and the trivial state $|\mathbf{0}\rangle =|00\cdots 0\rangle$. The orange dotted line indicates $S_A=1.236$, which is obtained from Eq. (49) for $L=10$.

Figure 17

$\langle H_{\mathrm{AKLT}}\rangle$ in all eigenstates of $H_1(h,\left\{D_j\right\})$ in Eq. (45) for $L=10$, $h=1$ in the symmetry sector $({\mathcal{S}}^z,\mathcal{F})=(0,1)$. Each $D_j$ is randomly chosen from $[-5,5]$. The density of data points is color coded. The red, blue, and green circles indicate $|{\overline{B}}_{L/2}\rangle$, $|{\psi }_s\rangle$, and $|00\cdots 0\rangle$, respectively.

Figure 18

The dynamics of the fidelity with $L=8$, $h=1$, and $D_j(j=1,2,...,L)$ chosen randomly from $[-1,1]$. Perfectly periodic revivals can be seen when the initial state is a coherent state, whereas for other generic states the fidelity decays rapidly to zero.

Figure 19

Dynamics of the half-chain entanglement entropies with the same setup as Fig. 18. The dashed line indicates the Page value $S_{\mathrm{Page}}$ [Eq. (60)]. Initial coherent states have constant entanglement entropy, but that of $|-0+-0+-0\rangle$ rapidly grows and saturates near $S_{\mathrm{Page}}$.

Figure 20

(a) Entanglement entropies in all eigenstates in Eq. (61) with $L=10,h=1$. Each $D_j$ is randomly chosen from $[-1,1]$. The density of data points is color coded. The states $|{\overline{A}}_n\rangle$ (red circles) and $|{\overline{B}}_n\rangle$ (blue circles) have relatively low entanglement entropy. Note that although $|{\overline{A}}_n\rangle$ and $|{\overline{B}}_n\rangle$ are eigenstates of $H_2(h,{\left\{D_j\right\}}_j)$, some circles do not overlap with corresponding data points because of degeneracies. (b) Entanglement entropies in all eigenstates of inhomogeneous model Eq. (61) for $L=10,h=1$ in the symmetry sector ${\mathcal{S}}^z=0$. Each $D_j$ is randomly chosen from $[-1,1]$. The density of data points is color coded. The red, blue and green circles indicate $|{\overline{A}}_L\rangle$, $|{\overline{B}}_{L/2}\rangle$, and $|000\cdots \rangle$, respectively.

Figure 21

The dynamics of the fidelity with $h=1$, $L=8$, and $D_j(j=1,2,...,L)$ chosen randomly from $[-1,1]$. The fidelity shows perfectly periodic revivals when the initial state is a coherent state, whereas it decays rapidly to zero for other generic states.

Figure 22

An example of the triangular lattice. The integers denote the site indices, and the black and white sites with the same index are identified by the periodic boundary conditions. We take the subsystem $A$ to be the set of sites enclosed by the dashed lines, which is used to calculate the entanglement entropy.

Figure 23

Entanglement entropies in all eigenstates of $H_1^{2\mathrm{d}}(h,\left\{D_j\right\})$ in Eq. (64) for $L=9,h=1$ in the symmetry sector ${\mathcal{S}}^z=1$. Each $D_j$ is randomly chosen from $[-1,1]$. The density of data points is color coded. The red circle indicate the scar state $|{\mathrm{\Psi }}_{n=4}\rangle$.

Figure 24

The dynamics of the fidelity of the superposition of two coherent states $|\alpha \rangle$ and $|\beta \rangle$ driven by $H_2$ [Eq. (61)] with $h=1,L=8$, and $D_j(j=1,2,...,L)$ chosen randomly from $[-1,1]$. The period of the revivals is $2\pi /h$.