Topological many-body scar states in dimensions one, two, and three

We propose an exact construction for atypical excited states of a class of nonintegrable quantum many-body Hamiltonians in one, two, and three dimensions that display area-law entanglement entropy. These examples of many-body “scar” states have, by design, other properties, such as topological degeneracies, usually associated with the gapped ground states of symmetry-protected topological phases or topologically ordered phases of matter.

Figure 1

Distribution of consecutive energy-level spacings $s_n^{}$ for the 1D Hamiltonian $H$ defined in Eq. (3) with $L=20,\alpha =0.3$, and $\beta =0.5$. The distributions for the $s_n^{}$ from all momentum sectors, except for $k=0,\pi$, have been joined. The middle 60% of the spectrum in each sector is taken. The distribution obtained can be seen to be well approximated by the Gaussian orthogonal ensemble (GOE) of random matrix theory.

Figure 2

Entanglement entropy of the eigenstates of Hamiltonian (3) for a real-space bipartition of the system into two equal halves. The parameters are set at $L=16,\beta =0.5$, and $\alpha =0.3$. The analytically obtained scar state has $E=0$ (red circle) and is well separated from the highly entangled states.

Figure 3

Example of a lattice structure of the 2D model. Dashed sites and lines are used to represent periodic boundary conditions. (a) Starting from a $\left(N_x^{}\times N_y^{}=2\times 4\right)$ square lattice ${\mathrm{\Lambda }}_{★}^{}$, we define the median and dual lattices ${\mathrm{\Lambda }}_{\circ }^{}$ and ${\mathrm{\Lambda }}_{\square }^{}$ in such a way that sites of ${\mathrm{\Lambda }}_{★}^{},{\mathrm{\Lambda }}_{○}$, and ${\mathrm{\Lambda }}_{\square }^{}$ are represented by the symbols $★,○$, and $\square$, respectively. The red (blue) path ${\mathcal{P}}_1^{}\left({\mathcal{P}}_2^{}\right)$ along the bonds of ${\mathrm{\Lambda }}_{★}^{}\left({\mathrm{\Lambda }}_{\square }^{}\right)$ goes through all sites $★\in {\mathrm{\Lambda }}_{★}^{}\left(\square \in {\mathrm{\Lambda }}_{\square }^{}\right)$ without intersecting itself. (b) The toric code assigns a local spin-$\frac{1}{2}$ degree of freedom to each site $○$ of the median lattice ${\mathrm{\Lambda }}_{○}$. To each site $★(\square )$ of the lattice ${\mathrm{\Lambda }}_{★}^{}\left({\mathrm{\Lambda }}_{\square }^{}\right)$, we assign the subset $s\left(p\right)$ consisting of the four sites of ${\mathrm{\Lambda }}_{○}$ on the red cross (blue square) at the site $★(\square )$ and define the star (plaquette) operator $A_s^{}:={\prod }_{i\in s}{X}_i^{}\left(B_p^{}:={\prod }_{i\in p}{Z}_i^{}\right)$. The two orthogonal green lines are the “electric” paths $l_x^{}$ and $l_y^{}$ needed to define two Wilson loops $W_{\mu }^{}:={\prod }_{i\in l_{\mu }^{}\cap {\mathrm{\Lambda }}_{○}}{Z}_i^{}$ with $\mu =x,y$, respectively.

Figure 4

The notation ${\mathrm{\Lambda }}_{★}^{},{\mathrm{\Lambda }}_{○}$, and ${\mathrm{\Lambda }}_{\square }^{}$ of Fig. 3 becomes ${\mathrm{\Lambda }}_{★}^{},{\mathrm{\Lambda }}_{○}$, and , where ${\mathrm{\Lambda }}_{★}^{}$ denotes the cubic lattice, ${\mathrm{\Lambda }}_{○}$ its median lattice, and its dual lattice. (a) The elementary unit cell of ${\mathrm{\Lambda }}_{★}^{}$ is cubic. Spin-$\frac{1}{2}$ degrees of freedom represented by $○$ are located on its midbonds. The $12○$'s on the bonds of a define a subset $c\subset {\mathrm{\Lambda }}_{○}$. The corners of define sites $★$ of ${\mathrm{\Lambda }}_{★}^{}$. The center of defines a site from . For any such , we define $B_c^{}$ by taking the product of all 12 Pauli matrices $Z_i^{}$ from the neighboring bonds with $i\in c\cap {\mathrm{\Lambda }}_{○}$. (b) The center of a cross $+$ joining its four nearest-neighbor sites from ${\mathrm{\Lambda }}_{○}$ defines a site from ${\mathrm{\Lambda }}_{★}^{}$ and the subset $s\subset {\mathrm{\Lambda }}_{○}$. There are three oriented crosses for any site from ${\mathrm{\Lambda }}_{★}^{}$. They are in one-to-one correspondence with the three oriented planes in the Cartesian coordinates of ${\mathbb{R}}^3$. For any such oriented cross, we define $A_s^{}$ by taking the product of all four Pauli matrices $X_i^{}$ with $i\in s$.

Figure 5

(a) Level statistics for the Hamiltonian given by Eq. (C1a) with system size $L=20$. As with Fig. 1, the middle 60% of the eigenvalues from all the symmetry sectors except for $k=0,\pi$ are used. The distribution follows the Gaussian unitary ensemble (GUE), which indicates nonintegrability. The absence of inversion symmetry in Eq. (C1a) removes the possible connection between two momentum sectors with opposite $k$, and makes the matrix elements nonreducible complex numbers. Hence, the trend of GUE, which is different from Fig. 1, is justified. (b) Entanglement entropy–energy plot for every eigenstate of Eq. (C1a) with $L=16$. Two scar states at $E=-L$ and $+L$ are marked by red circles. Parameters are set at $\beta =0.5,{\alpha }_i^{}={(-1)}^i$.

Figure 6

Examples of lattice structures for the 2D model. Dashed sites and lines are used to represent periodic boundary conditions. Any path ${\mathcal{P}}_1^{}$ that is colored in red starts and ends by definition on the sites of the lattice ${\mathrm{\Lambda }}_{★}^{}$. Any path ${\mathcal{P}}_2^{}$ colored in blue starts and ends by definition on the sites of the dual lattice ${\mathrm{\Lambda }}_{\square }^{}$. The spin degrees of freedom are located on the sites of the median lattice ${\mathrm{\Lambda }}_{○}$ denoted by open circles. (a)–(d) Example of the path ${\mathcal{P}}_1^{}$ colored in red and the path ${\mathcal{P}}_2^{}$ colored in blue for a square lattice of given aspect ratio. Only the sites $i$ of ${\mathrm{\Lambda }}_{○}$ represented by open circles are shown. With this choice for the paths ${\mathcal{P}}_1^{}$ and ${\mathcal{P}}_2^{}$, the condition ${\beta }_1^{},{\beta }_2^{}>0$ is sufficient to guarantee that the sum over $s$ in $H_1^{2\mathrm{D}}$ (the sum over $p$ in $H_2^{2\mathrm{D}})$ can never be arranged into the sum of two nonvanishing Hermitian operators that commute pairwise and commute with $H_2^{2\mathrm{D}}\left(H_1^{2\mathrm{D}}\right)$. (e) The choice made for the path ${\mathcal{P}}_1^{}$ colored in red and the path ${\mathcal{P}}_2^{}$ colored in blue fails to guarantee that the sum in $H_{\mathtt{a}}^{2\mathrm{D}}$ can be arranged into the sum of two nonvanishing Hermitian operators that commute pairwise and with $H_{\overline{\mathtt{a}}}^{2\mathrm{D}}$ when ${\beta }_{\mathtt{a}}^{},{\beta }_{\overline{\mathtt{a}}}^{}>0$. Indeed, of all Hermitian operators $B_p^{}$ entering $H_2^{2\mathrm{D}}$, those sites from the dual lattice ${\mathrm{\Lambda }}_{\square }^{}$ that are identified by the symbol $\square$ are not traversed by ${\mathcal{P}}_2^{}$. They give a set of operators $\left\{B_{\square }^{}\right\}$, whereby $B_{\square }^{}$ commutes with both $H_1^{2\mathrm{D}}$ and $H_2^{2\mathrm{D}}$.