Flat band quantum scar

We show that a quantum scar state, an atypical eigenstate breaking eigenstate thermalization hypothesis embedded in a many-body energy spectrum, can be constructed in flat band systems. The key idea of our construction is to make use of orthogonal compact localized states. We concretely discuss our construction scheme, taking a sawtooth flat lattice system as an example, and numerically demonstrate the presence of a quantum scar state. Examples of higher-dimensional systems are also addressed. Our construction method of quantum scar has broad applications to various flat band systems.

Figure 1

Schematic picture of sawtooth lattice. The yellow shaded object is a CLS. The CLSs do not spatially overlap each other due to the presence of a finite $\mu$. The minimum distance between the neighboring CLS is $d_m=2$.

Figure 2

Numerical results for the sawtooth lattice model. (a) Mean level spacing $\langle r\rangle$ averaged over all energy eigenvalues in the momentum sector $k=0$ (see [8, 55]). We set $L=5$ with five particles and $L=6$ with six particles. (b) Overlap to $|Z_4\rangle$ for all eigenstates. The red circle indicates $|{\mathrm{\Psi }}_{\mathrm{L}}\rangle$. (c) Distribution of EE. The EE is normalized by the number of sites in the subsystem. The red circle indicates $|{\mathrm{\Psi }}_{\mathrm{L}}\rangle$. The EE for the red circle is very small because the only single CLS is cut; the value is $s_{\min }=\frac{1}{2L}(2ln2-\frac{3}{4}ln3)\approx 0.05623$. For (b) and (c), we set $\mu =-2, V_0=3$, and $L=5$ with five particles. (d) Single-shot quench dynamics of EE: the disorder is ${\mu }_{A,j}, {\mu }_{B,j}\in [-w_0,w_0]$ with $V_0=3$. We set $L=5$ with five particles and two initial states, $|{\mathrm{\Psi }}_{\mathrm{L}}\rangle$ and $|{\mathrm{Z}}_4\rangle$.

Figure 3

(a) Schematic figure of the kagome model with real hoppings. The CLSs are located at the hexagons on which the yellow shaded balls are placed. (b) The band structure for $(t,{\mu }_1,{\mu }_2,{\mu }_3)=(1,0.5,0.75,-0.8)$. The high-symmetry points in the Brillouin zone are $\mathrm{\Gamma }=(0,0), K=(\frac{2\pi }{\sqrt{3}{a}_0},\frac{2\pi }{3a_0})$, and $M=(\frac{\pi }{\sqrt{3}{a}_0},\frac{\pi }{a_0})$, with $a_0$ being the lattice constant. The blue line represents the flat band. (c) Schematic figure of the kagome model with pure imaginary hoppings. The CLSs are located at the hexagons on which the yellow shaded balls are placed. (d) The band structure for $(t,{\mu }_1,{\mu }_2,{\mu }_3)=(1,0.5,0.75,-0.8)$. The blue line represents the flat band. The Chern number is shown beside each band. (e) Schematic figure of the square kagome model. The CLSs are located at the square plaquettes on which the yellow shaded balls are placed. (f) The band structure for $(t_1,t_2)=(1,1.2)$. The blue line represents the flat band.