Flat-band full localization and symmetry-protected topological phase on bilayer lattice systems

In this work, we present bilayer flat-band Hamiltonians, in which all bulk states are localized and specified by extensive local integrals of motion (LIOMs). The present systems are bilayer extension of the Creutz ladder, which is studied previously. In order to construct models, we employ building blocks, cube operators, which are linear combinations of fermions defined in each cube of the bilayer lattice. There are eight cubic operators, and the Hamiltonians are composed of the number operators of them, the LIOMs. A suitable arrangement of locations of the cube operators is needed to have exact projective Hamiltonians. The projective Hamiltonians belong to a topological classification class, the BDI class. With the open boundary condition, the constructed Hamiltonians have gapless edge modes, which commute with each other as well as the Hamiltonian. This result comes from a symmetry analogous to the one-dimensional chiral symmetry of the BDI class. These results indicate that the projective Hamiltonians describe a kind of symmetry-protected topological phase matter. Careful investigation of topological indexes, such as the Berry phase and string operator, is given. We also show that, by using the gapless edge modes, a generalized Sachdev-Ye-Kitaev model is constructed.

Figure 1

Schematic picture of fermion operators on the bilayer lattice. $c_{\vec{r}}$'s reside on the upper square lattice, and $d_{\vec{r}}$'s on the lower lattice, where $\vec{r}=(x,y)$.

Figure 2

Schematic picture of eight cube operators on the bilayer lattice, $Q_{\vec{r}}^{+}$ through ${\tilde{P}}_{\vec{r}}^{-}$. The eight cube operators are linear combinations of $\{{\omega }_{\vec{r},\widehat{i}}^A,{\tilde{\omega }}_{\vec{r},\widehat{i}}^A,{\overline{\omega }}_{\vec{r},\widehat{i}}^A\}$ (blue) and $\{{\omega }_{\vec{r},\widehat{i}}^B,{\tilde{\omega }}_{\vec{r},\widehat{i}}^B,{\overline{\omega }}_{\vec{r},\widehat{i}}^B\}$ (red). The numbers near vertices of the cube indicate the value of the coefficient of the cube operators.

Figure 3

Schematic picture of the Hamiltonian of the bilayer system given by Eq. (12). The unit cell is $2\times 2$, which contains four cubes.

Figure 4

Schematic picture of the Hamiltonian of a square prism given by Eq. (16). Each cube contains four cube operators, $Q_{\vec{r}}^{+}$ through ${\tilde{Q}}_{\vec{r}}^{-}$.

Figure 5

(a) Energy eigenvalues of $H_{\mathrm{SP}}$ for ${\tau }_0=2$ and ${\tau }_1=1$ for the periodic boundary condition. There exist four flat bands. The numbers near each band show ${\gamma }_M$, the Berry phase. All flat bands are topologically nontrivial. (b) Energy eigenvalues under the open boundary condition. Fourfold zero-energy edge states appear.

Figure 6

(a) Coarse-grained schematic picture of the bilayer system with open boundary condition of thin cylinder (disk) shape. (b) There emerge four edge modes, each of which is one dimensional and a linear combination of ${\left({\omega }^A\right)}^{†}$'s $\mathit{or} {\left({\omega }^B\right)}^{†}$'s.

Figure 7

Density profile of half-filled state $+$ two particles for various interaction strength $V$. We plot the total density of the unit cell, $N_r^{unit}=n_{r+y}^c+n_r^c+n_{r+y}^d+n_r^d$. Edge modes are quite stable against the repulsion. The system size is $L=5$ (20 sites) with 12 particles.

Figure 8

(a) Berry phase ${\gamma }_L$ as a function of $V$ for $1/4$ filling. For $V>V_c\simeq 2.7$, the Berry phase is unstable. (b) The energy gap $\mathrm{\Delta }E$ for $1/4$ filling (the difference between ground state and first excited state). Both results indicate the existence of some kind of crossover in the vicinity of $V_c$. (c) Energy of the ground state, and first and second excited states without twist. The energy gap of the ground state decreases as $V$ increases, but it keeps finite even for $V>V_c$. System size $L=4$ ($4\times 4$ sites) and 4 particles. (d) Berry phase ${\gamma }_L$ as a function of $V$ for half filling. For $V>V_c\simeq 2$, the Berry phase is unstable. (e) The energy gap $\mathrm{\Delta }E$ for half filling (the difference between ground state and first excited state). Both results indicate the existence of a phase transition in the vicinity of $V_c=2$. (f) Energy of the ground state, and first and second excited states without twist for half filling. The gapless ground state emerges for $V>V_c^{\prime }\simeq 2.0$. System size $L=4$ ($4\times 4$ sites) and 8 particles.