Exact projector Hamiltonian, local integrals of motion, and many-body localization with symmetry-protected topological order

In this work, we construct an exact projector Hamiltonian with interactions, which is given by a sum of mutually commuting operators called stabilizers. The model is based on the recently studied Creutz ladder of fermions, in which flat-band structure and strong localization are realized. These stabilizers are local integrals of motion from which many-body localization (MBL) is realized. All energy eigenstates are explicitly obtained even in the presence of local disorders. All states are MBL states, that is, this system is a full many-body localized (FMBL) system. We show that this system has a topological order and stable gapless edge modes exist under the open boundary condition. By the numerical study, we investigate stability of the FMBL and topological order.

Figure 1

Creutz ladder model: The yellow shade regime is a unit cell; the blue and magenta shade objects are compact localized states $W^{+}$ and $W^{-}$ defined by Eq. (6).

Figure 2

Single-shot disordered many-body spectra for different fillings; the data show the many-body energy from the lowest to the eighth states. Here, $2t_j=1+{\delta }_j$ with the uniform distribution random values ${\delta }_j\in [-0.5,0.5]$. (a) $u=0$. There exist fourfold degenerate states with different particle numbers indicating the existence of zero-energy edge modes. (b) $u=3$. The fourfold degeneracy is broken. The system size is $L=14$.

Figure 3

Short-time exact dynamics for the $H_{\mathrm{flat}}+H_{\mathrm{IR}1}$ system. (a) $P_0\left(t\right)$ in Eq. (42). (b) Dynamics of the $w$-particle density at the center of the system. System size $L=8$. (c) Short-time exact dynamics of $P_0\left(t\right)$ in the presence of interaction of $H_S$ and weak $H_{\mathrm{IR}1}$. Here, $v_1=0.1$, $L=8$. For all data, the unit of time is $\hslash /\left(2{\tau }_0\right)$, and unit of energy $2{\tau }_0$.

Figure 4

Short-time exact dynamics of the particle density at the center of the system. (a) Dynamics for the system $H_{\mathcal{T}}$. Data show perfect oscillating behavior of all values of $g$. Period of the oscillation is longer for smaller $g$. (b) Dynamics for the system $H_{\mathcal{T}}+H_{\mathrm{IR}1}$. Here, $v_1=0.1$. Particle density exhibits stable oscillation for larger value of $g$, whereas it decreases to $1/2$ for small $g$. For all data, $L=8$. The hopping amplitude in $H_{\mathcal{T}}$ is set to be uniform: $t_j={\tau }_0$. The unit of time is $\hslash /\left(2{\tau }_0\right)$ and unit of energy $2{\tau }_0$.

Figure 5

Energy spectra in the $w$-particle representation under the OBC. (a) Uniform coupling $2t_j=1$. Flat-band dispersion and zero-energy modes exist in the band center. (b) Single-shot disordered spectra for random coupling: $2t_j=1+{\delta }_j$ with the uniformly-distributed random values ${\delta }_j\in [-0.5,0.5]$. (c) $v_1=0.3$. Flat bands disappear, but zero-energy modes exist in the band center. (d) $v_2=0.3$. Flat bands exist, but zero-energy modes split into interband states with finite energy.