Topological band insulators without translational symmetry

Abstract

In the research of the topological band phases, the conventional wisdom is to start from the crystalline translational symmetry systems. Nevertheless, the translational symmetry is not always a necessary condition for the energy bands. Here we propose a systematic method of constructing the topological band insulators without translational symmetry in the amorphous systems. By way of the isospectral reduction approach from spectral graph theory, we reduce the structural-disordered systems formed by different multi-atomic cells into the isospectral effective periodic systems with the energy-dependent hoppings and potentials. We identify the topological band insulating phases with extended bulk states and topological in-gap edge states by the topological invariants of the reduced systems, density of states, and the commutation of the transfer matrix. In addition, when the building blocks of the two multi-atomic cells have different number of the lattice sites, our numerical calculations demonstrate that the existences of the flat band and the macroscopic bound states in the continuum in the amorphous systems. Our findings uncover an arena for the exploration of the topological band states beyond translational symmetry systems paradigm.

Figures (5)

• Figure 1: $(a)$ The systematic construction of the topological band insulators without translational symmetry. $(b)$ Amorphous chain formed by tetratomic and diamond-shaped cells. $(c)$ The four topological Bloch bands of the amorphous chain.
• Figure 2: The lattice structure and phase diagram of the diamond-tetratomic model. $(a)$ A typical configuration of the amorphous chain formed through the random arrangement of tetratomic and diamond cells. $(b)$ The reduced dimerized chain by ISR approach. $(c)$ Phase diagram of the original diamond-tetratomic amorphous system at $1/4$ filling in the $J_a-J_b$ parameter space.
• Figure 3: Real-space energy spectra and wave functions with the number of building cells $N=100$ and inter-cell hopping amplitude $t=1$. $(a)$ Energy spectrum as a function of intra-cell hopping amplitude $J_{a}$ ($J_{a}=J_{b}$) under OBC, $J_a$ from 0 to 1.5. $(b)$ The density profiles of the wave functions of two topological edge states at $1/4$ filling with energies $E_1=-0.1414$, $E_2 =-0.2613$ both marked as red circle in $(c)$. $(c)$ Energy spectrum of the original diamond-tetratomic system for intra-cell hopping amplitudes $J_{a}=J_{b} =0.1$.
• Figure 4: Energy spectrum and average DOS of the original diamond-tetratomic system under OBC with the system size $N = 100$, intra-cell couplings $J_a=J_b=0.3$, and the inter-cell hopping $t=1$. $(a)$ Energy spectrum with $m_a=\sqrt{2}J_a$ and $m_b=(\sqrt{2}+1)J_b$. $(b)$ Energy spectrum with $m_a=4J_a$ and $m_b=3.5J_b$ under one disorder configuration. Panels $(c)$ and $(d)$ are average DOS acting as a function of energy.
• Figure 5: The diamond-trimer model and its topological features. $(a)$ A typical realization of the diamond-trimer model. $(b)$ Numerically calculate winding number $w$ as a function of intre-cell hopping $t$, with the intra-cell hopping $J_a=1$, $J_b=\sqrt{2}$. $(c)$ The real-space energy spectrum of the original diamond-trimer model with the system parameters $J_a = 1$, $J_b=\sqrt{2}$, $t = 2$, and the number of building cells $N=100$. $(d)$ The probability density of the topological edge states with energy $E = -\sqrt2$, marked as red diamond in $(c)$, localized on the boundary of the system.