The disordered Su-Schrieffer-Heeger model

TL;DR

The paper investigates how diagonal and off-diagonal disorder affect localization and topology in the disordered SSH model by deriving an analytical expression for the Lyapunov exponent $\lambda(E)$. A non-perturbative approach based on a local-density recurrence is developed, yielding $\rho_{n+1}=\sum_{i=0}^2 D_n^{(i)}\rho_{n-i}$ and its disorder-averaged form, which leads to $\lambda(E)$ that matches extensive numerical simulations across disorder types and strengths. The analysis shows that at $E=0$ the Lyapunov exponent is $\lambda=\log(t_1/t_2)$ and remains invariant under off-diagonal disorder, reflecting chiral-symmetry protection of the zero-energy state. The work also characterizes topology under disorder via real-space winding numbers and density of states, demonstrating robust topological features against off-diagonal disorder with implications for quantum-topological applications.

Abstract

Quantum topology categorizes physical systems in integer invariants, which are robust to some deformations and certain types of disorder. A prime example is the Su-Schrieffer-Heeger (SSH) model, which has two distinct topological phases, the trivial phase with no edge states and the non-trivial phase with zero-energy edge states. The energy dispersion of the SSH model is dominated by a gap around zero energy, which suppresses the transmission. This exponential suppression of the transmission with system length is determined by the Lyapounov exponent. Here we find an analytical expression of the Lyapounov as a function of energy in the presence of both diagonal and off-diagonal disorder. We obtain this result by finding a recurrence relation for the local density, which can be averaged over different disorder configurations. There is excellent agreement between our analytical expression and the numerical results over a wide range of disorder strengths and disorder types. The real space winding number is evaluated as a function of off-diagonal and on-site disorder for possible applications of quantum topology.

Figures (5)

• Figure 1: Illustration of the SSH model with alternating hopping $t_1$ and $t_2$, and random corrections $\tau_n$. Further included are random onsite energies $v_n$. The corresponding tight binding equations are shown for the local amplitudes $\psi_n$.
• Figure 2: Lyapounov exponent as a function of energy. In (a) the plot shows different values of $t_1$ from 1 to 1.25 with $t_2=1/t_1$ and with off-diagonal disorder with $W=0.5$. In (b) different values of off-diagonal disorder strengths are shown for $W=0.1$ to $0.5$ and $t_1=1.25$ and $t_2=1/t_1$. The Lyapounov exponent is computed using $\lambda_N$ from equ. \ref{['lyapounovnum']}, where we used $N=2000$ and 1000 configurational averages. For $\lambda_T$ we used equ. \ref{['lypounovT']}.
• Figure 3: In (a) we show the topological $\lambda_N$ as a function of energy for different values of $N$ for $t_1=0.8$ and $t_2=1/t_1$. Also shown is the non-topological $\lambda_N$ for $N=2000$ and $t_1=1/0.8$ and $t_2=1/t_1$ as well as $\lambda_T$ for the same parameters. In b) $\Delta\lambda= |\lambda_N(E)-|\log(t_1/t_2)||$ is shown as a function of $N$ for $t_1=0.8$ and $t_2=1/t_1$ for 2 values of $E$. In c) and d) $\lambda_N$ is shown as a function of $E$ for different values of $N$ for the non-topological and topological cases, respectively. We used ($t_1=1/0.8$, $t_2=1/t_1$) and ($t_1=1/0.8$, $t_2=1/t_1$), respectively.
• Figure 4: The (a) and (b) panels show the density of states as a function of off-diagonal ($W$) (a) and diagonal ($V$) (b) disorder strength. The (c) and (d) panels show the DOS at $E=0$, the real space winding number, and the Lypounov exponent (the theoretical and numerical value) as a function of off-diagonal (a) and diagonal (b) disorder strength. The red line shows the winding number as a function of the disorder strength. Here $t_1=0.8$ and $t_2=1/t_1$.
• Figure 5: The top panels show the real space winding number $\gamma$ (color scale) as a function of off-diagonal disorder ($W$) and on-site disorder ($V$) for $t_1=0.9$ (left) and $t_1=1/0.9$ (right), respectively. In both cases $t_2=1/t_1$. The bottom plot depicts the two cases for $V=0$.