Localization with non-Hermitian off-diagonal disorder

TL;DR

This paper investigates a one-dimensional non-Hermitian tight-binding model with random off-diagonal hopping, where left-right and right-left amplitudes are unequal but the spectrum remains real due to a sign constraint on the hopping products. Using exact diagonalization, spectral statistics, DOS analysis, and extensive finite-size scaling of localization length, IPR, and the ground-state gap, the authors reveal a delocalization-localization crossover driven by off-diagonal disorder in finite systems, with localization emerging for infinitesimal disorder in the thermodynamic limit. A key finding is a mid-band anomaly: a singular DOS at $E=0$ accompanied by a delocalized mid-band state that persists for any disorder strength, while band-edge states localize as disorder grows. The scaling analysis yields a universal set of exponents ($\nu \approx 0.63$, $\delta \approx 2$, $\gamma = \nu$) consistent with 1D Anderson universality, and the results point to a non-Hermitian extension of chiral universality; the work also discusses experimental realizations in circuit networks and the broader implications for pseudo-Hermitian random matrices.

Abstract

In this work, we discuss a non-Hermitian system described via a one-dimensional single-particle tight-binding model, where the non-Hermiticity is governed by random nearest-neighbour tunnellings, such that the left-to-right and right-to-left hopping strengths are unequal. A physical situation of completely real eigenspectrum arises owing to the Hamiltonian's tridiagonal matrix structure under a simple sign conservation of the product of the conjugate nearest-neighbour tunnelling terms. The off-diagonal disorder leads the non-Hermitian system to a delocalization-localization crossover in finite systems. The emergent nature of the crossover is recognized through a finite-size spectral analysis. The system enters into a localized phase for infinitesimal disorder strength in the thermodynamic limit. We perform a careful scaling analysis of localization length, inverse participation ratio (IPR), and energy splitting and report the corresponding scaling exponents. Noticeably, in contrast to the diagonal disorder, the density of states (DOS) has a singularity at E=0 in the presence of the off-diagonal disorder and the corresponding wavefunction remains delocalized for any given disorder strength.

Figures (6)

• Figure 1: (a-d) Spectral Analysis at Band Edge: (a) represents a case of distribution for the level spacing $s$ between the ground and first excited state for $L=50$ and $W=0.1$. Fitting a function of the form given in Eq. \ref{['Wigner-Dyson']}, it turns out $\alpha=1.5$ and $\beta=2.5$, implying a Wigner-Dyson type distribution. Keeping the same disorder $W=0.1$ for $L=200$, the distribution shifts towards the origin as shown in (b). The same nature occurs in (c) for $L=50$ with $W=1.0$, which indicates that the spectral statistics move away from the Wigner-Dyson type distribution as we increase either the system size or the disorder. In the case of (d), where $L=200$ and $W=1.0$, it has become a Poisson type distribution having $\lambda=0.83$ as found by fitting a function of the form provided in Eq. \ref{['Poisson']}. (e-h) Spectral Analysis at Mid-Band: In the mid-band, the wavefunctions being least localized, for smaller system sizes, the spectral statistics is indeed Wigner-Dyson type even for $W=1.0$, and consequently, for $L=200$ with $W=1.0$ in (e), the spectral statistics is still Wigner-Dyson type. For the same disorder, with increasing system size $L=500$, $1000$, $2000$ in (f), (g), and (h), respectively, the nature of the spectral statistics slowly moves towards a Poisson-type distribution. This trend indicates that for all the energy levels, in the thermodynamic limit, the spectral statistics would collapse from Wigner-Dyson to Poisson-type distribution for an infinitesimal disorder.
• Figure 2: Density of States: The red solid line and blue dotted line correspond to the density of states for disorder strength $W=1$ and $W=10$, respectively. An increase in the disorder results in a larger peak at $E=0$ energy. The density of states is normalized in such a way that $\int \rho(E)~dE=1$. The presence of singularity in the mid-band is related to the delocalized nature of the wavefunction.
• Figure 3: Localization Length vs Energy: The red solid line, blue dashed line, and green dotted line correspond to system size $L=200$, $500$, and $1000$, respectively. For all of these, the disorder strength is kept fixed at $W=1.0$. In all three cases, there is a rise in the value of localization length at the middle of the spectrum, and this behaviour becomes stronger with an increase in system size. At the thermodynamic limit, only $E=0$ has a delocalized eigenstate.
• Figure 4: Localization Length: (a) shows the variation in localization length of the ground state, $\xi_g$, with increasing disorder $W$ for various system sizes $L$, where square, circle, triangle, diamond, and pentagon represent the cases for $L$ = 50, 100, 200, 500, and 1000 respectively. They are characterized by an initial flat region, where the corresponding wave function has an extended nature. Beyond a certain disorder $W^{*}$, the localization length has only a very weak dependence on the system size and decreases almost linearly. This $W^{*}$ is found to be proportional to $L^{-1/\nu}$ with $\nu=0.63(3)$, which is obtained via fitting as shown using the solid line in (b), implying the onset of localization in the thermodynamic limit even at infinitesimal disorder. (c) illustrates localization length at $W^*$, $\xi_g^*$, as a function of $L$. The circles stand for the data points, and the solid line is the best fit. It turns out that $\xi_g^* \propto L$ which suggests $\mu=-1$ in Eq. \ref{['Collapse']}. (d) depicts the collapse plot, where different plots corresponding to different system sizes in (a) merge for $\nu=0.63(1)$. In both (a) and (d), horizontal and vertical axes are in log$_{10}$ scale.
• Figure 5: Inverse Participation Ratio and Energy Splitting: The IPR for the ground state, $I_g$, is plotted as a function of $W$ in (a) for different $L$. Here also, as in the case of localization length, there is a flat region in $I_g$ upto the same $W^*$ for each system size that scales as $W^*\propto L^{-1/\nu}$. The corresponding collapse plot is shown in (b), where the perfect data collapse is found at $\gamma=\nu=0.63(1)$. This suggests that $I_g\propto L^{-1}$ near $W_c$. The average level spacing $\Delta E$ between the ground state and the $1$st excited state remains small for small values of $W$ and after a certain $W^*$ for each system it continues to increase as shown in (c). From the scaling analysis in (d) it is found that $\Delta E$ scales as $L^{-\delta}$ with $\delta=2$. This implies that although the system is gapless in the thermodynamic limit, the introduction of infinitesimal disorder opens up a gap. The same point-type scheme is used here as in Fig. \ref{['Localization Length']} to indicate different system sizes. For both the plots horizontal and vertical axes are in log$_{10}$ scale.