Disorder-driven exceptional points and concurrent topological phase transitions in non-Hermitian systems

TL;DR

This work shows that random disorder can intrinsically generate exceptional points (EPs) and concurrent topological phase transitions (TPTs) in a non-Hermitian lattice with nonreciprocal hopping. Using an Anderson-disorder model and an effective medium theory based on the self-consistent Born approximation, the authors reveal a competition between disorder-induced energy renormalization and nonreciprocity-driven interorbital hybridization that produces real–complex–real spectral transitions and band inversions. The phase diagram features extended EP lines emanating from the Hermitian TPT point, demonstrating that disorder acts as a robust control knob for EP-mediated topology across a broad parameter range. The findings provide a general framework for engineering EP-driven topology in realistic platforms such as photonic lattices, topolectrical circuits, and cold-atom systems, and point to future work on interactions, structured disorder, and 3D extensions.

Abstract

Exceptional points (EPs) are spectral degeneracies unique to non-Hermitian systems which underpin phenomena from enhanced sensing to unconventional topology. While disorder is usually viewed as detrimental, it can also drive topological phase transitions (TPTs). Here, we show that random disorder alone can generate EPs and concurrent TPTs in a multiorbital non-Hermitian lattice with nonreciprocal hopping. Increasing disorder induces successive real-complex-real spectral transitions accompanied by band inversion and quantized changes in the spin Bott index. Using effective medium theory and large-scale simulations, we trace these transitions to a competition between disorder-induced energy-level renormalization and nonreciprocity-driven hybridization. The resulting phase diagram reveals extended EP lines that emerge from the Hermitian TPT point and persist over a broad parameter range. Our results establish disorder as an active mechanism for engineering exceptional point mediated topology in non-Hermitian matter.

Figures (8)

• Figure 1: Exceptional points induced by nonreciprocal hopping in a two-dimensional lattice. (a) Schematic of a two-dimensional lattice model with three orbitals per site. Arrows indicate the couplings between atom $i$ and its neighboring atoms, with dark (light) green arrows representing hoppings to the right (left) of atom $i$. The cutoff radius is $R=1.9a$, where $a$ is the lattice constant. (b) and (c) Real and imaginary parts of the conduction ($E_c$) and valence ($E_v$) band energies as a function of non-reciprocal coupling strength $\kappa$ when $\lambda = 1.2$ and $2$, respectively. The other parameters are $a=1$, $\varepsilon_s = 1.8$, $\varepsilon_p= -6.5$, $V_{ss\sigma} = -0.256$, $V_{sp\sigma} = 0.576$, $V_{pp\sigma} = 1.152$, and $V_{pp\pi}= 0.032$. (d) Spin Bott index $\mathrm{B_s}$ as a function of $\kappa$, with the top and bottom panels shows corresponding to the results in (b) and (c), respectively. All calculations are performed on a $50\times50$ lattice with periodic boundary conditions.
• Figure 2: Disorder-driven EPs and TPTs. (a)-(c) Disorder-averaged real parts $\left\langle {\mathrm{Re}(E_{c/v})} \right\rangle$ and imaginary parts $\left\langle {\mathrm{Im}(E_{c/v})} \right\rangle$ and the spin Bott index $\left\langle {\mathrm{B_{s}}} \right\rangle$ as a functions of disorder strength w for $\kappa=0.05$ and $\lambda=1.2$, respectively. (d)-(f) Corresponding results for $\kappa=0.2$ and $\lambda=2$, respectively. Error bars indicate fluctuations across $100$ independent disorder realizations. All other parameters are identical to those in Fig. \ref{['Fig1']}. The results highlight the emergence of EPs and concurrent TPTs driven by increasing disorder.
• Figure 3: Two complementary cases of disorder-driven EPs and TPTs. (a)-(c) Disorder-averaged real parts $\left\langle {\mathrm{Re}(E_{c/v})} \right\rangle$ and imaginary parts $\left\langle {\mathrm{Im}(E_{c/v})} \right\rangle$ and the spin Bott index $\left\langle {\mathrm{B_{s}}} \right\rangle$ as a functions of disorder strength w for $\kappa=0.09$ and $\lambda=1.2$, respectively. (d)-(f) Corresponding results for $\kappa=0.05$ and $\lambda=2$, respectively. Error bars indicate fluctuations across $100$ independent disorder realizations. All other parameters are identical to those in Fig. \ref{['Fig1']}. The results highlight the emergence of EPs and concurrent TPTs driven by increasing disorder.
• Figure 4: Effective medium theory captures the emergence of disorder-induced EPs and TPTs. (a) and (e) Disordered-averaged self-energy components $\Sigma^{ij}_{\rm{And}}$ as a function of disorder strength w for $\lambda=1.2$ (case ①) and $\lambda=2$ (case ④), respectively. Diagonal terms renormalize orbital energies, while the off-diagonal term $\Sigma^{12}_{\rm{And}}$ induces inter-orbital coupling. (b) and (f) Evolution of the discriminant $y$ (left axis) and the energy gap $\Delta E_g$ (right axis) as a functions of w, showing the appearance of EPs (where $y=0$) and gap closing/reopening. For $\lambda=1.2$ a band inversion occurs at the second EP, indicating a TPT. In contrast, for $\lambda=2$, a TPT also takes place at the EP, marked by the reopening of a real bandgap. (c),(d),(g) and (h) Comparison between effective medium theory (EMT) and real-space numerical calculations (NC). (c) and (d) correspond to $\lambda=1.2$, while (g) and (h) correspond to $\lambda=2$. EMT accurately reproduces the emergence of EPs, real-complex-real spectral transitions, and the associated topological behavior across disorder regimes.
• Figure 5: Phase diagram of the energy spectrum and topological invariant in the presence of disorder and nonreciprocity. The disorder-averaged real and imaginary parts of the valence and conduction band energies, $\left\langle {\mathrm{Re}(E_{c/v})} \right\rangle$ and $\left\langle {\mathrm{Im}(E_{c/v})} \right\rangle$, and the spin Bott index $\left\langle {\mathrm{B_{s}}} \right\rangle$ are shown in the parameter space defined by nonreciprocal hopping strength $\kappa$ and disorder strength w. (a)-(c) correspond to case ①, while (d)-(f) correspond to case ④. Each data point is averaged over 100 independent disorder configurations. Black dotted lines denote the location of EP lines. All calculations are performed on a $20 \times 20$ with periodic boundary conditions.