Universality Classes of the Anderson Transitions Driven by Non-Hermitian Disorder

The interplay between non-Hermiticity and disorder plays an important role in condensed matter physics. Here, we report the universal critical behaviors of the Anderson transitions driven by non-Hermitian disorders for a three-dimensional (3D) Anderson model and 3D U(1) model, which belong to 3D class ${\mathrm{AI}}^{†}$ and 3D class A in the classification of non-Hermitian systems, respectively. Based on level statistics and finite-size scaling analysis, the critical exponent for the length scale is estimated as $\nu =0.99\pm 0.05$ for class ${\mathrm{AI}}^{†}$, and $\nu =1.09\pm 0.05$ for class A, both of which are clearly distinct from the critical exponents for 3D orthogonal and 3D unitary classes, respectively. In addition, spectral rigidity, level spacing distribution, and level spacing ratio distribution are studied. These critical behaviors strongly support that the non-Hermiticity changes the universality classes of the Anderson transitions.

Figure 1

Level spacing ratio $⟨r⟩$ as a function of the disorder strength $W$ for the class ${\mathrm{AI}}^{†}$ model. The circles are for raw data of $⟨r⟩$, where an error is smaller than the circle size. The curves are from the polynomial fitting results with $m_1,n_1,m_2,n_2=(3,3,0,1)$. Inset: the same plot for class A.

Figure 2

Number variance ${\mathrm{\Sigma }}_2$ as a function of averaged level number $⟨N⟩$ at the critical point ($W=6.3$ for class ${\mathrm{AI}}^{†}$ and $W=7.16$ for class A), and ${\mathrm{\Sigma }}_2=\chi ⟨N⟩$ with $\chi \approx 0.46$ for class ${\mathrm{AI}}^{†}$ and $\chi \approx 0.55$ for class A. The plot comes from ${10}^4$ samples for class ${\mathrm{AI}}^{†}$ and 6400 samples for class A with $L=24$. Variable $⟨N⟩$ is obtained by changing the energy window within the 10% eigenvalues around $E=0$. The linear relationship holds true for the system size $L\ge 8$ with a consistent $\chi$ [47].

Figure 3

Critical level spacing distribution $P(s)$ at large $s$ and small $s$ (inset) for class ${\mathrm{AI}}^{†}$ at $W_c=6.3$. The small-$s$ behavior of $P(s)$ is fitted by $P(s)\propto s^{{\beta }_c}$ with ${\beta }_c\approx 2.62$ (blue solid line in the inset), and the large-$s$ behavior of $P(s)$ is fitted by $P(s)\propto e^{-\alpha s}$ with $\alpha \approx 5.06$ (blue solid line). The distribution is obtained from 10% eigenvalues around $E=0$ of ${10}^4$ disorder realizations with $L=24$. Red crosses are data excluded from the linear fitting. Similar critical behaviors but with different ${\beta }_c$ and $\alpha$ are also observed for class A [47].