Transfer matrix study of the Anderson transition in non-Hermitian systems

During the last several years, physics community witnessed tremendous revivals of research interests on non-Hermitian (NH) quantum physics. Nontrivial topological band theories have been introduced in NH systems, including the breakdown of the bulk-boundary correspondence and the emergence of new topological invariants[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], enriched topological classifications according to symmetries [15, 16, 17], and NH skin effects [18, 19, 20, 21, 22, 23]. NH quantum phenomena occur not only in condensed matter systems, but also in photonic systems [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] and ultracold atoms [37, 38, 39]. Many of NH quantum systems have quenched disorders by their own generic nature. Canonical examples of this is random lasers in a region of random dissipation and amplification [40, 41, 42], nonequilibrium open systems with gain and/or loss[43, 44, 45, 46, 47], and correlated quantum many-particle systems of quasiparticles with finite life-time [48, 49]. Nonetheless, the Anderson transition [50] in the NH quantum systems has been poorly studied in higher spatial dimension [51, 52, 53, 54, 55, 56], in spite of the quarter century history since the Hatano-Nelson’s pioneering work of the one-dimensional (1D) Anderson model [57, 58]. Hatano and Nelson introduced the non-Hermiticity through asymmetric hopping terms in the 1D Anderson model. Recent works focused on the effect of on-site complex-valued random potentials in Anderson and U(1) models in the two and three dimensions [52, 53, 54, 55, 56]. A positive imaginary part of the on-site potential gives a gain of a wavefunction amplitude in time, while a negative value of the imaginary part of the potential gives a loss of the wavefunction amplitude. Such complex-valued random potentials can be experimentally implemented by random dissipation and amplification of light waves in random lasers [40, 41, 42].

There are three major shortcomings in the previous studies of the Anderson transition in the NH quantum systems. First of all, all the studies so far [54, 56] are based on eigenenergy level statistics [59, 60, 61], which suffers from an ambiguity of how to set a window for the single-particle energy. The level statistics analysis sets a finite width of the energy window over which a dimensionless quantity associated with the level statistics is averaged. Thus, when using a finite-size scaling analysis in favor for the universal critical exponent, the analysis with a nonlinear polynomial fitting assumes that any fitting parameters in a universal function for the dimensionless quantity have no variations within the energy window. Nonetheless, the parameters in the universal function include not only the universal critical exponent but also non-universal critical quantities such as critical disorder strength. These non-universal quantities are continuous functions of the energy. Thus, the energy window must be sufficiently narrow, such that all the single-particle states within the window could share the same values of these non-universal quantities. To take sufficient amount of the level statistics with such a narrow window, one needs to diagonalize single-particle Hamiltonian with many different disorder realizations. This often meets an upper limit from computational resources. Large statistical errors due to limited number of samples together with the systematic errors due to finite energy window make it hard for the precise investigation of the Anderson transition in NH quantum systems.

Eigenvalues of the NH systems are distributed in the complex Euler plane, except for a system with a special symmetry, such as ${\cal PT}$ symmetry [62]. Thereby, the complex plane can possibly host a novel phase diagram structure, which has no counterpart in the Hermitian physics. To uncover such unique NH quantum physics, one needs to clarify how critical nature of single-particle states varies as a function of their complex-valued eigenenergy. It is also unclear how the known universality classes of the Anderson transition in the Hermitian case cross over to new universality classes in the non-Hermitian case. To clarify this, one needs to study precisely how the critical properties change as a function of the system parameters.

The two-terminal conductance is one of the best physical quantities that characterize the Anderson transition in the experimental NH systems mentioned above. Nonetheless, it is unknown how the conductance as well as the localization length behave in localized, delocalized phases and at the critical point in the NH disordered systems. To know them, one needs a transfer matrix analyses of the conductance in the NH systems. In the NH systems, however, a symplectic structure of the transfer matrix is absent and so is the unitarity of a scattering matrix. Besides, the presence of the reciprocal symmetries of the conductance and Lyapunov exponents is not clear in the NH quantum systems. In fact, the reciprocal symmetry is completely absent in some NH systems such as in the 1D Hatano-Nelson model, while it exists in other NH systems in some ways.

In order to solve these shortcomings in theory of NH quantum systems, we carry out comprehensive transfer matrix analyses of localization length and conductance in the three-dimensional (3D) NH Anderson, U(1) and Peierls models. Here we list the major findings in this paper as follows,

• •

  The critical exponents of the NH Anderson, U(1) and Peierls models are accurately determined. The results unambiguously conclude that the critical exponent of the NH class-AI^† (Anderson) is $\nu=1.19\pm 0.01$ and the critical exponent of the NH class A (U(1) and Peierls) is $\nu=1.00\pm 0.04$. These high precision estimates enable us to distinguish the NH class A and NH class AI^†, which was impossible for the previous level statistics study [56]. This supports a strong relation between the universality classes of the Anderson transition and symmetry classification in the NH disordered systems. More importantly, they are distinctly different from the critical exponents in the corresponding universality classes in the Hermitian limit; $\nu=1.572\pm 0.003$ in the orthogonal class [63] (Hermitian class AI) and $\nu=1.443\pm 0.003$ in unitary class (Hermitian class A)  [64, 65].

• •

  Critical properties at different single-particle energy $E$ and system parameters are clarified in the NH Anderson model. This gives phase diagrams in a two-dimensional plane subtended by the disorder strength of the real part of the on-site random potential ($W_{r}$) and that of the imaginary part ($W_{i}$). Especially at the zero single-particle energy ($E=0$), the phase diagram as well as the critical properties across the phase boundary becomes completely symmetric with respect to an exchange of $W_{r}$ and $W_{i}$; Fig. 1(a). The symmetric structure at $E=0$ is a generic feature in any NH bipartite-lattice models with the on-site NH random potentials. Based on the results, we propose a renormalization group (RG) flow diagram in the two-dimensional plane at $E=0$; Fig. 1(b). At $E\neq 0$, the symmetry of the phase diagram and the critical properties is absent; see the RG flow diagram in Fig. 1(c).

• •

  The validity of the transfer matrix analyses in these 3D NH models as well as the 1D Hatano-Nelson model is clarified and demonstrated numerically. The presence or absence of the reciprocal symmetries in Lyapunov exponents and conductance are clarified in the three NH models. Based on a physically reasonable definition of the two-terminal conductance, we show numerically that a conductance distribution is not Gaussian, but it contains small numbers of extremely large conductance values due to rare events of strong amplifications by the NH disorders. We also demonstrate that the geometric mean of the conductance enables a feasible finite-size scaling (FSS) analysis and the results from the FSS analysis give the critical exponents consistent with those values by the localization length.

The organization of this paper is as follows. In the following section, we introduce the three NH models and argue the validity of the transfer matrix analyses. We show the presence or absence of the reciprocal symmetries in the Lyapunov exponents and two-terminal conductance. In Sec. III, we clarify the critical properties of the Anderson transition at different single-particle energies and system parameters in the NH Anderson models together with an accurate estimate of the critical exponent of the NH class AI^†. In Sec. IV, we clarify the critical properties of the Anderson transition in two NH class A models, i.e., U(1) and Peierls models, together with an estimate of the critical exponent of the NH class A. In Sec. V, we compare the transfer matrix analyses with the level statistics analyses[54, 56] and describe a possible reason for a discrepancy of the critical exponent of the NH class AI^† between the two analyses. The final section is devoted to summary and concluding remarks.

We study the following tight-binding models defined on a 3D cubic lattice,

where $c^{\dagger}_{\bm{i}}$ ($c_{\bm{i}}$) is the creation (annihilation) operator. $\bm{i}$ and $\bm{j}$ specify the cubic lattice site; ${\bm{i}}=(i_{x},i_{y},i_{z})$ and $i_{\mu}=1,2,\cdots,L_{\mu}$ ($\mu=x,y,z$). The models have Hermitian hoppings between nearest neighbor sites in the cubic lattice; $\langle{\bm{i}},{\bm{j}}\rangle$ means that $\bm{i}$ and $\bm{j}$ are the nearest neighbor sites with $\theta_{{\bm{i}},{\bm{j}}}=-\theta_{{\bm{j}},{\bm{i}}}$. The model is the Anderson model for $\theta_{{\bm{i}},{\bm{j}}}$ = 0 and the U(1) model with a random number $\theta_{{\bm{i}},{\bm{j}}}$ in a range of $[0,1)$ [66, 64]. When $\theta_{{\bm{i}},{\bm{j}}}=\Phi\!\ i_{x}$ for ${\bm{j}}={\bm{i}}+{\bm{e}}_{z}$, the model is the Peierls model [67, 68], where $2\pi\Phi$ is a magnetic gauge flux that penetrates through every square plaquette in the $z$-$x$ plane of the cubic lattice. In this paper, we study the Anderson transition driven by random complex on-site potentials, $\varepsilon_{\bm{i}}=w^{r}_{\bm{i}}+{\rm i}\!\ w^{i}_{\bm{i}}$ with the imaginary unit ${\rm i}$, where $w^{r}_{\bm{i}}$ and $w^{i}_{\bm{i}}$ are independent random numbers with the uniform distribution in a range of $[-W_{r}/2,W_{r}/2]$ and $[-W_{i}/2,W_{i}/2]$, respectively. Hence ${\cal H}^{\dagger}\neq{\cal H}$, where the non-zero $w^{r}_{\bm{i}}$ and $w^{i}_{\bm{i}}$ bring about the non-Hermiticity in the system. According to the symmetry classification, the Anderson model ($\theta_{{\bm{i}},{\bm{j}}}=0$) belongs to 3D NH class AI^† and the U(1) and Peierls models ($\theta_{{\bm{i}},{\bm{j}}}\neq 0$) belong to 3D NH class A. The time reversal symmetry (TRS) is broken (${\cal H}^{*}\neq{\cal H}$) in both classes, whereas the transposition symmetry (${\cal H}^{\mathrm{T}}={\cal H}$), namely TRS^†, holds true in the class AI^†[16, 69]. We note that not only a parameter region of $W_{i}=0$ but also a parameter region of $W_{r}=0$ and $E=0$ belong to the Hermitian universality class in these three NH models (Sec. III).

Transfer matrix method are widely used both in the Hermitian [70, 71, 72, 73] and non-Hermitian [74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84] systems. In order to study the Anderson transition of eigenstates of ${\cal H}$ with a complex-valued eigenenergy $E$, the quasi-1D localization length (Lyapunov exponent) and two-terminal conductance in cubic systems are evaluated by the transfer matrix method at the energy $E$ for different system sizes. The quantities are further analyzed by the finite size scaling, which determines the critical properties of the Anderson transition in the NH systems. Although the transfer matrix method has been widely used in the non-Hermitian optical systems during last several decades, the transfer matrix analysis on the Anderson transitions in non-Hermitian systems has not been carried out prior to this work. Thus, we first clarify the nature of the Lyapunov exponent and the two-terminal conductance in the NH systems studied in this paper as well as pseudo-Hermitian systems.

A phase diagram of the NH Anderson model at $E=0$ is determined in a two-dimensional plane subtended by $W_{r}$ and $W_{i}$; Fig. 1(a). The phase boundary between localized and delocalized phases is determined by the scale-invariant point of the normalized localization length $\Lambda_{z}\equiv\lambda/L$ with $L_{x}=L_{y}=L$. $\Lambda_{z}$ increases with the size $L$ in the delocalized phase, while decreases in the localized phase. The phase transition is nothing but the Anderson transition in the Hermitian limit ($W_{i}=0$), where the critical disorder is consistent with a literature value; $W_{r,c}\approx 16.5$ [63]. The phase diagram at $E=0$ is symmetric with respect to an exchange of $W_{r}$ and $W_{i}$.

There are two significant features in the phase diagram. Firstly, the phase boundary of $\{W^{2}_{r}+W^{2}_{i}\}^{1/2}$ bends in toward the smaller disorder value than the critical disorder strength in the Hermitian case. This means that states are more easily localized by the NH on-site disorder than by the Hermitian on-site disorder with the same modulus. This tendency in 3D Anderson model is consistent with the 2D Anderson model, which shows no localization-delocalization transition with the on-site NH disorder [54]. The second significant feature is that the phase boundary is symmetric along the line of $W_{r}=W_{i}$, indicating that the role of $W_{r}$ and that of $W_{i}$ are identical in the Anderson localization at $E=0$. In fact, the critical point along an axis of $W_{r}=0$ as well as along an axis of $W_{i}=0$ belongs to the 3D Hermitian orthogonal class; the critical exponent evaluated at $W_{i,c}\approx 16.5$ along the axis of $W_{r}=0$ is consistent with the critical exponent in the orthogonal class in the Hermitian case, $\nu\approx 1.57$ (TABLE 1).

The symmetric nature of the phase diagram in $W_{r}-W_{i}$-plane at $E=0$ is due to the bipartite lattice structure and the choice of the onsite potential the real and imaginary parts of which obey the same distribution. To see this, note that a diagonalization of these disordered Hamiltonians at $E=0$ is clearly inclusive of solving transfer matrices in favor for the Lyapunov exponents at $E=0$ for these models;

$[\mathbb{H}]$ is the $L_{x}L_{y}L_{z}\times L_{x}L_{y}L_{z}$ matrix that has complex-valued random numbers in its diagonal elements and Hermitian hoppings with the unit modulus in its off-diagonal elements. The cubic lattice is decomposed into A sublattice and B sublattice, where the off-diagonal elements appear only between these two sublattices. Let the $L_{x}L_{y}L_{z}$-component eigenvector $\Psi$ be transformed by a diagonal matrix $[\mathbb{B}]$ that has $-{\rm i}$ for B sublattice and $+1$ for A sublattice;

Since $E=0$, let another diagonal matrix $[\mathbb{A}]$ apply from the left of Eq. (73). $[\mathbb{A}]$ has $+{\rm i}$ for A sublattice and $+1$ for B sublattice;

Now that the hopping terms appear only between A and B sublattices, the off-diagonal terms in $\mathbb{H}^{\prime}\equiv\mathbb{A}\mathbb{H}\mathbb{B}$ are identical to those in $\mathbb{H}$. Meanwhile the real and imaginary parts of the diagonal elements in $\mathbb{H}^{\prime}$ are exchanged with each other, compared to those in $\mathbb{H}$;

$w_{\bm{i}}^{r}$ and $w_{\bm{i}}^{i}$ in $\mathbb{H}$ are uniformly distributed in a range of $[-W_{r}/2,W_{r}/2]$ and $[-W_{i}/2,W_{i}/2]$, respectively. Besides, on-site disorder potentials have no correlation between different lattice sites. Thus, an ensemble of different disorder realization for $\mathbb{H}$ with $(W_{r},W_{i})=(x,y)$ is the same as an ensemble of different disorder realization for $\mathbb{A}\mathbb{H}\mathbb{B}$ with $(W_{r},W_{i})=(y,x)$. Now that the zero-energy eigenfunction of $\mathbb{H}$ and that of $\mathbb{A}\mathbb{H}\mathbb{B}$ are related to each other by Eq. (74) and the transformation $\mathbb{B}$ does not change localization lengths of these two eigenfunctions, the phase diagram at $E=0$ must be symmetric with respect to an change between $W_{i}$ and $W_{r}$.

One could also repeat the same argument in the framework of the transfer matrix method. Thereby, one can introduce a product of another transfer matrix

Here $\mathbb{a}_{i_{z}}$ ($\mathbb{b}_{i_{z}}$) is a $L_{x}L_{y}\times L_{x}L_{y}$ diagonal matrix whose $(i_{x},i_{y})$ diagonal element takes $+{\rm i}$ ($-{\rm i}$) for $(i_{x},i_{y},i_{z})\in$ A (B) sublattice and takes $+1$ for $(i_{x},i_{y},i_{z})\in$ B (A) sublattice. Here we used $\mathbb{B}_{i_{z}}\mathbb{A}_{i_{z}-1}={\mathbb{1}}$. Eq. (84) dictates that the Lyapunov exponents of $M$ are identical to those of $M^{\prime}$. Note the hoppings terms in $M^{\prime}_{i_{z}}$ are the same as those in ${M}_{i_{z}}$, and the real and imaginary parts of the on-site NH disorder potentials in $M^{\prime}_{i_{z}}$ are exchanged with each other, compared to those in $M_{i_{z}}$; Eq. (78). We therefore obtain the same results at $E=0$ for $W_{r}=0$ and for $W_{i}=0$.

When $E\neq 0$, the phase diagram of the three NH models becomes asymmetric with respect to the exchange of $W_{r}$ and $W_{i}$. Besides, the critical point along an axis of $W_{r}=0$ at $E\neq 0$ belongs not to the 3D Hermitian orthogonal/unitary class, but to the new universality classes AI^†/A of the NH systems. In the following three subsections, we first clarify critical behaviors of the Anderson transition in the NH Anderson model.