Singular-Value Statistics of
Non-Hermitian Random Matrices and Open Quantum Systems

We first focus on the 9 symmetry classes where no symmetry or only one symmetry is present (Table 1) and provide the complete classification for all the 38 symmetry classes shortly in Sec. III.3 (Tables 2-5). Singular values of a matrix $H$ are the square roots of eigenvalues of $H^{{\dagger}}H$ or $HH^{{\dagger}}$. As a clear difference from eigenvalues, singular values are always real and nonnegative even in non-Hermitian matrices. Then, we calculate the distributions of singular-value-spacing ratios $r_{n}$ [Fig. 1 (a, b)], in a similar manner to the spacing ratios of real eigenvalues [96, 97]. Specifically, $r_{n}$ is defined as

for an ordered set of singular values $s_{n}$’s. In classes AII, AII^†, and D, all the singular values are two-fold degenerate, and we calculate $r_{n}$ by identifying the two degenerate singular values with the one singular value. While we can also study the statistical distributions of singular-value spacing $s_{n+1}-s_{n}$ (see also Appendix B), the statistical distributions of $r_{n}$ are independent of the local density of singular values and hence do not require the unfolding of singular values, which facilitates the calculations for physical open quantum systems.

The singular-value-spacing ratios exhibit the three different distributions in the presence of time-reversal symmetry in Eq. (1) or time-reversal symmetry^† in Eq. (4). Here, non-Hermitian matrices without symmetry belong to class A, and those with Eq. (1) [Eq. (4)] belong to classes AI and AII (AI^† and AII^†) [65]. Notably, time-reversal symmetry changes the level-spacing distributions even for arbitrary singular values. By contrast, while time-reversal symmetry^† changes the bulk statistics of complex eigenvalues [39], time-reversal symmetry changes the eigenvalue statistics only around the real axis [98, 99, 100, 58, 101, *2023arXiv230105022B]. Thus, the singular-value statistics give a clearer measure of time-reversal symmetry in non-Hermitian random matrices. It is also notable that all the level-spacing distributions of complex eigenvalues exhibit the cubic level repulsion [37, 43, 45] in contrast to Hermitian random matrices. Conversely, the singular-value-spacing distributions exhibit the 3-fold level repulsion even in non-Hermitian random matrices,

for

All the other symmetries also change the level-spacing-ratio distributions of singular values [Fig. 1 (b)]. Remarkably, in classes AIII and AIII^†, where chiral and sublattice symmetries in Eqs. (6) and (3) are respected by definition, the probability density is nonvanishing even for small level-spacing ratio $0\leq r\ll 1$, which implies the weaker level repulsion between singular values. Still, the level-spacing-ratio distributions deviate from the Poisson statistics

for uncorrelated singular values. We elucidate these unconventional level-spacing-ratio distributions through Hermitization in Sec. III.2.

While some symmetry classes (e.g., classes AI and AI^†) exhibit the same level-spacing distributions, they can be distinguished by the distributions of the minimum singular value $s_{\rm min}$ [Fig. 1 (c, d)]. Here, in contrast to the spacing ratio $r$, the minimum singular value $s_{\rm min}$ depends on the normalization of random matrices; in our numerical calculations in Fig. 1 (c, d), we normalize $s_{\rm min}$ such that its average is unity (i.e., $\braket{s_{\rm min}}=1$). For example, non-Hermitian random matrices in classes AI and AI^† follow the same level-spacing distributions but exhibit clearly different distributions of the minimum singular value. The former is quantified by the average singular-value-spacing ratio $\braket{r}$ while the latter by the average normalized variances $\braket{s_{\rm min}^{2}}/\braket{s_{\rm min}}^{2}$ of the minimum singular value, as summarized in Table 1. Additionally, each symmetry class can be distinguished between by the power-law decay of the probability distribution $p_{\rm min}$ for small $s_{\rm min}$,

with

We also note that class AIII is characterized by unique singular-value statistics, as discussed shortly. If matrices additionally respect Hermiticity or normality, they exhibit different singular-value statistics (see Appendices A and C for details).

We demonstrate that the singular-value statistics are captured by Hermitization. Previously, the Hermitization technique was applied to the complex-eigenvalue statistics [69]. It is also relevant to the classification of the Anderson transitions [77] and topological phases [80, 65] in non-Hermitian systems. For a given non-Hermitian matrix $H$, we introduce a Hermitized matrix

Singular values of $\tilde{H}$ are given as eigenvalues of $\sqrt{HH^{{\dagger}}}$ and $\sqrt{H^{{\dagger}}H}$, and hence coincide with singular values of $H$. Thus, the singular-value statistics of non-Hermitian matrices $H$ reduce to those of the corresponding Hermitian matrix $\tilde{H}$. We provide the singular-value statistics of Hermitian random matrices in Appendix A (Table 6). Importantly, Hermitized matrices $\tilde{H}$ respect additional chiral symmetry by construction,

which changes the relevant symmetry classes. For example, for non-Hermitian matrices that respect only time-reversal symmetry in Eq. (1) (i.e., classes AI or AII), the Hermitized matrices belong to classes BDI or CII. We also summarize the correspondence of symmetry classes in Table 1, as well as Tables 2-5.

Through Hermitization, the singular-value statistics of non-Hermitian random matrices $H$ with time-reversal symmetry in Eq. (1), particle-hole symmetry in Eq. (2), time-reversal symmetry^† in Eq. (4), or particle-hole symmetry^† in Eq. (5) reduce to the real-eigenvalue statistics of Hermitized matrices $\tilde{H}$ in the chiral [103, 104, 105, *Damgaard-01] and BdG [6, 107] classes. In fact, this correspondence is consistent with our numerical calculations of the singular-value statistics (compare Table 1 and Fig. 1 with Table 6 and Fig. 4, respectively). The joint probability distribution functions of singular values $s_{i}$’s are given as

with the random-matrix indices $\alpha$ in Eq. (11) and $\beta$ in Eq. (8).

By contrast, in the presence of chiral symmetry in Eq. (6), the Hermitized matrices $\tilde{H}$ satisfy

and hence the singular values of $\tilde{H}$ further reduce to the singular values of the Hermitian matrices $\text{i}H\mathcal{S}$ and $-\text{i}H\mathcal{S}$ without any symmetry (i.e., class A). When the non-Hermitian matrix $H$ is taken from the Gaussian ensemble, the Hermitian matrices $\text{i}H\mathcal{S}$ and $-\text{i}H\mathcal{S}$ also belong to the Gaussian ensemble. Since singular values of Hermitian matrices are given as the absolute values of their eigenvalues, the singular-value statistics without symmetry reduce to the real-eigenvalue statistics of two independent ensembles one of which corresponds to positive eigenvalues and the other of which corresponds to negative eigenvalues. There, uncorrelated levels can be close to each other, and the level repulsion is significantly weakened, which is the origin of $p_{\rm r}\left(r=0\right)\neq 0$ in Fig. 1 (b). Still, a part of the level repulsion survives, resulting in the deviation from the Poisson statistics.

Furthermore, non-Hermitian matrices $H$ with sublattice symmetry in Eq. (3) can be generally expressed as

with two non-Hermitian matrices $h_{1}$ and $h_{2}$, where we assume $\mathcal{S}=\sigma_{z}$. Hence, the singular values of $H$ consist of the singular values of the two independent non-Hermitian matrices $h_{1}$ and $h_{2}$. Consequently, the spacing distributions in the bulk of the singular-value spectra again reduce to those of Hermitian matrices without symmetry. On the other hand, the minimum singular value of $H$ reduces to the smaller one of the minimum singular values of $h_{1}$ and $h_{2}$, the distribution of which obeys that of Hermitian matrices with chiral symmetry (a subtle difference can arise in generic cases; see Sec. III.3 for details). In the absence of chiral or sublattice symmetry, positive and negative eigenvalues of Hermitized matrices $\tilde{H}$ are correlated owing to symmetry, leading to the level repulsion between singular values.

Based on Hermitization, we completely classify the singular-value statistics of non-Hermitian random matrices for all the 38 symmetry classes in Tables 2-5, 9 of which are summarized also in Table 1. In a similar manner to the 10-fold AZ symmetry classes for Hermitian matrices, time-reversal symmetry in Eq. (1), particle-hole symmetry in Eq. (2), and chiral symmetry in Eq. (6) form the 10-fold symmetry classes for non-Hermitian matrices (Tables 2 and 3). Furthermore, time-reversal symmetry^† in Eq. (4), particle-hole symmetry^† in Eq. (5), and chiral symmetry in Eq. (6) form additional 10-fold symmetry classes, which are the AZ^† symmetry classes for non-Hermitian matrices (Tables 2 and 4). Taking sublattice symmetry into consideration as additional symmetry (Tables 2 and 5), we have the 38-fold symmetry classification for non-Hermitian matrices [65]. Some symmetry classes in Tables 2-5 give the equivalent symmetry classes, which are not double counted in the 38-fold symmetry classification. For example, time-reversal symmetry in Eq. (1) and particle-hole symmetry^† in Eq. (5) lead to essentially the same universal spectral statistics of both eigenvalues and singular values [108]. Consequently, classes AI and AII are equivalent to classes D^† and C^†, respectively (Table 3).

In these classification tables, we provide the random-matrix indices $\beta$ for the singular-value statistics in the bulk of the spectra [i.e., Eq. (8)] and $\alpha$ for the statistics of the minimum singular value [Eq. (11)]. On the other hand, the symmetry classes for which the corresponding Hermitized matrices belong to the standard classes (i.e., classes A, AI, and AII) are not characterized by these random-matrix indices $\left(\alpha,\beta\right)$, as discussed above. In the symmetry classes for which the Hermitized matrices respect additional unitary symmetry and hence are block diagonalized, the singular-value statistics in the bulk of the spectra are no longer described by $\beta$, whereas the statistics of the minimum singular value can still be described by $\alpha$. Furthermore, we analytically derive the singular-value statistics of small random matrices in Appendix B. These small-$N$ analytical results well describe the large-$N$ results, in the similar spirit to the Wigner surmise [7, *Wigner-58, 9, 97]. This also contrasts with the complex-eigenvalue statistics of non-Hermitian random matrices, where a significant deviation arises between the large-$N$ and small-$N$ results [43, 58].

In the presence of chiral symmetry in Eq. (6), singular values of a non-Hermitian matrix $H$ generally reduce to singular values of Hermitian matrices $\text{i}H\mathcal{S}$ and $-\text{i}H\mathcal{S}$ with certain symmetry, as shown in Eq. (16). It should be noted that the spectral supports of $\text{i}H\mathcal{S}$ and $-\text{i}H\mathcal{S}$ can differ from each other for generic random matrices, as well as physical models such as Lindbladians and non-Hermitian Hamiltonians (see Sec. IV below). The two spectra may not overlap when all the eigenvalues of $\text{i}H\mathcal{S}$ ($-\text{i}H\mathcal{S}$) are positive (negative). In such a case, the level repulsion between eigenvalues is manifested also in the singular-value statistics, which contrasts with the weaker level repulsion discussed in Sec. III.2. Conversely, for the non-Hermitian matrix $H$ from the Gaussian ensemble, the two Hermitian matrices $\text{i}H\mathcal{S}$ and $-\text{i}H\mathcal{S}$ also belong to the Gaussian ensemble and share the same spectral support. Similarly, to obtain the universal results in the presence of chiral symmetry, the spectrum of the Hermitian matrices $\pm\text{i}H\mathcal{S}$ should be chosen to be statistically symmetric with respect to zero.

Furthermore, in the presence of sublattice symmetry in Eq. (3), singular values of a non-Hermitian matrix $H$ generally decompose into singular values of two independent non-Hermitian matrices $h_{1}$ and $h_{2}$ in Eq. (17). In particular, the minimum singular value of $H$ reduces to the smaller one of the minimum singular values of $h_{1}$ and $h_{2}$. In the sole presence of sublattice symmetry (i.e., class AIII^† in Table 2), the distribution of the minimum singular value in $H$ coincides with that in $h_{1}$ or $h_{2}$. However, this is not necessarily the case, in general. For example, in the additional presence of time-reversal symmetry that commutes with sublattice symmetry (i.e., class AI + $\mathcal{S}_{+}$ in Table 5), each non-Hermitian matrix $h_{1}$ or $h_{2}$ respects time-reversal symmetry and belongs to class AI. According to the analytical result, the distribution of the minimum singular value in $h_{1}$ or $h_{2}$ is [see Eq. (51) in Appendix B]

and the cumulative distribution function is

Then, the cumulative distribution function $F^{(H)}_{\rm min}\left(s\right)$ for the original non-Hermitian matrix $H$ is obtained as

leading to

Still, there is merely a slight difference between $p_{\rm min}^{(H)}\left(s\right)$ and $p_{\rm min}^{(h_{i})}\left(s\right)$. In fact, both of them do not vanish for $s\to 0$ and hence are characterized by $\alpha=0$. Additionally, we have $\braket{s_{\rm min}^{2}}/\braket{s_{\rm min}}^{2}=1.6862\cdots$ for $p_{\rm min}^{(H)}\left(s\right)$ and $\braket{s_{\rm min}^{2}}/\braket{s_{\rm min}}^{2}=1.6018\cdots$ for $p_{\rm min}^{(h_{i})}\left(s\right)$, which are nearly equal to each other. We also numerically obtain $\braket{s_{\rm min}^{2}}/\braket{s_{\rm min}}^{2}=1.6795\cdots$ from $5\times 10^{4}$ realizations of $10^{3}\times 10^{3}$ non-Hermitian random matrices in the Gaussian ensemble, which is consistent with the analytical result $\braket{s_{\rm min}^{2}}/\braket{s_{\rm min}}^{2}=1.6862\cdots$.

We investigate open quantum systems described by the quantum master equation $d\rho/dt=\mathcal{L}\left(\rho\right)$ with the Lindbladian [109, 110, 23, 24, 25]

where $H$ is a Hermitian Hamiltonian for the coherent dynamics, and $L_{n}$’s are dissipators for the coupling with the surrounding environment. Here, we choose the Hamiltonian as the quantum Ising model

where the open boundary conditions are imposed, and $\varepsilon_{n}$ is randomly chosen from $\left[-0.1,0.1\right]$ for each site $n$ to break unwanted symmetry. We choose the dissipators as damping

or dephasing

We study the singular-value statistics of the Lindbladian $\mathcal{L}$ by doubling the Hilbert space. Specifically, we map the density operator $\rho=\sum_{i,j}\rho_{ij}\ket{i}\bra{j}$ to a pure state $\ket{\rho}=\sum_{i,j}\rho_{ij}\ket{i}\ket{j}$ in the double Hilbert space. Through this vectorization procedure $\rho\to\ket{\rho}$, the Lindblad equation reduces to $\left(d/dt\right)\ket{\rho}=\mathcal{L}\ket{\rho}$, where $\mathcal{L}$ is the non-Hermitian operator in the double Hilbert space:

We calculate the spacing ratios of singular values for the Lindbladians $\mathcal{L}$ with the damping and dephasing (Fig. 2). There, we focus on singular values of the traceless non-Hermitian operator $\mathcal{L}-\left(\mathrm{tr}\,\mathcal{L}/\mathrm{tr}\,I\right)I$ by shifting the Lindbladian $\mathcal{L}$ by a constant $\mathrm{tr}\,\mathcal{L}/\mathrm{tr}\,I$.

Since arbitrary Lindbladians $\mathcal{L}$ are required to preserve Hermiticity of density operators, they are invariant under modular conjugation (see, for example, Ref. [63]),

where $\mathcal{J}$ is an antiunitary operator that exchanges the bra and ket degrees of freedom,

for arbitrary bosonic operators $O^{\pm}$ and complex numbers $z\in\mathbb{C}$. Consequently, singular values of the damped Ising model obey the random-matrix statistics in class AI [Fig. 2 (a)]. As also discussed above, time-reversal symmetry affects complex eigenvalues only around the real axis of the Lindbladian spectrum; conversely, time-reversal symmetry even changes the local correlations of arbitrary singular values and manifests itself more clearly in the singular-value statistics.

On the other hand, the dephased Ising model additionally respects time-reversal symmetry^†

leading to the singular-value statistics in class BDI^† [Fig. 2 (b)]. There, chiral symmetry arising from the combination of Eqs. (27) and (29) weakens the level repulsion between singular values even in the chaotic regime. We emphasize that the Lindbladians $\mathcal{L}$ are represented by sparse matrices because of the locality constraint, as opposed to random matrices. Nevertheless, the singular-value statistics of $\mathcal{L}$ follow the random-matrix statistics, providing a defining feature of chaos and nonintegrability in open quantum systems.

In addition to the Lindbladians, we also show that singular values of nonintegrable non-Hermitian Hamiltonians follow the random-matrix statistics in the corresponding symmetry classes. We study a non-Hermitian spin model [111, 43]

under the open boundary conditions. Here, $\varepsilon_{n}$ is randomly chosen from $\left[-1,1\right]$ for each site $n$ to break unwanted symmetry (e.g., translation symmetry), and $\vec{D}\coloneqq D\left(\vec{e}_{x}+\vec{e}_{z}\right)/\sqrt{2}$ is the degree of the Dzyaloshinskii-Moriya interaction. Owing to the nonintegrability of this non-Hermitian spin model $H$, singular values obey the random-matrix statistics (Fig. 3). Depending on the parameters, $H$ belongs to classes A, AI^†, and AII^†, as follows:

1. (i)

   For $h\neq 0$ and $D\neq 0$, the non-Hermitian spin model does not respect any symmetries and belongs to class A [Fig. 3 (a)].

2. (ii)

   For $h\neq 0$ and $D=0$, the non-Hermitian spin model respects time-reversal symmetry^†

   $\displaystyle H^{T}=H$

   (31)

   and belongs to class AI^† [Fig. 3 (b)].

3. (iii)

   For $h=0$ and $D\neq 0$, the non-Hermitian spin model respects time-reversal symmetry^†

   $\displaystyle\left(\prod_{n=1}^{L}\sigma_{n}^{y}\right)H^{T}\left(\prod_{n=1}^{L}\sigma_{n}^{y}\right)^{-1}=H$

   (32)

   and belongs to class AI^† for even $L$ and class AII^† for odd $L$ [Fig. 3 (c)]. For odd $L$, all the singular values are two-fold degenerate.

Time-reversal symmetry^† also changes the universal level-spacing statistics for complex eigenvalues [43]. Nevertheless, it does not impact the level repulsion for small level spacing but rather influences the peak of the probability distribution. Conversely, time-reversal symmetry^† even affects the level repulsion of singular values, as illustrated in Fig. 3. The distinctive singular-value statistics among the different symmetry classes are clearly discerned according to the power-law behavior for small level spacing, as in Eq. (8).