Symmetry of Open Quantum Systems: Classification of Dissipative Quantum Chaos

We develop a theory of symmetry in open quantum systems. Using the operator-state mapping, we characterize symmetry of Liouvillian superoperators for the open quantum dynamics by symmetry of operators in the double Hilbert space and apply the 38-fold internal-symmetry classification of non-Hermitian operators. We find rich symmetry classification due to the interplay between symmetry in the corresponding closed quantum systems and symmetry inherent in the construction of the Liouvillian superoperators. As an illustrative example of open quantum bosonic systems, we study symmetry classes of dissipative quantum spin models. For open quantum fermionic systems, we develop the ${\mathbb{Z}}_4$ classification of fermion parity symmetry and antiunitary symmetry in the double Hilbert space, which contrasts with the ${\mathbb{Z}}_8$ classification in closed quantum systems. We also develop the symmetry classification of open quantum fermionic many-body systems—a dissipative generalization of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation. We establish the periodic tables of the SYK Lindbladians and elucidate the difference from the SYK Hamiltonians. Furthermore, from extensive numerical calculations, we study its complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.

Popular Summary

Symmetry underlies a variety of phenomena and plays a pivotal role in physics. Fundamental symmetry of time reversal and charge conjugation enables a comprehensive understanding of quantum phases of matter, including topological materials. Another application of symmetry is the characterization of quantum chaos, which is fundamentally relevant to the foundations of thermodynamics and statistical physics. Despite the significant role of symmetry in physics, the role of symmetry in open quantum systems—quantum systems that exchange energy, particles, and information with the external environment—has yet to be fully understood. In view of the recent rapid development of quantum information science and technology, it is important to develop a theory of symmetry in open quantum systems and explore new open quantum phenomena.

In our work, we develop a general theory of symmetry in open quantum systems and find their rich symmetry classification. To show the significance of our theory, we also establish symmetry-enriched chaotic behavior in open quantum systems. We develop the periodic table of open quantum fermionic many-body systems—a dissipative generalization of the Sachdev-Ye-Kitaev model described by the quantum master equation. Furthermore, we study the complex-spectral statistics and demonstrate dissipative quantum chaos enriched by symmetry.

Our findings develop a unified understanding of open quantum physics, such as chaos and topological phenomena in open quantum systems and may lead to new quantum technology.

Figure 1

Dissipative $\mathrm{XYZ}$ model in one dimension with periodic boundaries and nearest-neighbor coupling ($L=6$, $J_{ij}^x=1.0$, $J_{ij}^y=0.7$, $J_{ij}^z=0.9$, ${\gamma }_i=0.5$). (a) Complex spectrum in the presence of a magnetic field $h_i^x=0.2$. (b) Complex spectrum in the presence of magnetic fields $h_i^x=0.2$, $h_i^z=0.4$.

Figure 2

Relationship between the four types of antiunitary symmetry operators $\mathcal{P}$, $\mathcal{Q}$, $\mathcal{R}$, $\mathcal{S}$ and the fermion parity symmetry operators $(-1{)}^{\mathcal{F}}$, $(-1{)}^{F^{\pm }}$. The product of $\mathcal{P}$ and $\mathcal{R}$ and that of $\mathcal{Q}$ and $\mathcal{S}$ are proportional to total fermion parity $(-1{)}^{\mathcal{F}}$ in the double Hilbert space. The product of $\mathcal{P}$ and $\mathcal{S}$ and that of $\mathcal{Q}$ and $\mathcal{R}$ are proportional to fermion parity $(-1{)}^{F^{+}}$ in the ket space. The product of $\mathcal{P}$ and $\mathcal{Q}$ and that of $\mathcal{R}$ and $\mathcal{S}$ are proportional to fermion parity $(-1{)}^{F^{-}}$ in the bra space.

Figure 3

Complex spectrum of a single realization of the Sachdev-Ye-Kitaev (SYK) Lindbladian with linear dissipators $p=1$ and complex coupling $K_{m;i}\in \mathbb{C}$. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=6$, $K=0.52$, (b) $N=7$, $K=1$, (c) $N=8$, $K=0.68$, (d) $N=9$, $K=1$, (e) $N=10$, $K=0.78$, (f) $N=11$, $K=1$, (g) $N=12$, $K=0.87$, and (h) $N=13$, $K=1$. The other parameters are taken as $J=1$ and $M=N$. The different colors represent eigenstates with even ($+1$; red dots) or odd ($-1$; blue dots) total fermion parity $(-1{)}^{\mathcal{F}}$.

Figure 4

Density of states across the real axis as a function of the imaginary part of the complex eigenvalue $\lambda$ for Sachdev-Ye-Kitaev (SYK) Lindbladians with linear dissipators $p=1$ and complex coupling $K_{m;i}\in \mathbb{C}$. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=10$, $K=0.78$ and (b) $N=12$, $K=0.87$. The other parameters are taken as $J=1$ and $M=N$. The double Hilbert space is divided into the two subspaces according to total fermion parity $(-1{)}^{\mathcal{F}}$, as shown by the blue curves ($+1$) and orange curves ($-1$). We take eigenvalues satisfying $|\mathrm{Im}\lambda |>ϵ/\sqrt{\dim \mathcal{L}}$ ($ϵ={10}^{-4}$) and exclude real eigenvalues.

Figure 5

Distributions for normalized nearest spectral spacings $s$ of Sachdev-Ye-Kitaev (SYK) Lindbladians with linear dissipators $p=1$ and complex coupling $K_{m;i}\in \mathbb{C}$, compared with the random-matrix results. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=10$, $K=0.78$ and (b) $N=12$, $K=0.87$. The other parameters are taken as $J=1$ and $M=N$. Each datum is averaged over $2000$ samples.

Figure 6

Marginal distributions for argument $\theta$ of the complex-spacing ratios of Sachdev-Ye-Kitaev (SYK) Lindbladians with linear dissipators $p=1$ and complex coupling $K_{m;i}\in \mathbb{C}$, compared with the random-matrix results. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=10$, $K=0.78$ and (b) $N=12$, $K=0.87$. The other parameters are taken as $J=1$ and $M=N$. Each datum is averaged over $2000$ samples.

Figure 7

Complex spectrum of a single realization of the Sachdev-Ye-Kitaev (SYK) Lindbladian with linear dissipators $p=1$ and real coupling $K_{m;i}\in \mathbb{R}$. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=6$, $K=0.73$, (b) $N=7$, $K=1$, (c) $N=8$, $K=0.96$, (d) $N=9$, $K=1$, (e) $N=10$, $K=1.10$, (f) $N=11$, $K=1$, (g) $N=12$, $K=1.23$, and (h) $N=13$, $K=1$. The other parameters are taken as $J=1$ and $M=N$. The different colors represent eigenstates with even ($+1$; red dots) or odd ($-1$; blue dots) total fermion parity $(-1{)}^{\mathcal{F}}$.

Figure 8

Density of states projected to (a),(b) the imaginary axis and (c),(d) the real axis as a function of the projected eigenvalues for Sachdev-Ye-Kitaev (SYK) Lindbladians with linear dissipators $p=1$ and real coupling $K_{ij}\in \mathbb{R}$. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a),(c) $N=10$, $K=1.10$ and (b),(d) $N=12$, $K=1.23$. The other parameters are taken as $J=1$ and $M=N$. The double Hilbert space is divided into the two subspaces according to total fermion parity $(-1{)}^{\mathcal{F}}$, as shown by the blue curves ($+1$) and orange curves ($-1$). In (a) and (b), we take eigenvalues satisfying $|\mathrm{Im}\lambda |>ϵ/\sqrt{\dim \mathcal{L}}$ ($ϵ={10}^{-4}$) and exclude real eigenvalues. In (c) and (d), on the other hand, we include the eigenvalues on the symmetric line $\mathrm{Re}\lambda =\mathrm{tr}\mathcal{L}/\mathrm{tr}I$. In (c), no eigenvalues appear on the symmetric line. The inset in (d) shows the delta-function peak of the density on the symmetric line.

Figure 9

Marginal distributions for argument $\theta$ of the complex-spacing ratios of Sachdev-Ye-Kitaev (SYK) Lindbladians with linear dissipators $p=1$ and real coupling $K_{m;i}\in \mathbb{R}$, compared with the random-matrix results. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=10$, $K=1.10$ and (b) $N=12$, $K=1.23$. The other parameters are taken as $J=1$ and $M=N$. Each datum is averaged over $3000$ samples.

Figure 10

Complex spectrum of a single realization of the Sachdev-Ye-Kitaev (SYK) Lindbladian with quadratic dissipators $p=2$ and complex coupling $K_{m;i}\in \mathbb{C}$. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=6$, $K=1.99$, (b) $N=7$, $K=1$, (c) $N=8$, $K=2.91$, (d) $N=9$, $K=1$, (e) $N=10$, $K=3.64$, (f) $N=11$, $K=1$, (g) $N=12$, $K=4.47$, and (h) $N=13$, $K=1$. The other parameters are taken as $J=1$ and $M=N$. For even $N$, the different colors represent eigenstates from even ($+1$) or odd ($-1$) fermion parity $(-1{)}^{F^{+}}$ and $(-1{)}^{F^{-}}$, shown by blue dots ($++$), orange dots ($+-$), green dots ($-+$), and red dots ($--$). For odd $N$, eigenstates with even (red dots) and odd (blue dots) total fermion parity $(-1{)}^{\mathcal{F}}$ are degenerate.

Figure 11

Density of states across the real axis as a function of the imaginary part of the complex eigenvalue $\lambda$ for Sachdev-Ye-Kitaev (SYK) Lindbladians with quadratic dissipators $p=2$ and complex coupling $K_{m;i}\in \mathbb{C}$. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=10$, $K=3.64$ and (b) $N=12$, $K=4.74$. The other parameters are taken as $J=1$ and $M=N$. The double Hilbert space is divided into the four subspaces according to fermion parity $(-1{)}^{F^{\pm }}$, as shown by the blue curves ($++$), orange curves ($+-$), green curves ($-+$), and red curves ($--$). We take eigenvalues satisfying $|\mathrm{Im}\lambda |>ϵ/\sqrt{\dim \mathcal{L}}$ ($ϵ={10}^{-4}$) and exclude real eigenvalues.

Figure 12

Distributions for normalized nearest spectral spacings $s$ of Sachdev-Ye-Kitaev (SYK) Lindbladians with quadratic dissipators $p=2$ and complex coupling $K_{m;i}\in \mathbb{C}$, compared with the random-matrix results. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=10$, $K=3.64$ and (b) $N=12$, $K=4.47$. The other parameters are taken as $J=1$ and $M=N$. Each datum is averaged over $2000$ samples.

Figure 13

Marginal distributions for argument $\theta$ of the complex-spacing ratio of Sachdev-Ye-Kitaev (SYK) Lindbladians with quadratic dissipators $p=2$ and complex coupling $K_{m;i}\in \mathbb{C}$, compared with the random-matrix results. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=10$, $K=3.64$ and (b) $N=12$, $K=4.47$. The other parameters are taken as $J=1$ and $M=N$. Each datum is averaged over 3000 samples.

Figure 14

Complex spectrum of a single realization of the Sachdev-Ye-Kitaev (SYK) Lindbladian with quadratic dissipators $p=2$ and real coupling $K_{m;i}\in \mathbb{R}$. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=6$, $K=2.82$, (b) $N=8$, $K=4.12$, (c) $N=10$, $K=5.15$, and (d) $N=12$, $K=6.33$. The other parameters are taken as $J=1$ and $M=N$. The double Hilbert space is divided into the four subspaces according to fermion parity $(-1{)}^{F^{+}}$ and $(-1{)}^{F^{-}}$, shown by the blue dots ($++$), orange dots ($+-$), green dots ($-+$), and red dots ($--$). In each subspace with fixed $(-1{)}^{F^{\pm }}$, modular conjugation symmetry is respected for $(-1{)}^{N/2}(-1{)}^{\mathcal{F}}=+1$, while pseudo-Hermiticity is respected for $(-1{)}^{\mathcal{F}}=+1$. For $N=6,10\equiv 2$ (mod $4$), eigenvalues with the same total fermion parity $(-1{)}^{\mathcal{F}}$ are twofold degenerate.

Figure 15

Complex spectral density of Sachdev-Ye-Kitaev (SYK) Lindbladians with quadratic dissipators $p=2$ and real coupling $K_{m;i}\in \mathbb{R}$ for (a),(b) $N=10$ and (c),(d) $N=12$. Fermion parity $(-1{)}^{F^{+}}$ and $(-1{)}^{F^{-}}$ is chosen as (a),(c) $++$ and (b),(d) $+-$. The other parameters are chosen to be $J=1$, $K=1$, and $M=N$. Each datum is averaged over (a),(b) $1024$ samples for $N=10$ and (c),(d) $166$ samples for $N=12$.

Figure 16

Density of states across the real axis as a function of the imaginary part of complex eigenvalues $\lambda$ for Sachdev-Ye-Kitaev (SYK) Lindbladians with quadratic dissipators $p=2$ and real coupling $K_{m;i}\in \mathbb{R}$. The number $N$ of fermion flavors is chosen to be (a) $N=10$ and (b) $N=12$. The other parameters are taken as $J=1$, $K=1$, and $M=N$. The double Hilbert space is divided into the four subspaces according to fermion parity $(-1{)}^{F^{\pm }}$, as shown by the blue curves ($++$), orange curves ($--$), green curves ($+-$), and red curves ($-+$). For $N\equiv 2$ (mod $4$) including $N=10$, the complex spectrum with the same total fermion parity $(-1{)}^{\mathcal{F}}$ is two-fold degenerate. We only consider eigenvalues away from the origin ($|\lambda |>{10}^{-10}$) and away from the real axis ($|\mathrm{Im}\lambda |>{10}^{-10}$).

Figure 17

Complex spectrum of a single realization of the Sachdev-Ye-Kitaev (SYK) Lindbladian with cubic dissipators $p=3$ and complex coupling $K_{m;i}\in \mathbb{C}$. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=6$, $K=5.21$, (b) $N=8$, $K=8.39$, (c) $N=10$, $K=11.8$, and (d) $N=12$, $K=15.4$. The other parameters are taken as $J=1$ and $M=2N$. The double Hilbert space is divided into the two subspaces according to total fermion parity $(-1{)}^{\mathcal{F}}$, as shown by the red dots (even) and blue dots (odd).

Figure 18

Density of states across the real axis as a function of the imaginary part of the complex eigenvalue $\lambda$ for Sachdev-Ye-Kitaev (SYK) Lindbladians with cubic dissipators $p=3$ and complex coupling $K_{m;i}\in \mathbb{C}$. The number $N$ of fermion flavors is chosen to be (a) $N=10$ and (b) $N=12$. The other parameters are taken as $J=1$, $K=10$, and $M=N$. The double Hilbert space is divided into the two subspaces according to total fermion parity $(-1{)}^{\mathcal{F}}$, as shown by the blue curves ($+$) and orange curves ($-$). We take eigenvalues satisfying $|\mathrm{Im}\lambda |>ϵ/\sqrt{\dim \mathcal{L}}$ ($ϵ={10}^{-4}$) and exclude real eigenvalues.

Figure 19

Distributions for normalized nearest spectral spacings $s$ of Sachdev-Ye-Kitaev (SYK) Lindbladians with cubic dissipators $p=3$ and complex coupling $K_{m;i}\in \mathbb{C}$, compared with the random-matrix results. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=10$, $K=11.8$ and (b) $N=12$, $K=15.4$. The other parameters are taken as $J=1$, and $M=N$. Each datum is averaged over (a) $3000$ and (b) $800$ samples.

Figure 20

Distribution of the complex-spacing ratios of the bulk spectrum for Sachdev-Ye-Kitaev (SYK) Lindbladians with cubic dissipators $p=3$ and complex coupling $K_{m;i}\in \mathbb{C}$ for (a),(b) $N=10$ and (c),(d) $N=12$. Total fermion parity $(-1{)}^{\mathcal{F}}$ is chosen as (a),(c) $+$ and (b),(d) $-$. The other parameters are taken as $J=1$, $K=10$, and $M=N$. The complex eigenvalues with $|\lambda |>0.01$ and $|\mathrm{Im}\lambda |>{10}^{-5}$ are taken. Each datum is averaged over (a),(b) $96$ samples for $N=10$ and (c),(d) $24$ samples for $N=12$.

Figure 21

Angle distribution of the complex-spacing ratios of Sachdev-Ye-Kitaev (SYK) Lindbladians with cubic dissipators $p=3$ and complex coupling $K_{m;i}\in \mathbb{C}$, compared with the random-matrix results. The number $N$ of fermion flavors and the strength $K$ of the dissipators are chosen to be (a) $N=10$, $K=11.8$ and (b) $N=12$, $K=15.4$. The other parameters are taken as $J=1$ and $M=N$. Each datum is averaged over (a) $3000$ and (b) $800$ samples.