Topological Phases of Non-Hermitian Systems

While Hermiticity lies at the heart of quantum mechanics, recent experimental advances in controlling dissipation have brought about unprecedented flexibility in engineering non-Hermitian Hamiltonians in open classical and quantum systems. Examples include parity-time-symmetric optical systems with gain and loss, dissipative Bose-Einstein condensates, exciton-polariton systems, and biological networks. A particular interest centers on the topological properties of non-Hermitian systems, which exhibit unique phases with no Hermitian counterparts. However, no systematic understanding in analogy with the periodic table of topological insulators and superconductors has been achieved. In this paper, we develop a coherent framework of topological phases of non-Hermitian systems. After elucidating the physical meaning and the mathematical definition of non-Hermitian topological phases, we start with one-dimensional lattices, which exhibit topological phases with no Hermitian counterparts and are found to be characterized by an integer topological winding number even with no symmetry constraint, reminiscent of the quantum-Hall insulator in Hermitian systems. A system with a nonzero winding number, which is experimentally measurable from the wave-packet dynamics, is shown to be robust against disorder, a phenomenon observed in the Hatano-Nelson model with asymmetric hopping amplitudes. We also unveil a novel bulk-edge correspondence that features an infinite number of (quasi)edge modes. We then apply the $K$ theory to systematically classify all the non-Hermitian topological phases in the Altland-Zirnbauer (AZ) classes in all dimensions. The obtained periodic table unifies time-reversal and particle-hole symmetries, leading to highly nontrivial predictions such as the absence of non-Hermitian topological phases in two dimensions. We provide concrete examples for all the nontrivial non-Hermitian AZ classes in zero and one dimensions. In particular, we identify a ${\mathbb{Z}}_2$ topological index for arbitrary quantum channels (completely positive trace-preserving maps). Our work lays the cornerstone for a unified understanding of the role of topology in non-Hermitian systems.

Popular Summary

Topological phases are exotic states of matter in which electrons, photons, or atoms are arranged and coupled in a way that is topologically distinct from a simple assembly of individuals. Materials with topological phases usually exhibit novel properties at their surface, such as the robust surface metallicity in a topological insulator. To help make sense of the various topological phases, researchers have developed “periodic tables” of phases, similar to the periodic table of elements. All of these tables assume that matter is not exchanged between the topological material and its environment, however, recent experimental developments have allowed one to create systems that are open to the external world. It is thus natural to ask whether new “elements” of topological phases exist in this largely unexplored regime. Here, we establish a general theoretical framework to address this question and discover a hitherto unknown periodic table.

Technically speaking, because of the exchange of matter, the systems we focus on are described by non-Hermitian Hamiltonians, which are typically used to describe dissipative systems and generally have complex energy spectrums. The spectrum itself allows for a distinction between topological phases. Surprisingly, a prototypical example of our newly discovered non-Hermitian topological family turns out to be the celebrated model proposed in 1996 by Hatano and Nelson that describes a one-dimensional lattice with imbalanced leftward and rightward hopping amplitudes.

We expect plenty of unprecedented non-Hermitian topological phases to be discovered in our framework, which would deepen our understanding of the role of topology in nonequilibrium physics and open up new possibilities for designing topological devices operating far from equilibrium.

Figure 1

(a) Energy spectrum (thick lines and dots) of a Hermitian insulator. We can always perform band flattening, i.e., continuously deform the spectrum into $\{E_{-},E_{+}\}$ with $E_{-}<E_F<E_{+}$, where $E_F$ (red dot) is the Fermi energy. In particular, we can choose $E_{\pm }=\pm 1$ for $E_F=0$. (b) Energy spectrum of a non-Hermitian system forming a loop that encircles a base point $E_B\in \mathbb{C}$. (In the figure, we set $E_B=0$ for simplicity.) While the shape can be deformed continuously, the loop can never shrink to a single point without crossing the base point.

Figure 2

(a) One-dimensional lattice with asymmetric hopping amplitudes $J_L\ne J_R^{*}$. Here, we show the case in which $|J_L|>|J_R|$, as indicated by the thickness of the arrows. (b) Phase diagram and typical complex-energy spectra for the model in (a), where $w$ is the winding number. A topological phase transition occurs at $|J_L|=|J_R|$ (purple dot), where the spectrum touches the origin, while the specific case of (a) (blue star) belongs to the $w=1$ phase, where the energy spectrum forms a loop encircling the origin. An arrow inside each loop indicates the direction of increasing $k$ which corresponds to the sign of the winding number $w$.

Figure 3

(a) Complex-energy spectra and (b) flows of $\mathrm{Arg}(\det H)$ with respect to the flux $\mathrm{\Phi }$ for typical realizations of the Hatano-Nelson Hamiltonian (11) with $L={10}^3$, $J_R=2$, $J_L=1$, and real on-site disorder $V_j\in [-W,W]$, where $W=1$, 3, 4, 5. (c) and (d) correspond to (a) and (b), respectively, with the same set of parameters except for the inclusion of a complex on-site disorder $V_j=|V_j|e^{i{\varphi }_j}$, where $|V_j|\in [0,W]$ with $W=2$, 3, 3.5, 4 and ${\varphi }_j\in [0,2\pi ]$. Note that the flows of $\mathrm{Arg}(\det H)$ almost overlap in (b) for $W=1$, 3, 4 and in (d) for $W=2$, 3, 3.5 and that they also overlap with each other between (b) and (d). We see that the transition occurs between $W=4$ and $W=5$ in (a) and between $W=3.5$ and $W=4$ in (c). In the nontrivial phase [$W=1$, 3, 4 in (a) and $W=2$, 3, 3.5 in (c)], the spectra encircle the base point at $E=0$, giving the winding number $w=-1$. In the trivial phase, the data points lie on the real axis in (a) and scatter in the complex-energy plane without forming a closed loop in (c).

Figure 4

(a) Energy spectrum of Eq. (7) with $J_L=2$ and $J_R=1$ under the periodic-boundary condition (PBC, blue ellipse) and the open-boundary condition (OBC, red line). For each energy $E$ inside the ellipse (light-blue region), there exists a $w=1$ edge state localized at the left boundary in the semi-infinite space. Three colored points show energies of the three quasiedge modes in (d). (b) An edge state in the semi-infinite space (magenta wave packet) eventually becomes unstable (orange wave packet) in a finite open chain with length $L$ after a time $t^{*}\sim (L/v)$, where $v$ is the Lieb-Robinson velocity. (c) Time evolution of the relative deviation $R(t)\equiv \parallel [e^{-i(H-E)t}-1]|\psi ⟩\parallel$ for the edge state $|\psi ⟩$ with $E=0.1i$ in an open chain with $L=100$. Inset: Time evolution (solid curves) of $|{\psi }_j(t){|}^2$ at the leftmost three sites ($j=1$, 2, 3) in comparison with that of $|{\psi }_j(t){|}^2=e^{2\mathrm{Im}Et}|{\psi }_j(0){|}^2$ (dashed lines). (d) Finite-size scaling of $t^{*}$ for three different quasiedge states with energies $E=0.1i$, 2.85 and $0.5-0.5i$ [marked in (a)]. We define $t^{*}$ by $R(t^{*})={10}^{-2}$, as indicated by the dashed red line in (c). (e) The same as (c) but in the presence of real on-site disorder $V_j\in [-W,W]$ with $W=3$. The green curves and the inset show three typical realizations, and the dashed purple curve gives the disorder average over $10^3$ realizations. The average is taken for $\ln R(t)$ and, thus, gives the geometric mean for $R(t)$. (f) The same as (d) but for different disorder strengths $W=3$, 3.5, 3.7 [compared to $W=0$, the same as $E=0.1i$ in (d)] with fixed $E=0.1i$.

Figure 5

(a) Gaussian wave packet in a lattice with asymmetric hopping amplitudes $J_L=2$ and $J_R=1$ and tilted by a potential gradient $F=0.4$. (b) Profiles of the wave packet in real space at $t=0$, $0.2T$, $0.4T$, $0.5T$, $0.7T$, and $0.9T$, with $T=[(2\pi )/F]$ for the lattice length $L=100$. (c) Numerical (“$+$” marks) and semiclassical [dashed curves, obtained from Eq. (24)] results for the wave-packet dynamics in real space. Here, $\mathrm{\Delta }⟨x{⟩}_t\equiv ⟨x{⟩}_t-⟨x{⟩}_0$ denotes the center-of-mass displacement at time $t$. (d) Complex eigenenergies reconstructed from (c) (dots) in comparison with the theoretical results (dashed curve). The arrows in (b) and (d) show the direction of time. Since the data are taken stroboscopically, the imaginary energies $\mathrm{Im}E$ are estimated from $\ln (⟨{\psi }_{t+\mathrm{\Delta }t}|{\psi }_{t+\mathrm{\Delta }t}⟩/⟨{\psi }_t|{\psi }_t⟩)/(2\mathrm{\Delta }t)$.

Figure 6

(a) Spectral flow (from red to green, guided by the arrows) in the course of the unitarization process of an invertible complex matrix with size 20. Note that the spectrum of the unitarized matrix locates on a unit circle (black dashed line). (b) The same as in (a) but for a time-reversal-symmetric matrix. The time-reversal symmetry, which manifests itself as the mirror symmetry of the spectrum with respect to the real axis, is kept in the unitarization process.

Figure 7

(a) Spectrum of Eq. (34) with $(\gamma ,m)=(0.25,1)$. The zero mode in the $\gamma =0$ limit (sparse dots) disappears due to the global spectrum shift along the imaginary energy axis (indicated by the arrows). (b) The same as (a) but with $(\gamma ,m)=(0,1+0.5i)$. The symmetry constraint given in Eq. (35) enforces the spectrum to be inversion symmetric, leading to a robust zero mode (gray dot). In both (a) and (b), the blue (red) dots correspond to the periodic- (open-) boundary condition, and the system size is $40\times 40$.

Figure 8

Spectrum deformation in a class AI system described by a $3\times 3$ matrix in zero dimension. The spectrum is always symmetric with respect to the real axis. Without touching $E_B=0$, the number of eigenvalues on the negative or positive real axis can change only by an even number, so a ${\mathbb{Z}}_2$ index ($s=-1$) can be defined as in Eq. (37).

Figure 9

(a) Pulse sequence of the stroboscopic qubit dynamics governed by two types of operations ${\mathcal{R}}_{\epsilon }{\mathcal{E}}_{xy}$ and ${\mathcal{R}}_{\epsilon }{\mathcal{E}}_x$. In the former case, $\pi$ pulses are applied randomly in the $x$ and $y$ directions with equal probability, leading to $s_{\mathrm{ms}}=-1$. In the latter case, $\pi$ pulses are applied in the $x$ direction, leading to $s_{\mathrm{ms}}=1$. (b) Starting from ${\rho }_0=|\uparrow ⟩⟨\uparrow |$, the dynamics of $⟨{\sigma }_z⟩$ for $\epsilon =0$ (red dots) are the same between the two cases. As for $\epsilon =0.05\pi$ (green dots), the dynamics governed by ${\mathcal{R}}_{\epsilon }{\mathcal{E}}_{xy}$ (left) exhibit a discrete time-crystalline-like behavior [160, 161, 162], but the dynamics governed by ${\mathcal{R}}_{\epsilon }{\mathcal{E}}_x$ do not. (c) Fourier transform of $⟨{\sigma }_z{⟩}_{t=nT}$ into the frequency domain. The single peak located at $\omega =0.5{\omega }_T$ (${\omega }_T\equiv [(2\pi )/T]$) stays robust for ${\mathcal{R}}_{\epsilon }{\mathcal{E}}_{xy}$ (left) but splits into two peaks for ${\mathcal{R}}_{\epsilon }{\mathcal{E}}_x$ (right).

Figure 10

Non-Hermitian open chains with unidirectional hoppings (indicated by the arrows) belonging to (a) class AIII and (b) class AII. In (a), the number of zero modes at the left boundary (enclosed by a red rectangle) is not the same as that on the right boundary. In (b), the zero modes form Kramers pairs, which interchange via the time-reversal operator $T$, and therefore the total number of the edge modes must be even.

Figure 11

(a) Energy spectrum of an infinite translation-invariant lattice described by Eq. (a7). The arrows indicate the flow of eigenenergy as the wave vector $k$ increases from 0 to $2\pi$. (b) The same as in (a) but for a finite ($L=30$) ring subjected to a flux $\mathrm{\Phi }$. The arrow indicates the spectral flow as $\mathrm{\Phi }$ changes from 0 to $2\pi$.

Figure 12

Spectral flow (right) and two representative eigenwave functions (left) of a Hatano-Nelson ring with complex disorder $W=2.5$, $L=30$, $J_L=2$, $J_R=1$, and threaded by a varying flux $\mathrm{\Phi }$. A delocalized wave function (left-upper panel) behaves flexibly, while a localized wave function (left-lower panel) exhibits rigidity.

Figure 13

(a) Disorder-averaged minimum absolute value of energy $⟨|E{|}_m⟩$ for the Hatano-Nelson model (11) with $J_L=1$, $J_R=0$, real on-site disorder $V_j\in [-W,W]$, and different system sizes ranging from $L=1000$ to 7000. (b) Disorder-averaged maximum $\zeta$ [defined in Eq. (b8)] for the same model but with complex disorder $V_j=|V_j|e^{i{\varphi }_j}$, where $|V_j|\in [0,W]$ and ${\varphi }_j\in [0,2\pi ]$, and different system sizes ranging from $L=100$ to 1600. In both (a) and (b), the red dashed line indicates the theoretical transition point $W_c=e=2.718\dots$. The number of disorder realizations ranges from thousands to hundreds, depending on the system size. The error bars denote twice the standard deviations of the mean.

Figure 14

(a) Complex-energy spectra of Eq. (11) with $L={10}^3$, $J_R=0$, $J_L=1$, and binary on-site disorder $V_j=\pm W$ with an equal probability of occurrence for $W$ and $-W$, where $W=0.5$, 0.9, 1.1, 1.5. (b) Disorder-averaged maximum $\zeta$ [see Eq. (b8)] for the same model but with different system sizes ranging from $L=100$ to 1600. The red dashed line indicates the theoretical topological transition point $W_c=1$.

Figure 15

Finite-size scaling for the logarithmic-disorder-averaged smallest singular value $⟨\ln {\lambda }_m⟩$ of the Hatano-Nelson Hamiltonian (11) with $J_L=2$, $J_R=1$, and complex disorder. Each point is obtained from $2.5\times {10}^5$ disorder realizations.

Figure 16

Dynamics in a Hermitian single-band lattice with $J=1$ and starting from a wave function ${\psi }_j\propto e^{-j/\ell }$ with $\ell =2$. (a) Relative deviation $R(t)$. Inset: Enlarged view of an initial behavior up to $t=2.5$ (orange region). The purple (red) line shows a linear fit [the lower bound in Eq. (d6)]. (b) Density profile $|{\psi }_j(t){|}^2$ at the leftmost three sites ($j=1$, 2, 3) in comparison with the initial values $|{\psi }_j(0){|}^2$ (dashed lines). The system size is $L=60$ in (a) and (b). (c) and (d) are the same as (a) and (b), except for $L=80$. In (a)–(d), the blue dashed line denotes the revival time $t_r=L/(2J)$.

Figure 17

Implementation of asymmetric hopping amplitudes in optical lattices. A stable (dissipative) optical lattice is applied to the ground (excited) state $|g⟩$ ($|e⟩$). A running wave parallel to an optical lattice couples $|g⟩$ to $|e⟩$, which undergoes rapid on-site loss at a rate $\gamma$. By making the wavelength of the running wave equal to that of the lattice constant, the phases of Rabi couplings can be adjusted to change by $\pi /2$ compared with the left nearest ones.

Figure 18

Snapshots of the energy spectra during the trivialization process for two coupled $PT$-symmetry Su-Schrieffer-Heeger chains (i3) under the open-boundary condition. The arrows indicate the direction of the spectral flow. The parameters $(J,J^{\prime },\gamma ,J_c)$ are given by (1,2,0.5,0) and (0,0,0,1) at the initial and final times, respectively.