Knots and Non-Hermitian Bloch Bands

Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum field theory discovered. Here we show that knots tied by the eigenenergy strings provide a complete topological classification of one-dimensional non-Hermitian (NH) Hamiltonians with separable bands. A ${\mathbb{Z}}_2$ knot invariant, the global biorthogonal Berry phase $Q$ as the sum of the Wilson loop eigenphases, is proved to be equal to the permutation parity of the NH bands. We show the transition between two phases characterized by distinct knots occur through exceptional points and come in two types. We further develop an algorithm to construct the corresponding tight-binding NH Hamiltonian for any desired knot, and propose a scheme to probe the knot structure via quantum quench. The theory and algorithm are demonstrated by model Hamiltonians that feature, for example, the Hopf link, the trefoil knot, the figure-8 knot, and the Whitehead link.

Figure 1

Four examples of links and knots in 1D NH Bloch bands. Braid operator ${\tau }_i$ (${\tau }_i^{-1}$) denotes the $i$th string crossing over (under) the ($i+1$)th string from the left. Colors label different knot components. $Q$ is the biorthogonal Berry phase defined in Eq. (3). The four knots are realized by NH Hamiltonians $T_2$, $T_3$ as defined in Eq. (6), $H_8$ and $H_w$ [81], respectively. The eigenenergy strings are shown in space ($\mathrm{Re}E$, $\mathrm{Im}E$, $k$).

Figure 2

Phase diagram and phase transitions of $H_{12}(k)$ defined in Eq. (7). (a) Schematic of knot transitions. Type-I (type-II) transition occurs by going through one (two) EP. (b) The phase diagram of $H_{12}$ with parameters $m_1$ and $m_2$. The blue, pink, and green regions label the Hopf link (${\tau }_1^2$), unknot (${\tau }_1$), and unlink phase (${\tau }_1^0$), respectively. In each region, a representative band structure is plotted. (c) and (e) Eigenenergy $|E(m_1,k)|$ along the cut labeled by I and II, respectively, in (b): an EP is visible at $(1/\sqrt{2},\pi )$ in (c), while there are two EPs at (1,0) and $(1,\pi )$ in (e). (d) and (f) The Wannier centers ${\nu }_n$ along the cut I and II. (g) Schematic of a periodic electric circuit that realizes $H_{12}$. The unit cell (oval) contains two “sites,” the red and blue nodes, connected by resistors $R$, inductors $L$, and negative impedance converters $C_{1,2,3}$; see Supplemental Material [81] for details.

Figure 3

Signatures of knots after quantum quench. The red and blue curves are the eigenvectors $|{\psi }_{1,2}(k)⟩$ of $H_{12}(k)$ on the Bloch sphere. From an initial state $|{\xi }_0⟩=(1,0{)}^T$ (north pole), the state evolves with $H_{12}(k)$ and after a long time falls into the solid line part of the eigenstates. The arrow denotes increasing $k$ from 0 to $2\pi$, and the orange (green) dots represent the $k=0$ ($k=\pi$) mode. The parameters are (a) $m_1=m_2=0.5$, the Hopf-link phase; (b) $m_1=m_2=0.9$, the unknot phase; and (c) $m_1=1.2$, $m_2=0$, the unlink phase.