Observation of knot topology of exceptional points

Exceptional points (EPs), which refer to degeneracies in non-Hermitian systems, have garnered significant attention in the last few years due to their potential applications in the fields of sensing, single-mode lasing, and others. Recently, the complete classification of isolated EPs based on homotopy theory has been proposed, where the topology of energy spectra around EPs can be fully characterized by the braiding group and knot topology. However, there have been few experimental observations of EPs with different knot topologies due to the need for tunable, nonlocal, and nonreciprocal couplings in the proposed non-Hermitian lattice model. Here, we report experimental observation of EPs with different types of knots/links by non-Hermitian electric circuits with voltage-tunable node couplings. Specifically, a second-order EP with trefoil knot topology and a third-order EP with $6_3^3$ link topology (according to Rolfsen's table) are achieved. Moreover, we also provide a method on the construction of higher-order Dirac EPs with nested-link topologies, and experimentally construct a sixth-order Dirac EP. Our work provides an artificial platform for exploring EPs with knot and link topologies, offering potential applications in designing EP-based electronic devices.

Figure 1

The theoretical results on the implement of knot topologies of EPs by electric circuits. (a) The illustration of the knot topology related to the braid group $B_n$ for an ${\mathrm{EP}}_n$. (b), (c) The schematic diagrams of designed electric circuits to fulfill the trefoil knot ${\mathrm{EP}}_2$ and the $6_3^3$ link ${\mathrm{EP}}_3$, respectively. Red and blue dashed blocks enclose the circuit structures of the voltage follower and the multiplier module. (d), (e) The real parts of eigenspectra for the circuit Laplacian of $L_{\left(2,3\right)}$ and $L_{\left(3,3\right)}$ in the 2D momentum space. (f), (g) Numerical results of the real part of eigenspectra for $L_{\left(2,3\right)}$ and $L_{\left(3,3\right)}$ with the 2D $k$ vector ($k_x,k_y$) evolving along a closed path $\mathrm{\Gamma }\left(\theta \right)$. Two insets plot 3D structures of energy strands in the form of a trefoil knot and a $6_3^3$ link, respectively.

Figure 2

Experimental results on the observation of trefoil knot and $6_3^3$ link topologies for exceptional points in electric circuits. (a) The photograph image of the fabricated circuit sample of $L_{\left(2,3\right)}$. (b), (c) The experimental impedance spectra of node 1 with external voltages being ($V_a=V_b=-5\mathrm{V}$), ($V_a=-V_b=-5\mathrm{V}$), between the closing paren and the opening square bracket and ($V_a=-V_b=5$ V) of the circuit sample with a trefoil knot ${\mathrm{EP}}_2$ and a $6_3^3$ link ${\mathrm{EP}}_3$. (d), (e) The marker points present the experimentally reconstructed admittance eigenspectra for circuit samples with $L_{\left(2,3\right)}$ and $L_{\left(3,3\right)}$ along a closed path in the 2D $k$-vector space. The colored lines plot the simulation results of reconstructed admittance eigenspectra.

Figure 3

The theory on the construction of EPs with nested knot topology. (a) The illustration of the hierarchically periodic braid with three generations. The bottom inset presents an enlarged view of the starting plane of the nested braid. (b), (c) Numerical results of the real and imaginary parts of eigenspectra for the non-Hermitian Hamiltonian sustaining a sixth-order EP with a two-generation nested knot in the 2D $k$ space. (d) Numerical results on the real part of eigenenergies with the evolution of ($k_x,k_y$) along a closed path $\mathrm{\Gamma }\left(\theta \right)$. (e) The illustration of the nested knot of eigenenergies around the sixth-order EP.

Figure 4

Experimental results on the observation of the nested link for a sixth-order Dirac EP in electric circuits. (a) The schematic diagram of the designed electric circuit for the realization of a nested knot related to the sixth-order exceptional point. (b) The photograph image of the fabricated electric circuits. (c) The measured impedance spectra of node 1 with the external voltages being ($V_a=V_b=-5\mathrm{V}$), ($V_a=-V_b=-5\mathrm{V}$), and ($V_a=-V_b=5$ V). (d) Experimental results of the real part of the recovered eigenspectrum, which is in the form of a nested periodic braid, along a closed line in the 2D $k$-vector space for the circuit sample.

Figure 5

Numerical results of eigenspectra with different knot and link topologies around EPs. (a)–(c) The eigenspectra of $H_{\left(p,q\right)}$ with ($p=2$, $q=1$), ($p=2$, $q=3$), and ($p=2$, $q=5$). (d)–(f) The eigenspectra of $H_{\left(p,q\right)}$ with ($p=2$, q = 2), ($p=3$, $q=2$), and ($p=4$, $q=2$). Insets display the real part of eigenspectra along a closed path in the 2D $k$ space, and the associated 3D knotted energy strands.

Figure 6

Numerical results of energy dispersions and phase rigidities around (2, $q$)-knot EP. (a)–(c) The energy dispersions around EPs with ($p=2$, $q=3$), ($p=2$, $q=2$), and ($p=2$, $q=1$). (d)–(f) The phase rigidities around EPs with ($p=2$, $q=3$), ($p=2$, $q=2$), and ($p=2$, $q=1$).

Figure 7

The movement of EPs with changing the perturbation term.

Figure 8

The photograph image of a fabricated circuit sample with the $6_3^3$ link EP.

Figure 9

The simulation results of impedance spectra. (a)–(c) The simulation impedance spectra of node 1 with the external voltages being ($V_a=V_b=-5\mathrm{V}$), ($V_a=-V_b=-5\mathrm{V}$), and ($V_a=-V_b=5$ V) of the circuit sample with a trefoil knot ${\mathrm{EP}}_2$, a $6_3^3$ link ${\mathrm{EP}}_3$, and a sixth-order EP.

Figure 10

The splitting behavior passing through the EPs. (a), (b) The real and imaginary parts of eigenspectra of the trefoil knot ${\mathrm{EP}}_2$. (c), (d) The real and imaginary parts of eigenspectra of ${\mathrm{EP}}_3$ with the $6_3^3$ link topology. (e), (f) The real and imaginary parts of eigenspectra of ${\mathrm{EP}}_6$ with the nested-link topology.

Figure 11

Numerical results of eigenspectra with different knot and link topologies around EPs. (a), (b) The eigenspectra and evolution of eigenenergies around EPs with different nested-link topologies with ($n_1=4$, $n_2=2$), ($n_1=2$, $n_2=2$, $n_3=2$).

Figure 12

(a), (c), (e) and (b), (d), (f) The real and imaginary parts of eigenspectra along three paths of $k_x=k_y$, $2k_x=k_y$, and $k_x=0$ through trefoil knot ${\mathrm{EP}}_2$, $6_3^3$ link ${\mathrm{EP}}_3$, and ${\mathrm{EP}}_6$ with the nested link.