Topological phase transition measured in a dissipative metamaterial

We construct a metamaterial from radio-frequency harmonic oscillators, and find two topologically distinct phases resulting from dissipation engineered into the system. These phases are distinguished by a quantized value of bulk energy transport. The impulse response of our circuit is measured and used to reconstruct the band structure and winding number of circuit eigenfunctions around a dark mode. Our results demonstrate that dissipative topological transport can occur in a wider class of physical systems than considered before.

Figure 1

(a) A topological metamaterial: Circuit nodes, indexed by an integer $m$, are connected to ground with capacitors $c$, and coupled to each other by variable inductors $l_A$ and $l_B$. $B$ sites contain a resistor $r$ to ground, while $A$ sites do not. (b) Circuit photograph. The inset shows one $A$ site and one $B$ site. Coupling inductances are variable by greater than a factor of 10 using switch networks. The solid pink line traces a chain of adjacent cells, and the dashed pink line connects the edges of this chain to form periodic boundary conditions. This asymmetric connection has a negligible effect on circuit dynamics; its parasitic capacitance is calculated to be about $3\%$ of the on-site capacitance, and its length (49 $\mathrm{cm}$) is less than about $1\%$ of the wavelength of light at frequencies of interest. (c), (d) The time-dependent voltages measured on $A$ and $B$ sites, $v_m^A\left(t\right)$ and $v_m^B\left(t\right)$, respectively, are plotted as a function of site index $m$ and time $t$ for the case where $l_A=l_B$. Wave forms are normalized to the voltage $v_0$, present at site $m_A=0$ at time $t=0$.

Figure 2

Fraction of the total energy dissipated, $P_m$ vs $m$, for (a) the topological phase and (b) the trivial phase of our circuit. Gray dashed lines indicate $m=0$, where an excitation is initialized. Two experimental data sets (black and purple distributions) are compared to simulated distributions (orange) of an “ideal” circuit with no disorder, no additional loss, and 501 unit cells. (c) Mean displacement $\delta$ as a function of $\alpha =l_A/(l_B+l_A)$ for the mean of both experimental data sets (solid red line) and a numerical simulation (dashed red line) of an ideal circuit. Black lines show the range of $\delta$ over both experimental runs, which is nearly zero for most values of $\alpha$. A phase transition between $\delta =1$ and $\delta =0$ occurs around $\alpha =0.5$. The behavior of $\delta$ approximates the winding number of ${\xi }_{j,k}$ around the dark state (blue line).

Figure 3

(a)–(c) Circuit band structure, exhibited in the Fourier-transformed $A$-sublattice voltages $\left|v_k^A\left(\omega \right)\right|$. A dark mode appears at $\alpha =l_A/(l_A+l_B)=0.5$ and $k=\pi$. The dashed white lines show the real parts of the circuit's complex eigenfrequencies, found by diagonalizing Eq. (2) with the same circuit parameter values that are used in each experiment. Eigenvector sublattice components ${\mathit{V}}_{j,k}^A$ and ${\mathit{V}}_{j,k}^B$ are extracted from line cuts across the lower ($j=1$) and upper ($j=2$) bands. (d)–(f) The complex ratio ${\xi }_{j,k}={\mathit{V}}_{j,k}^B/{\mathit{V}}_{j,k}^A$, whose winding number around the origin (the location of the dark mode, yellow square) distinguishes the topological phases of the circuit. Experimental points (dots) agree closely with results of a numerical simulation (lines).