Identifying Topological Phase Transitions in Experiments Using Manifold Learning

We demonstrate the identification of topological phase transitions from experimental data using diffusion maps: a nonlocal unsupervised machine learning method. We analyze experimental data from an optical system undergoing a topological phase transition and demonstrate the ability of this approach to identify topological phase transitions even when the data originates from a small part of the system, and does not even include edge states.

Figure 1

(a) The Haldane unit cell: the green and blue lattice sites indicate the two underlying sublattices whose potential may be detuned from one another. The red and blue inward and outward arrows are direction of the nonuniform magnetic field. (b)–(e) Examples of the evolution of an arbitrary initial wave packet made of bulk states only. (b) Intensity distribution (in arbitrary units) of the initial bulk wave packet at $t=0$. (c),(d) Intensity distribution at $t=T$ for the topological cases of $M=0$ and $M=4$. (e) Intensity distribution for the trivial case of $M=8$, at $t=T$. The intensity distribution looks similar for all 4 cases of evolution in the bulk. (f) Scatter plot of the first two principal components of diffusion maps. The points are colored by the detuning in the honeycomb potential (magnitude of the detuning given by the color map). The cusp in (f), found by applying diffusion maps to the bulk propagation data [as (c)–(e)], is at the phase transition predicted by theory.

Figure 2

(a) The spectrum of the Haldane model. The available data are associated with a small part of the spectrum around $E_1$, which is represented by the red shaded area. (b) Diffusion maps for the Haldane model, when observing the evolution of a single band eigenstate. The phase transition is at the sharp turn of the curve.

Figure 3

(a) Helical waveguide array in a diagonal lattice with shifted helicity (as in Ref. [33]). (b),(d) location of input beam when the edge states are excited (b) and when the edge states are not excited (d). (c),(e) Example of experimental output intensity images, when the input wave packet is on the upper edge (c), and when the input is in the middle of the lattice (e).

Figure 4

(a) Data distribution in parameter space for the case of edge excitation [Figs. 3]; each point represents an experimental image. The color bar represents the distance in parameter space from the theoretical phase transition that is marked with a red line. (b) Diffusion maps applied to the experimental data. The color of the data points is the same as in (a). (c)–(d) Same as (a)–(b) only for the case where the bulk is excited [Figs. 3]. A second cusp (obtuse) is visible in (d) for distance values (color) of around 4. The obtuse cusp points to the destruction of the topoligcal phase due to enhanced NNN coupling.