Unsupervised Topological Phase Discovery in Periodically Driven Systems via Floquet-Bloch State

Abstract

Floquet engineering offers an unparalleled platform for realizing novel non-equilibrium topological phases. However, the unique structure of Floquet systems, which includes multiple quasienergy gaps, poses a significant challenge to classification using conventional analytical methods. We propose a novel unsupervised machine learning framework that employs a kernel defined in momentum-time ($\boldsymbol{k},t$) space, constructed directly from Floquet-Bloch eigenstates. This approach is intrinsically data-driven and requires no prior knowledge of the underlying topological invariants, providing a fundamental advantage over prior methods that rely on abstract concepts like the micromotion operator or homotopic transformations. Crucially, this work successfully reveals the intrinsic topological characteristics encoded within the Floquet eigenstates themselves. We demonstrate that our method robustly and simultaneously identifies the topological invariants associated with both the $0$-gap and the $π$-gap across various symmetry classes (1D AIII, 1D D, and 2D A), establishing a robust methodology for the systematic classification and discovery of complex non-equilibrium topological matter.

Figures (3)

• Figure 1: (a) Two static topological systems possessing distinct topological numbers must necessarily exhibit band inversion points. The top and bottom figures illustrate the Bloch vector representations for the topologically trivial and non-trivial phases, respectively, in the SSH model. (b) The Bloch vector representations for the flattened Floquet operator for the modulated Floquet SSH model. The top configuration is sampled from the $0$-phase ($\theta_{\mathrm{Re}}=0.2\pi/T,\theta_{\mathrm{Im}}=0.2\pi/T$) and the bottom configuration is sampled from the $\pi$-phase ($\theta_{\mathrm{Re}}=0.6\pi/T,\theta_{\mathrm{Im}}=0.2\pi/T$). The colorbar describes the $z$-component of the Bloch vector. (c) The phase diagram is derived using the unsupervised clustering algorithm. Comparing the result with the theoretical topological numbers, we observe that Phase I has $W_0 = 1, W_\pi = 0$, Phase II has $W_0 = 1, W_\pi = 1$ and Phase III has $W_0 = 0, W_\pi = 1$. (d) The eigenvalues output by the diffusion map algorithm, where three eigenvalues close to 1 indicate that the data has been partitioned into three clusters. The inset shows the two-dimensional projection of the dataset in the principal component space after sufficient diffusion.
• Figure 2: (a) Schematic illustration of the 2D modulated Floquet model, where different colors indicate the coupling between lattice sites in different time segments. The inter-atomic couplings evolve in a clockwise sequence following the order red, green, blue, and orange. (b) Phase diagram obtained using an unsupervised clustering algorithm. By comparing the clustering results with the theoretical topological invariants, three distinct phases are identified: the trivial phase with $W_0 = 0, \; W_\pi = 0$; the $\pi$ phase with $W_0 = 0, \; W_\pi = 1$; and the $0\pi$ phase with $W_0 = 1, \; W_\pi = 1$. (c,d) Inverse images of two Bloch vectors $\mathbf{n}$ and $-\mathbf{n}$ of the Floquet–Bloch states. The first Brillouin zone is indicated by the dashed lines; the linking number is 0 in (c) and 1 in (d). (orientation of the Bloch vector $x=\cos(\theta)\sin(\phi)$, $y=\sin(\theta)\sin(\phi)$, $z=\cos(\phi)$, the red line: $\phi=\pi$, the blue line: $\theta=\pi,\phi=0.75\pi$.) (c) corresponds to the case with $J=0.4\pi/T$, $W_0 = 0, \; W_\pi = 0$, where the linking number of the preimages is 0; (d) corresponds to $J=2.7\pi/T$, $W_0 = 1, \; W_\pi = 1$, where the linking number of the preimages is 1.
• Figure 3: (a) (c)The Bloch vector representations for the FF0 for the modulated Floquet D model. The configuration in (a) is sampled from the Trivial-phase ($J_1=0,J_2=-\pi/T$) and the configuration in (c) is sampled from the $0\pi$-phase ($J_1=1.5\pi/T,J_2=\pi/T$). The colorbar describes the $z$-component of the Bloch vector.The red circles mark the flipping points identified by the FFO. (b) displays the phase diagram of the model, where the downward arrows indicate phases in which the Bloch vector at the high-symmetry point $k = 0$ point along the negative $y$ direction. (d) The eigenvalues output by the diffusion map algorithm, where eight eigenvalues close to 1 indicate that the data has been partitioned into eight clusters. The inset shows the two-dimensional projection of the dataset in the principal component space after sufficient diffusion. In the inset the triangles mark the phases in which the Bloch vector at the high-symmetry point $k = 0$ points along the negative $y$ direction.