Geometric quantum drives and topological dynamical responses: hyperbolically-driven quantum systems and beyond

TL;DR

This work introduces geometric quantum driving, a general framework where a classical particle traversing a smooth manifold steers a parent quantum Hamiltonian via H(t)=H_Q(x_t). By selecting compact geometries such as the hyperbolic Bolza surface and the Klein bottle, the authors define hyperbolically-driven quantum systems (HDQS) and nonorientably-driven quantum systems (NDQS), respectively, and show that in the fully gapped, adiabatic regime these drives exhibit quantized dynamical responses tied to topological invariants: the first Chern number $C_1^{(n)}$ for HDQS and the dipolar Chern number $D_y^{(n)}$ for NDQS. These responses are extracted from carefully constructed observables and long-time averages, with ergodicity of geodesic flows underpinning the connection between dynamics and topology. The framework unifies periodic, quasiperiodic, and these novel geometric drives, offering a versatile tool for exploring topological phases and enabling new quantum simulation capabilities on exotic effective lattices. Overall, geometric quantum driving provides a universal bridge between quantum dynamics, geometry, and topology, with potential applications in quantum simulators and the exploration of universal phase structures across time-dependent quantum systems.

Abstract

We introduce a geometrical framework to construct a large class of time-dependent quantum systems, in which the position of a classical particle moving autonomously on a smooth connected manifold is used to steer a quantum Hamiltonian over time. This results in quantum drives with structured temporal profiles and properties dependent on the local and global nature of the underlying choice of manifold. We show that our construction recovers the well-known classes of periodically-driven and quasiperiodically-driven quantum systems, but also unveils fundamentally new classes of quantum dynamics: by utilizing a compact 2d hyperbolic Bolza surface and a nonorientable Klein-bottle surface, we demonstrate examples of a hyperbolically-driven quantum system and a nonorientably-driven quantum system respectively. Furthermore, we demonstrate that these driven systems exhibit unusual quantized dynamical responses reflecting their different underlying topologies, under the condition of being fully gapped and in the adiabatic limit, and which have interpretations as quantized crystalline electromagnetic responses in certain exotic effective tight-binding lattice models. We envision geometric quantum driving as a general framework to chart the landscape of time-dependent quantum systems and investigate the universal phase structures they exhibit, as well as a useful tool to enhance the capabilities of modern day quantum simulators.

Figures (5)

• Figure 1: Geometric quantum drives. Geodesic flows on a circle $S^1$ (top left), a torus $T^2$ (top right), a genus-2 torus $\Sigma_2$ (bottom left) and a Klein-bottle surface $K^2$ (bottom right), yield periodically, quasiperiodically, hyperbolically and nonorientably driven quantum systems respectively. In this work, we present explicit constructions of the latter two systems and expound upon the novel physics they exhibit.
• Figure 2: (a) We take $\Sigma_2$ to be the Bolza surface, a compact Riemann surface with constant negative curvature. On the Poincairé disk, the fundamental domain is an octagon, with opposite sides identified in the orientation shown via Fuchsian translates $\gamma_i$. (b) Geodesic on the Bolza surface. (c) Temporal profile of the example hyperbolically driven qubit Hamiltonian considered in this work with $\epsilon = 0.5$.
• Figure 3: (a) Spin texture $\bm{d}_\epsilon(z)$ of the parent qubit Hamiltonian $H_Q(z)$ on the Bolza surface for $\epsilon = 1.5, 0.5$. When $\epsilon = 1.5(0.5)$, the texture is ferromagnetic(skyrmionic), pointing mostly up(inverting from up to down). (b) The difference in textures can be quantitatively and sharply captured by the Chern number of the upper band of $H_Q(z)$, which changes from $0$ (trivial) when $|\epsilon|>1$, to $1$ (topological) when $|\epsilon|<1$. (c) Dynamical response $w(T)$ beginning from the instantaneous eigenstate of the upper band. For small $\lambda$, $w(T)$ converges to the Chern number of the band at late times, in both trivial and topological regimes. (d) Fidelity between time evolving state and instantaneous eigenstate for different $\lambda$s and $\epsilon=0.5$.
• Figure 4: (a) Geodesics on the Klein bottle (gray region) are depicted by red straight lines. Every time the particle leaves the gray region, it gets mapped back to a point within the fundamental region following the identification rules of the boundaries (yellow and blue arrows). Equivalently, one may consider two gluing copies of the Klein bottle as shown (white and gray regions) which yield a standard torus; a Klein bottle geodesic is then equivalent to a standard geodesic on the torus (dotted trajectory). (b) Temporal profile of the example Klein-bottle driven qubit Hamiltonian Eq. \ref{['eqn:example_NDQS']} utilized for numerical simulations, with $m = 0.5$.
• Figure 5: (a) Spin texture $\bm{d}(\bm\theta)$ defining the parent qubit Hamiltonian $H_Q(\boldsymbol{\theta})$ on the Klein bottle surface (gray regions) for $m = 0.5, 2,4$ respectively. (b) The corresponding dipolar Chern number versus $m$ is shown. For generic $m$ it is quantized in units of $\pi/2$. (c) Dynamical response $v(T)$ starting from the instantaneous eigenstate of the upper band. For the small driving frequency $\bm\omega$ stated in the main text, $v(T)$ approaches the dipolar Chern number of the band at late times.