Session Y19: Floquet Topological Systems

Topological Phenomena in the Floquet-Lattice of Electrical Circuits

Floquet-topological states are conventionally studied in the context of quantum mechanics' Schrödinger equation, a first-order ordinary differential equation. However, recent research has shown that topological states are a fundamental aspect of all types of lattice-like wave systems, such as optical waveguides, coupled mechanical oscillators or electrical circuit lattices. Investigating periodically modulated electric circuit networks, we find that the rich structure of the differential equations governing them present fertile grounds for the discovery of novel kinds of Floquet-Topological phenomena. I will discuss how the differences between Quantum mechanics' Schrödinger equation and an electrical circuits' higher-order differential equations manifest in the Floquet description, and what this implies for the possibility of emerging topological effects.   

Author not Attending

Higher order topological insulators (HOTIs) are n-dimensional materials that exhibit gapless protected states having dimensionality n-2 or lower. While practical realizations are limited to three spatial dimensions, Floquet systems having periodic driving can be used to introduce synthetic dimensions that enable the exploration of higher dimensional physics. In this work we describe a general approach to producing Floquet HOTIs using time-varying topolectric circuits. Our approach enables dynamically reconfigurable time protocols and on-demand TI phases within the same material. As an explicit example, we discuss the realization of a Floquet quadrupole HOTI with anomalous dynamical polarization. This 2D material exhibits 0D corner modes analogous to a non-driven quadrupole HOTI, and additionally exhibits a new set of topologically protected states originating from interactions in the synthetic dimension.

    

Real-space multifold degeneracy in graphene irradiated by twisted circularly polarized light

We study monolayer graphene coherently driven by twisted light carrying orbital angular momentum and a radial vortex profile. Using Floquet theory, we characterize the real-space structure of quasienergies and Floquet modes. We obtain the effective real-space Floquet Hamiltonian and show it supports crossings of Floquet modes, associated with high-symmetry K and Γ points of graphene, at specific radial positions from the vortex center. This yields a ring of vortex bound states whose number is a topological invariant. These bound states form a multifold degenerate structure in real space that is purely dynamically generated and controlled by the frequency and intensity of twisted light.   

Decoherence of edge states in two-dimensional Floquet systems with temporal noise

Two-dimensional periodically driven systems exhibit interesting topological features which cannot be achieved in static systems.

Anomalous Floquet topological insulators (AFTI) are one such example.

AFTIs are known to withstand more spatial disorder than any other Floquet topological phases in a temporal noise-free case.

In this work, we show that a two-dimensional AFTI is more robust to temporal noise than a Floquet Chern Insulator (FCI).

We obtain our results by computing the survival probability and the time-averaged velocity expectation of an edge state.

We observe that the survival probability of an edge state is exponentially decaying in both AFTI and FCI phases.

In contrast, the velocity decays as a power law in an AFTI and decays exponentially in an FCI.

However, the exponent of the power law decay can be changed by altering the spatial disorder in an AFTI away from resonant driving, a feature not seen in an FCI.   

Dynamic melting and condensation of topological dislocation modes

Bulk dislocation lattice defects are instrumental to identify translationally active topological insulators (TATIs), featuring band inversion at a finite momentum (K_inv). They harbor robust gapless modes around the dislocation core, when the associated Burgers vector (b) satisfies K_inv · b = π (modulo 2π). From the time evolution of the appropriate density matrix, here we show that when a TATI via a ramp enters into a trivial or topological insulating phase, devoid of any gapless dislocation mode, at least weak signatures of the original defect modes survive for a long time. More intriguingly, as the system ramps into a TATI phase, signature of the dislocation mode dynamically builds up. Such evolutions of dislocation modes are more prominent for slow ramps. We exemplify these generic outcomes for two-dimensional time-reversal symmetry breaking insulators. Proposed dynamic responses at the core of dislocation lattice defects can be experimentally observed on quantum crystals, optical lattices and various metamaterials with time tunable band gap.   

Tunable Floquet-Bloch manipulation of surface states in the antiferromagnetic topological insulator MnBi2Te4

The Floquet-Bloch manipulation of materials using the time-periodic potential of a laser pulse is a promising route to generate and coherently control new phases of matter. So far, experimental realization has been limited to the manipulation of time-reversal symmetric and non-interacting systems such as graphene and the surface states of a 3D topological insulator. In this work, we realize Floquet-Bloch manipulation of a time-reversal broken system: the putative antiferromagnetic topological insulator MnBi₂Te₄. Using time- and angle-resolved photoemission spectroscopy (tr-ARPES) with a mid-infrared pump, we generate Floquet-Bloch sidebands and demonstrate a gap opening at the Dirac point with circularly polarized light. Combined with Floquet theory calculations, our results uncover a rich, tunable Floquet phase diagram that uniquely arises in MnBi₂Te₄ due to the intrinsically broken time-reversal symmetry and the coherent dressing of an unequal number of particle- and hole-like bands.   

Dynamic metal at the grain boundary of Floquet topological insulators

Driven quantum materials often feature emergent topology solely arising from the time periodic drive, otherwise absent in static crystals, among which the dynamic bulk-boundary correspondence encoded by nondissipative gapless modes residing near the Floquet zone center and/or boundaries are the most prominent ones. Here we show that such dynamic topological crystal can harbor topologically robust gapless matallic state along the grain boundary in its interior when the Floquet-Bloch band inversion occurs at a finite momentum (K_inv), such that the Burgers vector (b) of the constituting array of dislocations satisfies K_inv·b = π (modulo 2π). Such in-gap topological bulk metal, which can appear at the center and/or edge of the underlying periodic Floquet Brillouin zone, results from the hybridization among the localized modes bound to the core of individual dislocation. We exemplify these general outcomes by considering a paradigmatic representative of time-reversal symmetry breaking two-dimensional insulating system. Possible experimental setup underpinning such emergent driven topological metallic states in real materials are also discussed.   

Versatile Floquet Weyl semimetal phases in an anisotropic Dirac system

Topological electronic states induced by periodic external fields such as laser light have attracted much attention [1]. Recently, it has been theoretically predicted that the time-reversal symmetry is broken by irradiating the three-dimensional Dirac semimetal with a circularly polarized laser, and the Floquet-Weyl semimetal state (FWSM) appears due to the chiral gauge field [2]. While it is well known that conventional Weyl semimetals can show anomalous Hall effect originating from the Berry curvature between the two Weyl points with opposite chirality, the topological natures and transport properties in the FWSM have not been detailed in a periodically driven system.

In this work, we study an effective model for paramagnetic Dirac semimetals hosting two spin-degenerated Dirac cones and investigate the consequence of the periodically driven system by the Floquet theory. We clarify that when the spin-orbit coupling is finite, the FWSM can be realized by splitting the Dirac points into two Weyl point pairs by circularly polarized irradiation. We also find several topological phase transitions with realignments and pair annihilations of the Weyl points by varying strength, frequency, and direction of the incident laser, which accompany the drastic changes of the anomalous Hall conductivity.

[1] T. Oka and H. Aoki, Phys. Rev. B 79, 081406(R) (2009). [2] S. Ebihara, K. Fukushima, and T. Oka, Phys. Rev. B 93, 155107 (2016).   

Topological Edge Modes in a Floquet Hyperbolic System

Periodically driven (Floquet) systems can exhibit new topological properties absent in equilibrium systems. One example [1] is the existence of anomalous topological edge modes while the bulk bands have zero Chern numbers. In this work, we study a Floquet insulating system consisting of sites on a finite two-dimensional hyperbolic lattice that tessellates a finite portion of a negatively curved space. We show that in our Floquet system, while the bulk states have zero Chern numbers, there exists a chiral topological edge band with no analog in equilibrium topological insulators. Our study extends previous investigations [2, 3, 4] of topology in hyperbolic lattices. Due to the unique property of negatively curved spaces, a finite ratio of a hyperbolic lattice belongs to its boundary, even in the thermodynamic limit. As a result, the edge modes of a topological system on a hyperbolic lattice comprise a finite ratio of the total density of states. Unlike previous works, the topological edge modes in our system are achieved by a multi-step driving protocol without any need for complex hoppings, i.e., magnetic fluxes threaded through the lattice. Finally, we study the protection of the edge modes against disorder.

[1] Rudner, M. S., et al., Physical Review X 3.3 (2013): 031005

[2] Yu, S., et al., Physical Review Letters 125.5 (2020): 053901

[3] Liu, Z. R., et al., Physical Review B 105.24 (2022): 245301

[4] Urwyler, D. M., et al., arXiv preprint arXiv:2203.07292 (2022)   

Strain Induced Curvature in a Photonic Floquet Topological Insulator

The research on topological states of matter of the past few decades has mainly focused on Euclidean or flat lattice geometries. Only very recently, the topological phenomena in negatively curved or Hyperbolic lattices have attracted significant attention and Hyperbolic lattice tilings were studied especially in electronic circuits and circuit QED. In this talk, we present a different approach to Hyperbolic geometries which can be applied in optical systems such as photonic waveguides or fiber loop setups. By straining a topological Floquet- or time-periodically driven- graphene lattice, we simulate a pseudomagnetic field as well as negative Gaussian curvature. This allows us to move smoothly from Euclidean to Hyperbolic space and tune the curvature continuously, which is impossible by studying different Hyperbolic tilings. In this talk we explore the interplay between this curvature and the topology of the Floquet system and discuss phenomena unobserved in Euclidean lattice geometries.   

Topological phase transitions at finite temperature

The ground states of noninteracting fermions in one-dimension with chiral symmetry form a class of topological band insulators, described by a topological invariant that can be related to the Zak phase. Recently, a generalization of this quantity to mixed states - known as the ensemble geometric phase (EGP) - has emerged as a robust way to describe topology at non-zero temperature. By using this quantity, we explore the nature of topology allowed for dissipation beyond a Lindblad description, to allow for coupling to external baths at finite temperatures. We introduce two main aspects to the theory of mixed state topology. First, we discover topological phase transitions as a function of the temperature T, manifesting as changes in winding number of the EGP accumulated over a closed loop in parameter space. We characterize the nature of these transitions and reveal that the corresponding non-equilibrium steady state at the transition can exhibit a nontrivial structure - contrary to previous studies where it was found to be in a fully mixed state. Second, we demonstrate that the EGP itself becomes quantized when key symmetries are present, allowing it to be viewed as a topological marker which can undergo equilibrium topological transitions at non-zero temperatures.   

Quantized Uhlmann phase and edge-state suppression of quantum transport in Lindblad dynamics

The Uhlmann phase is a generalization of the Berry phase from pure states to mixed states. Quantized Uhlmann phase has been found in many quantum systems at finite temperatures, revealing its potential to be a finite-temperature topological indicator. We show that the quantization of the Uhlmann phase may survive quantum jumps from system-reservoir couplings by using the Lindblad equation to simulate quantum dynamics of topological systems. The robustness of the Uhlmann phase offers hope for finite-temperature quantum computation utilizing the Uhlmann holonomy. By introducing a density or temperature difference through the reservoirs and monitoring the Lindblad quantum dynamics, we show that the Su-Schrieffer-Heeger model exhibits edge-state suppression of quantum transport. In contrast, the bandwidth of the Kitaev chain dominates its dynamics, leaving no observable topological signature in transport at the single-particle level.   

low-energy effects of two dimensional Dirac semimetals with interactions.

We discuss the effects of interactions and incommensuration on the low-energy properties of two dimensional Dirac semimetals. We focus on charge neutrality and in a model that possesses a magic-angle (i.e. vanishing Dirac velocity) and study the evolution of the interaction induced magnetic transition in the presence of a magic-angle. The model is investigated analytically using perturbation theory and a mean field description as well as numerically using Hartree-Fock and an auxiliary field quantum Monte Carlo approach.