Session A10: Driven Topological Systems - I

Anomalous Localized Topological Phases

There is a well established principle and formalism for classifying ground state phases of gapped Hamiltonian systems––Hamiltonians which can be deformed into one another without closing the gap belong to the same topological phase. In the nonequilibrium context, topological phases of localized systems may be identified as sets of Hamiltonians which can be deformed into one another without going through a delocalization transition. For Floquet systems, it is known how to map the classification of such Hamiltonians to a classification of locality preserving unitaries––called quantum cellular automata (QCA). We show how to classify localized topological phases using QCA beyond the context of periodic driving, including static and (quasi)periodically driven systems, with or without symmetry. Further, we adapt many tools from the study of gapped ground states to the localized context, allowing for significant progress in the general classification. Some of the localized topological phases so discovered are characterized by eigenstate order––eigenstates of the model are ordered like ground states of nontrivial gapped Hamiltonians. However, there are also anomalous localized topological phases (ALT phases), for which each eigenstate is trivial when regarded individually, but the system as a whole still cannot be deformed into an atomic insulator.   

2D Topological Chiral Edge-States in a Synthetic Dimension of Atomic Trap States

Given the broad interest in topological physics, many powerful tools have been developed to induce such effects in a plethora of platforms, like cold atoms. One such technique is that of "synthetic dimensions", in which a set of states is externally coupled to engineer an effective spatial dimension. This approach is well-suited to investigating topological systems as the external coupling can naturally be designed to imprint a desired artificial magnetic field, and hence engineer quantum Hall-like models. In Birmingham, we have been developing a type of synthetic dimension that is based on coupling the harmonic trap states associated with cold atomic clouds. To first test this platform, a recent experiment demonstrated 1D Bloch oscillations along the synthetic dimension. We now theoretically propose how to realise a 2D quantum Hall system in this set-up by combining the synthetic dimension with a real spatial dimension and an artificial magnetic field. We demonstrate how to induce topological one-way chiral orbits with experimentally realistic parameters. We highlight, through numerical investigation, how to tune the length and add impurities to the synthetic dimension by using a digital micro-mirror device. Additionally, we utilise the length of the synthetic dimension to investigate bulk physics, specifically cyclotron orbits. This opens the way for future experiments on quantum Hall physics with atomic trap states.   

Light-driven high-order topological insulators

Light-matter interaction is a powerful tool to manipulate the properties of materials. In particular, the use of a periodic light beam, a technique generically known as Floquet engineering, has the appealing prospect of inducing novel phases of matter that are rare to find in equilibrium. In this work, we propose the use of an intense bicircular light (BCL) field to tune the bulk bands of a topological insulator. Bicircular light consists of a superposition of two circularly polarized light beams with opposite helicities and an integer frequency ratio. The resulting electric field traces a rose pattern in the polarization plane, which allows for a new pathway toward the ultrafast control of magnetic symmetries in the driven system. Using a realistic model for Bi₂Se₃, we theoretically show that a BCL-driven topological insulator undergoes a transition to a high-order topological insulator, which can be tuned by the light-field parameters.  We discuss the dependence of the magnetic symmetries of the driven system on the BCL parameters and its consequences on the bulk bands and hinge modes.   

Nonadiabatic dynamics of a parametrically modulated spin chain in a topologically nontrivial regime

We study the dynamics of a resonantly modulated spin chain in a strong magnetic field. The modulation of the spin-spin coupling close to twice the Larmor frequency leads to parametric resonance. The spin dynamics in the rotating frame maps on the Kitaev chain. By varying the modulation frequency, the chain can be bought into a topologically nontrivial regime.  We show that in this regime, even for a closed chain, the response to slow turning on the drive becomes nonadiabatic, leading to excitations of pairs of Jordan-Wigner fermions. The system displays a history-dependent behavior. It depends on the order in which the drive parameters are changed so that they cross the boundary of the topologically nontrivial regime or start from inside this regime, given that initially the spin system is in its ground state. We also analyze the dissipative dynamics of two coupled modulated spins and the stationary distribution over the Floquet states. For decay processes associated with the energy transfer to the thermal reservoir close to the Larmor frequency (in energy units), the states can be equally populated even where the temperature of the thermal reservoir is zero.   

Photo-induced Multiply Quantized Vortex States in Dirac-like Materials

Light polarization and intensity are regularly used to tune the interaction with matter to induce band hybridizations and topological phase transitions. Further control is achieved by manipulating the spatial dependence of the field’s phase, as demonstrated by vortex light beams carrying orbital angular momentum [1]. This work explores the consequences of light-matter interaction for a two-dimensional massive Dirac-like system subjected to a monochromatic vortex light beam. Utilizing Floquet’s formalism, we introduce the exact definition of the total angular momentum in Floquet space and identify the polarizations for its conservation. Using the one-photon approximation, we map the resulting Hamiltonian to the model of an s-wave superfluid hosting multiple quantized vortex core states [2]. The procedure allows us to analytically determine the number of vortex states that appear in our system at low energies. We extend these results by applying the Bessel decomposition method to numerically diagonalize the full Floquet Hamiltonian in the resonant regime. We present a complete description of the photon-dressed electronic vortex states that emerge from the irradiated system in terms of the angular momentum-dependent dispersion relation, vorticity, and real-space extension.

[1] Y. Shen, et al., Light: Science & Applications 8, 1–29 (2019)

[2] A. Prem, S. Moroz, V. Gurarie, and L. Radzihovsky, PRL 119, 067003 (2017)   

Characterizing Exceptional Topology Through Tropical Geometry

Non-Hermitian (NH) Hamiltonians describing open quantum systems have been widely explored in platforms ranging from photonics to electric circuits [1]. A defining feature of NH systems is exceptional points (EPs), where both eigenvalues and eigenvectors coalesce [3]. The study of EPs has become an exciting frontier at the crossroads of optics, photonics, acoustics, and quantum physics [4]. Tropical geometry is an emerging field of mathematics at the interface between algebraic geometry and polyhedral geometry, with diverse applications to science [5]. Here, we introduce Newton's polygon method and adopt the notion of a geometrical object known as amoeba in developing a unified tropical geometric framework to characterize different facets of NH systems [6]. We illustrate the versatility of our approach using several examples and demonstrate that it can be used to select from a spectrum of higher-order EPs in gain and loss models, predict the skin effect in the NH Su-Schrieffer-Heeger model, and extract universal properties in the presence of disorder in the Hatano-Nelson model. Our work puts forth a new framework for studying NH physics and unveils a novel connection of tropical geometry to this field.

[1] Bergholtz et al., Rev. Mod. Phys. 93, 015005 (2021).

[2] T. Kato, Vol. 132 (Springer Science & Business Media, 2013).

[3] Miri et al.,  Science 363, eaar7709 (2019).

[4] Maclagan et al., Graduate Stud. Math.161, 75–91(2009).

[5] Banerjee et al., Proc. Natl. Acad. Sci. U.S.A. 120, e2302572120 (2023).   

Schwinger effect in a symmetry protected topological phase

The Schwinger effect, classically understood as the particle-antiparticle pair creation in electric fields, has a many-body analog in Mott insulators where doublon-hole pairs lead to dielectric breakdown [1,2]. In this talk, we extend this concept to systems in the symmetry protected topological (SPT) phase. Our focus centers on the S=1 1D Heisenberg model within the Haldane phase under spin-electric fields (or gradient magnetic fields). We'll discuss how these fields give rise to the formation of up and down triplon pairs, leading to an intriguing non-linear generation of spin current. Beyond the creation mechanism, we'll also explore the ramifications for SPT order, including insights into the entanglement spectrum.

[1] T. Oka, R. Arita and H. Aoki, Phys. Rev. Lett. 91, 066406 (2003)

[2] T. Oka, Phys. Rev. B 86, 075148 (2012)   

Nonperturbative nonlinear transport in a topological semimetal

Ultrafast light-matter interaction can engineer topological responses in quantum materials via the creation of photon-dressed Floquet-Bloch states. Here, we report on the time-resolved transport properties of T_d-MoTe_2, a type-II Weyl semimetal, driven by a femtosecond pulse of mid-infrared laser light. Measurements were performed using an ultrafast optoelectronic device architecture based on laser-triggered photoconductive switches and waveguides. We observed a helicity-dependent injection current and an anomalous Hall effect, both of which scaled linearly with the applied laser field, in violation of the perturbative description of nonlinear optics and transport. Analysis of the results indicates that such nonperturbative nonlinear transport is a direct probe of the emergence and hybridization of Floquet-Bloch states. This work may provide deeper understanding toward programming technologically relevant responses in quantum materials with light.   

Driving a topological insulator using a Gaussian pulse

A perfectly periodic time-dependent perturbation, such as radiation, when applied to a system at equilibrium can be understood

using Floquet theory and may result in generating topologically non-trivial phases in a system which was trivial to begin with.

However, in a real experiment that relies on pump-probe techniques to study materials, the external perturbation is never perfectly

periodic and has a pulse shape/envelope function. Using actual time-evolved states, we calculate the optical conductivity in presence

of such a drive and compare it to the response calculated using Floquet theory. The conductivity is found to bear a memory of the initial

equilibrium state. This hold even with a slow turning on of the pump and the measurement taken well after the ramp. The response of

the time-evolved system is interpreted as a result of the population of Floquet bands being determined by their overlap with the initial

equilibrium state. In particular, at band inversion points in the Brillouin zone the population of the Floquet bands is inverted as well.   

Prethermal higher-order topological time crystals protected by emergent symmetry

In this talk, we introduce a prethermal higher-order topological time crystal in a two-dimensional many-body interacting spin system. We find that by adding high frequency driving, the static Hamiltonian, which is fully stabilized and topologically trivial, will arise higher-order corner modes protected by an emergent global $mathbb{Z}_2 imes mathbb{Z}_2$ symmetry in the prethermal region. The sub-harmonic oscillations of the magnetization at corners, in contrast to those rapidly vanish in the bulk and edges, characterizing the breaking of discrete time-translation symmetry at higher-order topological corner modes. With sufficiently high driving frequency, these modes exhibit robustness against small local perturbations, even for those that anti-commute with the global symmetry. Before thermalization, a transient beating behaviour of the magnetization in the bulk and edge spins is observed and explained. In addition, we show that the lifetime of such a phase scales exponentially with the driving frequency. We further propose an experiment to realize this phase with superconducting qubits, where five-body interactions inherent to our model can be implemented through digital simulation in an analytical fashion.    

Floquet topological superconductivity induced by chiral many-body interaction

Floquet theory allows the design of quantum phases of matter with exotic properties via non-trivial modulation of the effective static Hamiltonian by time-periodic driving. 

In particular, by controlling the band topology, a variety of topological phases can be realized dynamically, including Chern insulators. 

While extending this idea to topological superconductivity is an important application, the same mechanism cannot be straightforwardly applied to typical superconductors.

The gap function is not directly coupled to the electromagnetic field, making it difficult to modulate the pairing symmetry and induce topological phase transitions.

In this study, we find that this difficulty can be overcome by taking into account correlation effects. 

Namely, we show that the d-wave superconductivity of the doped Hubbard model can be dynamically transformed to topological d+id superconductivity by irradiation of circularly polarized light.

To this end, we derive the Floquet t-J model by combining Floquet theory and the Schrieffer-Wolff transformation. 

The resulting Hamiltonian has emergent chiral many-body interactions with broken time-reversal symmetry, with which we can actually modulate the pairing symmetry. We demonstrate the topological phase transition within the framework of Gutzwiller approximation.   

Revealing the hidden Dirac gap in a topological antiferromagnet using Floquet-Bloch manipulation

Manipulating solids using the time-periodic drive of a laser pulse is a promising route to generate new phases of matter. Whether such `Floquet-Bloch' manipulation can be achieved in topological magnetic systems with disorder has so far been unclear. In this work, we realize Floquet-Bloch manipulation of the Dirac surface-state mass of the topological antiferromagnet (AFM) MnBi₂Te₄. Using time- and angle-resolved photoemission spectroscopy (tr-ARPES), we show that opposite helicities of mid-infrared circularly polarized light result in substantially different Dirac mass gaps in the AFM phase, despite the equilibrium Dirac cone being massless. We explain our findings in terms of a Dirac fermion with a random mass. Our results underscore Floquet-Bloch manipulation as a powerful tool for controlling topology even in the presence of disorder, and for uncovering properties of materials that may elude conventional probes.   

Achieving quantized transport in Floquet topological insulators via energy filters

Due to photon-assisted transport (PAT) processes, chiral edge modes induced by periodic driving do not directly mediate quantized transport. In the case of particle transport, PAT creates additional transport pathways that break the conductance quantization. In thermal transport, those processes break the energy conservation and lead to heating. Here we show how narrow bandwidth "energy filters'' can restore quantization of both particle and thermal conductance by suppressing PAT through Floquet sidebands. We derive a Floquet Landauer type equation to describe transport through such an energy-filtered setup, and show how the filter can be integrated out to yield a sharply energy-dependent renormalized system-lead coupling. We show analytically and through numerical simulations that a nearly quantized electrical and thermal conductance can be achieved in both off-resonantly and resonantly induced quasienergy gaps when filters are introduced. We introduce a "Floquet distribution function'' and show both analytically and numerically that it approaches the equilibrium Fermi-Dirac form when narrow-band filters are introduced, highlighting the mechanism that restores quantized transport.