Session K13: Topological States in AMO Systems

Topologically protected dynamical quantum phase transitions

A sudden quantum quench of a Bloch band from one topological phase toward another has been shown to exhibit an intimate connection with the notion of a dynamical quantum phase transition (DQPT), where the returning probability of the quenched state to the initial state—i.e. the Loschmidt echo—vanishes at critical times $\{t^*\}$. Analytical results to date are limited to two-band models, leaving the exact relation between topology and DQPT unclear. In this work, we show that for a general multi-band system, a robust DQPT relies on the existence of nodes (i.e. zeros) in the wavefunction overlap between the initial band and the post-quench energy eigenstates. These nodes are topologically protected if the two participating wavefunctions have distinctive topological indices. We demonstrate these ideas in detail for both one and two spatial dimensions using a three-band generalized Hofstadter model. We also discuss possible experimental observations.   

Floquet topological insulator in an optical lattice with modulated lattice depth

We propose a simple scheme to realize a Floquet topological insulator in an optical lattice by weakly modulating the lattice depth. When the modulation frequency resonantly couples the s and p bands, the Floquet Hamiltonian becomes topologically nontrivial. We map out the topological transition as a function of frequency and amplitude. We also confirm the bulk topology by finding edge states in a lattice with open boundary conditions. An advantage of our scheme is that the modulation amplitude can be relatively small, so the heating can be minimal.   

Self-similarity in Floquet topological insulators at low frequencies

We study theoretically the low-frequency regime of Floquet topological insulators. Specifically, we consider a periodically-driven one-dimensional Su-Schrieffer-Heeger (SSH) model, for which we calculate, analytically and numerically, the quasi-energy spectrum. We study the behavior of the quasi-energy gap as a function of drive frequency and other parameters and find self-similar spectral patterns. We also study the topological phase transitions, finding that they are present for arbitrarily small frequencies. We obtain the topological invariants as a function of the system's parameters, and compare with the explicit calculation of localized edge states for systems with open boundary conditions. Finally, we discuss the relevance of our results for the understanding of the long-time adiabatic limit in Floquet systems.   

Occupation probabilities and current densities of bulk and edge states of a Floquet topological insulator

Results are presented for the occupation probabilities and current densities of bulk and edge states of half-filled graphene in a cylindrical geometry, and irradiated by a circularly polarized laser. It is assumed that the system is closed, and that the laser has been switched on as a quench. Laser parameters corresponding to some representative topological phases are studied: one where the Chern number of the Floquet bands equals the number of chiral edge modes, a second where anomalous edge states appear in the Floquet Brillouin zone boundaries, and a third where the Chern number is zero, yet topological edge states appear at the center and boundaries of the Floquet Brillouin zone. Qualitative differences are found for the high frequency off-resonant and low frequency on-resonant laser with edge states arising due to resonant processes occupied with a high effective temperature on the one hand, while edge states arising due to off-resonant processes occupied with a low effective temperature on the other. Finally, we study the effects of inversion symmetry and particle-hole symmetry on the net current density and occupation probabilities in a half-filled system.   

Symmetry-protected edge states in periodically driven band insulators

The symmetry-protected edge states in topologically non-trivial band insulators are robust to any perturbations that are localized on the edges and preserve the relevant symmetries. In recent work we have found that the edge states in driven (Floquet) systems may be resistant to a much broader class of perturbations as compared to the time-independent case. We illustrate our finding by a numerical computation on the harmonically driven SSH model. A proposal for an experimental test using cold atoms is presented.   

Quantized magnetization density in periodically driven systems

We identify a new bulk quantized observable – the magnetization density -- that serves as a topological order parameter for periodically driven systems in which all bulk Floquet eigenstates are localized by disorder. While all Floquet states are localized when considered stroboscopically over a full period, the micromotion within the driving period may carry a nontrivial orbital magnetization. We find that the time-averaged magnetization density when the system is filled with fermions is quantized in units of the inverse driving period. We furthermore show that a quantized current flows around the boundary of any filled region of finite extent. The quantization has a topological origin: we relate the time-averaged magnetization density to the winding number characterizing the new phase identified in Phys. Rev. X 6, 021013 (2016). We thus establish that the winding number invariant can be accessed directly in bulk measurements, and propose an experimental protocol to do so using interferometry in a system of cold atoms in an optical lattice.   

Driven Phases of Quantum Matter

Clean and interacting periodically driven quantum systems are believed to exhibit a single, trivial ``infinite-temperature'' Floquet-ergodic phase.~By contrast, I will show that their disordered Floquet many-body localized counterparts can exhibit distinct ordered phases with spontaneously broken symmetries delineated by sharp transitions. Some of these are analogs of equilibrium states, while others are genuinely new to the Floquet setting. I will show that a subset of these novel phases are~\textit{absolutely stable~}to all weak local deformations of the underlying Floquet drives, and spontaneously break Hamiltonian dependent~\textit{emergent}~symmetries.~Strikingly, they simultaneously also break the underlying time-translation symmetry of the Floquet drive and the order parameter exhibits oscillations at multiples of the fundamental period. This ``time-crystallinity'' goes hand in hand with spatial symmetry breaking and, altogether, these phases exhibit a novel form of simultaneous long-range order in space and time. I will describe how this spatiotemporal order can be detected in experiments involving quenches from a broad class of initial states.   

Hofstadter's Butterfly in One-dimensional Driven Quantum Systems

A novel way to produce Hofstadter's butterfly is proposed in one-dimensional driven quantum systems. The system, which is modeled as a periodically kicked harmonic oscillator, creates various lattice structures in phase space. We develop the band theory of the square lattice and show that the Hofstadter's butterfly appears in the fractal quasienergy spectrum as a consequence of both the periodic lattice structure and the noncommutative geometry of phase space. We further introduce the concept pseudospin to distinguish degenerate states. Our proposal opens up the possibility to observe fractal structure in one-dimensional quantum systems and may bring a new way for quantum simulation.   

Quadratic band touching points and flat bands in two-dimensional topological Floquet systems

In this work we theoretically study, using Floquet-Bloch theory, the influence of circularly and linearly polarized light on two-dimensional band structures with Dirac and quadratic band touching points, and flat bands, taking the nearest neighbor hopping model on the kagome lattice as an example. We find circularly polarized light can invert the ordering of this three band model, while leaving the flat-band dispersionless. We find a small gap is also opened at the quadratic band touching point by 2-photon and higher order processes. By contrast, linearly polarized light splits the quadratic band touching point (into two Dirac points) by an amount that depends only on the amplitude and polarization direction of the light, independent of the frequency, and generally renders dispersion to the flat band. The splitting is perpendicular to the direction of the polarization of the light. We derive an effective low-energy theory that captures these key results. Finally, we compute the frequency dependence of the optical conductivity for this 3-band model and analyze the various interband contributions of the Floquet modes. Our results suggest strategies for optically controlling band structure and interaction strength in real systems.   

Topological Bloch oscillations

We propose a new way to characterize topological crystalline insulators without robust surface signatures. A topological insulator in an electric field is characterized by a recurrence time which is an integer multiple of the usual Bloch period. This same integer is a topological invariant protected by crystal symmetries, and divides n for n-fold rotationally-symmetric crystals. We explain the origin of topological Bloch oscillations from two dual perspectives: from symmetric parallel transport of Bloch states in the Brillouin zone, and from Wannier functions which are fixed to Wyckoff positions. By considering deformations of energy bands, we estimate how long topological Bloch oscillations survive.   

Protocols for probing topological edge modes and dimerization with atomic fermions in optical potentials

We propose protocols to probe localized edge states in a dimerized chain using cold Fermi gases. Standard trapping methods impose a confining harmonic potential preventing detection of edge states because addition of the trapping potential fuses zero-energy eigenstates into the bulk energy spectrum. An alternative trapping method with atoms confined in ring lattice, whose boundary conditions are transformed from periodic to open using an off resonant laser sheet, will induce topological modes under suitable conditions. Addition of a time-dependent artificial gauge field along the circumference of the ring results in mass transport mainly from bulk modes; measurement of the density demonstrates the remaining edge state. Signatures of dimerization in the presence of onsite interactions can be found in correlations as the system transforms from periodic to open boundary conditions. Persistence of finite correlations when the system undergoes a boundary transformation reveals a memory effect of the dimerized, initial structure.   

Measurement Protocol for the Topological Uhlmann Phase

Topological insulators and superconductors at finite temperature can be characterised by the topological Uhlmann phase. However, the direct experimental measurement in condensed matter systems has remained elusive. We explicitly demonstrate that the topological Uhlmann phase can be measured with the help of ancilla states in systems of entangled qubits that simulate a topological insulator. We propose a novel state-independent measurement protocol, which does not involve prior knowledge of the system state. With this construction, otherwise unobservable phases carrying topological information about the system become accessible. This enables the measurement of a complete phase diagram including environmental effects. We explicitly consider a realization of our scheme using a circuit of superconducting qubits. This measurement scheme is extendible to interacting particles and topological models with a large number of bands. arXiv: 1607.08778 (2016)