Quasi-periodically driven quantum systems are predicted to exhibit quantized topological properties, in analogy with the quantized transport properties of topological insulators. We use a single nitrogen-vacancy center in diamond to experimentally study a synthetic quantum Hall effect with a two-tone drive. We measure the evolution of trajectories of two quantum states, initially prepared at nearby points in synthetic phase space. We detect the synthetic Hall effect through the predicted overlap oscillations at a quantized fundamental frequency proportional to the Chern number, which characterizes the topological phases of the system. We further observe half-quantization of the Chern number at the transition between the synthetic Hall regime and the trivial regime, and the associated concentration of local Berry curvature in synthetic phase space. Our work opens up the possibility of using driven qubits to design and study higher-dimensional topological insulators and semi-metals in synthetic dimensions.
Nonequilibrium quantum states can be controlled via the driving field in periodically driven systems. Such control, which is called Floquet engineering, has opened various phenomena, such as the light-induced anomalous Hall effect. There are expected to be some essential differences between the anomalous Hall and spin Hall effects of periodically driven systems because of the difference in time-reversal symmetry. However, these differences remain unclear due to the lack of Floquet engineering of the spin Hall effect. Here we show that when the helicity of circularly polarized light is changed …
The hydrodynamical phase-slippage geometry and a semi-classical Josephson-RCSJ formulism are combined to make an analogy between the periodically driven Josephson ac effect and the quantum Hall effect. Both of these two macroscopic quantum effects are postulated to be originated from the one-dimensional linear vortex-electron relative motion. The mechanisms of the quantized Hall resistance of IQHE and FQHE are unified under the phase-locking dynamics of a Josephson-like oscillation which is periodically driven by the electron wave of the coherent Hall current, with the time-averaged frequency …
We theoretically report the emergence of $Z_4$ parafermion edge modes in a periodically driven spinful superconducting chain with modest fermionic Hubbard interaction. These parafermion edge modes represent $\pm π/(2T)$ quasienergy excitations ($T$ being the driving period), which have no static counterpart and arise from the interplay between interaction effect and periodic driving. At special parameter values, these exotic quasiparticles can be analytically and exactly derived. Strong numerical evidence of their robustness against variations in parameter values and spatial disorder is furthe…
Topological energy pumping in a quasi-periodically driven four-level system
SISSA（的里雅斯特国际高等研究学校）；Harish-Chandra Research Institute, a CI of Homi Bhabha National Institute（哈里什-昌德拉研究所，霍米·巴哈国家研究所下属中心）；Department of Physics, BITS Pilani-Hyderabad Campus（比拉理工学院海得拉巴校区物理系）
The use of ultra-low temperature cooling and of high hydrostatic pressure techniques has significantly expanded our understanding of the two-dimensional electron gas confined to GaAs/AlGaAs structures. This chapter reviews a selected set of experiments employing these specialized techniques in the study of the fractional quantum Hall states and of the charged ordered phases, such as the reentrant integer quantum Hall states and the quantum Hall nematic. Topics discussed include a successful cooling technique used, novel odd denominator fractional quantum Hall states, new transport results on e…
We study intrinsic DC spin and valley Hall conductivity in doped ferromagnetic silicene in the presence of an electric filed applied perpendicular to silicene sheet. By calculating its energy spectrum and wavefunction and by making use of Kubo formalism, we obtain a general relation for the transverse Hall conductivity which can be used to obtain spin- and valley-Hall conductivity. Our results, in the zero limit of the exchange field, reduces to the previous results. Furthermore we discuss electrically tunable spin and valley polarized transport in ferromagnetic silicene and obtain the necessa…
Recent theoretical work on time-periodically kicked Hofstadter model found robust counter-propagating edge modes. It remains unclear how ubiquitously such anomalous modes can appear, and what dictates their robustness against disorder. Here we shed further light on the nature of these modes by analyzing a simple type of periodic driving where the hopping along one spatial direction is modulated sinusoidally with time while the hopping along the other spatial direction is kept constant. We obtain the phase diagram for the quasienergy spectrum at flux 1/3 as the driving frequency $ω$ and the hop…
Chirally stacked $N$-layer graphene is a semimetal with $\pm p^N$ band-touching at two nonequivalent corners in its Brillioun zone. We predict that an off-resonant circularly polarized light (CPL) drives chirally stacked $N$-layer graphene into a Floquet Chern Insulators (FCIs), a.k.a. quantum anomalous Hall insulators, with tunable high Chern number $C_F=\pm N$ and large gaps. A topological phase transition between such a FCI and a valley Hall (VH) insulator with high valley Chern number $C_v=\pm N$ induced by a voltage gate can be engineered by the parameters of the CPL and voltage gate. We …
In this talk I present a summary of recent work on tunnel junctions of a fractional quantum Hall fluid and an electron reservoir, a Fermi liquid. I consider first the case of a single point contact. This is a an exactly solvable problem from which much can be learned. I also discuss in some detail how these solvable junction problems can be used to understand many aspects of the recent electron tunneling experiments into edge states. I also give a detailed picture of the unusual behavior of these junctions in their strong coupling regime. A pedagogical introduction to the theories of edge stat…
Three dimensional Weyl semimetals exhibit open Fermi arcs on their sample surfaces connecting the projection of bulk Weyl points of opposite chirality. The canonical interpretation of these surfaces states is in terms of chiral edge modes of a layer quantum Hall effect: The two-dimensional momentum-space planes perpendicular to the momentum connecting the two Weyl points are characterized by a non-zero Chern number. It might be interesting to note, that in analogy to the known two-dimensional Floquet anomalous chiral edge states, one can realize open Fermi arcs in the absence of Chern numbers …
The discovery of topological states of matter has profoundly augmented our understanding of phase transitions in physical systems. Instead of local order parameters, topological phases are described by global topological invariants and are therefore robust against perturbations. A prominent example thereof is the two-dimensional integer quantum Hall effect. It is characterized by the first Chern number which manifests in the quantized Hall response induced by an external electric field. Generalizing the quantum Hall effect to four-dimensional systems leads to the appearance of a novel non-line…
The quantum Hall effect occuring in two-dimensional electron gases was first explained by Laughlin, who envisioned a thought experiment that laid the groundwork for our understanding of topological quantum matter. His proposal is based on a quantum Hall cylinder periodically driven by an axial magnetic field, resulting in the quantized motion of electrons. We realize this milestone experiment with an ultracold gas of dysprosium atoms, the cyclic dimension being encoded in the electronic spin and the axial field controlled by the phases of laser-induced spin-orbit couplings. Our experiment prov…
$Z_{2}$ and Chern topological phases such as newly discovered quantum spin Hall and original quantum Hall states hardly both co--exist in a single material due to their contradictory requirement on the time--reversal symmetry (TRS). We show that although the TRS is broken in systems with a periodically driving ac-field, an effective TRS can still be defined provided the ac--field is linearly polarized or certain other conditions are satisfied. The controllable TRS provides us with a route to manipulate $Z_{2} $ and Chern topological phases in a single material by tuning the polarization of the…
We construct a periodically time-dependent Hamiltonian with a phase transition in the quantum Hall universality class. One spatial dimension can be eliminated by introducing a second incommensurate driving frequency, so that we can study the quantum Hall effect in a one-dimensional (1D) system. This reduction to 1D is very efficient computationally and would make it possible to perform experiments on the 2D quantum Hall effect using cold atoms in a 1D optical lattice.
Exploring a classical solution of the non-linear sigma model for a quantum Hall ferromagnet, a skyrmion-magnon effective hamiltonian is obtained via the collective coordinates method. Using the Feynman-Vernon functional integral formalism for this model we find the temperature dependent transport coefficients which characterize a single skyrmion dynamics.
Using the hierarchy picture of the fractional quantum Hall effect, we study the the ground state periodicity of a finite size quantum Hall droplet in a quantum Hall fluid of a different filling factor. The droplet edge charge is periodically modulated with flux through the droplet and will lead to a periodic variation in the conductance of a nearby point contact, such as occurs in some quantum Hall interferometers. Our model is consistent with experiment and predicts that superperiods can be observed in geometries where no interfering trajectories occur. The model may also provide an experimen…
Weyl semimetals and nodal line semimetals are characterized by linear band touching at zero-dimensional points and one-dimensional lines, respectively. We predict that a circularly polarized light drives nodal line semimetals into Weyl semimetals. The Floquet Weyl points thus obtained are tunable by the incident light, which enables investigations of them in a highly controllable manner. The transition from nodal line semimetals to Weyl semimetals is accompanied by the emergence of a large and tunable anomalous Hall conductivity. Our predictions are experimentally testable by transport measure…
It has been demonstrated numerically, mainly by considering ground state properties, that fractional quantum Hall physics can appear in lattice systems, but it is very difficult to study the anyons directly. Here, I propose to solve this problem by using conformal field theory to build semi-analytical fractional quantum Hall lattice models having anyons in their ground states, and I carry out the construction explicitly for the family of bosonic and fermionic Laughlin states. This enables me to show directly that the braiding properties of the anyons are those expected from analytical continua…
Akin to zero-mode anomalies, such as the chiral anomaly of edge states in quantum Hall effect, in this work, a pi-mode anomaly is proposed in a 1+1 dimensional periodically-driven topological/normal insulator (TI/NI) heterostructure. Usually, when coupling in a background gauge field, the zero modes on domain walls would provide an anomalous current term that is eventually canceled by additional boundary contributions from the topological bulk, via the Callan-Harvey mechanism. This anomaly cancellation associated with the generalization of bulk-boundary correspondence is called anomaly inflow.…
A periodically driven quantum Hall system in a fixed magnetic field is found to exhibit a series of phases featuring anomalous edge modes with the "wrong" chirality. This leads to pairs of counter-propagating chiral edge modes at each edge, in sharp contrast to stationary quantum Hall systems. We show that the pair of Floquet edge modes are protected by the chiral (sublattice) symmetry, and that they are robust against static disorder. The existence of distinctive phases with the same Chern and winding numbers but very different edge state spectra points to the important role played by symmetr…
Quantum dots in the fractional quantum Hall regime are studied using a Hartree formulation of composite fermion theory. Under appropriate conditions the chemical potential of the dots will oscillate periodically with B due to the transfer of composite fermions between quasi-Landau bands. This effect is analogous to the addition spectrum oscillations which occur in quantum dots in the integer quantum Hall regime. Period phi_0 oscillations are found in sharply confined dots with filling factors nu=2/5 and nu=2/3. Period 3*phi_0 oscillations are found in a parabolically confined nu=2/5 dot. More …
In condensed matter physics, one of the major topics is to find out and classify interesting novel topological matters and phenomena. Topologically nontrivial systems can also be achieved by using periodical driving fields. On the other hand, ratchet effect can be used to collect useful work from fluctuations or realize directed transport in periodically-driven systems. Here, we promote a dimer chain by applying two mutually-perpendicular ac electric fields, and obtain an effective two-dimensional Hamiltonian in the low-frequency regime. We thereby derive quantum Floquet anomalous Hall (QFAH) …
National Laboratory of Solid State Microstructures, School of Physics, Nanjing University（南京大学固体微结构物理学院）；College of Physics and Mechatronic Engineering, Guizhou Minzu University（贵州民族大学物理与机电工程学院）；Department of Physics, Sichuan Normal University（四川师范大学物理系）；College of Physics, Sichuan University（四川大学物理学院）；Institute for Functional Intelligent Materials, National University of Singapore（新加坡国立大学功能智能材料研究所）；Department of Physics and Astronomy, University of Manchester（曼彻斯特大学物理与天文学系）；National Graphene Institute, University of Manchester（曼彻斯特大学国家石墨烯研究所）；Research Center for Functional Materials, National Institute for Material Science（日本物质材料研究机构功能材料研究中心）；International Center for Materials Nanoarchitectonics, National Institute for Material Science（日本物质材料研究机构国际纳米材料组装中心）；Collaborative Innovation Center of Advanced Microstructures, Nanjing University（南京大学先进微结构协同创新中心）；Jiangsu Physical Science Research Center（江苏省物理科学研究中心）
Since the experimental realisation of the integer quantised Hall effect in a two dimensional electron system subject to strong perpendicular magnetic fields in 1980, a central question has been the interrelation between the conductance quantisation and the topological properties of the system. It is conjectured that if the electron system is described by a Bloch hamiltonian, then the system is insulating in the bulk of the sample throughout the quantised Hall plateau due to magnetic field induced energy gap. Meanwhile, the system is conducting at the edges resembling a 2+1 dimensional topologi…
We consider a two-dimensional electron gas (2DEG) in the Quantum Hall regime in the presence of a Zeeman field, with the Fermi level tuned to filling factor $ν=1$. We show that, in the presence of spin-orbit coupling, contacting the 2DEG to a narrow strip of an s-wave superconductor produces a topological superconducting gap along the contact as a result of crossed Andreev reflection (CAR) processes across the strip. The sign of the topological gap, controlled by the CAR amplitude, depends periodically on the Fermi wavelength and strip width and can be externally tuned. An interface between tw…
Chiral Majorana edge modes are theoretically proposed to perform braiding operations for the potential quantum computation. Here, we suggest a scheme to regulate trajectories of the chiral Majorana fermion based on a quantum anomalous Hall insulator (QAHI)-topological superconductor heterostructure. An applied external gate voltage to the QAHI region introduces a dynamical phase so that the outgoing Majorana fermions can be prominently tuned to different leads. The trajectory is mechanically analyzed and the electrical manipulation is represented by the oscillating transmission coefficients ve…
For cold atomic systems, varying the optical lattice potential periodically provides a general and simple way to drive the system into phases with nontrivial topology. Besides its simplicity, this driving approach, compared to the usual driving approach by exerting an external electromagnetic field to the static system, has the merit that it does not break the original static system's time-reversal symmetry at any given time. Based on this approach, we find that a trivial insulator with time-reversal symmetry can be driven into a Floquet quantum spin Hall insulator. This novel state of matter …
We study anisotropic stripe models of interacting electrons in the presence of magnetic fields in the quantum Hall regime with integer and fractional filling factors. The model consists of an infinite strip of finite width that contains periodically arranged stripes (forming supercells) to which the electrons are confined and between which they can hop with associated magnetic phases. The interacting electron system within the one-dimensional stripes are described by Luttinger liquids and shown to give rise to charge and spin density waves that lead to periodic structures within the stripe wit…
We report on an effective gauge theory of double-layer quantum Hall systems, that is constructed via bosonization from the response of incompressible states without referring to composite bosons and fermions. It is pointed out that dipole-active excitations, both elementary and collective, govern the long-wavelength features of quantum Hall systems, and the single-mode approximation is used to study them. The effective theory consists of three vector fields representing one interlayer collective mode and two cyclotron modes, and properly incorporates the spectrum of collective excitations on t…
Moiré systems have emerged in recent years as a rich platform to study strong correlations. Here, we will discuss a simple, experimentally feasible setup based on periodically strained graphene that reproduces several key aspects of twisted moiré heterostructures -- but without introducing a twist. We consider a monolayer graphene sheet subject to a $C_2$-breaking periodic strain-induced psuedomagnetic field (PMF) with period $L_M \gg a$, along with a scalar potential of the same period. This system has {\it almost ideal} flat bands with valley-resolved Chern number $\pm 1$, where the deviatio…
We propose that a tunable generalized three-dimensional Hofstadter Hamiltonian can be realized by engineering the Raman-assisted hopping of ultracold atoms in a cubic optical lattice. The Hamiltonian describes a periodic lattice system under artificial magnetic fluxes in three dimensions. For certain hopping configurations, the bulk bands can have Weyl points and nodal loops, respectively, allowing the study of both the two nodal semimetal states within this system. Furthermore, we illustrate that with proper rational fluxes and hopping parameters, the system can exhibit the three-dimensional …
We numerically study the three-dimensional (3D) quantum Hall effect (QHE) and magnetothermoelectric transport of Weyl semimetals in the presence of disorder. We obtain a bulk picture that the exotic 3D QHE emerges in a finite range of Fermi energy near the Weyl points determined by the gap between the $n=-1$ and $n=1$ Landau levels (LLs). The quantized Hall conductivity is attributable to the chiral zeroth LLs traversing the gap, and is robust against disorder scattering for an intermediate number of layers in the direction of the magnetic field. Moreover, we predict several interesting charac…
Quantized Hall conductance and de Haas van Alphen (dHvA) oscillation are studied theoretically in the tight-binding model for (TMTSF)$_2$NO$_3$, in which there are small pockets of electron and hole due to the periodic potentials of anion ordering in the $a$-direction. The magnetic field is treated by hoppings as complex numbers due to the phase caused by the vector potential, i.e. Peierls substitution. In realistic values of parameters and the magnetic field, the energy as a function of a magnetic field (Hofstadter butterfly diagram) is obtained. It is shown that energy levels are broadened a…
We investigate the ground state properties of a bosonic Harper-Hofstadter model with local interactions on a finite cylindrical lattice with filling fraction $ν=1/2$. We find that our system supports topologically ordered states by calculating the topological entanglement entropy, and its value is in good agreement with the theoretical value for the $ν=1/2$ Laughlin state. By exploring the behaviour of the density profiles, edge currents and single-particle correlation functions, we find that the ground state on the cylinder shows all signatures of a fractional quantum Hall state even for larg…
We propose a theoretical scenario for pumping of fractionally charged quasi-particle in the context of $ν=1/3$ fractional quantum Hall liquid. We consider quasi-particle pumping across an anti-dot level tuned close to the resonance. Fractional charge pumping is achieved by slow and periodic modulation of coupling of the anti-dot level to left and right moving edges of a Hall bar set-up. This is attained by periodically modulating the gate voltages controlling the couplings. In order to obtain quantization of pumped charge in the unit of the electronic charge fraction ($νe$) per pumping cycle i…
The recent experimental observations of the quantum Hall effect in 3D topological semimetals have attracted great attention, but there are still debates on its origin. We systematically study the dependence of the quantum Hall effect in topological semimetals on the thickness, Fermi energy, and growth direction, taking into account the contributions from the Fermi-arc surface states, confinement-induced bulk subbands, and helical side-surface edge states. In particular, we focus on the intensively studied Dirac semimetal Cd$_{3}$As$_{2}$ and its slabs grown along experimentally accessible dire…
The close theoretical analogy between the physics of rapidly rotating atomic Bose condensates and the quantum Hall effect (i.e., a two dimensional electron gas in a strong magnetic field) was first pointed out ten years ago. As a consequence of this analogy, a large number of strongly correlated quantum Hall-type states have been predicted to occur in rotating Bose systems, and suggestions have been made how to manipulate and observe their fractional quasiparticle excitations. Due to a very rapid development in experimental techniques over the past years, experiments on BEC now appear to be cl…
The electromagnetic characteristics of bilayer quantum Hall systems in the presence of interlayer coherence and tunneling are studied by means of a pseudospin-texture effective theory and an algebraic framework of the single-mode approximation, with emphasis on clarifying the nature of the low-lying neutral collective mode responsible for interlayer tunneling phenomena. A long-wavelength effective theory, consisting of the collective mode as well as the cyclotron modes, is constructed. It is seen explicitly from the electromagnetic response that gauge invariance is kept exact, this implying, i…
We propose a method by which the quantization of the Hall conductance can be directly measured in the transport of a one-dimensional atomic gas. Our approach builds on two main ingredients: (1) a constriction optical potential, which generates a mesoscopic channel connected to two reservoirs, and (2) a time-periodic modulation of the channel, specifically designed to generate motion along an additional synthetic dimension. This fictitious dimension is spanned by the harmonic-oscillator modes associated with the tightly-confined channel, and hence, the corresponding "lattice sites" are intimate…
Periodically driven quantum systems provide a novel and versatile platform for realizing topological phenomena. Among these are analogs of topological insulators and superconductors, attainable in static systems; however, some of these phenomena are unique to the periodically driven case. Here, we show that disordered, periodically driven systems admit an "anomalous" two dimensional phase, whose quasi-energy spectrum consists of chiral edge modes that coexist with a fully localized bulk - an impossibility for static Hamiltonians. This unique situation serves as the basis for a new topologicall…
The quantum kicked rotator is a periodically driven dynamical system with a metal-insulator transition. We extend the model so that it includes phase transitions between a metal and a topological insulator, in the universality class of the quantum spin Hall effect. We calculate the Z2 topological invariant using a scattering formulation that remains valid in the presence of disorder. The scaling laws at the phase transition can be studied efficiently by replacing one of the two spatial dimensions with a second incommensurate driving frequency. We find that the critical exponent does not depend…
We present in this paper experimental results on the transport hysteresis in electron double quantum well structures. Exploring the measurement technique of fixing the magnetic field and sweeping a front gate voltage (Vg), we are able to study the hysteresis by varying the top layer Landau level fillings while maintaining a relatively constant filling factor in the bottom layer, allowing us to tackle the question of the sign of Rxx(up)-Rxx(down), where Rxx(up) is the magnetoresistance when Vg is swept up and Rxx(down) when Vg swept down. Furthermore, we observe that hysteresis is generally str…
We propose a device consisting in an antidot periodically driven in time by a magnetic field as a fractional quantum Hall counterpart of the celebrated mesoscopic capacitor-based single electron source. We fully characterize the setup as an ideal emitter of individual quasiparticles and electrons into fractional quantum Hall edge channels of the Laughlin sequence. Our treatment relies on a master equation approach and identifies the optimal regime of operation for both types of sources. The quasiparticle/quasihole emission regime involves in practice only two charge states of the antidot, allo…
