Quantized Hall Response Research Articles

Connecting the Many-Body Chern Number to Luttinger's Theorem through Středa's Formula.

Relating the quantized Hall response of correlated insulators to many-body topological invariants is a key challenge in topological quantum matter. Here, we use Středa's formula to derive an expression for the many-body Chern number in terms of the single-particle interacting Green's function and its derivative with respect to a magnetic field. In this approach, we find that this many-body topological invariant can be decomposed in terms of two contributions, N_{3}[G]+ΔN_{3}[G], where N_{3}[G] is known as the Ishikawa-Matsuyama invariant and where the second term involves derivatives of Green's function and the self-energy with respect to the magnetic perturbation. As a by-product, the invariant N_{3}[G] is shown to stem from the derivative of Luttinger's theorem with respect to the probe magnetic field. These results reveal under which conditions the quantized Hall conductivity of correlated topological insulators is solely dictated by the invariant N_{3}[G], providing new insight on the origin of fractionalization in strongly correlated topological phases.

Artificial magnetic field for synthetic quantum matter without dynamical modulation

We propose an all-static method to realize an artificial magnetic field for charge neutral particles without introducing any time modulation. Our proposal consists of one-dimensional tubes subject to harmonic trapping potentials with shifted centers. We show that this setup realizes an artificial magnetic field in a hybrid real-momentum space. We discuss how characteristic features of particles in a magnetic field, such as chiral edge states and the quantized Hall response, can be observed in this setup. We find that the mean-field ground state of bosons in this setup in the presence of long-range interactions in physical real space can have quantized vortices in the hybrid real-momentum space; such a state with vortices exhibits a supersolid structure in the physical real space. Our method can be applied to a variety of synthetic quantum matter, including ultracold atomic gases, coupled photonic cavities, coupled waveguides, and exciton-polariton lattices.

Ambipolar magneto-optical response of ultralow carrier density topological insulators

We have investigated the THz range magneto-optical response of ultralow carrier density films of Sb$_2$Te$_3$ using time-domain THz polarimetry. Undoped Sb$_2$Te$_3$ has a chemical potential that lies inside the bulk valence band. Thus its topological response is masked by bulk carriers. However, with appropriate buffer layer engineering and chemical doping, Sb$_2$Te$_3$ thin films can be grown with extremely low electron or hole densities. The ultralow carrier density samples show unusual optical properties and quantized response in the presence of magnetic fields. Consistent with the expectations for Dirac fermions, a quantized Hall response is seen even in samples where the zero field conductivity falls below detectable levels. The discontinuity in the Faraday angle with small changes in the filling fraction across zero is manifestation of the parity anomaly in 2D Dirac systems with broken time reversal symmetry.

Noncommutative field theory and composite Fermi liquids in some quantum Hall systems

Composite Fermi liquid metals arise at certain special filling fractions in the quantum Hall regime and play an important role as parent states of gapped states with quantized Hall response. They have been successfully described by the Halperin-Lee-Read (HLR) theory of a Fermi surface of composite fermions coupled to a $U(1)$ gauge field with a Chern-Simons term. However, the validity of the HLR description when the microscopic system is restricted to a single Landau has not been clear. Here for the specific case of bosons at filling $\nu = 1$, we build on earlier work from the 1990s to formulate a low energy description that takes the form of a {\em non-commutative} field theory. This theory has a Fermi surface of composite fermions coupled to a $U(1)$ gauge field with no Chern-Simons term but with the feature that all fields are defined in a non-commutative spacetime. An approximate mapping of the long wavelength, small amplitude gauge fluctuations yields a commutative effective field theory which, remarkably, takes the HLR form but with microscopic parameters correctly determined by the interaction strength. Extensions to some other composite fermi liquids, and to other related states of matter are discussed.

Chiral twist on the high- Tc phase diagram in moiré heterostructures

We show that the large orbital degeneracy inherent in Moir\'e heterostructures naturally gives rise to a `high-$T_c$' like phase diagram with a chiral twist - wherein an exotic $\textit{quantum anomalous Hall}$ insulator phase is flanked by chiral $d+id$ superconducting domes. Specifically, we analyze repulsively interacting fermions on hexagonal (triangular or honeycomb) lattices near Van Hove filling, with an ${\rm SU}(N_f)$ flavor degeneracy. This model is inspired by recent experiments on graphene Moir\'e heterostructures. At this point, a nested Fermi surface and divergent density of states give rise to strong ($\ln^2$) instabilities to correlated phases, the competition between which can be controllably addressed through a combination of weak coupling parquet renormalization group and Landau-Ginzburg analysis. For $N_f=2$ (i.e. spin degeneracy only) it is known that chiral $d+id$ superconductivity is the unambiguously leading weak coupling instability. Here we show that $N_f\geq4$ leads to a richer (but still unambiguous and fully controllable) behavior, wherein at weak coupling the leading instability is to a fully gapped and chiral $\textit{Chern insulator}$, characterized by a spontaneous breaking of time reversal symmetry and a quantized Hall response. Upon doping this phase gives way to a chiral $d+id$ superconductor. We further consider deforming this minimal model by introducing an orbital splitting of the Van Hove singularities, and discuss the resulting RG flow and phase diagram. Our analysis thus bridges the minimal model and the practical Moir\'e band structures, thereby providing a transparent picture of how the correlated phases arise under various circumstances. Meanwhile, a similar analysis on the square lattice predicts a phase diagram where (for $N_f>2$) a nodal staggered flux phase with `loop current' order gives way upon doping to a nodal $d$-wave superconductor.

Cavity quantum electrodynamical Chern insulator: Towards light-induced quantized anomalous Hall effect in graphene

We show that an energy gap is induced in graphene by light-matter coupling to a circularly polarized photon mode in a cavity. Using many-body perturbation theory we compute the electronic spectra which exhibit photon-dressed sidebands akin to Floquet sidebands for laser-driven materials. In contrast with Floquet topological insulators, in which a strictly quantized Hall response is induced by light only for off-resonant driving in the high-frequency limit, the photon-dressed Dirac fermions in the cavity show a quantized Hall response characterized by an integer Chern number. Specifically for graphene we predict that a Hall conductance of $2 e^2/h$ can be induced in the low-temperature limit.

Scheme to equilibrate the Hall response of topological systems from coherent dynamics

Two-dimensional topologically distinct insulators are separated by topological gapless points, which exist as Weyl points in three-dimensional momentum space. Slowly varying parameters in the two-dimensional Hamiltonian across two distinct phases therefore necessarily experiences the gap closing process, which prevents the intrinsic physical observable, the Hall response, from equilibrating. To equilibrate the Hall response, engineered laser noises were introduced at the price of destroying the quantum coherence. Here we demonstrate a new scheme to equilibrate the quantized Hall response from pure coherent dynamics as the Hamiltonian is slowly tuned from the topologically trivial to nontrivial regimes. We show the elements that affect the process of equilibration including the sequence when the electric field is switched on, its strength and the band dispersion of the final Hamiltonian. We further apply our method to Weyl semimetals in three dimensions and find the equilibrated Hall response despite the underlying gapless band structure. Our finding not only lays the theoretical foundation for observing the two-dimensional topological phase transition but also for observing and controlling Weyl semimetals in ultracold atomic gases.

Interface effects on the magnetic-proximity-induced quantized Hall response in heterostructures based on three-dimensional topological insulators

We report a theoretical study of the spin-dependent transport properties in heterostructures containing a three-dimensional topological insulator (TI) thin film and ferromagnetic normal insulator (FMNI) slab. Within the framework of a continual approach for the FMNI/TI/FMNI trilayer model, we reveal how the magnetic proximity effect at the TI/FMNI interface can influence an intrinsic Hall response of the system. We predict that the FMNI/TI/FMNI trilayer undergoes a transition into the quantum anomalous Hall phase either from the topologically trivial phase or from the quantum spin Hall phase, which is controlled by tuning the proximity-induced exchange field, the TI film thickness, and the TI band structure parameters. We draw the corresponding phase diagram of the FMNI/TI/FMNI trilayer. Moreover, we argue that the roughness at the TI/FMNI interfaces can cause the decomposition of the TI film into topologically distinct domains, which affects the Hall conductivity. We discuss the specifics of manifestation and complexities of observation of quantized conductivity in realistic TI/FMNI heterostructures.

Topological insulation in a ladder model with particle-hole and reflection symmetries

A two-legged ladder model, one dimensional, exhibiting the parity anomaly is constructed. The model belongs to the C and CI symmetry classes, depending on the parameters, but, due to reflection, it exhibits topological insulation. The model consists of two superimposed Creutz models with onsite potentials. The topological invariants of each Creutz model sum to give the mirror winding number, with winding numbers which are nonzero individually but equal and opposite in the topological phase, and both zero in the trivial phase. We demonstrate the presence of edge states and quantized Hall response in the topological region. Our model exhibits two distinct topological regions, distinguished by the different types of reflection symmetries.

Exploring 4D quantum Hall physics with a 2D topological charge pump.

The discovery of topological states of matter has greatly improved our understanding of phase transitions in physical systems. Instead of being described by local order parameters, topological phases are described by global topological invariants and are therefore robust against perturbations. A prominent example is the two-dimensional (2D) integer quantum Hall effect: it is characterized by the first Chern number, which manifests in the quantized Hall response that is induced by an external electric field. Generalizing the quantum Hall effect to four-dimensional (4D) systems leads to the appearance of an additional quantized Hall response, but one that is nonlinear and described by a 4D topological invariant-the second Chern number. Here we report the observation of a bulk response with intrinsic 4D topology and demonstrate its quantization by measuring the associated second Chern number. By implementing a 2D topological charge pump using ultracold bosonic atoms in an angled optical superlattice, we realize a dynamical version of the 4D integer quantum Hall effect. Using a small cloud of atoms as a local probe, we fully characterize the nonlinear response of the system via in situ imaging and site-resolved band mapping. Our findings pave the way to experimentally probing higher-dimensional quantum Hall systems, in which additional strongly correlated topological phases, exotic collective excitations and boundary phenomena such as isolated Weyl fermions are predicted.

Anisotropic Harper-Hofstadter-Mott model: Competition between condensation and magnetic fields

We derive the reciprocal cluster mean-field method to study the strongly-interacting bosonic Harper-Hofstadter-Mott model. The system exhibits a rich phase diagram featuring band insulating, striped superfluid, and supersolid phases. Furthermore, for finite hopping anisotropy we observe gapless uncondensed liquid phases at integer fillings, which are analyzed by exact diagonalization. The liquid phases at fillings 1 and 3 exhibit the same band fillings as the fermionic integer quantum Hall effect, while the phase at filling 2 is CT-symmetric with zero charge response. We discuss how these phases become gapped on a quasi-one-dimensional cylinder, leading to a quantized Hall response, which we characterize by introducing a suitable measure for non-trivial many-body topological properties. Incompressible metastable states at fractional filling are also observed, indicating competing fractional quantum Hall phases. The combination of reciprocal cluster mean-field and exact diagonalization yields a promising method to analyze the properties of bosonic lattice systems with non-trivial unit cells in the thermodynamic limit.

Dynamical Buildup of a Quantized Hall Response from Nontopological States

We consider a two-dimensional system initialized in a topologically trivial state before its Hamiltonian is ramped through a phase transition into a Chern insulator regime. This scenario is motivated by current experiments with ultracold atomic gases aimed at realizing time-dependent dynamics in topological insulators. Our main findings are twofold. First, considering coherent dynamics, the nonequilibrium Hall response is found to approach a topologically quantized time-averaged value in the limit of slow but nonadiabatic parameter ramps, even though the Chern number of the state remains trivial. Second, adding dephasing, the destruction of quantum coherence is found to stabilize this Hall response, while the Chern number generically becomes undefined. We provide a geometric picture of this phenomenology in terms of the time-dependent Berry curvature.

Trajectory of the anomalous Hall effect towards the quantized state in a ferromagnetic topological insulator

Topological insulators are bulk electronic insulators with symmetry-protected gapless modes on their surfaces1,2,3,4,5. Breaking time-reversal symmetry in these systems via ferromagnetism induces a gapped state characterized by a zero magnetic field quantized Hall response—the quantum anomalous Hall (QAH) state6,7,8. A key question that has remained experimentally unexplored is the relationship of this new type of quantum Hall (QH) state with the previously known orbitally driven QH states. Here, we show experimentally that a ferromagnetic topological insulator exhibiting the QAH state is well described by the global phase diagram of the QH effect. By mapping the conductivity tensor in the vicinity of the QAH phase, we find evidence for quantum criticality and delocalization behaviour9,10 that can quantitatively be described by the renormalization group properties of the QH ground state11,12. This result demonstrates that this QAH state can be understood within the context of the law of corresponding states that governs the QH state13,14.