Four-dimensional Quantum Hall Research Articles

Coupled layer construction for synthetic Hall effects in driven systems

Quasiperiodically driven fermionic systems can support topological phases not realized in equilibrium. The fermions are localized in the bulk, but support quantized energy currents at the edge. These phases were discovered through an abstract classification, and few microscopic models exist. We develop a coupled layer construction for tight-binding models of these phases in $d\in\{1,2\}$ spatial dimensions, with any number of incommensurate drive frequencies $D$. The models exhibit quantized responses associated with synthetic two- and four-dimensional quantum Hall effects in the steady state. A numerical study of the phase diagram for $(d+D) = (1+2)$ shows: (i) robust topological and trivial phases separated by a sharp phase transition; (ii) charge diffusion and a half-integer energy current between the drives at the transition; and (iii) a long-lived topological energy current which remains present when weak interactions are introduced.

Topological invariants in two-dimensional quasicrystals

We provide a topological concept to characterize energy gaps in general two-dimensional quasiperiodic systems. We show that every single gap is uniquely characterized by a set of integers, which quantize the area of the momentum space in units of multiple Brillouin zones. These integers are found to be equivalent to the second Chern numbers by considering a formal relationship between an adiabatic charge pumping under a potential sliding and the four-dimensional quantum Hall effect. The integers are independent of commensurability, and invariant under an arbitrary continuous deformation, such as a relative rotation of a twisted bilayer system.

Revealing the Boundary Weyl Physics of the Four-Dimensional Hall Effect via Phason Engineering in Metamaterials

Quantum Hall physics has been theoretically predicted in four dimensions and higher. In hypothetical 2n dimensions, the topological characters of both the bulk and the boundary are manifested as quantized nonlinear transport coefficients that respectively connect to the $n\mathrm{th}$ Chern number of the bulk gap projection and to the $n\mathrm{th}$ winding number of the Weyl spectral singularities on the ($2n\ensuremath{-}1$)-dimensional boundaries. Here, we introduce the concept of phason engineering in metamaterials and use it as a vehicle to access and apply the quantum Hall physics in arbitrary dimensions. Using these specialized design principles, we fabricate a reconfigurable two-dimensional aperiodic acoustic crystal with a phason living on a 2-torus, giving us access to the four-dimensional quantum Hall physics. Also, we supply a direct experimental confirmation that the topological boundary spectrum assembles in a Weyl singularity when mapped as a function of the quasimomenta. We also demonstrate topological wave steering enabled by the Weyl physics of the three-dimensional boundaries.

Analytic non-Abelian gravitating solitons in the Einstein\u2013Yang\u2013Mills\u2013Higgs theory and transitions between them

Two analytic examples of globally regular non-Abelian gravitating solitons in the Einstein–Yang–Mills–Higgs theory in (3 + 1)-dimensions are presented. In both cases, the space-time geometries are of the Nariai type and the Yang–Mills field is completely regular and of meron type (namely, proportional to a pure gauge). However, while in the first family (type I) X_{0} = 1/2 (as in all the known examples of merons available so far) and the Higgs field is trivial, in the second family (type II) X_{0} = 1/2 is not 1/2 and the Higgs field is non-trivial. We compare the entropies of type I and type II families determining when type II solitons are favored over type I solitons: the VEV of the Higgs field plays a crucial role in determining the phases of the system. The Klein–Gordon equation for test scalar fields coupled to the non-Abelian fields of the gravitating solitons can be written as the sum of a two-dimensional D’Alembert operator plus a Hamiltonian which has been proposed in the literature to describe the four-dimensional Quantum Hall Effect (QHE): the difference between type I and type II solutions manifests itself in a difference between the degeneracies of the corresponding energy levels.

Second Chern Number and Non-Abelian Berry Phase in Topological Superconducting Systems

Topology ultimately unveils the roots of the perfect quantization observed in complex systems. The two-dimensional quantum Hall effect is the celebrated archetype. Remarkably, topology can manifest itself even in higher-dimensional spaces in which control parameters play the role of extra, synthetic dimensions. However, so far, a very limited number of implementations of higher-dimensional topological systems have been proposed, a notable example being the so-called four-dimensional quantum Hall effect. Here we show that mesoscopic superconducting systems can implement higher-dimensional topology and represent a formidable platform to study a quantum system with a purely nontrivial second Chern number. We demonstrate that the integrated absorption intensity in designed microwave spectroscopy is quantized and the integer is directly related to the second Chern number. Finally, we show that these systems also admit a non-Abelian Berry phase. Hence, they also realize an enlightening paradigm of topological non-Abelian systems in higher dimensions.4 MoreReceived 9 July 2020Accepted 15 December 2020DOI:https://doi.org/10.1103/PRXQuantum.2.010310Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasTopological materialsTopological phases of matterTopological superconductorsPhysical SystemsQuantum dotsSuperconductorsCondensed Matter, Materials & Applied Physics

Dissipative analog of four-dimensional quantum Hall physics

In this work, the authors show that it is possible to realize an anomalous configuration of Weyl cones in a three-dimensional, non-Hermitian Weyl semimetal. It is thus possible to probe the boundary physics of four-dimensional quantum Hall models in three-dimensional setups with gain and loss.

Three-Dimensional Chiral Lattice Fermion in Floquet Systems.

We show that the Nielsen-Ninomiya no-go theorem still holds on a Floquet lattice: there is an equal number of right-handed and left-handed Weyl points in a three-dimensional Floquet lattice. However, in the adiabatic limit, where the time evolution of the low-energy subspace is decoupled from the high-energy subspace, we show that the bulk dynamics in the low-energy subspace can be described by Floquet bands with extra left- or right-handed Weyl points, despite the no-go theorem. Assuming adiabatic evolution of two bands, we show that the difference of the number of right-handed and left-handed Weyl points equals twice the winding number of the adiabatic Floquet operator over the Brillouin zone. Based on these findings, we propose a realization of purely left- or right-handed Weyl particles on a 3D lattice using a Hamiltonian obtained through dimensional reduction of a four-dimensional quantum Hall system. We argue that the breakdown of the adiabatic approximation on the surface facilitates unusual closed orbits of wave packets in an applied magnetic field, which traverse alternatively through the low-energy and high-energy sector of the spectrum.

Measurement of Chern numbers through center-of-mass responses

Probing the center-of-mass of an ultracold atomic cloud can be used to measure Chern numbers, the topological invariants underlying the quantum Hall effects. In this work, we show how such center-of-mass observables can have a much richer dependence on topological invariants than previously discussed. In fact, the response of the center of mass depends not only on the current density, typically measured in a solid-state system, but also on the particle density, which itself can be sensitive to the topology of the band structure. We apply a semiclassical approach, supported by numerical simulations, to highlight the key differences between center-of-mass responses and more standard conductivity measurements. We illustrate this by analyzing both the two- and the four-dimensional quantum Hall effects. These results have important implications for experiments in engineered topological systems, such as ultracold gases and photonics.

Synthetic dimensions in integrated photonics: From optical isolation to four-dimensional quantum Hall physics

Recent technological advances in integrated photonics have spurred on the study of topological phenomena in engineered bosonic systems. Indeed, the controllability of silicon ring-resonator arrays has opened up new perspectives for building lattices for photons with topologically nontrivial bands and integrating them into photonic devices for practical applications. Here, we push these developments even further by exploiting the different modes of a silicon ring resonator as an extra dimension for photons. Tunneling along this synthetic dimension is implemented via an external time-dependent modulation that allows for the generation of engineered gauge fields. We show how this approach can be used to generate a variety of exciting topological phenomena in integrated photonics, ranging from a topologically-robust optical isolator in a spatially one-dimensional (1D) ring-resonator chain to a driven-dissipative analog of the 4D quantum Hall effect in a spatially 3D resonator lattice. Our proposal paves the way towards the use of topological effects in the design of novel photonic lattices supporting many frequency channels and displaying higher connectivities.

Four-Dimensional Quantum Hall Effect with Ultracold Atoms

We propose a realistic scheme to detect the 4D quantum Hall effect using ultracold atoms. Based on contemporary technology, motion along a synthetic fourth dimension can be accomplished through controlled transitions between internal states of atoms arranged in a 3D optical lattice. From a semiclassical analysis, we identify the linear and nonlinear quantized current responses of our 4D model, relating these to the topology of the Bloch bands. We then propose experimental protocols, based on current or center-of-mass-drift measurements, to extract the topological second Chern number. Our proposal sets the stage for the exploration of novel topological phases in higher dimensions.

Four-Dimensional Quantum Hall Effect in a Two-Dimensional Quasicrystal

One-dimensional (1D) quasicrystals exhibit physical phenomena associated with the 2D integer quantum Hall effect. Here, we transcend dimensions and show that a previously inaccessible phase of matter-the 4D integer quantum Hall effect-can be incorporated in a 2D quasicrystal. Correspondingly, our 2D model has a quantized charge-pump accommodated by an elaborate edge phenomena with protected level crossings. We propose experiments to observe these 4D phenomena, and generalize our results to a plethora of topologically equivalent quasicrystals. Thus, 2D quasicrystals may pave the way to the experimental study of 4D physics.

The Topological Structure of the SU(2) Chern–Simons Topological Current in the Four-Dimensional Quantum Hall Effect

In the light of the decomposition of the SU(2) gauge potential for I = 1/2, we obtain the SU(2) Chern-Simons current over S4, i.e. the vortex current in the effective field for the four-dimensional quantum Hall effect. Similar to the vortex excitations in the two-dimensional quantum Hall effect (2D FQH) which are generated from the zero points of the complex scalar field, in the 4D FQH, we show that the SU(2) Chern–Simons vortices are generated from the zero points of the two-component wave functions Ψ, and their topological charges are quantized in terms of the Hopf indices and Brouwer degrees of φ-mapping under the condition that the zero points of field Ψ are regular points.

The effective action for edge states in higher-dimensional quantum Hall systems

We show that the effective action for the edge excitations of a quantum Hall droplet of fermions in higher dimensions is generically given by a chiral bosonic action. We explicitly analyze the quantum Hall effect on complex projective spaces CP k , with a U(1) background magnetic field. The edge excitations are described by Abelian bosonic fields on S 2 k−1 with only one spatial direction along the boundary of the droplet relevant for the dynamics. Our analysis also leads to an action for edge excitations for the case of the Zhang–Hu four-dimensional quantum Hall effect defined on S 4 with an SU(2) background magnetic field, using the fact that CP 3 is an S 2-bundle over S 4.

Metallic nature of the four-dimensional quantum hall edge

We study the density response of the four-dimensional quantum Hall fluid found by Zhang and Hu, which has been advanced as a model of emergent relativity. We calculate the density–density correlation function along the edge at half-filling, and show that it is similar to the three-dimensional free electron gas. This indicates that the edge of the four-dimensional quantum Hall fluid behaves like a metal.

Collective excitations at the boundary of a four-dimensional quantum Hall droplet

In this work we investigate collective excitations at the boundary of a recently constructed four-dimensional (4D) quantum Hall state. Local bosonic operators for creating these collective excitations can be constructed explicitly. Massless relativistic wave equations with helicity S can be derived exactly for these operators from their Heisenberg equation of motion. For the $S=1$ and $S=2$ cases these equations reduce to the free Maxwell and linearized Einstein equations respectively. These collective excitations can be interpreted as hydrodynamical modes at the boundary of the 4D quantum Hall effect droplet. Outstanding issues are critically discussed.