Experimental Observation of Large Chern Numbers in Photonic Crystals

Abstract

Despite great interest in the quantum anomalous Hall phase and its analogs, all experimental studies in electronic and bosonic systems have been limited to a Chern number of one. Here, we perform microwave transmission measurements in the bulk and at the edge of ferrimagnetic photonic crystals. Band gaps with large Chern numbers of 2, 3, and 4 are present in the experimental results, which show excellent agreement with theory. We measure the mode profiles and Fourier transform them to produce dispersion relations of the edge modes, whose number and direction match our Chern number calculations. DOI: 10.1103/PhysRevLett.115.253901 PACS numbers: 42.79.Ta, 42.70.Qs, 42.79.Fm, 42.79.Gn The Chern number [1] is an integer defining the topological phase in the quantum Hall effect (QHE) [2], which determines the number of topologically protected chiral edge modes. The quantum anomalous Hall effect (QAHE) possesses these same properties as an intrinsic property of the band structure with time reversal symmetry breaking [3,4]. Recent experiments have discovered the QAHE and its analogs in ferrimagnetic photonic crystals [5], magnetically doped thin films [6], and ultracold fermion lattices [7]. However, the Chern numbers observed in all of these systems were limited to AE1. Finding larger Chern numbers would fundamentally expand the known topological phases [8][9][10][11][12].Here, we provide the first explicit experimental observation of Chern numbers of magnitude 2, 3, and 4, by measuring bulk transmission, edge transmission, and the edge-mode dispersion relations in a ferrimagnetic photonic crystal. The excellent agreement between the experiment and the modeling allows us to identify various topological band gaps and map out the dispersion relations of one-way edge modes for the first time in any QHE or QAHE system in nature.In a 2D system one can realize bands with nonzero Chern numbers, and generate the QAHE, by applying a T-breaking perturbation [13][14][15]. The Chern number is defined as the integral of the Berry flux over the entire Brillioun zone. When connected bands are gapped by a T-breaking perturbation, the bands will exchange equal and opposite Berry flux at each degenerate point, with the total Berry flux exchanged determining the Chern number. For instance, two isolated bands connected by one pair of Dirac points gapped by T-breaking will acquire AE2π Berry flux (π from each Dirac point) and a Chern number associated with the band gap ("gap Chern number") of AE1. A general way to calculate the gap Chern number (C gap ¼ ΣC i ) is to sum the Chern numbers of all the bands below the band gap [16]. A band gap with C gap ¼ 0 is trivial, while a band gap with C gap ≠ 0 is topologically nontrivial.In our previous theoretical study, we found that the magnitude of the gap Chern number can be increased above one by simultaneously gapping multiple sets of Dirac and quadratic degeneracies. If Berry flux from the gapped degeneracies adds constructively, C gap can be large. In Fig. 1(a) we present a theoretical to...

Experimental measurement of the Berry curvature from anomalous transport

“…In photonics, for example, topological edge states have been studied in a wide variety of (effectively) 1D set-ups, such as quantum walks 14 and pumping in optical quasicrystals 13,29,30 , as well as in 2D quantum Hall-like systems of photonic crystals [9][10][11] , propagating waveguides 12 and silicon ring resonators 15,16 . However, while the existence of such edge states is linked to the non-trivial global topological properties of bulk bands, a direct measurement of the Berry curvature itself has so far not been realized.…”

Topologically Enabled Optical Nanomotors

“…In recent years, there has been an increased interest in topological aspects of optical phenomena, including topological order in photonic structures (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15), optical vortices (16)(17)(18), and other robust phenomena in optics [for example, optical bound states in the continuum (19)(20)(21)]. Shaping the topology of a beam of light can enable unique abilities of light to manipulate matter.…”

Omnidirectional spin Hall effect in a Weyl spin-orbit-coupled atomic gas

“…On the other hand, when it comes to effective models, many quantum-mechanical systems with adiabatically varying parameters are naturally described in terms of Abelian gauge theories [2][3][4]. This geometric approach based on the Berry phase has paved the way to a multitude of both theoretical and experimental developments covering molecular [3][4][5], solid-state [6][7][8][9][10][11][12], photonic [13][14][15][16][17][18][19][20][21], mechanical [22,23] and electric [24,25] systems. Although the corresponding non-Abelian gauge structure in the presence of degenerate quantum states has been noticed promptly after the discovery of the Berry phase [26], a set of experimentally observed signatures of the non-Abelian geometrical phases remains limited [27,28].…”