Electromagnetic response of quantum Hall systems in dimensions five and six and beyond

Abstract

Quantum Hall (QH) states are arguably the most ubiquitous examples of nontrivial topological order, requiring no special symmetry and elegantly characterized by the first Chern number. Their higher dimension generalizations are particularly interesting from both mathematical and phenomenological perspectives, and have attracted recent attention due to high profile experimental realizations [1, 2]. In this work, we derive from first principles the electromagnetic response of QH systems in arbitrary number of dimensions, and elaborate on the crucial roles played by their modified phase space density of states under the simultaneous presence of magnetic field and Berry curvature. We provide new mathematical results relating this phase space modification to the noncommutativity of phase space, and show how they are manifested as a Hall conductivity quantized by a higher Chern number. When a Fermi surface is present, additional response currents unrelated to these Chern numbers also appear. This unconventional response can be directly investigated through a few minimal models with specially chosen fluxes. These models, together with more generic 6D QH systems, can be realized in realistic 3D experimental setups like cold atom systems through possibly entangled synthetic dimensions.

Four-dimensional semimetals with tensor monopoles: From surface states to topological responses

“…These systems support higher-spin quasiparticles, where the Weyl-like cones become multifold degenerate [31][32][33][34][35][36][37][38][39][40]. Interestingly, the parity anomaly can also exist in four and six dimensions [41,42], which suggests relevant implications in the context of higher-dimensional synthetic * giandomenico.palumbo@gmail.com topological matter [43][44][45]. In addition, it has been shown that a three-band model in four dimensions can give rise to a gapless topological phase characterized by a tensor monopole [46,47], whose topological nature is established by the Dixmier-Douady (DD) invariant [48,49].…”

Progress in Topological Mechanics

“…The synthetic dimensions refer to exploring higher-dimensional physics with virtual dimensions in lowerdimensional systems. Up to now, the synthetic dimensions has been used to relate the 2D, 4D and 6D QHE with 1D pump, 2D pump and 3D pump via 1D [119,120,125,127,128,132], 2D [24,129,133] and 3D [21,22] lattices The topological nature of the 2D (4D) quantum Hall systems are characterized by the first Chern number (the second Chern number) [129,134]. The Chern number dictates a quantization of particle transport in (meta)material during one cycle of an adiabatical pump.…”

“…This work showed the relationship between the 1D quantization of particle transport and 2D QHE. Recently, a topological pump was connected to 6D QHE [21,22] via synthetic dimensions. More recently, a topological pump has been observed in higherorder topological systems [23].…”

Fractional Quantum Hall Effect for Extended Objects: from Skyrmionic Membranes to Dyonic Strings

“…Higher-dimensional topological matter represents a very active field of research due to the possibility to engineer synthetic dimensions in suitable artificial setups [1][2][3][4][5][6]. In fact, topological invariants in four and higher dimensions and novel quantum effects such as the higher-dimensional Thouless pumping and anomalous quantum transports can be experimentally detected and measured in synthetic matter [7][8][9][10][11][12][13][14][15][16][17][18][19]. At microscopic level, these phases deal with point-like quasiparticles.…”