Extending the 2D topological charge pump beyond the integer 4D quantum Hall response: remaining nonlinear 4D responses, fractional 4D phases, and 4D boundary physics

The 2018 experiment of Lohse, Schweizer, Price, Zilberberg and Bloch realized a dynamical version of the four-dimensional (4D) integer quantum Hall effect using a 2D topological charge pump of ultracold atoms in an angled optical superlattice, and measured the associated second Chern number. The authors posed three follow-up directions: measuring the additional density-type nonlinear Hall responses implied by the intrinsic 4D symmetry of the pump; including and controlling interactions to search for fractional 4D quantum Hall phases; and realizing a quantum Hall system with four extended dimensions via synthetic dimensions, including its boundary phenomena. Later literature has advanced parts of this program — notably a genuine 4D quantum Hall system with two real and two synthetic dimensions (Bouhiron et al., 2024) and the theory of higher-dimensional transport coefficients — but no fractional 4D quantum Hall phase has been realized in any platform, the additional density-type responses of the pump platform remain unmeasured, and four extended spatial dimensions are still out of reach. The problem is open, with the interacting (fractional) regime as the clearest surviving core.

Background

The integer quantum Hall (QH) effect arises when charged particles move in two spatial dimensions (2D) under a strong perpendicular magnetic field: the Hall conductance is quantized in integer multiples of $e^2/h$ and is protected by a topological invariant, the (first) Chern number, which counts the winding of the occupied band's Berry curvature (the momentum-space geometric curvature of the single-particle energy eigenstates) over the 2D Brillouin zone (the unit cell of crystal momentum). Zhang and Hu proposed in 2001 that the QH effect generalizes to four spatial dimensions (4D), where a new quantized Hall response appears: it is nonlinear (second-order) in the applied fields, and it is governed by a 4D topological invariant, the second Chern number, integrated over the 4D Brillouin zone. In 4D, an electric-field-like perturbation $E$ and a magnetic-like perturbation $B$ can act simultaneously and couple two orthogonal 2D subspaces; the resulting transverse drift, the analogue of a 4D Lorentz force, cannot occur in 2D or 3D and is linear in $B$ and in $E$. Zhang and Hu's proposal also predicted strongly correlated generalizations, i.e. fractional 4D QH liquids, whose ground state has an internal $\mathrm{SO}(5)$ symmetry (the group of rotations of the four-dimensional sphere on which their model lives, acting as an internal symmetry of the liquid); these have remained purely theoretical.

Because no physical system has four extended spatial dimensions, engineered platforms emulate 4D physics by 'dimensional reduction': a lower-dimensional system whose parameters vary periodically in time can encode Fourier components of a higher-dimensional band structure. The Thouless pump is the prototypical example — a 1D lattice whose parameters are cycled adiabatically transports an integer number of particles per cycle, equal to a first Chern number, exactly as a 2D QH cylinder threaded by one flux quantum. Kraus, Ringel and Zilberberg showed that a 2D quasicrystal similarly encodes the 4D QH system and exhibits the quantized nonlinear 4D response. A complementary route uses 'synthetic dimensions': internal degrees of freedom such as atomic spin sublevels are treated as extra lattice coordinates, so that $m$ real dimensions plus $n$ synthetic ones emulate a $(m+n)$-dimensional lattice (Price et al. proposed a 4D QH system with ultracold atoms on this basis in 2015).

Lohse, Schweizer, Price, Zilberberg and Bloch combined these ideas into a 2D topological charge pump: bosonic $^{87}\mathrm{Rb}$ atoms in an angled optical superlattice whose two superlattice phases $\varphi_x, \varphi_y$ play the role of two extra (parameter-space) dimensions, so that the 2D driven system is the dimensional reduction of a 4D lattice QH model built from two independent 2D Harper–Hofstadter-type planes (lattice electrons in uniform magnetic flux). Adiabatically cycling $\varphi_x$ pumps particles along $x$; a tiny geometric tilt $\theta$ between the long and short lattices along $y$ makes the transverse phase position-dependent, emulating the weak magnetic-like perturbation $B$ of the 4D model. The experiment observed the resulting nonlinear transverse drift, reversed its sign with $\theta$ and with the pumping direction, and extracted the second Chern number of the lowest band, $\nu_2 = 1.01(8)$ from the linear scaling of the nonlinear response with the tilt $\theta$ (and $1.07(8)$ from an independent fit of the integrated nonlinear response versus $\varphi_y$), consistent with $\nu_2 = +1$. The quantitative analysis was based on a non-interacting band theory of a 2D Rice–Mele-type superlattice model (a 1D chain with alternating hopping amplitudes and alternating on-site energies, here extended to 2D); interactions entered only through corrections for doubly occupied sites, the perturbation was kept weak to protect the band gap, the pumping adiabatic, and the number of cycles limited by heating.

Within these limits, the experiment established only the integer, weak-perturbation, effectively single-particle sector of 4D QH physics. The paper's own outlook poses three directions for completing the program, and its abstract adds that the 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' — Weyl fermions being linearly dispersing band singularities that, at the boundary of a 4D QH system, appear as isolated degeneracy points connecting hyperedge modes, the 4D analogue of chiral edge states.

Problem Statement

The 2D topological charge pump has so far demonstrated only the integer, weak-perturbation, effectively single-particle sector of 4D quantum Hall physics: one measured topological invariant (the second Chern number $\nu_2 = +1$ of the lowest band) and one nonlinear response, under a non-interacting band description, weak magnetic-like perturbation $\theta$, adiabatic driving and a limited number of pump cycles. How far can this dynamical route to 4D quantum Hall physics be pushed to the fuller phenomena that the 4D theory predicts? Concretely, as posed in the source paper's outlook: (1) measure the additional density-type nonlinear Hall responses implied by the intrinsic 4D symmetry of the 2D charge pump — responses that, as the source states, manifest differently in the centre-of-mass drift but are characterized by the same topological invariant — and thereby map out the full set of 4D transport coefficients of the pump; (2) include and control inter-atomic interactions in the 2D pump so as to search for fractional 4D quantum Hall phases, i.e. strongly correlated many-body topological states in the higher-dimensional parameter space, including their fractional quantization of the nonlinear response; and (3) realize a quantum Hall system with four extended spatial dimensions using synthetic-dimension techniques (the route the source paper envisages), in order to access the 4D boundary phenomena predicted for finite 4D quantum Hall systems, such as isolated Weyl points and hyperedge modes (chiral boundary modes of the 4D system, the analogue of edge states). These three directions share one objective: extending the dynamical 4D quantum Hall program opened by the 2018 experiment from a single integer invariant to the complete set of intrinsically 4D responses, to the correlated fractional regime, and to genuine 4D boundary physics. Of these, the interaction-driven fractional regime is the least explored and constitutes the clearest open core, since later work has partially addressed the nonlinear-response and synthetic-dimension directions (refs. 6 and 11) while no realization of a fractional 4D quantum Hall phase exists in any platform. Within this shared objective, the target is to realize and characterize a fractional 4D quantum Hall phase in this pump platform, with the remaining density-type nonlinear responses and the 4D boundary phenomena as related parts of the same program.

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