Modern condensed matter physics relies on the concept of topology to classify matter, from quantum Hall systems to topological insulators. Engineered systems, benefiting from synthetic dimensions, can potentially give access to novel topological states predicted in dimensions $D > 3$ . We report the realization of an atomic quantum Hall system evolving in four dimensions (4D), with two spatial dimensions and two synthetic ones encoded in the large spin of dysprosium atoms. The non-trivial topology is evidenced by measuring a quantized electromagnetic non-linear response and observing anisotropic hyperedge modes. We also excite non-planar cyclotron motion, contrasting with its circular equivalents in $D\leq3$ . Our work opens to the investigation of strongly-correlated topological liquids in 4D generalizing fractional quantum Hall states.Modern condensed matter physics relies on the concept of topology to classify matter, from quantum Hall systems to topological insulators. Engineered systems, benefiting from synthetic dimensions, can potentially give access to novel topological states predicted in dimensions $D > 3$. We report the realization of an atomic quantum Hall system evolving in four dimensions (4D), with two spatial dimensions and two synthetic ones encoded in the large spin of dysprosium atoms. The non-trivial topology is evidenced by measuring a quantized electromagnetic non-linear response and observing anisotropic hyperedge modes. We also excite non-planar cyclotron motion, contrasting with its circular equivalents in $D\leq3$. Our work opens to the investigation of strongly-correlated topological liquids in 4D generalizing fractional quantum Hall states.

arXiv:2210.06322Oct 12, 20221 citation

Realizing the entanglement Hamiltonian of a topological quantum Hall system

Quentin Redon, Qi Liu, Jean-Baptiste Bouhiron +4 more

Topological quantum many-body systems are characterized by a hidden order encoded in the entanglement between their constituents. While entanglement is often quantified using the entanglement entropy, its full description relies on the entanglement Hamiltonian, which is commonly used to identify complex phases arising in numerical simulations, but whose measurement remains an outstanding challenge. Here, we map entanglement to spectral properties by realizing a physical system whose single-particle dynamics is governed by the entanglement Hamiltonian of a quantum Hall system. We use a synthetic dimension, encoded in the electronic spin of dysprosium atoms, to implement spatially deformed dynamics, as suggested by the Bisognano-Wichmann prediction. The realized Hamiltonian, probed with bosonic atoms with negligible interactions, exhibits a chiral dispersion akin to a topological edge mode, revealing the fundamental link between entanglement and boundary physics. We numerically show that our protocol could be extended to interacting systems in fractional quantum Hall states. Spatial entanglement in many-body systems is fully characterized by the entanglement Hamiltonian, but its measurement has remained challenging. Here the authors realize this for a topological quantum Hall system using ultracold gases of dysprosium atoms.Topological quantum many-body systems are characterized by a hidden order encoded in the entanglement between their constituents. While entanglement is often quantified using the entanglement entropy, its full description relies on the entanglement Hamiltonian, which is commonly used to identify complex phases arising in numerical simulations, but whose measurement remains an outstanding challenge. Here, we map entanglement to spectral properties by realizing a physical system whose single-particle dynamics is governed by the entanglement Hamiltonian of a quantum Hall system. We use a synthetic dimension, encoded in the electronic spin of dysprosium atoms, to implement spatially deformed dynamics, as suggested by the Bisognano-Wichmann prediction. The realized Hamiltonian, probed with bosonic atoms with negligible interactions, exhibits a chiral dispersion akin to a topological edge mode, revealing the fundamental link between entanglement and boundary physics. We numerically show that our protocol could be extended to interacting systems in fractional quantum Hall states. Spatial entanglement in many-body systems is fully characterized by the entanglement Hamiltonian, but its measurement has remained challenging. Here the authors realize this for a topological quantum Hall system using ultracold gases of dysprosium atoms.

arXiv:2307.06251Nature CommunicationsNov 21, 20249 citations

Chiral edge dynamics and quantum Hall physics in synthetic dimensions with an atomic erbium Bose-Einstein condensate

Roberto Vittorio Roell, Arif Warsi Laskar, Franz Huybrechts +1 more

Quantum Hall physics is at the heart of research on both matter and artificial systems, such as cold atomic gases, with nontrivial topological order. We report on the observation of a chiral edge current by transferring atomic wave packets simultaneously to opposite edges of a synthetic Hall system realized in the two-dimensional state space formed by one spatial and one synthetic dimension encoded in the $J=6$ electronic spin of erbium atoms. To characterize the system, the Hall drift of the employed atomic Bose-Einstein condensate in the lowest Landau-like level is determined. The topological properties are verified by determining the local Chern marker, and upon performing low-lying excitations both cyclotron and skipping orbits are observed in the bulk and edges, respectively. Future prospects include studies of novel topological phases in cold-atom systems.Quantum Hall physics is at the heart of research on both matter and artificial systems, such as cold atomic gases, with nontrivial topological order. We report on the observation of a chiral edge current by transferring atomic wave packets simultaneously to opposite edges of a synthetic Hall system realized in the two-dimensional state space formed by one spatial and one synthetic dimension encoded in the $J=6$ electronic spin of erbium atoms. To characterize the system, the Hall drift of the employed atomic Bose-Einstein condensate in the lowest Landau-like level is determined. The topological properties are verified by determining the local Chern marker, and upon performing low-lying excitations both cyclotron and skipping orbits are observed in the bulk and edges, respectively. Future prospects include studies of novel topological phases in cold-atom systems.

arXiv:2210.09874Physical Review AApr 3, 202311 citations

A two-dimensional realization of the parity anomaly

Nehal Mittal, Tristan Villain, Mathis Demouchy +4 more

Quantum anomalies arise when symmetries of a classical theory cannot be preserved upon quantization, leading to unconventional topological responses. A prominent example is the parity anomaly of a single two-dimensional Dirac fermion, which enforces a half-quantized Hall response. Anomaly inflow mechanism allows this effect to be observed at the surfaces of three-dimensional topological insulators, however, its realization in a genuinely two-dimensional system has remained elusive. Here we report the observation of a parity-anomalous Hall response at the critical point of a quantum Hall topological phase transition in a synthetic two-dimensional system of ultracold dysprosium atoms. By coupling a continuous spatial dimension to a finite synthetic dimension encoded in atomic spin states, we engineer tunable Chern bands with C = 0 and 1. At the transition, the bulk gap closes at a single Dirac point, where we observe a robust half-quantized Hall drift despite strong non-adiabatic excitations. We show that this response originates from the global structure of the band topology, is protected by an emergent parity symmetry at criticality, and disappears when parity is explicitly broken. Our work establishes synthetic quantum systems as a powerful platform to probe quantum anomalies and their interplay with topology and non-equilibrium dynamics.Quantum anomalies arise when symmetries of a classical theory cannot be preserved upon quantization, leading to unconventional topological responses. A prominent example is the parity anomaly of a single two-dimensional Dirac fermion, which enforces a half-quantized Hall response. Anomaly inflow mechanism allows this effect to be observed at the surfaces of three-dimensional topological insulators, however, its realization in a genuinely two-dimensional system has remained elusive. Here we report the observation of a parity-anomalous Hall response at the critical point of a quantum Hall topological phase transition in a synthetic two-dimensional system of ultracold dysprosium atoms. By coupling a continuous spatial dimension to a finite synthetic dimension encoded in atomic spin states, we engineer tunable Chern bands with C = 0 and 1. At the transition, the bulk gap closes at a single Dirac point, where we observe a robust half-quantized Hall drift despite strong non-adiabatic excitations. We show that this response originates from the global structure of the band topology, is protected by an emergent parity symmetry at criticality, and disappears when parity is explicitly broken. Our work establishes synthetic quantum systems as a powerful platform to probe quantum anomalies and their interplay with topology and non-equilibrium dynamics.

cond-mat.quant-gasCross-listedarXiv:2603.22173Mar 24, 2026

Four-Dimensional Quantum Hall Effect with Ultracold Atoms

Hannah M. Price, Oded Zilberberg, Tomoki Ozawa +2 more

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.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.

arXiv:1505.04387Physical Review LettersNov 3, 2015257 citations

Quantum anomalous Hall phase in a one-dimensional optical lattice

Sheng Liu, L. B. Shao, Qizhe Hou +1 more

We propose to simulate and detect quantum anomalous Hall phase with ultracold atoms in a one-dimensional optical lattice, with the other synthetic dimension being realized by modulating spin-orbit coupling. We show that the system manifests a topologically nontrivial phase with two chiral edge states which can be readily detected in this synthetic two-dimensional system. Moreover, it is interesting that at the phase transition point there is a flat energy band and this system can also be in a topologically nontrivial phase with two Fermi zero modes existing at the boundaries by considering the synthetic dimension as a modulated parameter. We also show how to measure these topological phases experimentally in ultracold atoms. Another model with a random Rashba and Dresselhaus spin-orbit coupling strength is also found to exhibit topological nontrivial phase, and the impact of the disorder to the system is revealed.We propose to simulate and detect quantum anomalous Hall phase with ultracold atoms in a one-dimensional optical lattice, with the other synthetic dimension being realized by modulating spin-orbit coupling. We show that the system manifests a topologically nontrivial phase with two chiral edge states which can be readily detected in this synthetic two-dimensional system. Moreover, it is interesting that at the phase transition point there is a flat energy band and this system can also be in a topologically nontrivial phase with two Fermi zero modes existing at the boundaries by considering the synthetic dimension as a modulated parameter. We also show how to measure these topological phases experimentally in ultracold atoms. Another model with a random Rashba and Dresselhaus spin-orbit coupling strength is also found to exhibit topological nontrivial phase, and the impact of the disorder to the system is revealed.

Journal of Physics Condensed MatterJan 30, 20182 citations

D > 3

. We report the realization of an atomic quantum Hall system evolving in four dimensions (4D), with two spatial dimensions and two synthetic ones encoded in the large spin of dysprosium atoms. The non-trivial topology is evidenced by measuring a quantized electromagnetic non-linear response and observing anisotropic hyperedge modes. We also excite non-planar cyclotron motion, contrasting with its circular equivalents in

D\leq3

. Our work opens to the investigation of strongly-correlated topological liquids in 4D generalizing fractional quantum Hall states.

J=6

electronic spin of erbium atoms. To characterize the system, the Hall drift of the employed atomic Bose-Einstein condensate in the lowest Landau-like level is determined. The topological properties are verified by determining the local Chern marker, and upon performing low-lying excitations both cyclotron and skipping orbits are observed in the bulk and edges, respectively. Future prospects include studies of novel topological phases in cold-atom systems.