Second Chern number of a quantum-simulated non-Abelian Yang monopole

Realization of an atomic quantum Hall system in four dimensions

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 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. We measure the nontrivial topological index of the ground band through a full characterization of the nonlinear electromagnetic response and observe the associated anisotropic hyperedge modes. We also excite nonplanar cyclotron motion, in contrast to the planar orbits in D ≤ 3. Our work may enable the investigation of strongly correlated topological liquids in 4D, generalizing fractional quantum Hall states.

Observation of vortex-string chiral modes in metamaterials

As hypothetical topological defects in the geometry of spacetime, vortex strings could have played many roles in cosmology, and their distinct features can provide observable clues about the early universe's evolution. A key feature of vortex strings is that they can interact with Weyl fermionic modes and support massless chiral-anomaly states along strings. To date, despite many attempts to detect vortex strings in astrophysics or to emulate them in artificially created systems, observation of these vortex-string chiral modes remains experimentally elusive. Here we report experimental observations of vortex-string chiral modes using a metamaterial system. This is implemented by inhomogeneous perturbation of Yang-monopole phononic metamaterials. The measured linear dispersion and modal profiles confirm the existence of topological modes bound to and propagating along the string with the chiral anomaly. Our work provides a platform for studying diverse cosmic topological defects in astrophysics and offers applications as topological fibres in communication techniques.

Non-Abelian physics in light and sound

Non-Abelian phenomena arise when the sequence of operations on physical systems influences their behaviors. By possessing internal degrees of freedom such as polarization, light and sound can be subjected to various manipulations, including constituent materials, structured environments, and tailored source conditions. These manipulations enable the creation of a great variety of Hamiltonians, through which rich non-Abelian phenomena can be explored and observed. Recent developments have constituted a versatile testbed for exploring non-Abelian physics at the intersection of atomic, molecular, and optical physics; condensed matter physics; and mathematical physics. These fundamental endeavors could enable photonic and acoustic devices with multiplexing functionalities. Our review aims to provide a timely and comprehensive account of this emerging topic. Starting from the foundation of matrix-valued geometric phases, we address non-Abelian topological charges, non-Abelian gauge fields, non-Abelian braiding, non-Hermitian non-Abelian phenomena, and their realizations with photonics and acoustics and conclude with future prospects.

Gauge Field Induced Chiral Zero Mode in Five-Dimensional Yang Monopole Metamaterials

Owing to the chirality of Weyl nodes characterized by the first Chern number, a Weyl system supports one-way chiral zero modes under a magnetic field, which underlies the celebrated chiral anomaly. As a generalization of Weyl nodes from three-dimensional to five-dimensional physical systems, Yang monopoles are topological singularities carrying nonzero second-order Chern numbers c_{2}=±1. Here, we couple a Yang monopole with an external gauge field using an inhomogeneous Yang monopole metamaterial and experimentally demonstrate the existence of a gapless chiral zero mode, where the judiciously designed metallic helical structures and the corresponding effective antisymmetric bianisotropic terms provide the means for controlling gauge fields in a synthetic five-dimensional space. This zeroth mode is found to originate from the coupling between the second Chern singularity and a generalized 4-form gauge field-the wedge product of the magnetic field with itself. This generalization reveals intrinsic connections between physical systems of different dimensions, while a higher-dimensional system exhibits much richer supersymmetric structures in Landau level degeneracy due to the internal degrees of freedom. Our study offers the possibility of controlling electromagnetic waves by leveraging the concept of higher-order and higher-dimensional topological phenomena.

Hyperbolic band topology with non-trivial second Chern numbers

Topological band theory establishes a standardized framework for classifying different types of topological matters. Recent investigations have shown that hyperbolic lattices in non-Euclidean space can also be characterized by hyperbolic Bloch theorem. This theory promotes the investigation of hyperbolic band topology, where hyperbolic topological band insulators protected by first Chern numbers have been proposed. Here, we report a new finding on the construction of hyperbolic topological band insulators with a vanished first Chern number but a non-trivial second Chern number. Our model possesses the non-abelian translational symmetry of {8,8} hyperbolic tiling. By engineering intercell couplings and onsite potentials of sublattices in each unit cell, the non-trivial bandgaps with quantized second Chern numbers can appear. In experiments, we fabricate two types of finite hyperbolic circuit networks with periodic boundary conditions and partially open boundary conditions to detect hyperbolic topological band insulators. Our work suggests a new way to engineer hyperbolic topological states with higher-order topological invariants.

Artificial Non-Abelian Lattice Gauge Fields for Photons in the Synthetic Frequency Dimension

Non-Abelian gauge fields give rise to nontrivial topological physics. Here we develop a scheme to create an arbitrary SU(2) lattice gauge field for photons in the synthetic frequency dimension using an array of dynamically modulated ring resonators. The photon polarization is taken as the spin basis to implement the matrix-valued gauge fields. Using a non-Abelian generalization of the Harper-Hofstadter Hamiltonian as a specific example, we show that the measurement of the steady-state photon amplitudes inside the resonators can reveal the band structures of the Hamiltonian, which show signatures of the underlying non-Abelian gauge field. These results provide opportunities to explore novel topological phenomena associated with non-Abelian lattice gauge fields in photonic systems.

Observation of Spin-Tensor Induced Topological Phase Transitions of Triply Degenerate Points with a Trapped Ion

Triply degenerate points (TDPs), which correspond to new types of topological semimetals, can support novel quasiparticles possessing effective integer spins while preserving Fermi statistics. Here by mapping the momentum space to the parameter space of a three-level system in a trapped ion, we experimentally explore the transitions between different types of TDPs driven by spin-tensor-momentum couplings. We observe the phase transitions between TDPs with different topological charges by measuring the Berry flux on a loop surrounding the gap-closing lines, and the jump of the Berry flux gives the jump of the topological charge (up to a 2π factor) across the transitions. For the Berry flux measurement, we employ a new method by examining the geometric rotations of both spin vectors and tensors, which lead to a generalized solid angle equal to the Berry flux. The controllability of a multilevel ion offers a versatile platform to study high-spin physics, and our Letter paves the way to explore novel topological phenomena therein.

Synthetic Topological Vacua of Yang-Mills Fields in Bose-Einstein Condensates

Topological vacua are a family of degenerate ground states of Yang-Mills fields with zero field strength but nontrivial topological structures. They play a fundamental role in particle physics and quantum field theory, but have not yet been experimentally observed. Here we report the first theoretical proposal and experimental realization of synthetic topological vacua with a cloud of atomic Bose-Einstein condensates. Our setup provides a promising platform to demonstrate the fundamental concept that a vacuum, rather than being empty, has rich spatial structures. The Hamiltonian for the vacuum of topological number n=1 is synthesized and the related Hopf index is measured. The vacuum of topological number n=2 is also realized, and we find that vacua with different topological numbers have distinctive spin textures and Hopf links. Our Letter opens up opportunities for exploring topological vacua and related long-sought-after instantons in tabletop experiments.

Observation of Non-Abelian Thouless Pump

Thouless pump provides robust ways to realize quantized transport of waves and particles, and it casts the static 2D quantum Hall effect onto 1D dynamic systems where one of the momentum dimensions is replaced by the evolution time or path parameter. In the past few decades, various types of Abelian Thouless pump have been achieved theoretically and experimentally. However, the study of non-Abelian Thouless pump is scarce, which tells us that the order of two evolution loops with the same base point cannot be changed, and there has been no experimental observation of non-Abelian Thouless pump. Here we report the observation of a non-Abelian Thouless pump in coupled acoustic waveguide array. The non-Abelian property originates from the noncommutative combination of two different ℤ_{3} pump cycles that traverse across multiple band degeneracies in the parameter space in a three-band system. Moreover, we can pump a specific initial state to any state on any lattice site by applying these two ℤ_{3} pump cycles multiple times in a well-designed sequence. Our study paves the way for exploring and utilizing non-Abelian dynamical effects in classical wave systems and may offer different recipes for quantum walking, quantum optics, and quantum computation.

A synthetic monopole source of Kalb-Ramond field in diamond

Magnetic monopoles play a central role in areas of physics that range from electromagnetism to topological matter. String theory promotes conventional vector gauge fields of electrodynamics to tensor gauge fields and predicts the existence of more exotic tensor monopoles. Here, we report the synthesis of a tensor monopole in a four-dimensional parameter space defined by the spin degrees of freedom of a single solid-state defect in diamond. Using two complementary methods, we characterized the tensor monopole by measuring its quantized topological charge and its emanating Kalb-Ramond field. By introducing a fictitious external field that breaks chiral symmetry, we further observed an intriguing spectral transition, characterized by spectral rings protected by mirror symmetries. Our work demonstrates the possibility of emulating exotic topological structures inspired by string theory.

Phonon Magnetic Moment from Electronic Topological Magnetization

The traditional theory of magnetic moments for chiral phonons is based on the picture of the circular motion of the Born effective charge, typically yielding a small fractional value of the nuclear magneton. Here we investigate the adiabatic evolution of electronic states induced by the lattice vibration of a chiral phonon and obtain an electronic orbital magnetization in the form of a topological second Chern form. We find that the traditional theory needs to be refined by introducing a k resolved Born effective charge, and identify another contribution from the phonon-modified electronic energy together with the momentum-space Berry curvature. The second Chern form can diverge when there is a Yang's monopole near the parameter space of interest as illustrated by considering a phonon at the Brillouin zone corner in a gapped graphene model. We also find large magnetic moments for the optical phonon in bulk topological materials where nontopological contribution is also important. Our results agree with recent observations in experiments.

Measurement of Spin Chern Numbers in Quantum Simulated Topological Insulators

The topology of quantum systems has become a topic of great interest since the discovery of topological insulators. However, as a hallmark of the topological insulators, the spin Chern number has not yet been experimentally detected. The challenge to directly measure this topological invariant lies in the fact that this spin Chern number is defined based on artificially constructed wave functions. Here we experimentally mimic the celebrated Bernevig-Hughes-Zhang model with cold atoms, and then measure the spin Chern number with the linear response theory. We observe that, although the Chern number for each spin component is ill defined, the spin Chern number measured by their difference is still well defined when both energy and spin gaps are nonvanished.

Nonlinear Dynamics in a Synthetic Momentum-State Lattice

The scope of analog simulation in atomic, molecular, and optical systems has expanded greatly over the past decades. Recently, the idea of synthetic dimensions-in which transport occurs in a space spanned by internal or motional states coupled by field-driven transitions-has played a key role in this expansion. While approaches based on synthetic dimensions have led to rapid advances in single-particle Hamiltonian engineering, strong interaction effects have been conspicuously absent from most synthetic dimensions platforms. Here, in a lattice of coupled atomic momentum states, we show that atomic interactions result in large and qualitative changes to dynamics in the synthetic dimension. We explore how the interplay of nonlinear interactions and coherent tunneling enriches the dynamics of a one-band tight-binding model giving rise to macroscopic self-trapping and phase-driven Josephson dynamics with a nonsinusoidal current-phase relationship, which can be viewed as stemming from a nonlinear band structure arising from interactions.

Linked Weyl surfaces and Weyl arcs in photonic metamaterials

Generalization of the concept of band topology from lower-dimensional to higher-dimensional (n > 3) physical systems is expected to introduce new bulk and boundary topological effects. However, theoretically predicted topological singularities in five-dimensional systems-Weyl surfaces and Yang monopoles-have either not been demonstrated in realistic physical systems or are limited to purely synthetic dimensions. We constructed a system possessing Yang monopoles and Weyl surfaces based on metamaterials with engineered electromagnetic properties, leading to the observation of several intriguing bulk and surface phenomena, such as linking of Weyl surfaces and surface Weyl arcs, via selected three-dimensional subspaces. The demonstrated photonic Weyl surfaces and Weyl arcs leverage the concept of higher-dimension topology to control the propagation of electromagnetic waves in artificially engineered photonic media.

Experimental Observation of Tensor Monopoles with a Superconducting Qudit

Monopoles play a center role in gauge theories and topological matter. There are two fundamental types of monopoles in physics: vector monopoles and tensor monopoles. Examples of vector monopoles include the Dirac monopole in three dimensions and Yang monopole in five dimensions, which have been extensively studied and observed in condensed matter or artificial systems. However, tensor monopoles are less studied, and their observation has not been reported. Here we experimentally construct a tunable spin-1 Hamiltonian to generate a tensor monopole and then measure its unique features with superconducting quantum circuits. The energy structure of a 4D Weyl-like Hamiltonian with threefold degenerate points acting as tensor monopoles is imaged. Through quantum-metric measurements, we report the first experiment that measures the Dixmier-Douady invariant, the topological charge of the tensor monopole. Moreover, we observe topological phase transitions characterized by the topological Dixmier-Douady invariant, rather than the Chern numbers as used for conventional monopoles in odd-dimensional spaces.

Non-Abelian generalizations of the Hofstadter model: spin-orbit-coupled butterfly pairs

The Hofstadter model, well known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum Hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment, we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin-orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and symmetric gauges, the corresponding non-Abelian generalizations exhibit distinct spectra due to the non-commutativity of the gauge fields. We derive the genuine (necessary and sufficient) non-Abelian condition for the two models from the commutativity of their arbitrary loop operators. At zero energy, the models are gapless and host Weyl and Dirac points protected by internal and crystalline symmetries. Double (8-fold), triple (12-fold), and quadrupole (16-fold) Dirac points also emerge, especially under equal hopping phases of the non-Abelian potentials. At other fillings, the gapped phases of the models give rise to topological insulators. We conclude by discussing possible schemes for experimental realization of the models on photonic platforms.

Circuit implementation of a four-dimensional topological insulator

The classification of topological insulators predicts the existence of high-dimensional topological phases that cannot occur in real materials, as these are limited to three or fewer spatial dimensions. We use electric circuits to experimentally implement a four-dimensional (4D) topological lattice. The lattice dimensionality is established by circuit connections, and not by mapping to a lower-dimensional system. On the lattice's three-dimensional surface, we observe topological surface states that are associated with a nonzero second Chern number but vanishing first Chern numbers. The 4D lattice belongs to symmetry class AI, which refers to time-reversal-invariant and spinless systems with no special spatial symmetry. Class AI is topologically trivial in one to three spatial dimensions, so 4D is the lowest possible dimension for achieving a topological insulator in this class. This work paves the way to the use of electric circuits for exploring high-dimensional topological models.

Realization of a deeply subwavelength adiabatic optical lattice

We propose and describe our realization of a deeply subwavelength optical lattice for ultracold neutral atoms using N resonantly Raman-coupled internal degrees of freedom. Although counterpropagating lasers with wavelength λ provided two-photon Raman coupling, the resultant lattice period was λ/2N, an N-fold reduction as compared to the conventional λ/2 lattice period. We experimentally demonstrated this lattice built from the three F = 1 Zeeman states of a 87Rb Bose-Einstein condensate, and generated a lattice with a λ/6 = 132 nm period from λ = 790 nm lasers. Lastly, we show that adding an additional rf-coupling field converts this lattice into a superlattice with N wells uniformly spaced within the original λ/2 unit cell.

Emergent Periodic and Quasiperiodic Lattices on Surfaces of Synthetic Hall Tori and Synthetic Hall Cylinders

Synthetic spaces allow physicists to bypass constraints imposed by certain physical laws in experiments. Here, we show that a synthetic torus, which consists of a ring trap in the real space and internal states of ultracold atoms cyclically coupled by Laguerre-Gaussian Raman beams, could be threaded by a net effective magnetic flux through its surface-an impossible mission in the real space. Such a synthetic Hall torus gives rise to a periodic lattice in real dimensions, in which the periodicity of the density modulation of atoms fractionalizes that of the Hamiltonian. Correspondingly, the energy spectrum is featured by multiple bands grouping into clusters with nonsymmorphic-symmetry-protected band crossings in each cluster, leading to swaps of wave packets in Bloch oscillations. Our scheme allows physicists to glue two synthetic Hall tori such that localization may emerge in a quasicrystalline lattice. If the Laguerre-Gaussian Raman beams and ring traps were replaced by linear Raman beams and ordinary traps, a synthetic Hall cylinder could be realized and deliver many of the aforementioned phenomena.

Decay of a Quantum Knot

We experimentally study the dynamics of quantum knots in a uniform magnetic field in spin-1 Bose-Einstein condensates. The knot is created in the polar magnetic phase, which rapidly undergoes a transition toward the ferromagnetic phase in the presence of the knot. The magnetic order becomes scrambled as the system evolves, and the knot disappears. Strikingly, over long evolution times, the knot decays into a polar-core spin vortex, which is a member of a class of singular SO(3) vortices. The polar-core spin vortex is stable with an observed lifetime comparable to that of the condensate itself. The structure is similar to that predicted to appear in the evolution of an isolated monopole defect, suggesting a possible universality in the observed topological transition.

Braiding and Fusion of Non-Abelian Vortex Anyons

We have studied topology and dynamics of quantum vortices in spin-2 Bose-Einstein condensates. By computationally modeling controllable braiding and fusion of these vortices, we have demonstrated that certain vortices in such spinor condensates behave as non-Abelian anyons. We identify these anyons as fluxon, chargeon, and dyon quasiparticles. The pertinent anyon models are defined by the quantum double of the underlying discrete non-Abelian symmetry group of the condensate ground state order parameter.

Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit

A Berry curvature is an imaginary component of the quantum geometric tensor (QGT) and is well studied in many branches of modern physics; however, the quantum metric as a real component of the QGT is less explored. Here, by using tunable superconducting circuits, we experimentally demonstrate two methods to directly measure the quantum metric tensor for characterizing the geometry and topology of underlying quantum states in parameter space. The first method is to probe the transition probability after a sudden quench, and the second one is to detect the excitation rate under weak periodic driving. Furthermore, based on quantum metric and Berry-curvature measurements, we explore a topological phase transition in a simulated time-reversal-symmetric system. The work opens up a unique approach to explore the topology of quantum states with the QGT.

Topological one-way fiber of second Chern number

One-way waveguides have been discovered as topological edge states in two-dimensional (2D) photonic crystals. Here, we design one-way fiber modes in a 3D magnetic Weyl photonic crystal realizable at microwave frequencies. We first obtain a 3D Chern crystal with a non-zero first Chern number by annihilating the Weyl points through supercell modulation. When the modulation becomes helixes, one-way modes develop along the winding axis, with the number of modes determined by the spatial frequency of the helix. These single-polarization single-mode and multi-mode one-way fibers, having nearly identical group and phase velocities, are topologically-protected by the second Chern number in the 4D parameter space of the 3D wavevectors plus the winding angle of the helix. This work suggests a unique way to utilize high-dimensional topological physics using topological defects.

Topological one-way fibers are promising candidates for novel fiber devices. Here, Lu et al. propose that one-way fiber modes are topologically protected by the second Chern number in a four-dimensional parameter space, which develop in a helically-modulated magnetic Weyl photonic crystal.

Revealing Tensor Monopoles through Quantum-Metric Measurements

Monopoles are intriguing topological objects, which play a central role in gauge theories and topological states of matter. While conventional monopoles are found in odd-dimensional flat spaces, such as the Dirac monopole in three dimensions and the non-Abelian Yang monopole in five dimensions, more exotic objects were predicted to exist in even dimensions. This is the case of "tensor monopoles," which are associated with tensor (Kalb-Ramond) gauge fields, and which can be defined in four-dimensional flat spaces. In this work, we investigate the possibility of creating and measuring such a tensor monopole in condensed-matter physics by introducing a realistic three-band model defined over a four-dimensional parameter space. Our probing method is based on the observation that the topological charge of this tensor monopole, which we relate to a generalized Berry curvature, can be directly extracted from the quantum metric. We propose a realistic three-level atomic system, where tensor monopoles could be generated and revealed through quantum-metric measurements.