Chiral-symmetric higher-order topological phases protected by multipole winding number invariants.
Abstract not provided.
Cerjan, Alexander↗
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Results for “Topological Protection”
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From trivial to topological paramagnets: The case of Z 2 and Z 2 3 symmetries in two dimensions
Using quantum Monte Carlo simulations, we map out the phase diagram of Hamiltonians interpolating between trivial and nontrivial bosonic symmetry-protected topological phases, protected by Z 2 and Z 2 3 symmetries, in two dimensions. In all cases, we find that the trivial and the topological phases are separated by an intermediate phase in which the protecting symmetry is spontaneously broken. Depending on the model, we identify a variety of magnetic orders on the triangular lattice, including ferromagnetism, 3 × 3 order, and stripe orders (both commensurate and incommensurate). Critical properties are determined through a finite-size scaling analysis. Possible scenarios regarding the nature of the phase transitions are discussed.
2-dimensional systems↗
Dynamical topological phase realized in a trapped-ion quantum simulator
Nascent platforms for programmable quantum simulation offer unprecedented access to new regimes of far-from-equilibrium quantum many-body dynamics in almost isolated systems. Here achieving precise control over quantum many-body entanglement is an essential task for quantum sensing and computation. Extensive theoretical work indicates that these capabilities can enable dynamical phases and critical phenomena that show topologically robust methods to create, protect and manipulate quantum entanglement that self-correct against large classes of errors. However, so far, experimental realizations have been confined to classical (non-entangled) symmetry-breaking orders. In this work, we demonstrate an emergent dynamical symmetry-protected topological phase, in a quasiperiodically driven array of ten 171 Yb + hyperfine qubits in Quantinuum’s System Model H1 trapped-ion quantum processor. This phase shows edge qubits that are dynamically protected from control errors, cross-talk and stray fields. Crucially, this edge protection relies purely on emergent dynamical symmetries that are absolutely stable to generic coherent perturbations. Here, this property is special to quasiperiodically driven systems: as we demonstrate, the analogous edge states of a periodically driven qubit array are vulnerable to symmetry-breaking errors and quickly decohere. Our work paves the way for implementation of more complex dynamical topological orders that would enable error-resilient manipulation of quantum information.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Interface-Driven Chiral Magnetism in Ultrathin Metallic Ferromagnets: Towards Skyrmion Spintronics
Magnetic skyrmions are localized chiral magnetic textures in the form of nanoscale vortices or bubbles that are topologically protected from being ‘unwound’. Their topological nature gives rise to rich behaviors including ordered lattice formation, emergent electrodynamics and robust current-driven displacement at remarkably low current densities. However, magnetic skyrmions have so far been restricted to just a few materials and observed only at low temperatures, limiting the experimental accessibility and technological application of these unique topological objects. This project aimed to realize magnetic skyrmions at room temperature in a new class of engineered chiral ferromagnets that exploit broken mirror symmetry at interfaces to generate helical magnetic order. Ultrathin magnetic heterostructures were engineered and patterned into laterally-confined nanostructures in which skyrmions were stabilized, manipulated, and detected for the first time under ambient conditions. Static and dynamic properties were examined using advanced x-ray imaging to provide a fundamental understanding of topological magnetic phases. The interactions between magnetic skyrmions and electron charge and spin currents were studied to identify fundamental new physics and enable device applications. These experimental studies, supported by detailed modeling, will help launch a new subfield of skyrmion-based spintronics in which individual skyrmions can be used to encode, store, and transport information in high-performance, low power memory and logic devices.
36 MATERIALS SCIENCE↗
Symmetry-protected self-correcting quantum memory in three space dimensions
Whether self-correcting quantum memories can exist at nonzero temperature in a physically reasonable setting remains a great open problem. Furthermore, it has recently been argued that symmetry-protected topological (SPT) systems in three space dimensions subject to a strong constraint—that the quantum dynamics respect a 1-form symmetry—realize such a quantum memory. We illustrate how this works in Walker-Wang codes, which provide a specific realization of these desiderata. In this setting we show that it is sufficient for the 1-form symmetry to be enforced on a subvolume of the system. This strongly suggests that the SPT character of the state is not essential. We confirm this by constructing an explicit example with a trivial (paramagnetic) bulk that realizes a self-correcting quantum memory. We therefore show that the enforcement of a 1-form symmetry on a measure-zero subvolume of a three-dimensional system can be sufficient to stabilize a self-correcting quantum memory at nonzero temperature.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Anyonic Membranes and Pontryagin Statistics
Anyons, unique to two spatial dimensions, underlie extraordinary phenomena such as the fractional quantum Hall effect, but their generalization to higher dimensions has remained elusive. The topology of Eilenberg-MacLane spaces constrains the loop statistics to be only bosonic or fermionic in any dimension. In this work, we introduce the novel anyonic statistics for membrane excitations in four dimensions. Analogous to the $\mathbb{Z}_N$-particle exhibiting $\mathbb{Z}_{N\times \gcd(2,N)}$ anyonic statistics in two dimensions, we show that the $\mathbb{Z}_N$-membrane possesses $\mathbb{Z}_{N\times \gcd(3,N)}$ anyonic statistics in four dimensions. Given unitary volume operators that create membrane excitations on the boundary, we propose an explicit 56-step unitary sequence that detects the membrane statistics. We further analyze the boundary theory of $(5{+}1)$D 1-form $\mathbb{Z}_N$ symmetry-protected topological phases and demonstrate that their domain walls realize all possible anyonic membrane statistics. We then show that the $\mathbb{Z}_3$ subgroup persists in all higher dimensions. In addition to the standard fermionic $\mathbb{Z}_2$ membrane statistics arising from Stiefel-Whitney classes, membranes also exhibit $\mathbb{Z}_3$ statistics associated with Pontryagin classes. We explicitly verify that the 56-step process detects the nontrivial $\mathbb{Z}_3$ statistics in 5, 6, and 7 spatial dimensions. Furthermore, in 7 and higher dimensions, the statistics of membrane excitations stabilize to $\mathbb{Z}_{2} \times \mathbb{Z}_{3}$, with the $\mathbb{Z}_3$ sector consistently captured by this process.
Abstract algebra↗
Bulk‐Boundary Correspondence of Semimetal Ru 3 Sn 7 and Topological Surface States on Chemically Realistic Terminations
Ru 3 Sn 7 is experimentally demonstrated as an efficient catalyst, with potential utilization of topological surface states for hydrogen evolution reaction. Despite its promising catalytic performance, the topological nature of Ru 3 Sn 7 remains uncertain. Particularly, the bulk-boundary correspondence has not yet been established, hence hindering a rigorous justification of its topologically-protected surface states. In this work, the bulk topology of Ru 3 Sn 7 is detailed using first-principles calculations and the topological quantum chemistry formalism. Ru3Sn7 turns out to be an enforced semimetal possessing symmetry-protected crossings within a set of bands near the Fermi level, which are enforced and prescribed by the violations of symmetry-prescribed compatibility relations. Moreover, the surface states and the associated origin from the same set of entangled bands are identified, thereby establishing the bulk-boundary correspondence. To evaluate the effects of chemical modifications, the response of topological surface states to various surface terminations, stoichiometry, and oxidation is examined. The surface structures are globally optimized, and the phase diagrams for various experimental conditions are built. It is shown that, due to changes in the local chemical environment, the original surface states are significantly altered. Modified surface bands can be observed near the Fermi level on surface terminations that preserve the C 4v symmetry.
36 MATERIALS SCIENCE↗
Topological prethermal strong zero modes on superconducting processors
Abstract Symmetry-protected topological phases 1–4 cannot be described by any local order parameter and are beyond the conventional symmetry-breaking model 5 . They are characterized by topological boundary modes that remain stable under symmetry respecting perturbations 1–4,6–8 . In clean, gapped systems without disorder, the stability of these edge modes is restricted to the zero-temperature manifold; at finite temperatures, interactions with mobile thermal excitations lead to their decay 9–11 . Here we report the observation of a distinct type of topological edge mode 12–14 , which is protected by emergent symmetries and persists across the entire spectrum, in an array of 100 programmable superconducting qubits. Through digital quantum simulation of a one-dimensional disorder-free stabilizer Hamiltonian, we observe robust long-lived topological edge modes over up to 30 cycles for a wide range of initial states. We show that the interaction between these edge modes and bulk excitations can be suppressed by dimerizing the stabilizer strength, leading to an emergent U(1) × U(1) symmetry in the prethermal regime of the system. Furthermore, we exploit these topological edge modes as logical qubits and prepare a logical Bell state, which exhibits persistent coherence, despite the system being disorder-free and at finite temperature. Our results establish a viable digital simulation approach 15–18 to experimentally study topological matter at finite temperature and demonstrate a potential route to construct long-lived, robust boundary qubits in disorder-free systems.
Science & Technology - Other Topics↗
Local probe of bulk and edge states in a fractional Chern insulator
The fractional quantum Hall effect is a key example of topological quantum many-body phenomena, arising from the interplay between strong electron correlation, topological order and time-reversal symmetry breaking. Recently, a lattice analogue of the fractional quantum Hall effect at zero magnetic field has been observed, confirming the existence of a zero-field fractional Chern insulator (FCI). Despite this, the bulk–edge correspondence—a hallmark of a FCI featuring an insulating bulk with conductive edges—has not been directly observed. In fact, this correspondence has not been visualized in any system for fractional states owing to experimental challenges. Here we report the imaging of FCI edge states in twisted MoTe 2 (t-MoTe 2 ) using microwave impedance microscopy. By tuning the carrier density, we observe the system evolving between metallic and FCI states, the latter of which exhibits insulating bulk and conductive edges, as expected from the bulk–boundary correspondence. Further analysis suggests the composite nature of the FCI edge states. We also observe the evolution of edge states across the topological phase transition as a function of interlayer electric field and reveal exciting prospects of neighbouring domains with different fractional orders. Furthermore, these findings pave the way for research into topologically protected one-dimensional interfaces between various anyonic states at zero magnetic field, such as gapped one-dimensional symmetry-protected phases with non-zero topological entanglement entropy, Halperin–Laughlin interfaces and the creation of non-abelian anyons.
Imaging techniques↗
Topological orders of monopoles and hedgehogs: From electronic and magnetic spin-orbit coupling to quarks
Topological states of matter are, generally, quantum liquids of conserved topological defects. We establish this by constructing and analyzing topological field theories which introduce gauge fields to describe the dynamics of singularities in the original field configurations. In this work, homotopy groups are utilized to identify topologically protected singularities, and the conservation of their protected number is captured by a topological action term that unambiguously obtains from the given set of symmetries. Stable phases of these theories include quantum liquids with emergent massless Abelian and non-Abelian gauge fields, as well as topological orders with long-range quantum entanglement, fractional excitations, boundary modes, and unconventional responses to external perturbations. This paper focuses on the derivation of topological field theories and basic phenomenological characterization of topological orders associated with homotopy groups π n (S n ), n ≥ 1 . These homotopies govern monopole and hedgehog topological defects in d = n + 1 dimensions, and enable the generalization of both weakly interacting and fractional quantum Hall liquids of vortices to d > 2 . Hedgehogs have not been in the spotlight so far, but they are particularly important defects of magnetic moments because they can be stimulated in realistic systems with spin-orbit coupling, such as chiral magnets and d = 3 topological materials. We predict topological orders in systems with U(1) × Spin(d) symmetry in which fractional electric charge attaches to hedgehogs. Monopoles, the analogous defects of charge or generic U(1) currents, may bind to hedgehogs via Zeeman effect, or effectively emerge in purely magnetic systems. The latter can lead to spin liquids with different topological orders than that of the resonant valence bond spin liquid. Charge fractionalization of quarks in atomic nuclei is also seen as possibly arising from the charge-hedgehog attachment.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Custodial Chiral Symmetry in a Su-Schrieffer-Heeger Electrical Circuit with Memory
Custodial symmetries are common in the standard model of particle physics. They arise when quantum corrections to a parameter are proportional to the parameter itself. Here, we show that a custodial symmetry of the chiral type is also present in a classical Su-Schrieffer-Heeger (SSH) electrical circuit with memory. In the absence of memory, the SSH circuit supports a symmetry-protected topological edge state. Memory induces nonlinearities that break chiral symmetry explicitly and spread the state across the circuit. However, the resulting state is still protected against perturbations by the ensuing custodial chiral symmetry. Furthermore, these predictions can be verified experimentally and demonstrate the interplay between symmetry and memory.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Classification of (2 + 1) D invertible fermionic topological phases with symmetry
Here we provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups G f and general values of the chiral central charge c - . Here G f is a central extension of a bosonic symmetry group G b by fermion parity, (-1) F , specified by a second cohomology class [ω 2 ]∈ $\mathscr{H}^2$(Gb,$\mathbb{Z}_2$). Our approach proceeds by gauging fermion parity and classifying the resulting G b symmetry-enriched topological orders while keeping track of certain additional data and constraints. We perform this analysis through two perspectives, using G-crossed braided tensor categories and Spin(2c - ) 1 Chern-Simons theory coupled to a background G gauge field. These results give a way to characterize and classify invertible fermionic topological phases in terms of a concrete set of data and consistency equations, which is more physically transparent and computationally simpler than the more abstract methods using cobordism theory and spectral sequences. Our results also generalize and provide a different approach to the recent classification of fermionic symmetry-protected topological phases by Wang and Gu, which have chiral central charge c - = 0. We show how the tenfold way classification of topological insulators and superconductors fits into our scheme, along with general nonperturbative constraints due to certain choices of c - and G f . Mathematically, our results also suggest an explicit general parametrization of deformation classes of (2 + 1)D invertible topological quantum field theories with G f symmetry.
36 MATERIALS SCIENCE↗
Topological order and entanglement dynamics in the measurement-only XZZX quantum code
We examine the dynamics of a ( 1 + 1 ) -dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a D 2 = Z 2 × Z 2 symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a twofold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent ν = 4 3 and the dynamical critical exponent z = 1 are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take noninteger multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of L qubits. It reveals a modified dynamical critical exponent z * ≠ z , which is observable up to times t ~ L z * and is reminiscent of the relaxation of critical systems into a prethermal state.
36 MATERIALS SCIENCE↗
Fabrication of Arrays of Topological Solitons in Patterned Chiral Liquid Crystals for Real-Time Observation of Morphogenesis
Topological solitons have knotted continuous field configurations embedded in a uniform background, and occur in cosmology, biology, and electromagnetism. However, real-time observation of their morphogenesis and dynamics is still challenging because their size and timescale are enormously large or tiny. Liquid crystal (LC) structures are promising candidates for a model-system to study the morphogenesis of topological solitons, enabling direct visualization due to the proper size and timescale. We report a new way is found to rationalize the real-time observation of the generation and transformation of topological solitons using cholesteric LCs confined in patterned substrates. The experimental demonstration shows the topologically protected structures arise via the transformation of topological defects. Numerical modeling based on minimization of free energy closely reconstructs the experimental findings. The fundamental insights obtained from the direct observations pose new theoretical challenges in understanding the morphogenesis of different types of topological solitons within a broad range of scales.
36 MATERIALS SCIENCE↗
Observation of Persistent Zero Modes and Superconducting Vortex Doublets in UTe 2
Superconducting vortices can reveal electron pairing details and nucleate topologically protected states. Yet, vortices of bulk spin-triplet superconductors have never been visualized at the atomic scale. Recently, UTe 2 has emerged as a prime spin-triplet superconductor, but its superconducting order parameter is elusive, and whether time-reversal symmetry (TRS) is broken remains unsettled. Here, in this work, we visualize vortices on the (011) surface of ultraclean UTe 2 single crystals (T c = 2.1 K) using scanning tunneling microscopy (STM). We introduce $\frac{d^{2}𝐼}{d𝑉^{2}}$ imaging as an effective technique for vortex visualization in superconductors with substantial residual zero-energy density of states (DOS), as in UTe 2 . Anisotropic single-flux-quantum vortices, with coherence lengths of ∼12 nm (4 nm) parallel (perpendicular) to the a -axis, form a triangular vortex lattice (VL) under a small out-of-plane magnetic field. The invariance of vortex structures and VL under changes of field polarity and cooling history strongly supports time-reversal invariant superconductivity under zero field. At vortex cores (VCs), nonsplit, spectrally sharp zero-bias conductance peaks (ZBPs) persist up 8 T that are consistent with symmetry-protected Majorana zero modes (MZMs) in a topological vortex line. Close examination of vortex structures reveals a mirror-asymmetric doublet─one with ZBPs and another with an enhanced apparent gap, possibly originating from a field-induced multicomponent order parameter.
UTe2↗
Topological nanospaser
We propose a nanospaser made of an achiral plasmonic–metal nanodisk and a two-dimensional chiral gain medium – a monolayer nanoflake of a transition-metal dichalcogenide (TMDC). When one valley of the TMDC is selectively pumped (e.g. by a circular-polarized radiation), the spaser (surface plasmon amplification by stimulated emission of radiation) generates a mode carrying a topological chiral charge that matches that of the gain valley. There is another, chirally mismatched, time-reversed mode with exactly the same frequency but the opposite topological charge; it is actively suppressed by the gain saturation and never generates, leading to a strong topological protection for the generating matched mode. This topological spaser is promising for use in nano-optics and nanospectroscopy in the near field especially in applications to biomolecules that are typically chiral. Another potential application is a chiral nanolabel for biomedical applications emitting in the far field an intense circularly polarized coherent radiation.
36 MATERIALS SCIENCE↗
Hofstadter Topology: Noncrystalline Topological Materials at High Flux
he Hofstadter problem is the lattice analog of the quantum Hall effect and is the paradigmatic example of topology induced by an applied magnetic field. Conventionally, the Hofstadter problem involves adding ~10 4 T magnetic fields to a trivial band structure. In this work, we show that when a magnetic field is added to an initially topological band structure, a wealth of possible phases emerges. Remarkably, we find topological phases that cannot be realized in any crystalline insulators. We prove that threading magnetic flux through a Hamiltonian with a nonzero Chern number or mirror Chern number enforces a phase transition at fixed filling and that a 2D Hamiltonian with a nontrivial Kane–Mele invariant can be classified as a 3D topological insulator (TI) or 3D weak TI phase in periodic flux. We then study fragile topology protected by the product of twofold rotation and time reversal and show that there exists a higher order TI phase where corner modes are pumped by flux. We show that a model of twisted bilayer graphene realizes this phase. Our results rely primarily on the magnetic translation group that exists at rational values of the flux. The advent of Moiré lattices renders our work relevant experimentally. Due to the enlarged Moiré unit cell, it is possible for laboratory-strength fields to reach one flux per plaquette and allow access to our proposed Hofstadter topological phase.
36 MATERIALS SCIENCE↗
Obstructions to gapped phases from noninvertible symmetries
Quantum systems in 3+1 dimensions that are invariant under gauging a one-form symmetry enjoy novel noninvertible duality symmetries encoded by topological defects. These symmetries are renormalization group invariants which constrain dynamics. We show that such noninvertible symmetries often forbid a symmetry-preserving vacuum state with a gapped spectrum. In particular, we prove that a self-dual theory with $\mathbb{Z}^{(1)}_N$ one-form symmetry is gapless or spontaneously breaks the self-duality symmetry unless N = k 2 ℓ where –1 is a quadratic residue modulo . We also extend these results to noninvertible symmetries arising from invariance under more general gauging operations including, e.g., triality symmetries. Along the way, we discover how duality defects in symmetry-protected topological phases have a hidden time-reversal symmetry that organizes their basic properties. These noninvertible symmetries are realized in lattice gauge theories, which serve to illustrate our results.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗