Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions

Chen, S.Y.; Bouhon, Adrien; Slager, Robert-Jan; Monserrat, Bartomeu

doi:10.1103/physrevb.105.l081117

Cited by 29 publications

(16 citation statements)

References 78 publications

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“…Utilizing previous, albeit slightly technically involved, work [50] that addresses multi-gap topological parametrizations using homotopy perspectives, we here derive explicit models that can be readily used as a benchmark for experimental and theoretical pursuits. Recent interest on both these fronts, exemplified by trapped-ion experiments [52] that verified predicted multi-gap topological signatures [51] and an ever increasing interest in theoretical predictions and characterizations [54,55,57,60,61,66], suggests that these results may be anticipated to be of general interest as well as of use to further progress this nascent research field, for example by considering different additional symmetries or (non-Hermitian) extensions. We here take an illustrative first step in uncovering this rich panorama by discussing possible descendant Chern-valued phases upon including specific symmetry-breaking terms, which could for example also flourish in the context of magnetism [58].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…In that case however, the system must be spinless.) Because the anti-unitary symmetry squares to the identity, it can be shown (through the Takagi factorization of the symmetric unitary matrix that represents C 2 T in the Bloch orbital basis, see below) that there exists a special basis for which the Bloch Hamiltonian matrix is real and symmetric [44,57]. In the following we call it the reality condition of Euler phases.…”

Section: Geometric and Homotopic Modeling Of Orientable Euler Phasesmentioning

confidence: 99%

“…Such Euler class models are increasingly becoming of importance and have for example been proposed to induce monopole-antimonople generation in quench setups [51], while the observation of this physical observable has just been reported in trapped-ion experiments [52]. In addition, the braiding and emerging of such non-Abelian charges and its relation to Euler class have also been inspiring pursuits in other experimental contexts that range from from phononic systems [53][54][55][56] and electronic systems [38,44,[57][58][59][60][61] to acoustic, photonic and electric circuit metamaterials [62][63][64][65][66].…”

Section: Introductionmentioning

confidence: 99%

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Multi-gap topological conversion of Euler class via band-node braiding: minimal models, $PT$-linked nodal rings, and chiral heirs

Bouhon¹,

Slager²

2022

Preprint

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The past few years have seen rapid progress in characterizing topological band structures using symmetry eigenvalue indicated methods. Recently, however, there has been increasing theoretical and experimental interest in multi-gap dependent topological phases that cannot be captured by this paradigm. These topologies arise by braiding band degeneracies that reside between different bands and carry non-Abelian charges due to the presence of either C2T or P T symmetry, culminating in different invariants such as Z-valued Euler class. Here, we present a universal formulation for Euler phases motivated by their homotopy classification that is related to the Skyrmion-profile of a single unit-vector in three-level systems, and that of two unit-vectors in four-level systems. In addition, upon employing the strategy of systematically building 3D models from a pair of sub-dimensional Euler phases, we show that phase transitions between any two inequivalent Euler phases are mediated by the presence of adjacent (in-gap) nodal rings linked with sub-gap nodal lines, forming trajectories corresponding to the braiding or debraiding of nodal points. The stability of the linked adjacent nodal rings is furthermore demonstrated to be indicated by an Euler class monopole charge matching with its Z-valued linking numbers. We finally also systematically address the conversion of Euler phases into descendant Chern phases upon breaking the C2T or P T symmetry. All the topological phases discussed in this work are corroborated with explicit minimal lattice models. These models can themselves directly serve as an extra impetus for experimental searches or be employed for theoretical studies, thereby underpinning the upcoming of this nascent pursuit.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Section: Geometric and Homotopic Modeling Of Orientable Euler Phasesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multi-gap topological conversion of Euler class via band-node braiding: minimal models, $PT$-linked nodal rings, and chiral heirs

Bouhon¹,

Slager²

2022

Preprint

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“…This phenomenon was recently observed in trapped ion experiments [34]. Similarly, Euler class and non-trivial braiding was predicted in phononic systems [35][36][37][38] and electronic systems that are strained [25,39], that undergo a structural phase transition [40,41], or are submitted to an external magnetic field [42]. The most immediate playground for these uncharted topological phases of matter is however the context of metamaterials [43][44][45][46][47][48].…”

mentioning

confidence: 57%

Experimental observation of meronic topological acoustic Euler insulators

Jiang¹,

Bouhon²,

Wu³

et al. 2022

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We report on the experimental observation of a gapped nontrivial Euler topology in an acoustic metamaterial. The topological invariant in this case is the Euler class, which characterizes the twodimensional geometry of the acoustic Bloch wave function of multiple bands in the Brillouin zone-a feature that cannot be captured by conventional topological classification schemes. Starting from a carefully designed acoustic system, we retrieve a rich phase diagram that includes the gapped Euler phase on top of non-Abelian topological nodal phases. In particular we find that the gapped phase is characterized by a meron (half-skyrmion) type topology in which the cancellation of the orbital Zak phases by Zak phases of the bands contributes in forming the characterizing winding number of the acoustic bands. Using pump-probe techniques, we are able to extract the wave functions of the acoustic Bloch bands and demonstrate the non-trivial Euler topology via a direct observation of the meron geometry of these bands in the Brillouin zone, in addition to the detection of the acoustic bulk and edge dispersions. We finally observe a topological edge state in the gapped Euler phase which is a direct manifestation of the underlying cancellation of the Zak phases. These experimental features reveal the unprecedented topological Euler phase as well as setting a benchmark for probing acoustic wave function geometry in the Brillouin zone.

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“…That is, a non-zero integer-valued Euler class indicates that the band touchings in the patch are topologically obstructed to annihilate. Multi-gap topologies have been predicted to culminate in new effects such as specific monopole-antimonopole generation [30] that was recently observed in trapped ion experiments [31] as well as signatures in structural phase transitions [32,33], strained and magnetic electronic systems [24,34], and phononic systems [27,28,[35][36][37].…”

mentioning

confidence: 99%

Phase transitions of non-Abelian charged nodal links in a spring-mass system

Park,

Wong,

Bouhon

et al. 2022

Preprint

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Although a large class of topological materials have uniformly been identified using symmetry properties of wave functions, the past two years have seen the rise of multi-gap topologies beyond this paradigm. Given recent reports of unexplored features of such phases, platforms that are readily implementable to realize them are therefore desirable. Here, we demonstrate that multi-gap topological phase transitions of non-Abelian charged nodal lines arise in classical phonon waves. By adopting a simple spring-mass system, we construct nodal lines of a three-band system. The braiding process of the nodal lines is readily performed by adjusting the spring constants. The generation and annihilation of the nodal lines are then analyzed using Euler class. Finally, we retrieve topological transitions from trivial nodal lines to a nodal link. Our work provides a simple platform that can offer diverse insights to not only theoretical but also experimental studies on multi-gap topology.

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Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions

Cited by 29 publications

References 78 publications

Multi-gap topological conversion of Euler class via band-node braiding: minimal models, $PT$-linked nodal rings, and chiral heirs

Multi-gap topological conversion of Euler class via band-node braiding: minimal models, $PT$-linked nodal rings, and chiral heirs

Experimental observation of meronic topological acoustic Euler insulators

Phase transitions of non-Abelian charged nodal links in a spring-mass system

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