Intertwined Weyl phases emergent from higher-order topology and unconventional Weyl fermions via crystalline symmetry

www.nature.com/npjquantmats ARTICLE OPEN Intertwined Weyl phases emergent from higher-order topology and unconventional Weyl fermions via crystalline symmetry W. B. Rui 1✉ , Zhen Zheng 1 , Moritz M. Hirschmann 2 , Song-Bo Zhang3, Chenjie Wang 1✉ and Z. D. Wang1 ✉ We discover three-dimensional intertwined Weyl phases, by developing a theory to create topological phases. The theory is based on intertwining existing topological gapped and gapless phases protected by the same crystalline symmetry. The intertwined Weyl phases feature both unconventional Weyl semimetallic (monopole charge>1) and higher-order topological phases, and more importantly, their exotic intertwining. While the two phases are independently stabilized by the same symmetry, their intertwining results in the specific distribution of them in the bulk. The construction mechanism allows us to combine different kinds of unconventional Weyl semimetallic and higher-order topological phases to generate distinct phases. Remarkably, on 2D surfaces, the intertwining causes the Fermi-arc topology to change in a periodic pattern against surface orientation. This feature provides a characteristic and feasible signature to probe the intertwining Weyl phases. Moreover, we provide guidelines for searching candidate materials, and elaborate on emulating the intertwined double-Weyl phase in cold-atom experiments. 1234567890():,; npj Quantum Materials (2022)7:15 ; https://doi.org/10.1038/s41535-022-00422-0 INTRODUCTION Owing to fertile ground of crystalline symmetries, topological gapless phases, characterized by nontrivial band degeneracies, have been undergoing rapid development in condensed matter physics1–3. A representative example is the diversification of prototypical Weyl phases. Beyond conventional Weyl fermions with monopole charge ± 14–9, unconventional Weyl fermions, which possess a higher monopole charge due to crystalline symmetry, were discovered10–19. For instance, threefold Weyl fermions with charge ± 2 can be stabilized by a nonsymmorphic symmetry10–13, and double(triple)-Weyl fermions with a quadratic (cubic) twofold degeneracy can be protected by a rotation symmetry14–19. On the other hand, crystalline symmetries have largely enriched topological gapped phases20–31. Among these phases, higher-order topological (HOT) insulators, which feature anomalous boundary states, have drawn particular attention recently32–45. In contrast to conventional bulk-boundary correspondence, i.e., a d-dimensional bulk topology corresponds to (d − 1)-dimensional boundary states, the boundary states of HOT insulators are further restricted by crystalline symmetries and exhibit boundary states in a lower dimension, e.g., corner or hinge states. Recently, it has been found that HOT semimetals (e.g., Dirac, Weyl, and nodal-line) can be realized by nontrivially stacking higher-order insulators46–58. Specifically, higher-order Weyl semimetals were discovered by stacking two-dimensional (2D) HOT insulators, with broken time-reversal and/ or inversion symmetry, in a third dimension50–55. Note that in this process, for electronic systems, the Kramers degeneracy of the original HOT insulators, if any, has to be broken before stacking50. In this regard, the higher-order Weyl semimetal phase depends on the 2D sub higher-order phases. The phase transition between these phases results in the higher-order Weyl points, which are typically of conventional type with charge ± 1. Though topological gapless phases characterized by nontrivial band degeneracies, and topological gapped phases featuring 1 anomalous boundary states, can be stabilized by the same crystalline symmetry, so far, their interplay remains to be explored. In this work, we develop a theory to generate topological phases of matter, by intertwining existing ones protected by a crystalline symmetry. We discover intertwined Weyl phases, which feature the exotic interplay of unconventional Weyl fermions and higher-order topology. In the bulk, the two phases are independently stabilized by the crystalline symmetry: the Weyl fermions have a monopole charge larger than 1 by the symmetry, and a HOT phase was further superposed by the symmetry. The intertwining results in a specific distribution of the two phases in the bulk, i.e., the higher-order topology exists in the region outside pairs of unconventional Weyl points with opposite charges. Due to their independence, the two phases are separately tunable. The combination of different unconventional Weyl semimetallic and HOT phases results in distinct intertwined Weyl phases. Note that this mechanism is different from the higher-order Weyl semimetal which is determined by the higherorder topology as discussed above. The intertwined Weyl phases are characterized by the drastic change of Fermi-arc topology on 2D surfaces upon rotating surface termination. This phenomenon comes from the intertwining between the two constituent phases. Even though HOT phases feature boundary states on 1D hinges, they can intertwine with the unconventional Weyl phase on 2D surfaces of the system: Fermi arcs form due to the underlying unconventional Weyl phases, while the topology of these Fermi arcs is drastically changed by the HOT phase. Thus, we identify a prominent topological feature of the intertwined phase: the change of Fermiarc topology against surface orientation in a periodic pattern. The period is determined by rotation symmetry, e.g., a 2n-fold rotation symmetry leads to a period of π/n. Specifically, we discuss intertwined double-Weyl phases, where double-Weyl fermions and higher-order topology are intertwined due to a fourfold rotation symmetry. The topological phase is characterized by the Department of Physics and HKU-UCAS Joint Institute for Theoretical and Computational Physics at Hong Kong, The University of Hong Kong, Pokfulam Road, Hong Kong, China. Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany. 3Institute for Theoretical Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany. ✉email: ; ; 2 Published in partnership with Nanjing University W.B. Rui et al. 2 periodic change of Fermi-arc topology with period π/2. The periodic behavior can be perfectly explained by an effective boundary Hamiltonian, in accordance with our theory. Finally, we show that intertwined double-Weyl phases are realizable in coldatom experiments, which can serve as a promising platform to implement our theory in experiment. RESULTS Intertwining topological phases by crystalline symmetry We begin by showing how to intertwine topological phases by crystalline symmetry. The essential physics can be captured by the following simple but generic Hamiltonian in 3D momentum space, HðkÞ ¼ HWeyl þ mΓh 1234567890():,; ¼ k nþ τ 3 σ  þ k n τ 3 σ þ þ k z τ 3 σ 3 þ mΓh ; (1) where k± = kx ± iky, σ± = (σ1 ± iσ2)/2, and σi and τi (i = 1, 2, 3) are Pauli matrices for (pseudo)spin and or (...truncated)