Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Qi, Yang, and Liang Fu. "Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators." Phys. Rev. Lett. 115, 236801 (December 2015). © 2015 American Physical Society As Published http://dx.doi.org/10.1103/PhysRevLett.115.236801 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/100230 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 115, 236801 (2015) PHYSICAL REVIEW LETTERS 4 DECEMBER 2015 Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators Yang Qi1,2 and Liang Fu3 1Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China 2Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 9 June 2015; revised manuscript received 28 October 2015; published 2 December 2015) The surface of a three-dimensional topological electron system often hosts symmetry-protected gapless surface states. With the effect of electron interactions, these surface states can be gapped out without symmetry breaking by a surface topological order, in which the anyon excitations carry anomalous symmetry fractionalization that cannot be realized in a genuine two-dimensional system. We show that for a mirror-symmetry-protected topological crystalline insulator with mirror Chern number n ¼ 4, its surface can be gapped out by an anomalous Z2 topological order, where all anyons carry mirror-symmetry fractionalization M2 ¼ −1. The identification of such anomalous crystalline symmetry fractionalization 2 implies that in a two-dimensional Z2 spin liquid, the vison excitation cannot carry M ¼ −1 if the spinon carries M2 ¼ −1 or a half-integer spin. DOI: 10.1103/PhysRevLett.115.236801 PACS numbers: 73.20.-r, 03.65.Vf, 05.30.Pr, 75.10.Kt The advent of topological insulators (TIs) [1–3] and the mirror symmetry [18,19], confirming the mechanism topological superconductors (TSCs) [4] has greatly broad- of crystalline protection unique to TCIs [14]. ened our understanding of topological phases in quantum The study of interacting TCIs has just begun. A recent systems. While the concepts of TIs and TSCs originate work by Isobe and Fu [20] shows that in the presence of from topological band theory of noninteracting electrons interactions, the classification of 3D TCIs protected by or quasiparticles, recent theoretical breakthroughs [5–10] mirror symmetry (i.e., the SnTe class) reduces from being have found that interactions can, in principle, change the characterized by an integer known as the mirror Chern fundamental properties of these topological phases dra- number [21] (hereafter denoted by n) to its Z8 subgroup. matically, thus, creating a new dimension to explore. In This implies that interactions can turn the n ¼ 8 surface particular, interactions can drive the gapless Dirac fermion states, which consist of eight copies of 2D massless surface states of three-dimensional (3D) TIs and TSCs Dirac fermions with the same chirality, into a completely into topologically ordered phases that are gapped and trivial phase that is gapped, mirror symmetric, and symmetry preserving. Nonetheless, such a surface mani- without intrinsic topological order. It remains an open fests the topological property of the bulk in a subtle but question what interactions can do to TCIs with unambiguous way: its anyon excitations have anomalous n ≠ 0 mod 8. In this work, we take the first step to study symmetry transformation properties, which cannot be strongly interacting TCI surface states for the case realized in any two-dimensional (2D) system with the n ¼ 4 mod 8. same symmetry. Our main result is that the surface of a 3D TCI with Given the profound consequences of interactions in TIs mirror Chern number n ¼ 4 mod 8 can become a gapped and TSCs, the effect of interactions in topological phases and mirror-symmetric state with Z2 topological order. protected by spatial symmetries of crystalline solids, Remarkably, the mirror symmetry acts on this state in commonly referred to as topological crystalline insulators an anomalous way that all three types of anyons carry 2 (TCIs) [11], is now gaining wide attention. A wide array of fractionalized mirror quantum number M~ ¼ −1 (in this TCI phases with various crystal symmetries have been Letter we use M~ to represent the projective representation found in the framework of topological band theory [12,13]. of mirror symmetry M acting on an anyon), which cannot One class of TCIs has been predicted and observed in the be realized in a purely 2D system. Furthermore, the IV-VI semiconductors SnTe, Pb1−xSnxSe, and Pb1−xSnxTe anomalous mirror-symmetry fractionalization protects a [14–17]. The topological nature of these materials is twofold degeneracy between two mirror-symmetry-related warranted by a particular mirror symmetry of the under- edges. Such anomalous mirror-symmetry fractionalization lying rocksalt crystal and is manifested by the presence cannot be realized in a 2D system, including a 2D Z2 spin of topological surface states on mirror-symmetric crystal liquid state [22]. Hence, our finding constrains the possible faces. Remarkably, there surface states were found to ways of fractionalizing the mirror symmetry in a 2D Z2 become gapped under structural distortions that break spin liquid [23,24]. Brief reviews of 3D TCIs, 2D Z2 spin 0031-9007=15=115(23)=236801(5) 236801-1 © 2015 American Physical Society week ending PRL 115, 236801 (2015) PHYSICAL REVIEW LETTERS 4 DECEMBER 2015 liquids, and their edge theory are available in the (this is demonstrated with a 2D lattice model in Sec. V of Supplemental Material [25]. the Supplemental Material [25]). Noninteracting TCIs.—We begin by considering non- Uð1Þ Higgs phase and Z2 topological order.—In this interacting TCIs protected by the mirror symmetry x → −x. work, we study interacting surface states of TCIs with With the mirror symmetry, the extra Uð1Þ symmetry in a n ¼ 4. Starting from four copies of Dirac fermions in the TCI does not change the classification in 3D compared to a noninteracting limit, we will introduce microscopic inter- mirror-protected topological crystalline superconductor. actions and explicitly construct a Z2 topologically ordered Hence, for convenience, we choose a TCI as our starting phase on the TCI surface, which is gapped and mirror point, although the Uð1Þ symmetry plays no role in this symmetric. work. As we will explain in Sec. II of the Supplemental Our construction is inspired by the work of Senthil and Material [25], in order to produce an anomalous Z2 surface Fisher [30] and Senthil and Motrunich [30,31] on frac- topological order, the mirror operation must be defined as a tionalized insulators. We construct on the surface of an 2 Z2 symmetry with the property M ¼ 1. In our previous n ¼ 4 TCI a Higgs phase with an xy-order parameter works on spin-orbit coupled systems, the mirror operation hbi ≠ 0, which is odd under the mirror symmetry and gaps M0 acts on the electron’s spin in addition to its spatial the Dirac fermions. Next, we couple these gapped fermions 02 coordinate, which leads to M ¼ −1. Nonetheless, one to additional degrees of freedom aμ that are introduced to 0 can redefine the mirror operation by combining M with mimic a Uð1Þ gauge field. This gauge field aμ plays three the Uð1Þ symmetry of charge conservation c → ic, which crucial roles: (i) the coupling between matter and aμ restores the property M2 ¼ 1. We note that without the restores the otherwise broken Uð1Þ symmetry and, thus, Uð1Þ symmetry, only M satisfying M2 ¼þ1 protects the mirror symmetry along with it; (ii) the Goldstone mode nontrivial topological crystalline superconductors. is eaten by the gauge boson and becomes massive; (iii) since The mirror TCIs are classified by the mirror Chern the xy-order parameter carries Uð1Þ charge 2, the Uð1Þ number n defined for single-particle states on the mirror- gauge group is broken to the Z2 subgroup in the Higgs symmetric plane kx ¼ 0 in the 3D Brillouin zone. The phase. Because of these properties, the Higgs phase thus states with mirror eigenvalues 1 and −1 form two different constructed is a gapped and mirror-symmetric phase with subspaces, each of which has a Chern number denoted by Z2 topological order. nþ and n−, respectively. This leads to two independent We now elaborate on the construction (details of this topological invariants for noninteracting systems with construction can be found in Sec. III of the Supplemental mirror symmetry: the total Chern number nT ¼ nþ þ n− Material [25]). First, we relabel the fermion flavors and the mirror Chern number n ¼ nþ − n−. A ¼ 1; …; 4 using a spin index s ¼ ↑; ↓ and a Uð1Þ-charge The TCI with a nontrivial mirror Chern number n has index a ¼ (unrelated to the electric charge). We take gapless surface states consisting of n copies of massless fermion interactions that are invariant under both the SUð2Þ Dirac fermions described by the following surface spin rotation and the Uð1Þ rotation Hamiltonian iaθ Xn Uð1Þ∶ ψ as → e ψ as;a¼: ð3Þ ¼ ψ † ð σ − σ Þψ ð Þ Hs v A kx y ky x A; 1 ¼1 A Moreover, we introduce a boson field bðx; yÞ that carries Uð1Þ-charge 2 and is odd under mirror symmetry, where the two-dimensional fermion fields ψ Aðx; yÞ trans- form as the following under mirror operation: Uð1Þ∶ b → ei2θb; M∶ bðx; yÞ → −bð−x; yÞ; ð4Þ M∶ψ Aðx; yÞ → σxψ Að−x; yÞ: ð2Þ The presence of mirror symmetry (2) forbids any Dirac and couple this boson to the massless Dirac fermions as ψ † σ ψ follows: mass term A z B.