Anomalous Crystal Symmetry Fractionalization on the Surface of Topological Crystalline Insulators

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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

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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 transform 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. As a result, the surface states described by Eq. (1) cannot be gapped by fermion bilinear terms, for any flavor number n. † † − H ¼ Vb ψ τ σ ψ þ H:c: ð5Þ We emphasize that the above Dirac fermions on the bf as ab z bs surface of a 3D TCI cannot be realized in any 2D system with mirror symmetry, as expected for symmetry-protected When these bosons condense, hbi ≠ 0 spontaneously topological phases in general. According to the breaks both the Uð1Þ and mirror symmetry, and gaps Hamiltonian (1), the surface states with kx ¼ 0 within a out the fermions. given mirror subspace are chiral as they all move in the Finally, we introduce another boson vector field aμðx; yÞ, same direction [14]. In contrast, in any 2D system single- which couples to b and ψ as through minimal coupling. particle states within a mirror subspace cannot be chiral An effective theory of this system has the following form,

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† μ † þ L ¼ −iψ s α ð∂μ þ iaμτzÞψ s þðbψ s τ σzψ s þ H:c:Þ fermions. As the Higgs field gaps out four Dirac fermions, 1 there are four Majorana fermions or two complex fermion þ jð∂ − 2 Þ j2 þ j j2 þ j j4 þ μν ð Þ zero modes in each vortex core. Hence, there are two types 2 μ iaμ b r b u b FμνF ; 6 g of vortices whose core has even or odd fermion parity, respectively. In the case of n ¼ 4, it can be shown that α0 ¼ 1 αx ¼ σ αy ¼ σ where the matrices , y, and x. the vortices carry Bose statistics (see Sec. IV of the Furthermore, we add to the effective action an interation Supplemental Material [25] for details), and they are 2 ¼ ψ †τ ψ þ 2 † − ∇ E ¼ ¼ term UN , where N z b b · (Ei F0i mapped to the m and e anyons in the Z2 topological order, ∂ − ∂ 0ai ia0 is the electric field strength). In the limit of respectively. U → ∞, this enforces the local constraint N ¼ 0.Asa Mirror-symmetry fractionalization.—Now we consider ψ result, the bare fermion s and boson b are no longer low- how the mirror symmetry acts in the Z2 spin liquid energy excitations, since adding them to the ground state phase described by Eq. (7). In this effective theory, the violates the constraint N ¼ 0 and costs an energy U. ψ~ field is the fermionic anyon ϵ. Equation (9) implies that ψ ~ 2 Therefore, in the low-energy effective model, s and b it carries M ¼ −1. must be screened by the gauge field aμ and become Next, we consider how the mirror symmetry acts on the iaθ ~ 2iθ quasiparticles ψ~ as ¼ ψ ase and b ¼ be , where the m anyon, which is a vortex of the Higgs field where all core inθ operator e creates n gauge charge of aμ and restores states are empty. Since the mirror symmetry preserves the ~ the constraint N ¼ 0. In terms of these quasiparticles, the Higgs field hbi but maps x to −x, it maps a vortex to an effective theory becomes antivortex. Therefore, we consider a mirror-symmetric configuration with one vortex and one antivortex, as shown † μ ~ † þ ~ 2 ¼1 L ¼ −iψ~ s α ð∂μ þ iaμτzÞψ~ s þðbψ~ s τ σzψ~ s þ H:c:Þ in Fig. 1(a). The symmetry fractionalization of M 1 can be detected from the M parity of the fermion wave ~ 2 ~ 2 ~ 4 μν þ jð∂μ − 2iaμÞbj þ rjbj þ ujbj þ FμνF : ð7Þ function with such a vortex-antivortex pair [34]: for two 2g 2 bosonic vortices, the mirror parity is equal to M~ . From Eq. (7), we get the fermion Hamiltonian Furthermore, a Uð1Þ gauge symmetry emerges in the low- N ¼ 0 energy Hilbert space defined by the local constraint ¼ ψ~ †ð σ − σ Þψ~ þ ψ~ †ðh~iτþ þh~iτ−Þσ ψ~ [32]. Specifically, the constraint is the Gauss law, and it H v s kx y ky x s s b b z s: restricts the low-energy Hilbert space to states that are ð10Þ invariant under the gauge transformation It has a particle-hole symmetry Ξ∶ ψ~ → σzψ~ which maps iϕτz ~ ~ 2iϕ H to ΞHΞ ¼ −H. This implies that its spectrum is Uϕ∶ ψ~ s → e ψ~ s; b → be ;aμ → aμ − ∂μϕ: symmetric with respect to zero. Assume that the dimension ð Þ 8 of the whole Hilbert space is 4N; there are 2N states with 2 ð1Þ positive energy and N states with negative energy. For the In this effective theory with the emergent U gauge vortex configuration in Fig. 1(a), there are four complex field, condensing the boson b~ no longer breaks the global ð1Þ zero modes, two from each vortex core, which are all U and the mirror symmetries, as it instead breaks the unoccupied. Therefore, excluding these four states, there Uð1Þ gauge symmetry to Z2. Naively, the mirror symmetry maps hb~i to −hb~i. However these two symmetry breaking vacuua are equivalent because they are related by the ~ ~ gauge symmetry transformation Uπ=2∶ b → −b. This resto- ration of mirror symmetry becomes clearly manifested if ~ we assume that b and ψ~ s transform projectively under mirror symmetry with the additional Uð1Þ gauge transformation Uπ=2, (a) (b)

~ 0 ~ ~ 0 FIG. 1 (color online). (a) A vortex and an antivortex. The M∶ ψ~ sðrÞ → iτz ⊗ σxψ~ sðr Þ; bðrÞ → bðr Þ: ð9Þ direction of the arrow represents the phase of the Higgs field h~i This Higgs phase obtained by condensing the charge-2 b~ b . The dotted line is the mirror axis. (b) Illustration of the fermion spectrum flow from the left to the right as we create a field indeed has a Z2 topological order when the number of vortex-antivortex pair from the vacuum and move them Dirac fermions is n ¼ 4 [10,33]. This can be understood by ψ~ far apart. In this process, four vortex core states are identifying the Bogoliubov quasiparticle and vortices as separated from the bulk spectrum, two from the conducting the anyons e, m, and ϵ (see Sec. II of the Supplemental band and two from the valence band, and become degenerate zero Material [25] for the definition of the notation) in the Z2 modes. As illustrated in the inset, the core states are twofold topological order. ψ~ becomes the ϵ anyon as both are degenerate with spin s ¼1.

236801-3 week ending PRL 115, 236801 (2015) PHYSICAL REVIEW LETTERS 4 DECEMBER 2015 are 2N − 2 states with negative energy, which are all anyon on the edge costs a finite energy. In a generic Z2 occupied in the fermion wave function. state, this degeneracy can be further lifted by tunneling an 2 − 2 ¼ λ † † þ Next, we consider the mirror eigenvalues of these N e anyon between the two edges, Ht eLeR H:c:, where ~ 2 † occupied fermion states. Since ψ~ carries M ¼ −1, each eL;R creates two e anyons on the two edges, respectively. ~ 2 ¼ −1 state has mirror eigenvalue λM ¼i. Because both H, Ξ However, the e anyon carries M ; therefore, the and M are diagonal in pseudospin s ¼ ↑; ↓, all occupied tunneling term Ht is odd under mirror and, thus, forbidden states are pseudospin doublets, and two states in each by M. As a result, this twofold topological degeneracy is doublet have the same λM. Hence, the mirror eigenvalue protected by the mirror symmetry even in the limit of of all occupied states organized as N − 1 doublets is L → 0. This argument is formulated using the effective ð−1ÞN−1 ¼ −1. Therefore, the wave function of two empty edge Lagrangian in the Supplemental Material [25]. vortices is odd under mirror symmetry, which implies that In the limit of L → 0, this topological degeneracy the e particle has the symmetry fractionalization M~ 2 ¼ −1. becomes a local degeneracy protected by the mirror 2 symmetry. Therefore, if the Z2 topological order is killed Combining the results that both e and ϵ carry M~ ¼ −1,we ~ 2 ¼ −1 by collapsing two gapped edges, the ground state is either conclude that the m anyon also has M (see the gapless or mirror-symmetry breaking, and this cannot discussion in Sec. II of the Supplemental Material [25]). Z be avoided regardless of edge types because all types In summary, the gapped 2 surface state we constructed of anyons have M~ 2 ¼ −1. This topological degeneracy has an anomalous mirror-symmetry fractionalization that ~ 2 ¼ −1 reveals the anomalous nature of this mirror-symmetry both types of anyons carry M , which cannot be fractionalization. Furthermore, if we collapse two gapless realized in a genuine 2D system. For comparison, it was edges of the Z2 state, the edges remain gapless because the shown in Sec. II of the Supplemental Material [25] that anyon tunneling is forbidden by M. Hence, we get a gapless applying the same construction to a 2D lattice model results ¼ 1 þ 1 ¼ 2 Z domain wall with central charge c , which in a 2 phase with a different mirror-symmetry fraction- recovers the edge with four chiral fermion modes in the alization which is not anomalous. — aforementioned free-fermion limit. This is explained in Mirror anomaly. The anomalous crystal symmetry more detail in Sec. VI of the Supplemental Material [25]. fractionalization presented in the surface topological order Conclusion.—In this Letter, we show that the surface implies a symmetry-protected topological degeneracy asso- of a 3D mirror TCI with mirror Chern number n ¼ 4, ciated with the edges of the surface topological ordered containing four gapless Dirac fermion modes in the free region. This mirror anomaly is a remnant of the anomalous limit, can be gapped out without breaking the mirror surface fermion modes in the free-fermion limit. To see symmetry by a Z2 topological order. This surface Z2 this, we consider the setup presented in Fig. 2, in which Z topological order has an anomalous mirror-symmetry frac- the 2 surface topologically ordered state is terminated at tionalization in which all three types of anyons carry two edges symmetric with respect to the mirror plane, by ~ 2 ¼ −1 h i¼1 fractionalized mirror-symmetry quantum number M , two regions with opposite b on either side of the and such a topological order cannot be realized in a purely mirror plane, respectively. 2D system. This setup itself does not break the mirror symmetry, and Our finding also puts constraints on possible ways to all local excitations can be gapped everywhere on the Z Z fractionalize the mirror symmetry in a 2D 2 quantum spin surface. Particularly, since the 2 topological order is not liquid [24,36]. The result of this work indicates that the chiral, its edge can be gapped out by condensing either e or combination that both the e and m carry the fractionalized m anyons on the edge [35]. The edges next to an ordered ~ 2 M ¼ −1 is anomalous and cannot be realized in a 2D Z2 phase with hbi ≠ 0 are e edges, as condensing e breaks the spin liquid. Furthermore, our result can be easily general- global Uð1Þ symmetry. A Z2 spin liquid state on an infinite ized to also rule out the combination that the e anyon cylinder has four degenerate ground states jΨ i, each has a 1 ~ 2 ¼ −1 one type of anyon flux a ¼ 1;e;m;ϵ going through the carries spin-2 and m anyon carries M [37], because ~ 2 cylinder. On a finite cylinder with two e edges, only jΨ1i if e carries M ¼þ1, we can define a new mirror 0 ¼ iπSz and jΨei remain degenerate, because adding an m or ϵ symmetry M Me , for which both e and m carry ðM~ 0Þ2 ¼ −1, and, therefore, this combination is also anomalous. In summary, our finding implies that the vison ~ 2 must carry M ¼þ1 in a Z2 spin liquid where the spinon carries a half-integer spin. We thank Chen Fang and Senthil Todadri for invaluable FIG. 2 (color online). Two mirror-symmetric edges of a Z2 surface topological order. The mirror symmetry maps x to −x discussions. Y. Q. is supported by National Basic Research with respect to the mirror plane marked by the red disk at x ¼ 0. Program of China through Grant No. 2011CBA00108 The surface topological order marked by the shade terminates at and by NSFC Grant No. 11104154. L. F. is supported edges against two ordered regions with hbi¼1, respectively. by the DOE Office of Basic Energy Sciences, Division of

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