Weyl points and Dirac lines protected by multiple screw rotations

In three-dimensional noncentrosymmetric materials two-fold screw rotation symmetry forces electron's energy bands to have Weyl points at which two bands touch. This is illustrated for space groups No. 19 ($P2_12_12_1$) and No. 198 ($P2_13$), which have three orthogonal screw rotation axes. In the case of space groups No. 61 ($Pbca$) and No. 205 ($P$a-3) that have extra inversion symmetry, Weyl points are promoted to four-fold degenerate line nodes in glide-invariant planes. The three-fold rotation symmetry present in the space groups No. 198 and No. 205 allows Weyl and Dirac points, respectively, to appear along its rotation axes in the Brillouin zone and generates four-fold and six-fold degeneracy at the $\Gamma$ point and R point, respectively.Comment: 8 pages, v2 added references; v3 corrected typos, sec.IIB slightly expande

Correlation effects on topological crystalline insulators

We study interaction effects on the topological crystalline insulators protected by time-reversal ($T$) and reflection symmetry ($R$) in two and three spatial dimensions. From the stability analysis of the edge states with bosonization, we find that the classification of the two-dimensional SPT phases protected by Z_2\times[\mbox{U(1)}\rtimes T] symmetry is reduced from $\mathbb{Z}$ to $\mathbb{Z}_4$ by interactions, where the $Z_2$ symmetry denotes the reflection whose mirror plane is the two-dimensional plane itself. By extending the approach recently proposed by Isobe and Fu, we show that the classification of the three-dimensional SPT phases (i.e., topological crystalline insulators) protected by R\times[\mbox{U(1)}\rtimes T] symmetry is reduced from $\mathbb{Z}$ to $\mathbb{Z}_8$ by interactions.Comment: 9 pages, 1 figures, v2: typos correcte

Stability of surface states of weak Z2\mathbb{Z}_2 topological insulators and superconductors

We study the stability against disorder of surface states of weak $\mathbb{Z}_2$ topological insulators (superconductors) which are stacks of strong $\mathbb{Z}_2$ topological insulators (superconductors), considering representative Dirac Hamiltonians in the Altland-Zirnbauer symmetry classes in various spatial dimensions. We show that, in the absence of disorder, surface Dirac fermions of weak $\mathbb{Z}_2$ topological insulators (superconductors) can be gapped out by a Dirac mass term which couples surface Dirac cones and leads to breaking of a translation symmetry (dimerization). The dimerization mass is a unique Dirac mass term in the surface Dirac Hamiltonian, and the two dimerized gapped phases which differ in the sign of the Dirac mass are distinguished by a $\mathbb{Z}_2$ index. In other words the dimerized surfaces can be regarded as a strong $\mathbb{Z}_2$ topological insulator (superconductor). We argue that the surface states are not localized by disorder when the ensemble average of the Dirac mass term vanishes.Comment: 8 page

Topological classification with additional symmetries from Clifford algebras

We classify topological insulators and superconductors in the presence of additional symmetries such as reflection or mirror symmetries. For each member of the 10 Altland-Zirnbauer symmetry classes, we have a Clifford algebra defined by operators of the generic (time-reversal, particle-hole, or chiral) symmetries and additional symmetries, together with gamma matrices in Dirac Hamiltonians representing topological insulators and superconductors. Following Kitaev's approach, we classify gapped phases of non-interacting fermions under additional symmetries by examining all possible distinct Dirac mass terms which can be added to the set of generators of the Clifford algebra. We find that imposing additional symmetries in effect changes symmetry classes and causes shifts in the periodic table of topological insulators and superconductors. Our results are in agreement with the classification under reflection symmetry recently reported by Chiu et al. Several examples are discussed including a topological crystalline insulator with mirror Chern numbers and mirror superconductors.Comment: 18 page

Weyl and Dirac semimetals with Z_2 topological charge

We study the stability of gap-closing (Weyl or Dirac) points in the three-dimensional Brillouin zone of semimetals using Clifford algebras and their representation theory. We show that a pair of Weyl points with $\mathbb{Z}_2$ topological charge are stable in a semimetal with time-reversal and reflection symmetries when the square of the product of the two symmetry transformations equals minus identity. We present toy models of $\mathbb{Z}_2$ Weyl semimetals which have surface modes forming helical Fermi arcs. We also show that Dirac points with $\mathbb{Z}_2$ topological charge are stable in a semimetal with time-reversal, inversion, and SU(2) spin rotation symmetries when the square of the product of time-reversal and inversion equals plus identity. Furthermore, we briefly discuss the topological stability of point nodes in superconductors using Clifford algebras.Comment: 14 pages, 2 figure

Chain of Majorana States from Superconducting Dirac Fermions at a Magnetic Domain Wall

We study theoretically a strongly type-II s-wave superconducting state of two-dimensional Dirac fermions in proximity to a ferromagnet having in-plane magnetization. It is shown that a magnetic domain wall can host a chain of equally spaced vortices in the superconducting order parameter, each of which binds a Majorana fermion state. The overlap integral of neighboring Majorana states is sensitive to the position of the chemical potential of the Dirac fermions. This leads to a characteristic V-shaped dependence of thermal conductivity of Majorana fermions on the chemical potential.Comment: 4+ pages, 2 figure

Bosonic symmetry protected topological phases with reflection symmetry

We study two-dimensional bosonic symmetry protected topological (SPT) phases which are protected by reflection symmetry and local symmetry [$Z_N\rtimes R$, $Z_N\times R$, U(1)$\rtimes R$, or U(1)$\times R$], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer-$S$ spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by $Z_N\rtimes R$ symmetry are classified as $\mathbb{Z}_2\times\mathbb{Z}_2$ for even $N$ and 0 (no SPT phase) for odd $N$ while those protected by U(1)$\rtimes R$ symmetry are $\mathbb{Z}_2$. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of $S=2$ spins on the square lattice is a $\mathbb{Z}_2$ SPT phase protected by reflection and $\pi$-rotation symmetries.Comment: 11+epsilon pages, 2 figures; v3: typos correcte

Breakdown of the topological classification Z for gapped phases of noninteracting fermions by quartic interactions

The conditions for both the stability and the breakdown of the topological classification of gapped ground states of noninteracting fermions, the tenfold way, in the presence of quartic fermion-fermion interactions are given for any dimension of space. This is achieved by encoding the effects of interactions on the boundary gapless modes in terms of boundary dynamical masses. Breakdown of the noninteracting topological classification occurs when the quantum nonlinear sigma models for the boundary dynamical masses favor quantum disordered phases. For the tenfold way, we find that (i) the noninteracting topological classification $\mathbb{Z}^{\,}_{2}$ is always stable, (ii) the noninteracting topological classification $\mathbb{Z}$ in even dimensions is always stable, (iii) the noninteracting topological classification $\mathbb{Z}$ in odd dimensions is unstable and reduces to $\mathbb{Z}^{\,}_{N}$ that can be identified explicitly for any dimension and any defining symmetries. We also apply our method to the three-dimensional topological crystalline insulator SnTe from the symmetry class AII$+R$, for which we establish the reduction $\mathbb{Z}\to\mathbb{Z}^{\,}_{8}$ of the noninteracting topological classification.Comment: 29 page

Unconventional Neel and dimer orders in a spin-1/2 frustrated ferromagnetic chain with easy-plane anisotropy

We study the ground-state phase diagram of a one-dimensional spin-1/2 easy-plane XXZ model with a ferromagnetic nearest-neighbor (NN) coupling $J_1$ and a competing next-nearest-neighbor (NNN) antiferromagnetic coupling $J_2$ in the parameter range $0<J_2/|J_1|<0.4$. When $J_2/|J_1|\lesssim1/4$, the model is in a Tomonaga-Luttinger liquid phase which is adiabatically connected to the critical phase of the XXZ model of $J_2=0$. On the basis of the effective (sine-Gordon) theory and numerical analyses of low-lying energy levels of finite-size systems, we show that the NNN coupling induces phase transitions from the Tomonaga-Luttinger liquid to gapped phases with either Neel or dimer order. Interestingly, these two types of ordered phases appear alternately as the easy-plane anisotropy is changed towards the isotropic limit. The appearance of the antiferromagnetic (Neel) order in this model is remarkable, as it is strongly unfavored by both the easy-plane ferromagnetic NN coupling and antiferromagnetic NNN coupling in the classical-spin picture. We argue that emergent trimer degrees of freedom play a crucial role in the formation of the Neel order.Comment: 10 pages, 7 figures. To be published in Phys. Rev. B (Editor's Suggestion

Quantum impurity spin in Majorana edge fermions

We show that Majorana edge modes of two-dimensional spin-triplet topological superconductors have Ising-like spin density whose direction is determined by the d-vector characterizing the spin-triplet pairing symmetry. Exchange coupling between an impurity spin (S=1/2) and Majorana edge modes is thus Ising-type. Under external magnetic field perpendicular to the Ising axis, the system can be mapped to a two-level system with Ohmic dissipation, which is equivalent to the anisotropic Kondo model. The magnetic response of the impurity spin can serve as a local experimental probe for the order parameter.Comment: 4+ pages, 2 figure