Abstract. We present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermionsa simplified Su-Schrieffer-Heeger (SSH) model-with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a scaling analysis of these data for essentially semi-infinite chains we obtain the fidelity susceptibility and show that it contains a boundary contribution which is different in the topologically ordered than in the topologically trivial phase. For the entanglement spectrum and entropy we confirm predictions from massive field theory for a block in the middle of an infinite chain but also consider blocks containing the edge of the chain. For the latter case we show that in the SPT phase additional entanglement-as compared to the trivial phase-is present which is localized at the boundary. Finally, we extend our study to the dimerized chain with a nearest-neighbour interaction using exact diagonalization, Arnoldi, and densitymatrix renormalization group methods and show that a phase transition into a topologically trivial charge-density wave phase occurs.

We study the Loschmidt echo for quenches in open one-dimensional lattice models with symmetry protected topological phases. For quenches where dynamical quantum phase transitions do occur we find that cusps in the bulk return rate at critical times tc are associated with sudden changes in the boundary contribution. For our main example, the Su-Schrieffer-Heeger model, we show that these sudden changes are related to the periodical appearance of two eigenvalues close to zero in the dynamical Loschmidt matrix. We demonstrate, furthermore, that the structure of the Loschmidt spectrum is linked to the periodic creation of long-range entanglement between the edges of the system. arXiv:1712.03618v1 [cond-mat.stat-mech]

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We study spin-Hall effects in time-reversal-symmetry-͑TRS-͒ broken systems such as triplet chiral superconductors and TRS-preserved ones such as graphene. For chiral triplet superconductors, we show that the edge states carry a quantized spin-Hall current in response to an applied Zeeman magnetic field B along the d vector ͓A. J. Leggett, Rev. Mod. Phys. 47, 331 ͑1975͔͒, whereas the edge spin current for B Ќ d is screened by the condensate. We also derive the bulk spin-Hall current for chiral triplet superconductors for arbitrary relative orientation of B and d and discuss its relation with the edge spin current. For TRS-invariant system graphene, we show that the bulk effective action, unlike its TRS-broken counterparts, does not support a SU͑2͒ Hopf term but allows a crossed Hopf term in the presence of an external electromagnetic field, which yields a quantized bulk spin-Hall current in response to an electric field. We also present an analytical solution of the edge problem for armchair edges of graphene and contrast the properties of these edge states with their timereversal-symmetry-broken counterparts in chiral superconductors. We propose possible experiments to test our results.