We show that the Haldane phase of S=1 chains is characterized by a double degeneracy of the entanglement spectrum. The degeneracy is protected by a set of symmetries (either the dihedral group of $\pi$-rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry), and cannot be lifted unless either a phase boundary to another, "topologically trivial", phase is crossed, or the symmetry is broken. More generally, these results offer a scheme to classify gapped phases of one dimensional systems. Physically, the degeneracy of the entanglement spectrum can be observed by adiabatically weakening a bond to zero, which leaves the two disconnected halves of the system in a finitely entangled state.Comment: 11 pages, 4 figures, references added, minor corrections, meta data update

Recently, several authors have investigated topological phenomena in periodically driven systems of noninteracting particles. These phenomena are identified through analogies between the Floquet spectra of driven systems and the band structures of static Hamiltonians. Intriguingly, these works have revealed phenomena that cannot be characterized by analogy to the topological classification framework for static systems. In particular, in driven systems in two dimensions (2D), robust chiral edge states can appear even though the Chern numbers of all the bulk Floquet bands are zero. Here, we elucidate the crucial distinctions between static and driven 2D systems, and construct a new topological invariant that yields the correct edge-state structure in the driven case. We provide formulations in both the time and frequency domains, which afford additional insight into the origins of the ''anomalous'' spectra that arise in driven systems. Possibilities for realizing these phenomena in solid-state and cold-atomic systems are discussed.

λJ J J λJ J J λJ J J

Topological superconductors have a full pairing gap in the bulk and gapless surface Andreev bound states. In this Letter, we provide a sufficient criterion for realizing time-reversal-invariant topological superconductors in centrosymmetric superconductors with odd-parity pairing. We next study the pairing symmetry of the newly discovered superconductor CuxBi2Se3 within a two-orbital model, and find that a novel spin-triplet pairing with odd parity is favored by strong spin-orbit coupling. Based on our criterion, we propose that CuxBi2Se3 is a good candidate for a topological superconductor. We close by discussing experimental signatures of this new topological phase.

We discuss the characterization and stability of the Haldane phase in integer spin chains on the basis of simple, physical arguments. We find that an odd-S Haldane phase is a topologically non-trivial phase which is protected by any one of the following three global symmetries: (i) the dihedral group of π-rotations about x, y and z axes; (ii) time-reversal symmetry S x,y,z → −S x,y,z ; (iii) link inversion symmetry (reflection about a bond center), consistently with previous results [Phys. Rev. B 81, 064439 (2010)]. On the other hand, an even-S Haldane phase is not topologically protected (i.e., it is indistinct from a trivial, site-factorizable phase). We show some numerical evidence that supports these claims, using concrete examples.

The quantum walk was originally proposed as a quantum mechanical analogue of the classical random walk, and has since become a powerful tool in quantum information science. In this paper, we show that discrete time quantum walks provide a versatile platform for studying topological phases, which are currently the subject of intense theoretical and experimental investigation. In particular, we demonstrate that recent experimental realizations of quantum walks simulate a non-trivial one dimensional topological phase. With simple modifications, the quantum walk can be engineered to realize all of the topological phases which have been classified in one and two dimensions. We further discuss the existence of robust edge modes at phase boundaries, which provide experimental signatures for the non-trivial topological character of the system. Quantum walks, the quantum analogues of classical random walks [1], form the basis of efficient quantum algorithms [2,3], and provide a universal platform for quantum computation [4]. Much like their classical counterparts, quantum walks can be used to model a wide variety of physical processes including photosynthesis [5,6], quantum diffusion [7], optical/spin pumping and vortex transport [8], and electrical breakdown [9,10]. Motivated by the prospect of such an array of applications, several groups have recently realized quantum walks in experiments using ultracold atoms in optical lattices [11], trapped ions[12], photons [13], and nuclear magnetic resonance [14]. These systems offer the possibility to study quantum dynamics of single or many particles in a precisely controlled experimental setting.Here we show that quantum walks can be used to explore dynamics in a wide range of topological phases [15][16][17]. Interest in topological phases was first sparked by the discovery of the integer quantized Hall effect [17,18], and has rapidly increased in recent years following the prediction [19][20][21] and experimental realization [22,23] of a new class of materials called "topological insulators." Unlike more familiar states of matter such as the ferromagnetic and superconducting phases, which break SU(2) (spin-rotation) and U(1) (gauge) symmetries, respectively, topological phases do not break any symmetries and cannot be described by any local order parameters. Rather, these phases are described by topological invariants which characterize the global structures of their ground state wavefunctions. Topological phases are known to host a variety of exotic phenomena such as fractional charges and magnetic monopoles [24,25].The class of topological phases which can be realized in a system of non-interacting particles is determined by the dimensionality of the system and the underlying symmetries of its Hamiltonian. Figure 1 shows the ten classes of topological phases which can arise in one dimensional (1D) and two dimensional (2D) systems with and without time-reversal symmetry (TRS) and particle-hole symmetry (PHS) (see Refs.[26,27] and discussion below). If both symmetries are ab...

The effect of interactions on topological insulators and superconductors remains, to a large extent, an open problem. Here, we describe a framework for classifying phases of one-dimensional interacting fermions, focusing on spinless fermions with time-reversal symmetry and particle number parity conservation, using concepts of entanglement. In agreement with an example presented by Fidkowski et. al. (Ref. [1]), we find that in the presence of interactions there are only eight distinct phases, which obey a Z8 group structure. This is in contrast to the Z classification in the non-interacting case. Each of these eight phases is characterized by a unique set of bulk invariants, related to the transformation laws of its entanglement (Schmidt) eigenstates under symmetry operations, and has a characteristic degeneracy of its entanglement levels. If translational symmetry is present, the number of distinct phases increases to 16.

Topological phases exhibit some of the most striking phenomena in modern physics. much of the rich behaviour of quantum Hall systems, topological insulators, and topological superconductors can be traced to the existence of robust bound states at interfaces between different topological phases. This robustness has applications in metrology and holds promise for future uses in quantum computing. Engineered quantum systems-notably in photonics, where wavefunctions can be observed directly-provide versatile platforms for creating and probing a variety of topological phases. Here we use photonic quantum walks to observe bound states between systems with different bulk topological properties and demonstrate their robustness to perturbations-a signature of topological protection. Although such bound states are usually discussed for static (time-independent) systems, here we demonstrate their existence in an explicitly time-dependent situation. moreover, we discover a new phenomenon: a topologically protected pair of bound states unique to periodically driven systems.