60 research outputs found

    Topological complexity of the relative closure of a semi-Pfaffian couple

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    Gabrielov introduced the notion of relative closure of a Pfaffian couple as an alternative construction of the o-minimal structure generated by Khovanskii's Pfaffian functions. In this paper, use the notion of format (or complexity) of a Pfaffian couple to derive explicit upper-bounds for the homology of its relative closure. Keywords: Pfaffian functions, fewnomials, o-minimal structures, Betti numbers.Comment: 12 pages, 1 figure. v3: Proofs and bounds have been slightly improve

    Topology of definable Hausdorff limits

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    Let ARn+rA\sub \R^{n+r} be a set definable in an o-minimal expansion §\S of the real field, ARrA' \sub \R^r be its projection, and assume that the non-empty fibers AaRnA_a \sub \R^n are compact for all aAa \in A' and uniformly bounded, {\em i.e.} all fibers are contained in a ball of fixed radius B(0,R).B(0,R). If LL is the Hausdorff limit of a sequence of fibers Aai,A_{a_i}, we give an upper-bound for the Betti numbers bk(L)b_k(L) in terms of definable sets explicitly constructed from a fiber Aa.A_a. In particular, this allows to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the {\em relative closure} to construct the o-minimal structure \S_\pfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure (X,Y)0(X,Y)_0 in the special case where YY is empty.Comment: Latex, 23 pages, no figures. v2: Many changes in the exposition and notations in an attempt to be clearer, references adde

    Quantitative study of semi-Pfaffian sets

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    We study the topological complexity of sets defined using Khovanskii's Pfaffian functions, in terms of an appropriate notion of format for those sets. We consider semi- and sub-Pfaffian sets, but more generally any definable set in the o-minimal structure generated by the Pfaffian functions, using the construction of that structure via Gabrielov's notion of limit sets. All the results revolve around giving effective upper-bounds on the Betti numbers (for the singular homology) of those sets. Keywords: Pfaffian functions, fewnomials, o-minimal structures, tame topology, spectral sequences, Morse theory.Comment: Author's PhD thesis. Approx. 130 pages, no figure

    Knots, links, and long-range magic

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    We study the extent to which knot and link states (that is, states in 3d Chern-Simons theory prepared by path integration on knot and link complements) can or cannot be described by stabilizer states. States which are not classical mixtures of stabilizer states are known as "magic states" and play a key role in quantum resource theory. By implementing a particular magic monotone known as the "mana" we quantify the magic of knot and link states. In particular, for SU(2)kSU(2)_k Chern-Simons theory we show that knot and link states are generically magical. For link states, we further investigate the mana associated to correlations between separate boundaries which characterizes the state's long-range magic. Our numerical results suggest that the magic of a majority of link states is entirely long-range. We make these statements sharper for torus links.Comment: 36 pages; 64 knots; 34 links; v2 to match published versio

    Wallpaper Fermions and the Nonsymmorphic Dirac Insulator

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    Recent developments in the relationship between bulk topology and surface crystal symmetry have led to the discovery of materials whose gapless surface states are protected by crystal symmetries. In fact, there exists only a very limited set of possible surface crystal symmetries, captured by the 17 "wallpaper groups." We show that a consideration of symmetry-allowed band degeneracies in the wallpaper groups can be used to understand previous topological crystalline insulators, as well as to predict new examples. In particular, the two wallpaper groups with multiple glide lines, pggpgg and p4gp4g, allow for a new topological insulating phase, whose surface spectrum consists of only a single, fourfold-degenerate, true Dirac fermion. Like the surface state of a conventional topological insulator, the surface Dirac fermion in this "nonsymmorphic Dirac insulator" provides a theoretical exception to a fermion doubling theorem. Unlike the surface state of a conventional topological insulator, it can be gapped into topologically distinct surface regions while keeping time-reversal symmetry, allowing for networks of topological surface quantum spin Hall domain walls. We report the theoretical discovery of new topological crystalline phases in the A2_2B3_3 family of materials in SG 127, finding that Sr2_2Pb3_3 hosts this new topological surface Dirac fermion. Furthermore, (100)-strained Au2_2Y3_3 and Hg2_2Sr3_3 host related topological surface hourglass fermions. We also report the presence of this new topological hourglass phase in Ba5_5In2_2Sb6_6 in SG 55. For orthorhombic space groups with two glides, we catalog all possible bulk topological phases by a consideration of the allowed non-abelian Wilson loop connectivities, and we develop topological invariants for these systems. Finally, we show how in a particular limit, these crystalline phases reduce to copies of the SSH model.Comment: Final version, 6 pg main text + 29 pg supplement, 6 + 13 figure

    Entanglement in Many-Body Systems

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    The recent interest in aspects common to quantum information and condensed matter has prompted a prosperous activity at the border of these disciplines that were far distant until few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermionic and bosonic model systems. Both bipartite and multipartite entanglement will be considered. At equilibrium we emphasize on how entanglement is connected to the phase diagram of the underlying model. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.Comment: 61 pages, 29 figure
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