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In general, we know that there are four states of matter solid, liquid, gas and plasma. But there are much more states of matter. For e. g. there are ferromagnetic states of matter as revealed by the phenomenon of magnetization and superfluid states defined by the phenomenon of zero viscosity. The various phases in our colorful world are so rich that it is amazing that they can be understood systematically by the symmetry breaking theory of Landau. Topological phenomena define the topological order at macroscopic level. Topological order need new mathematical framework to describe it. More recently it is found that at microscopic level topological order is due to the long range quantum entanglement, just like the fermions fluid is due to the fermion-pair condensation. Long range quantum entanglement leads to many amazing emergent phenomena, such as fractional quantum numbers, non- Abelian statistics ad perfect conducting boundary channels. It can even provide a unified origin of light and electron i.e. gauge interactions and Fermi statistics. Light waves (gauge fields) are fluctuations of long range entanglement and electron (fermion) are defect of long range entanglements.The Himalayan Physics Vol. 6 & 7, April 2017 (108-111)

In this paper we compute topological entropy with prescribed accuracy for different economic models, showing the existence of a topologically chaotic regime for them. In order to make the paper self-contained, a general overview on the topological entropy of continuous interval maps is given. More precisely, we focus on piecewise monotone maps which often appear as dynamical models in economy, but also in population growth and physics. Our main aim is to show that when topological entropy can be approximated up to a given error, it is a useful tool which helps to analyze the chaotic dynamics in one dimensional models.

In this paper, we study the relation between an anomaly-free n+1 D topological order, which are often called n+1 D topological order in physics literature, and its n D gapped boundary phases. We argue that the n+1 D bulk anomaly-free topological order for a given n D gapped boundary phase is unique. This uniqueness defines the notion of the “ bulk ” for a given gapped boundary phase. In this paper, we show that the n+1 D “ bulk ” phase is given by the “center” of the n D boundary phase. In other words, the geometric notion of the “ bulk ” corresponds precisely to the algebraic notion of the “center”. We achieve this by first introducing the notion of a morphism between two (potentially anomalous) topological orders of the same dimension, then proving that the notion of the “ bulk ” satisfies the same universal property as that of the “center” of an algebra in mathematics, i.e. “ bulk = center”. The entire argument does not require us to know the precise mathematical description of a (potentially anomalous) topological order. This result leads to concrete physical predictions., National Science Foundation (U.S.) (Grant DMR-1506475), National Science Foundation (U.S.) (Grant NSFC11274192)

We investigate the topological properties of a ladder model of the Kitaev superconductor chains with a periodically modulated chemical potential. It is found that the system has two different topological classes, the class BDI characterized by the Z index and the class D characterized by the Z 2 index. The topological class of the system is determined by the relative phase θ between the inter- and intra-chain superconducting pairing. Based on the calculation of the topological numbers of the class BDI and the class D, we present the topological phase diagrams of the system for two different topological classes. It is shown that the topological phase transition accompanying the variation of the number of Majorana zero modes could be induced by periodical potential in both two classes. Finally, we calculate the differential conductance of normal metal-superconductor junction at zero bias, and find that it matches the topological phase diagrams in both class BDI and class D.

Left Bol loops with the A l -property or gyrogroups are generalization of groups which do not explicitly have associativity. In this work, we define topological gyrogroups and study some properties of them. In spite of having a weaker algebraic form, topological gyrogroups carry almost the same basic properties owned by topological groups. In particular, we prove that being T 0 and T 3 are equivalent in topological gyrogroups. Furthermore, a topological gyrogroup is first countable if and only if it is premetrizable. Finally, a direct product of topological gyrogroups is a topological gyrogroup.

The aim of this paper is to examine the topological entropy for a free semigroup action defined by Bufetov using separated and spanning sets. First, this study reveals that such entropy is a topological conjugacy invariant and also can be equivalently defined using open covers. Furthermore, a quantitative analogue of Bowen's theorem for semiconjugacy is provided and we compared the topological entropies presented by Bufetov and Biś. Finally, a formula for the entropy of skew-product transformation with respect to the subshift is obtained.

Topological derivatives for quasilinear elliptic equations have not been studied yet. Such results are needed to apply topological asymptotic methods in shape optimization to nonlinear elasticity equations and in imaging to detect sets with codimensions ≥2 (e.g. points in 2D or curves in 3D). Our contribution is to provide for the first time topological derivatives for a class of quasilinear elliptic equations. The topological derivative can be split into a classical linear term and a new term which accounts for the nonlinearity of the principal part. Shifting from linear equations to nonlinear ones requires to significantly change the asymptotic analysis method. One core issue lies in the ability to approximate the variation of the direct state at scale 1 in R N . Accordingly we construct dedicated function spaces and rely on comparison techniques. Moreover one has to set up a duality scheme between the direct and adjoint states that applies in the nonlinear framework at each stage of the analysis.

The phenomena associated with topological defects have had an enormous impact in condensed matter physics for more than 50 years. Beginning with an understanding of topological defects in ordered phases, the field is now sharply focused on defects in topological phases of matter. In this review, we cover aspects of defects in conventional ordered media, bound states on defects in strong topological insulators (TIs) and topological superconductors (TSCs), and bound states on defects in crystalline topological phases protected by spatial symmetries. As a unifying theme, we present the structure of many types of index theorems that relate the existence of topological bound states to the bulk topology of the host phase of matter and the topological charge of the relevant defects.

Mechanical systems can display topological characteristics similar to that of topological insulators. Here we report a large class of topological mechanical systems related to the BDI symmetry class. These are self-assembled chains of rigid bodies with an inversion centre and no reflection planes. The particle-hole symmetry characteristic to the BDI symmetry class stems from the distinct behaviour of the translational and rotational degrees of freedom under inversion. This and other generic properties led us to the remarkable conclusion that, by adjusting the gyration radius of the bodies, one can always simultaneously open a gap in the phonon spectrum, lock-in all the characteristic symmetries and generate a non-trivial topological invariant. The particle-hole symmetry occurs around a finite frequency, and hence we can witness a dynamical topological Majorana edge mode. Contrasting a floppy mode occurring at zero frequency, a dynamical edge mode can absorb and store mechanical energy, potentially opening new applications of topological mechanics., Properties of topological insulators can be realized in mechanical systems, opening potential applications of topological mechanics. Here, Prodan et al. report a dynamical topological Majorana edge mode in self-assembled chains of rigid bodies with particle-hole symmetry.

We discuss the topological equivalence between evolution families with a generalized exponential dichotomy. These can occur for example when all Lyapunov exponents are infinite or all Lyapunov exponents are zero. In particular, we show that any evolution family admitting a generalized exponential dichotomy is topologically equivalent to a certain normal form, in the which the exponential behavior in the stable and unstable directions are multiples of the identity. Moreover, we show that the topological equivalence between two evolution families admitting generalized exponential dichotomies, possibly with different growth rates, can be completely characterized in terms of a new notion of equivalence between these rates.

In this paper, we investigate the four-qubit spin-\(\frac {1}{2}\) XXZ Heisenberg chain with Dzyaloshinskii-Moriya interaction by topological basis method, and research the relationship between the topological basis states and the ground states. In order to study the Hamiltonian system beyond XXZ model, we introduce two Temperley-Lieb algebra generators and two other generalized generators. Then we investigate the relationship between topological basis and Heisenberg XXZ model with Dzyaloshinskii-Moriya interaction. The results show that the ground state of this model falls on the topological basis state for anti-ferromagnetic case and gapless phase case.

We study the topological entropy for dynamical systems with discrete or continuous multiple time. Due to the generalization of a well-known one time-dimensional result we show that the definition of topological entropy, using the approach for subshifts, leads to the zero entropy for many systems different from subshift. We define a new type of relative topological entropy to avoid this phenomenon. The generalization of Bowen’s power rule allows us to define topological and relative topological entropies for systems with continuous multiple time. As an application, we find a relation between the relative topological entropy and controllability of linear systems with continuous multiple time.

We extend to a possibly infinite chain the conformally invariant mechanical system that was introduced earlier as a toy model for understanding the topological Yang-Mills theory. It gives a topological quantum model that has interesting and computable zero modes and topological invariants. It confirms the recent conjecture by several authors that supersymmetric quantum mechanics may provide useful tools for understanding robotic mechanical systems (Vitelli et al.) and condensed matter properties (Kane et al.), where trajectories of effective models are allowed or not by the conservation of topological indices. The absences of ground state and mass gaps are special features of such systems., This work will appear in a special Nuclear Physics issue "Memoriam Raymond Stora"

This paper deals with higher-order topological sensitivity analysis for the Laplace operator with respect to the presence of a Dirichlet geometry perturbation. Two main results are presented in this work. In the first one, we discuss the influence of the considered geometry perturbation on the Laplace solution. The second one is devoted to the higher-order topological derivatives. We derive a higher-order topological sensitivity analysis for a large class of shape functions.

Second-order continuous phase transitions are characterized by symmetry breaking with order parameters. Topological orders of electrons, characterized by the topological index defined in momentum space, provide a distinct perspective for phase transitions, which are categorized as quantum phase transitions not being accompanied by symmetry breaking. However, there are still limited observations of counterparts in real space. Here we show a real-space topological phase transition in a chiral magnet MnGe, hosting a periodic array of hedgehog and antihedgehog topological spin singularities. This transition is driven by the pair annihilation of the hedgehogs and antihedgehogs acting as monopoles and antimonopoles of the emergent electromagnetic field. Observed anomalies in the magnetoresistivity and phonon softening are consistent with the theoretical prediction of critical phenomena associated with enhanced fluctuations of emergent field near the transition. This finding reveals a vital role of topology of the spins in strongly correlated systems., Phase transitions in topologically non-trivial systems are characterized by changes of topological invariants, rather than conventional order parameters. Here, the authors propose a real-space topological phase transition upon pair annihilation of emergent monopoles inherent in chiral magnet MnGe.

Topological conjugacy for continuous skew product flows on topological group bundles and semidirect products is studied resulting in a splitting theorem. The results are used to discuss fiber entropy.

We prove recurrence and multiple recurrence results for topological dynamical systems indexed by an arbitrary directed partial semigroup with respect to a coideal basis suitable for this semigroup, but otherwise arbitrary. These results are then applied to topological dynamical systems indexed by semigroups possessing digital representation. Our theory includes recurrence and multiple recurrence results for topological dynamical systems, indexed by natural numbers, or by finite non-empty subsets of natural numbers, proved by Furstenberg and Weiss, and analogous results for topological dynamical systems indexed by words.

This paper is devoted to the study of the topological pressure dimension for almost additive sequences, which is an extension of topological entropy dimension. We investigate fundamental properties of the topological pressure dimension for almost additive sequences. In particular, we study the relationships among different types of topological pressure dimension and identifies an inequality relating them. Also, we show that the topological pressure dimension is always equal to or greater than 1 for certain special almost additive sequence., 23 pages. arXiv admin note: text overlap with arXiv:1104.2195 by other authors

Crystal symmetries may protect single Dirac cones on the surface of a photonic crystal, creating a photonic analogue of a three-dimensional solid-state topological insulator.

Recent developments in the search for topological superconductivity have brought lattices of magnetic adatoms on a superconductor into intense focus. In this work we will study ferromagnetic chains of adatoms on superconducting surfaces with Rashba spin-orbit coupling. Generalising the deep-impurity approach employed extensively in previous works to arbitrary subgap energies, we formulate the theory of the subgap spectrum as a nonlinear matrix eigenvalue problem. We obtain an essentially analytical description of the subgap spectrum, allowing an efficient study of the topological properties. Employing a flat-band Hamiltonian sharing the topological properties of the chain, we evaluate the $\mathbb{Z}$-valued winding number and discover five distinct topological phases. Our results also confirm that the topological band formation does not require the decoupled Shiba energies to be fine-tuned to the gap centre. We also study the properties of Majorana bound states in the system., Comment: 11 pages, 16 figures

We present a unified approach to the asymptotic topological indices of various lattices.We propose the topological indices per vertex problem for lattice systems.The explicit asymptotic values of Laplacian energies for various lattices are obtained.We deduce the Laplacian energies of many types of lattices are independent of various boundary conditions. In this paper, we present a unified approach to the asymptotic topological indices of various lattices. Moreover, we propose the various topological indices per vertex problem for lattice systems and show that the various topological indices per vertex of lattices are independent of the toroidal, cylindrical, and free boundary conditions. Our result is a generalization of some earlier results.

We study the topology of two-dimensional open systems in terms of the Green's function. The Ishikawa-Matsuyama formula for the integer topological invariant is applied in open systems, which indicates the number difference of gapless edge bands arising from the poles and zeros of the Green's function. Meanwhile, we define another topological invariant via the single-particle density matrix, which works for general gapped systems and is equivalent to the former for the case of weak coupling to an environment. We also discuss two applications. For time-reversal-invariant insulators, the ${Z}_{2}$ index can be expressed by the invariant of each spin subsystem. As a second application, we consider the proximity effect when an ordinary insulator is coupled to a topological insulator., Comment: 9 pages, 3 figures (This version is the published one.)

The ground states of topological orders condense extended objects and support topological excitations. This nontrivial property leads to nonzero topological entanglement entropy $S_{topo}$ for conventional topological orders. Fracton topological order is an exotic class of models which is beyond the description of TQFT. With some assumptions about the condensates and the topological excitations, we derive a lower bound of the nonlocal entanglement entropy $S_{nonlocal}$ (a generalization of $S_{topo}$). The lower bound applies to Abelian stabilizer models including conventional topological orders as well as type \Rom{1} and type \Rom{2} fracton models, and it could be used to distinguish them. For fracton models, the lower bound shows that $S_{nonlocal}$ could obtain geometry-dependent values, and $S_{nonlocal}$ is extensive for certain choices of subsystems, including some choices which always give zero for TQFT. The stability of the lower bound under local perturbations is discussed., Comment: 13 pages, 14 figures. Modified in accordance to the published version

We show how to define a dynamical topological invariant for general one-dimensional topological systems after a quantum quench. Focusing on two-band topological insulators, we demonstrate that the reduced momentum-time manifold can be viewed as a series of submanifold $S^2$, and thus we are able to define a dynamical topological invariant on each of the sphere. We also unveil the intrinsic relation between the dynamical topological invariant and the difference of topological invariant of the initial and final static Hamiltonian. By considering some concrete examples, we illustrate the calculation of the dynamical topological invariant and its geometrical meaning explicitly., Comment: 5 pages, 3 figures + supplementary material

We analyze scaling behaviors of simulated annealing carried out on various classical systems with topological order, obtained as appropriate limits of the toric code in two and three dimensions. We first consider the three-dimensional $\mathbb{Z}_2$ (Ising) lattice gauge model, which exhibits a continuous topological phase transition at finite temperature. We show that a generalized Kibble-Zurek scaling ansatz applies to this transition, in spite of the absence of a local order parameter. We find perimeter-law scaling of the magnitude of a non-local order parameter (defined using Wilson loops) and a dynamic exponent $z=2.70 \pm 0.03$, the latter in good agreement with previous results for the equilibrium dynamics (autocorrelations). We then study systems where (topological) order forms only at zero temperature---the Ising chain, the two-dimensional $\mathbb{Z}_2$ gauge model, and a three-dimensional star model (another variant of the $\mathbb{Z}_2$ gauge model). In these systems the correlation length diverges exponentially, in a way that is non-smooth as a finite-size system approaches the zero temperature state. We show that the Kibble-Zurek theory does not apply in any of these systems. Instead, the dynamics can be understood in terms of diffusion and annihilation of topological defects, which we use to formulate a scaling theory in good agreement with our simulation results. We also discuss the effect of open boundaries where defect annihilation competes with a faster process of evaporation at the surface., 10 pages, 8 figures

The anomalous surface states of symmetry protected topological (SPT) phases are usually thought to be only possible in conjunction with the higher dimensional topological bulk. However, it has recently been realized that a class of anomalous SPT surface states can be realized in the same dimension if symmetries are allowed to act in a nonlocal fashion. An example is the particle-hole symmetric half filled Landau level, which effectively realizes the anomalous surface state of a 3D chiral Topological Insulator (class AIII). A dual description in terms of Dirac composite fermions has also been discussed. Here we explore generalizations of these constructions to multicomponent quantum Hall states. Our results include a duality mapping of the bilayer case to composite bosons with Kramers degeneracy and the possibility of a particle hole symmetric integer quantum Hall state when the number of components is a multiple of eight. Next, we make a further extension by half filling other classes of topological bands and imposing particle hole symmetry. When applied to time-reversal invariant topological insulators we realize a different chiral class (CII) topological surface state. Notably, half-filling a 3D TI band allows for the realization of the surface of the otherwise inaccessible 4D topological insulator, which supports an anomalous 3D Dirac cone. Surface topological orders equivalent to the 3D Dirac cone (from the global anomaly standpoint) are constructed and connections to Witten's SU(2) anomaly are made. These observations may also be useful for numerical simulations of topological surface states and of Dirac fermions without fermion doubling., Comment: 22 pages, 2 Figures, 3 Tables; published version

We develop methods to probe the excitation spectrum of topological phases of matter in two spatial dimensions. Applying these to the Fibonacci string nets perturbed away from exact solvability, we analyze a topological phase transition driven by the condensation of non-Abelian anyons. Our numerical results illustrate how such phase transitions involve the spontaneous breaking of a topological symmetry, generalizing the traditional Landau paradigm. The main technical tool is the characterization of the ground states using tensor networks and the topological properties using matrix-product-operator symmetries. The topological phase transition manifests itself by symmetry breaking in the entanglement degrees of freedom of the quantum transfer matrix.

Dynamical phases with novel topological properties are known to arise in driven systems of free fermions. In this paper, we obtain a `periodic table' to describe the phases of such time-dependent systems, generalizing the periodic table for static topological insulators. Using K-theory, we systematically classify Floquet topological insulators from the ten Altland-Zirnbauer symmetry classes across all dimensions. We find that the static classification scheme described by a group $G$ becomes $G\times G$ in the time-dependent case, and interpret the two factors as arising from the bipartite decomposition of the unitary time-evolution operator. Topologically protected edge modes may arise at the boundary between two Floquet systems, and we provide a mapping between the number of such edge modes and the topological invariant of the bulk., 16 pages, 2 tables, 2 figures; v2 includes minor changes/corrections and an improved Appendix E

The topological tail pressure and the conditional pressure are defined for sub-additive upper semi-continuous potentials via separated subsets and refining sequences of partitions in topological dynamical systems with finite topological entropy respectively. By proving the equivalence of the topological tail pressure and the conditional pressure with respect to the refining sequence of essential partitions, we present a variational principle for the topological tail pressure without any additional assumptions. The properties of the topological tail pressure including the power rule, the product rule and the natural extension are also discussed.

The goal of this paper is to investigate topological conditional pressure of a continuous transformation as defined for sub-additive potentials. This study presents a variational inequality for sub-additive topological conditional pressure on a closed subset, which is the other form of the variational principle for the sub-additive topological pressure presented by Cao, Feng, and Huang in [9]. Moreover, under some additional assumptions, this result can be generalized to the non-compact case.

This paper introduces a new simple system with a butterfly chaotic attractor. This system has rich and complex dynamics. With some typical parameters, its Lyapunov dimension is greater than other known three dimensional chaotic systems. It exhibits chaotic behavior over a large range of parameters, and the divergence of flow of this system is not a constant. The dynamics of this new system are analyzed via Lyapunov exponent spectrum, bifurcation diagrams, phase portraits and the Poincaré map. The compound structures of this new system are also analyzed. By means of topological horseshoe theory and numerical computation, the Poincaré map defined for the system is proved to be semi-conjugate to 3-shift map, and thus the system has positive topological entropy.

Classical topological relation expressions and computations are primarily based on abstract algebra. In this article, the representation and computation of geometry-oriented topological relations (GOTR) are developed. GOTR is the integration of geometry and topology. The geometries are represented by blades, which contain both algebraic expressions and construction structures of the geometries in the conformal geometric algebra space. With the meet, inner, and outer products, two topology operators, the MeetOp and BoundOp operators, are developed to reveal the disjoint/intersection and inside/on-surface/outside relations, respectively. A theoretical framework is then formulated to compute the topological relations between any pair of elementary geometries using the two operators. A multidimensional, unified and geometry-oriented algorithm is developed to compute topological relations between geometries. With this framework, the internal results of the topological relations computation are geometries. The topological relations can be illustrated with clear geometric meanings; at the same time, it can also be modified and updated parametrically. Case studies evaluating the topological relations between 3D objects are performed. The result suggests that our model can express and compute the topological relations between objects in a symbolic and geometry-oriented way. The method can also support topological relation series computation between objects with location or shape changes.

The topological embedding of a topological semigroup S, commutative with the property of cancelation, into the group G = S × S / R , (R the equivalence ( a , b ) R ( a ′ , b ′ ) ⇔ ab ′ = a ′ b ) to which S is algebraically embedded, was the subject of the search for the mathematicians of a long period. It was based on the fact that S must naturally be a uniform topological space, as every topological group was. The present paper is devoted to the fact that a quasi-uniformity is defined to any topological space, thus to any topological semigroup.

We give a new formula for the topological entropy of a continuous map f of the interval to itself with f2 topologically transitive. This leads to stronger forms of well-known results about horseshoes and periodic orbit patterns for continuous maps of the interval in the special case of topologically transitive maps.

In the paper we provide exact lower bounds of topological entropy in the class of transitive and mixing maps preserving the Lebesgue measure which are nowhere monotone (with dense knot points).

The aim of this paper is to introduce the concept of generalized topological molecular lattices briefly GTMLs as a generalization of Wang’s topological molecular lattices TMLs, Csaszar’s setpoint generalized topological spaces and lattice valued generalized topological spaces. Some notions such as continuous GOHs, convergence theory and separation axioms are introduced. Moreover, the relations among them are investigated.

We study the properties of the quantum states in the one-dimensional system with a shifted periodic potential in both the discrete model and the continuous model. With open boundary conditions, the edge states appear in the energy gaps which indicate non-trivial topological structures. The Chern numbers with respect to the Bloch vector and the potential shift angle are computed. In the limit of the continuous model, the Chern number of each band is exactly one. We demonstrate the particle number pumped by the adiabatically shift of the potential is directly related to the topological invariants., 5 pages, 3 figures, 1 table

We discuss a new type of solutions of the Yang-Baxter equation, called type-II solutions. They are related to quantum entanglements. The action of the corresponding braiding operator on the topological basis associated with a topological quantum field theory generates a (2J+1)-dimensional matrix form of the R-matrix for spin J, i.e., the Wigner function D with the spectral parameter θ denoting the entanglement degree. We present concrete examples for J = 1/2 and J = 1 in an explicit form. We show that the Hamiltonian related to the type-II R-matrix is Kitaev’s toy model.

The topological state of matter depends on its dimension. Remarkably, topological properties of quasiperiodic systems are found to emerge from higher dimensions.

We address the conditions required for a $\mathbb{Z}$ topological classification in the most general form of the non-Hermitian Su-Schrieffer-Heeger (SSH) model. Any chirally-symmetric SSH model will possess a "conjugated-pseudo-Hermiticity" which we show is responsible for a quantized "complex" Berry phase. Consequently, we provide the first example where the complex Berry phase of a band is used as a quantized invariant to predict the existence of gapless edge modes in a non-Hermitian model. The chirally-broken, $PT$-symmetric model is studied; we suggest an explanation for why the topological invariant is a global property of the Hamiltonian. A geometrical picture is provided by examining eigenvector evolution on the Bloch sphere. We justify our analysis numerically and discuss relevant applications., 8 pages (PRB, in press)

We study, by means of the density-matrix renormalization group (DMRG) technique, the evolution of the ground state in a one-dimensional topological insulator, from the non-interacting to the strongly-interacting limit, where the system can be mapped onto a topological Kondo-insulator model. We focus on a toy model Hamiltonian (i.e., the interacting "$sp$-ladder" model), which could be experimentally realized in optical lattices with higher orbitals loaded with ultra-cold fermionic atoms. Our goal is to shed light on the emergence of the strongly-interacting ground state and its topological classification as the Hubbard-$U$ interaction parameter of the model is increased. Our numerical results show that the ground state can be generically classified as a symmetry-protected topological phase of the Haldane-type, even in the non-interacting case $U=0$ where the system can be additionally classified as a time-reversal $\mathbb{Z}_{2}$-topological insulator, and evolves adiabatically between the non-interacting and strongly interacting limits., Comment: 9 pages, 3 figures, 1 appendix

We generalize the hierarchy construction to generic 2+1D topological orders (which can be non-Abelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a new equivalence relation between topological orders. We refer to the corresponding equivalent class (the orbit of the hierarchy construction) as "the non-Abelian family". Each non-Abelian family has one or a few root topological orders with the smallest number of anyon types. All the Abelian topological orders belong to the trivial non-Abelian family whose root is the trivial topological order. We show that Abelian anyons in root topological orders must be bosons or fermions with trivial mutual statistics between them. The classification of topological orders is then greatly simplified, by focusing on the roots of each family: those roots are given by non-Abelian modular extensions of representation categories of Abelian groups., 4+20 pages with a series of tables, comments welcome

The authors examine the phenomenon of ``Cheshire charge'' in topological phases of matter in three spatial dimensions. A looplike excitation is said to carry Cheshire charge if the charge is not locally detectable, that is, it can only be observed by a nonlocal process such as shrinking the loop to a point. The authors show that ``Cheshire charge'' is a generic feature of three-dimensional topological phases. They relate it to other features of these phases, such as three-loop braiding, as well as to higher-category theory that is hypothesized to be the general mathematical framework describing three-dimensional topological phases.

The periodically driven quantum Ising chain has recently attracted a large attention in the context of Floquet engineering. In addition to the common paramagnet and ferromagnet, this driven model can give rise to new topological phases. In this work we systematically explore its quantum phase diagram, by examining the properties of its Floquet ground state. We specifically focus on driving protocols with time-reversal invariant points, and demonstrate the existence of an infinite number of distinct phases. These phases are separated by second-order quantum phase transitions, accompanied by continuous changes of local and string order parameters, as well as sudden changes of a topological winding number and of the number of protected edge states. When one of these phase transitions is adiabatically crossed, the correlator associated to the order parameter is nonvanishing over a length scale which shows a Kibble-Zurek scaling. In some phases, the Floquet ground state spontaneously breaks the discrete time-translation symmetry of the Hamiltonian. Our findings provide a better understanding of topological phases in periodically driven clean integrable models., Comment: 19 pages, 12 figures, version published in PRB

Topological Matter Computing the electric polarization of a crystal is surprisingly tricky, but it can be tackled with the help of a topological concept, the so-called Berry phase. Extensions to higher multiple moments, such as quadrupole and octupole, are even trickier. Benalcazar et al. built a