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Condensed matter physics is currently undergoing a revolution through the introduction of concepts arising from topology that are used to characterize physical states, fields and properties from a completely different perspective. With the introduction of topology, the perspective is changed from describing complex systems in terms of local order parameters to a characterization by global quantities, which are measured nonlocally and which endow the systems with a global stability to perturbations. Prominent examples are topological insulators, skyrmions and Majorana fermions. Since topology translates into quantization, and topological order to entanglement, this ongoing revolution has impact on fields like mathematics, materials science, nanoelectronics and quantum information resulting in new device concepts enabling computations without dissipation of energy or enabling the possibility of realizing platforms for topological quantum computation, and ultimately reaching out into applications. Thus, these new exciting scientific developments and their applications are closely related to the grand challenges in information and communication technology and energy saving. Topology is the branch of mathematics that deals with properties of spaces that are invariant under smooth deformations. It provides newly appreciated mathematical tools in condensed matter physics that are currently revolutionizing the field of quantum matter and materials. Topology dictates that if two different Hamiltonians can be smoothly deformed into each other they give rise to many common physical properties and their states are homotopy invariant. Thus, topological invariance, which is often protected by discrete symmetries, provides some robustness that translates into the quantization of properties; such a robust quantization motivates the search and discovery of new topological matter. So far, the mainstream of modern topological condensed matter physics relies on two profoundly different scenarios: the emergence of the complex topology either in real space, as manifested e.g. in non-trivial magnetic structures or in momentum space, finding its realization in such materials as topological and Chern insulators. The latter renowned class of solids attracted considerable attention in recent years owing to its fascinating properties of spin-momentum locking, emergence of topologically protected surface/edge states governed by Dirac physics, as well as the quantization of Hall conductance and the discovery of the quantum spin Hall effect. Historically, the discovery of topological insulators gave rise to the discovery of a whole plethora of topologically non-trivial materials such asWeyl semimetals or topological superconductors, relevant in the context of the realization of Majorana fermions and topological quantum computation. [...]

Creating and measuring topological matter – with non-local order deeply embedded in the global structure of its quantum mechanical eigenstates – presents unique experimental challenges. Since this order has no signature in local correlation functions, it might seem experimentally inaccessible in any macroscopic system; however, as the precisely quantized Hall plateaux in integer and fractional quantum Hall systems show, topology can have macroscopic signatures at the system's edges. Ultracold atoms provide new experimental platforms where both the intrinsic topology and the edge behavior can be directly measured. This article reviews, using specific examples, how non-interacting topological matter may be created and measured in quantum gases.

We explore interesting possibilities for the realization, research and discovery of new topological states of matter in quantum gases with synthetic magnetic fields: namely, Weyl fermions and semimetals, and systems with fractional statistics (anyons).

Physics of graphene and physics of superfluid phases of 3He have many common features. Both systems are topological materials where quasiparticles behave as relativistic massless (Weyl, Majorana or Dirac) fermions. We formulate the points where these features are overlapping. This will allow us to use graphene for study the properties of superfluid 3He, to use superfluid 3He for study the properties of graphene, and to use the combination to study the physics of topological quantum vacuum. We suggest also some particular experiments with superfluid 3He using graphene as an atomically thin membrane impenetrable for He atoms but allowing for spin, momentum and energy transfer.

We consider the dimensional crossover in the topological matter, which involves the transformation of different types of topologically protected zeroes in the fermionic spectrum. In the considered case, the multiple Dirac (Fermi) point in quasi 2-dimensional system evolves into the flat band on the surface of the 3-dimensional system when the number of atomic layers increases. This is accompanied by formation of the spiral nodal lines in the bulk. We also discuss the topological quantum phase transition at which the surface flat band shrinks and changes its chirality, while the nodal spiral changes its helicity., 13 pages, 7 figures

Topological media are systems whose properties are protected by topology and thus are robust to deformations of the system. In topological insulators and superconductors the bulk-surface and bulk-vortex correspondence gives rise to the gapless Weyl, Dirac or Majorana fermions on the surface of the system and inside vortex cores. In gapless topological media, the bulk-surface and bulk-vortex correspondence produce topologically protected gapless fermions without dispersion - the flat band. Fermion zero modes forming the flat band are localized on the surface of topological media with protected nodal lines and in the vortex core in systems with topologically protected Fermi points (Weyl points). Flat band has an extremely singular density of states, and this property may give rise in particular to surface superconductivity which in principle could exist even at room temperature., 4 pages, 3 figures, version accepted for proceedings of the 4th International Conference "Fundamental Problems of High-Temperature Superconductivity" (FPS'11), Zvenigorod, October 2011

International audience

Topological materials and their unusual transport properties are now at the focus of modern experimental and theoretical research. Their topological properties arise from the bandstructure determined by the atomic composition of a material and as such are difficult to tune and naturally restricted to ≤3 dimensions. Here we demonstrate that n-terminal Josephson junctions with conventional superconductors may provide novel realizations of topology in n−1 dimensions, which have similarities, but also marked differences with existing 2D or 3D topological materials. For n≥4, the Andreev subgap spectrum of the junction can accommodate Weyl singularities in the space of the n−1 independent superconducting phases, which play the role of bandstructure quasimomenta. The presence of these Weyl singularities enables topological transitions that are manifested experimentally as changes of the quantized transconductance between two voltage-biased leads, the quantization unit being 4e2/h, where e is the electric charge and h is the Planck constant., Materials with topologically nontrivial band structures possess exotic electronic transport properties however they are naturally constrained below three dimensions. Here, the authors demonstrate how analogous systems with n−1 dimensions may be constructed from Josephson junctions of n-terminals.

The massless solutions to the Dirac equation are described by the so-called Weyl Hamiltonian. The Weyl equation requires a particle to have linear dispersion in all three dimensions while being doubly degenerate at a single momentum point. These Weyl points are topological monopoles of quantized Berry flux exhibiting numerous unusual properties. We performed angle-resolved microwave transmission measurements through a double-gyroid photonic crystal with inversion-breaking where Weyl points have been theoretically predicted to occur. The excited bulk states show two linear dispersion bands touching at four isolated points in the three-dimensional Brillouin zone, indicating the observation of Weyl points. This work paves the way to a variety of photonic topological phenomena in three dimensions.

A Weyl semimetal is a new state of matter that hosts Weyl fermions as emergent quasiparticles and admits a topological classification that protects Fermi arc surface states on the boundary of a bulk sample. This unusual electronic structure has deep analogies with particle physics and leads to unique topological properties. We report the experimental discovery of a Weyl semimetal, tantalum arsenide (TaAs). Using photoemission spectroscopy, we directly observe Fermi arcs on the surface, as well as the Weyl fermion cones and Weyl nodes in the bulk of TaAs single crystals. We find that Fermi arcs terminate on the Weyl fermion nodes, consistent with their topological character. Our work opens the field for the experimental study of Weyl fermions in physics and materials science.

We present a complete quantization of Lorentzian D=1+2 gravity with cosmological constant, coupled to a set of topological matter fields. The approach of Loop Quantum Gravity is used thanks to a partial gauge fixing leaving a residual gauge invariance under a compact semi-simple gauge group, namely Spin(4) = SU(2) x SU(2). A pair of quantum observables is constructed, which are non-trivial despite of being null at the classical level., 18 pages, Latex. Two references added

We describe an efficient numerical approach to calculate the longitudinal and transverse Kubo conductivities of large systems using Bastin's formulation. We expand the Green's functions in terms of Chebyshev polynomials and compute the conductivity tensor for any temperature and chemical potential in a single step. To illustrate the power and generality of the approach, we calculate the conductivity tensor for the quantum Hall effect in disordered graphene and analyze the effect of the disorder in a Chern insulator in Haldane's model on a honeycomb lattice., 5 pages, 3 figures and a supplementary material (3 pages)

A magnetic domain boundary on the surface of a three-dimensional topological insulator is predicted to host a chiral edge state, but direct demonstration is challenging. We used a scanning superconducting quantum interference device to show that current in a magnetized topological insulator heterostructure (EuS/Bi2Se3) flows at the edge when the Fermi level is gate-tuned to the surface band gap. We further induced micrometer-scale magnetic structures on the heterostructure and detected a chiral edge current at the magnetic domain boundary. The chirality of the current was determined by magnetization of the surrounding domain, and its magnitude by the local chemical potential rather than the applied current. Such magnetic structures provide a platform for detecting topological magnetoelectric effects and may enable progress in quantum information processing and spintronics.

Majorana fermions are predicted to localize at the edge of a topological superconductor, a state of matter that can form when a ferromagnetic system is placed in proximity to a conventional superconductor with strong spin-orbit interaction. With the goal of realizing a one-dimensional topological superconductor, we have fabricated ferromagnetic iron (Fe) atomic chains on the surface of superconducting lead (Pb). Using high-resolution spectroscopic imaging techniques, we show that the onset of superconductivity, which gaps the electronic density of states in the bulk of the Fe chains, is accompanied by the appearance of zero-energy end-states. This spatially resolved signature provides strong evidence, corroborated by other observations, for the formation of a topological phase and edge-bound Majorana fermions in our atomic chains.

Perturbing usual type B topological matter with vector $(0,1)$-forms we find a topological theory which contains explicitly Kodaira-Spencer deformation theory. It is shown that, in genus zero, three-point correlation functions give the Yukawa couplings for a generic point in the moduli space of complex structures. This generalization of type B topological matter seems to be the correct framework to understand mirror symmetry in terms of two-dimensional topological field theories., Comment: 17 pages, phyzzx, US-FT/7-94

N=2 super high-order Virasoro algebra and topological high-order Virasoro algebra are constructed. We then present the super high-order Virasoro gravity and high-order Virasoro topological matter.

We introduce a universally applicable method, based on the bond-algebraic theory of dualities, to search for generalized order parameters in disparate systems including non-Landau systems with topological order. A key notion that we advance is that of {\em holographic symmetry}. It reflects situations wherein global symmetries become, under a duality mapping, symmetries that act solely on the system's boundary. Holographic symmetries are naturally related to edge modes and localization. The utility of our approach is illustrated by systematically deriving generalized order parameters for pure and matter-coupled Abelian gauge theories, and for some models of topological matter., Comment: v2, 10 pages, 3 figures. Accepted for publication in Physical Review B Rapid Communications

Topological matter is characterized by the presence of a topological BF term in its long-distance effective action. Topological defects due to the compactness of the U(1) gauge fields induce quantum phase transitions between topological insulators, topological superconductors and topological confinement. In conventional superconductivity, due to spontaneous symmetry breaking, the photon acquires a mass due to the Anderson-Higgs mechanism. In this paper we derive the corresponding effective actions for the electromagnetic field in topological superconductors and topological confinement phases. In topological superconductors magnetic flux is confined and the photon acquires a topological mass through the BF mechanism: no symmetry breaking is involved, the ground state has topological order and the transition is induced by quantum fluctuations. In topological confinement, instead, electric charge is linearly confined and the photon becomes a massive antisymmetric tensor via the St\"uckelberg mechanism. Oblique confinement phases arise when the string condensate carries both magnetic and electric flux (dyonic strings). Such phases are characterized by a vortex quantum Hall effect potentially relevant for the dissipationless transport of information stored on vortices., Comment: 5 pages, no figures

The goal of this proposal is to model the stability of quantum states of matter derived from topological insulators against two types of corrections: strong inter-particle interactions and heating. I will examine interacting atoms in square optical lattices with spin orbit coupling, and more generally, gauge fields, as a route to building Hubbard models hosting fractional topological insulators. I will analyze these models by combining numerical exact diagonalization on small lattice clusters with an analytic variational theory. I also propose to study a new model of finite temperature topological superconductors of dipoles placed in an optical lattice. I will construct and analyze a model using a combination of mean field theory and quantum Monte Carlo. The proposed work will foster new directions in experiments with optical lattices containing cold atomic gases. These new states of matter should exhibit new particles as excitations. Analyses of the stability of these new topological phases will thus play a crucial role in advancing fundamentally new directions in physics.

We propose to use a permutation symmetric sample of multi-level atoms to simulate the properties of topologically ordered states. The Rydberg blockade interaction is used to prepare states of the sample which are equivalent to resonating valence bond states, Laughlin states, and string-net condensates and to create and study the properties of their quasi-particle-like fundamental excitations., 9 pages, 5 figures

We show how topological $G_k/G_k$ models can be embedded into the topological matter models that are obtained by perturbing the twisted $N=2$ supersymmetric, hermitian symmetric, coset models. In particular, this leads to an embedding of the fusion ring of $G$ as a sub-ring of the perturbed, chiral primary ring. The perturbation of the twisted $N=2$ model that leads to the fusion ring is also shown to lead to an integrable $N=2$ supersymmetric field theory when the untwisted $N=2$ superconformal field theory is perturbed by the same operator and its hermitian conjugate., Comment: 24 pages

We derive directly from the N=2 LG superpotential the differential equations that determine the flat coordinates of arbitrary topological CFT's., Comment: 34 p

Topological matter is an unconventional form of matter: it exhibits a global hidden order which is not associated with the spontaneous breaking of any symmetry. The defects of this exotic type of order are anyons, quasiparticles with fractional statistics. Moreover, when living on a surface with non-trivial topology, like a plane with a hole or a torus, this type of matter develops a number of degenerate states which are locally indistinguishable and could be used to build a quantum memory naturally resistant to errors. Except for the fractional quantum Hall effect there is no experimental evidence as to the existence of topologically ordered phases, and it remains a huge challenge to develop theoretical techniques to look for them in realistic models and find them in the laboratory. Here we show how to use ultracold atoms in optical lattices to create and detect different instances of topological order in the minimum nontrivial system: four spins in a plaquette. By combining different techniques we show how to prepare these spins in mimimum versions of topical topological liquids like resonant valence bond or Laughlin states, probe their fractional quasiparticle excitations, and exploit them to build a mini-topological quantum memory.

We show how the N = 2 super-W ∞ algebra can be twisted to give a topological algebra, which we call W ∞ top . This can be realised by a matter system comprising a complex scalar and a complex spinor field. We construct a topological BRST charge for W ∞ top , and use this to obtain a topological W ∞ gravity theory. This can then be coupled to the W-topological matter system. The total lagrangian arises as a BRST variation.

Motivated by the prospect of attaining Majorana modes at the ends of nanowires, we analyze interacting Majorana systems on general networks and lattices in an arbitrary number of dimensions, and derive various universal spin duals. Such general complex Majorana architectures (other than those of simple square or other crystalline arrangements) might be of empirical relevance. As these systems display low-dimensional symmetries, they are candidates for realizing topological quantum order. We prove that (a) these Majorana systems, (b) quantum Ising gauge theories, and (c) transverse-field Ising models with annealed bimodal disorder are all dual to one another on general graphs. As any Dirac fermion (including electronic) operator can be expressed as a linear combination of two Majorana fermion operators, our results further lead to dualities between interacting Dirac fermionic systems. The spin duals allow us to predict the feasibility of various standard transitions as well as spin-glass type behavior in {\it interacting} Majorana fermion or electronic systems. Several new systems that can be simulated by arrays of Majorana wires are further introduced and investigated: (1) the {\it XXZ honeycomb compass} model (intermediate between the classical Ising model on the honeycomb lattice and Kitaev's honeycomb model), (2) a checkerboard lattice realization of the model of Xu and Moore for superconducting $(p+ip)$ arrays, and a (3) compass type two-flavor Hubbard model with both pairing and hopping terms. By the use of dualities, we show that all of these systems lie in the 3D Ising universality class. We discuss how the existence of topological orders and bounds on autocorrelation times can be inferred by the use of symmetries and also propose to engineer {\it quantum simulators} out of these Majorana networks., Comment: v3,19 pages, 18 figures, submitted to Physical Review B. 11 new figures, new section on simulating the Hubbard model with nanowire systems, and two new appendices

We consider the coupling of scalar topological matter to (2+1)-dimensional gravity. The matter fields consist of a 0-form scalar field and a 2-form tensor field. We carry out a canonical analysis of the classical theory, investigating its sectors and solutions. We show that the model admits both BTZ-like black-hole solutions and homogeneous/inhomogeneous FRW cosmological solutions.We also investigate the global charges associated with the model and show that the algebra of charges is the extension of the Kac-Moody algebra for the field-rigid gauge charges, and the Virasoro algebrafor the diffeomorphism charges. Finally, we show that the model can be written as a generalized Chern-Simons theory, opening the perspective for its formulation as a generalized higher gauge theory., Comment: 40 pages