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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…

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…

Topology in momentum space is the main characteristics of the ground states of a system at zero temperature, the quantum vacua. The gaplessness of fermions in bulk, on the surface or inside the vortex core is protected by topology.…

Three-dimensional topological gapless matters with gapless degeneracies protected by a topological invariant defined over a closed manifold in momentum space have attracted considerable interest in various fields ranging from condensed…

We show that the topologically protected flat band emerging on a surface of a nodal fermionic system promotes the surface superconductivity due to an infinitely large density of states associated with the flat band. The critical temperature…

Topological phase transitions in condensed matters accompany emerging singularities of the electronic wave function, often manifested by gap-closing points in the momentum space. In conventional topological insulators in three dimensions…

The importance of models with an exact solution for the study of materials with non-trivial topological properties has been extensively demonstrated. Among these, the Kitaev model of a one-dimensional $p$-wave superconductor plays a guiding…

Topological semimetals, such as the Weyl and Dirac semimetals, represent one of the most active research fields in modern condensed matter physics. The peculiar physical properties of these systems mainly originate from their underlying…

Topological one-dimensional superconductors can sustain in their extremities zero energy modes that are protected by different kinds of symmetries. The observation of these excitations in the form of Majorana fermions is one of the most…

We review recent theoretical progress in the understanding and prediction of novel topological semimetals. Topological semimetals define a class of gapless electronic phases exhibiting topologically stable crossings of energy bands.…

Topological semimetals with different types of band crossings provide a rich platform to realize novel fermionic excitations, known as topological fermions. In particular, some fermionic excitations can be direct analogues of elementary…

Many quantum condensed matter systems are strongly correlated and strongly interacting fermionic systems, which cannot be treated perturbatively. However, topology allows us to determine generic features of their fermionic spectrum, which…

Most superconducting materials are well-understood and conventional in the sense that the pairs of electrons that cause the superconductivity by their condensation have the highest possible symmetry. Famous exceptions are the enigmatic…

Topological insulators and topological semimetals are both new classes of quantum materials, which are characterized by surface states induced by the topology of the bulk band structure. Topological Dirac or Weyl semimetals show linear…

It remains an open problem if there are universal scaling functions across a topological quantum phase transition (TPT) without an order parameter, but with extended Fermi surfaces (FS ). Here, we study a simple system of fermions hopping…

Topological semimetals are gapless states of matter which have robust and unique electromagnetic responses and surface states. In this paper, we consider semimetals which have point like Fermi surfaces in various spatial dimensions…

Quantum vacua are characterized by the topological structure of their fermion zero modes. The vacua are distributed into universality classes protected by topology in momentum space. The vacua whose manifold of fermion zero modes has…

Topological phase transitions in band models are usually associated to the gap closing between the highest valance band and the lowest conduction band, which can give rise to different types of nodal structures, such as Dirac/Weyl points,…

Study of the Weyl and Dirac topological materials (topological semimetals, insulators, superfluids and superconductors) opens the route for the investigation of the topological quantum vacua of relativistic fields. The symmetric phase of…

Granular matter at the jamming transition is poised on the brink of mechanical stability, and hence it is possible that these random systems have topologically protected surface phonons. Studying two model systems for jammed matter, we find…