In (1+1)D topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group $G_f$ of the ground state splits as $G_f = G_b \times \mathbb{Z}_2^f$, where $\mathbb{Z}_2^f$ is generated by fermion parity, $(-1)^F$. In contrast, (2+1)D topological superconductors (TSC) can host unpaired MZMs at defects even when $G_f$ is not of the form $G_b \times \mathbb{Z}_2^f$. In this paper we study how $G_f$ together with the chiral central charge $c_-$ strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a non-perturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as $\mathrm{U}(1)^f\rtimes H,\mathrm{SU}(2)^f \times H, \mathrm{U}(2)^f\rtimes H $, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as Sr$_2$RuO$_4$, transition metal dichalcogenides and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models.
Many clever routes to Majorana fermions have been discovered by exploiting the interplay between superconductivity and band topology in metals and insulators. However, realizations in semimetals remain less explored. We ask, ``under what conditions do superconductor vortices in time-reversal symmetric Weyl semimetals -- three-dimensional semimetals with only time-reversal symmetry -- trap Majorana fermions on the surface?'' If each constant-$k_{z}$ plane, where $z$ is the vortex axis, contains equal numbers of Weyl nodes of each chirality, we predict a generically gapped vortex and derive a to…
Majorana fermions are a fascinating medium for discovering new phases of matter. However, the standard analytical tools are very limited in probing the non-perturbative aspects of interacting Majoranas in more than one dimensions. Here, we employ the holographic correspondence to determine the specific heat of a two-dimensional interacting gapless Majorana system. To perform our analysis we first describe the interactions in terms of a pseudo-scalar torsion field. We then allow fluctuations in the background curvature thus identifying our model with a $(2+1)$-dimensional Anti-de Sitter (AdS) g…
Quantum spin liquids realize massive entanglement and fractional quasiparticles from localized spins, proposed as an avenue for quantum science and technology. In particular, topological quantum computations are suggested in the non-abelian phase of Kitaev quantum spin liquid with Majorana fermions, and detection of Majorana fermions is one of the most outstanding problems in modern condensed matter physics. Here, we propose a concrete way to identify the non-abelian Kitaev quantum spin liquid by magnetic field angle dependence. Topologically protected critical lines exist on a plane of magnet…
We study the low energy spectrum of a correlated quantum dot embedded between the normal conducting and superconducting reservoirs and hybridized with the topological superconducting nanowire, hosting the Majorana end-modes. We investigate the leaking Majorana quasiparticle and inspect its interplay with the proximity induced on-dot pairing and correlations. In particular, we focus on the subgap Kondo effect near the quantum phase transition/crossover from the spinfull (doublet) to the spinless (BCS-type singlet) configurations. Treating the correlations perturbatively and within the NRG appro…
Mechanical strain can generate a pseudo-magnetic field, and hence Landau levels (LL), for low energy excitations of quantum matter in two dimensions. We study the collective state of the fractionalised Majorana fermions arising from residual generic spin interactions in the central LL, where the projected Hamiltonian reflects the spin symmetries in intricate ways: emergent U(1) and particle-hole symmetries forbid any bilinear couplings, leading to an intrinsically strongly interacting system; also, they allow the definition of a filling fraction, which is fixed at 1/2. We argue that the result…
Geometric phases, generated by cyclic evolutions of quantum systems, offer an inspiring playground for advancing fundamental physics and technologies, alike. Intriguingly, the exotic statistics of anyons realised in physical systems can be interpreted as a topological version of geometric phases. However, non-Abelian statistics has not yet been demonstrated in the laboratory. Here we employ an all optical quantum system that simulates the statistical evolution of Majorana fermions. As a result we experimentally realise non-Abelian Berry phases with the topological characteristic that they are …
We analyze the effects of electron-electron interactions and disorder on a Rashba double-nanowire setup coupled to an s-wave superconductor, which has been recently proposed as a versatile platform to generate Kramers pairs of Majorana bound states in the absence of magnetic fields. We identify the regime of parameters for which these Kramers pairs are stable against interaction and disorder effects. We use bosonization, perturbative renormalization group, and replica techniques to derive the flow equations for various parameters of the model and evaluate the corresponding phase diagram with t…
The Kitaev model on the honeycomb lattice is a paradigmatic system known to host a wealth of nontrivial topological phases and Majorana edge modes. In the static case, the Majorana edge modes are nondispersive. When the system is periodically driven in time, such edge modes can disperse and become chiral. We obtain the full phase diagram of the driven model as a function of the coupling and the driving period. We characterize the quantum criticality of the different topological phase transitions in both the static and driven model via the notions of Majorana-Wannier state correlation functions…
Majorana Diagrammatics for Quantum Spin-1/2 Models
Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée（索邦大学）；Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Magnetic orders characterized by multiple ordering vectors harbor noncollinear and noncoplanar spin textures and can be a source of unusual electronic properties through the spin Berry phase mechanism. We theoretically show that such multiple-$Q$ states are stabilized in itinerant magnets in the form of superpositions of collinear up-up-down-down (UUDD) spin states, which accompany the density waves of vector and scalar chirality. The result is drawn by examining the ground state of the Kondo lattice model with classical localized moments, especially when the Fermi surface is tuned to be parti…
We prove that quantum information encoded in some topological excitations, including certain Majorana zero modes, is protected in closed systems for a time scale exponentially long in system parameters. This protection holds even at infinite temperature. At lower temperatures the decay time becomes even longer, with a temperature dependence controlled by an effective gap that is parametrically larger than the actual energy gap of the system. This non-equilibrium dynamical phenomenon is a form of prethermalization, and occurs because of obstructions to the equilibriation of edge or defect degre…
