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We find a series of topological phase transitions of increasing order, beyond the more standard second-order phase transition in a one-dimensional topological superconductor. The jumps in the order of the transitions depend on the range of the pairing interaction, which is parametrized by an algebraic decay with exponent $alpha$. Remarkably, in the limit $alpha = 1$ the order of the topological transition becomes infinite. We compute the critical exponents for the series of higher-order transitions in exact form and find that they fulfill the hyperscaling relation. We also study the critical behaviour at the boundary of the system and discuss potential experimental platforms of magnetic atoms in superconductors.
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Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional $p$-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the $k$-space reduced density matrix. We show that in the topologically nontrivial phases the highest weight eigenvector is fully aligned with the triplet pairing state. A signature of the various phase transitions between two points on the parameter space is encoded in the $k$-space fidelity operator.
We investigate the order of the topological quantum phase transition in a two dimensional quadrupolar topological insulator within a thermodynamic approach. Using numerical methods, we separate the bulk, edge and corner contributions to the grand potential and detect different phase transitions in the topological phase diagram. The transitions from the quadrupolar to the trivial or to the dipolar phases are well captured by the thermodynamic potential. On the other hand, we have to resort to a grand potential based on the Wannier bands to describe the transition from the trivial to the dipolar phase. The critical exponents and the order of the phase transitions are determined and discussed in the light of the Josephson hyperscaling relation.
Higher order topological insulators are a new class of topological insulators in dimensions $rm d>1$. These higher-order topological insulators possess $rm (d - 1)$-dimensional boundaries that, unlike those of conventional topological insulators, do not conduct via gapless states but instead are themselves topological insulators. Precisely, an $rm n^{rm th}$-order topological insulator in $rm m$ dimensions hosts $rm d_{c} = (m - n)$-dimensional boundary modes $rm (n leq m)$. For instance, a three-dimensional second (third) order topological insulator hosts gapless modes on the hinges (corners), characterized by $rm d_{c} = 1 (0)$. Similarly, a second order topological insulator in two dimensions only has gapless corner states ($rm d_{c} = 0$) localized at the boundary. These higher order phases are protected by various crystalline symmetries. Moreover, in presence of proximity induced superconductivity and appropriate symmetry breaking perturbations, the above mentioned bulk-boundary correspondence can be extended to higher order topological superconductors hosting Majorana hinge or corner modes. Such higher-order systems constitute a distinctive new family of topological phases of matter which has been experimentally observed in acoustic systems, multilayer $rm WTe_{2}$ and $rm Bi_{4}Br_{4}$ chains. In this general article, the basic phenomenology of higher order topological insulators and higher order topological superconductors are presented along with some of their experimental realization.
Topological insulators (TIs) having intrinsic or proximity-coupled s-wave superconductivity host Majorana zero modes (MZMs) at the ends of vortex lines. The MZMs survive up to a critical doping of the TI at which there is a vortex phase transition that eliminates the MZMs. In this work, we show that the phenomenology in higher-order topological insulators (HOTIs) can be qualitatively distinct. In particular, we find two distinct features. (i) We find that vortices placed on the gapped (side) surfaces of the HOTI, exhibit a pair of phase transitions as a function of doping. The first transition is a surface phase transition after which MZMs appear. The second transition is the well-known vortex phase transition. We find that the surface transition appears because of the competition between the superconducting gap and the local $mathcal{T}$-breaking gap on the surface. (ii) We present numerical evidence that shows strong variation of the critical doping for the vortex phase transition as the center of the vortex is moved toward or away from the hinges of the sample. We believe our work provides new phenomenology that can help identify HOTIs, as well as illustrating a promising platform for the realization of MZMs.
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 role in the search for Majorana modes in condensed matter systems. Also, the $sp$ chain, with an anti-symmetric mixing among the $s$ and $p$ bands provides a paradigmatic example of a topological insulator with well understood properties. There is an intimate relation between these two models and in particular their topological quantum phase transitions share the same universality class. Here we consider a two-band $sp$ model of spinless fermions with an attractive (inter-band) interaction. Both the interaction and hybridization between the $s$ and $p$ fermions are anti-symmetric. The zero temperature phase diagram of the model presents a variety of phases including a Weyl superconductor, topological insulator and trivial phases. The quantum phase transitions between these phases can be either continuous or discontinuous. We show that the transition from the topological superconducting phase to the trivial one has critical exponents different from those of an equivalent transition in Kitaevs model.
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