Number Of Majorana Research Articles

Tuning the topological state of a helical atom chain via a Josephson phase

By solving the Bogoliubov-de Gennes equations, we study the quasiparticle spectrum of a magnetic atom chain placed inside a short constriction-type Josephson junction. A helical magnetic order of the atomic spins is assumed, so that a topologically nontrivial state with Majorana edge modes at the ends of the chain can appear. It is found that in the presence of a nonzero Josephson phase the subgap spectrum of an infinite chain consists of four branches (Shiba bands). This spectrum is almost certainly gapped if the atomic spins form a coplanar spiral. The Majorana number of the given system is calculated analytically. It is demonstrated that a Josephson phase shift can be used to drive the system into a topologically nontrivial state and to tune the size of the topological gap. The spatial structure of Majorana edge modes is studied as well. We generalize the effective model based on discrete Bogoliubov-de Gennes equations developed by Pientka et al. for a bulk superconductor to the case of a Josephson junction. Using these discrete equations, the wave functions of Majorana zero modes are calculated analytically for a coplanar atomic spin spiral with arbitrary pitch. The wave functions exhibit an intermediate asymptotic behavior with a nonexponential fall-off with distance from the edge of the atomic chain.

Symmetry breaking to Majorana Brown-Susskind metric

In parts I [1] and II [2] of our earlier work, we studied how metrics gij on mathfrak{su} (n) may spontaneously break symmetry and crystallize into a form which is kaq, knows about qubits. We did this for n = 2N and then away from powers of 2. Here we address the Fermionic version and find kam metrics, these know about Majoranas. That is, there is a basis of principal axes {Hk} of which is of homogeneous Majorana degree. In part I, we searched unsuccessfully for functional minima representing crystallized metrics exhibiting the Brown-Susskind penalty schedule, motivated by their study of black hole scrambling time. Here, by segueing to the Fermionic setting we find, to good approximation, kam metrics adhering to this schedule on both mathfrak{su} (4) and mathfrak{su} (8). Thus, with this preliminary finding, our toy model exhibits two of the three features required for the spontaneous emergence of spatial structure: (1) localized degrees of freedom and, (2) a preference for low body-number (or low Majorana number) interactions. The final feature, (3) constraints on who may interact with whom, i.e. a neighborhood structure, must await an effective analytic technique, being entirely beyond what we can approach with classical numerics.

Reconstructed conventional bulk-boundary correspondence in the non-Hermitian s -wave nodal-ring superconductor

Recently, the interplay between the non-Hermitian skin effect (NHSE) and topological phases has attracted a lot of attention. When the NHSE occurs in a topological system, the conventional bulk-boundary correspondence (BBC) based on the Bloch band theory is destroyed. In this work, we investigate a non-Hermitian nodal-ring semimetal with the $s$-wave superconducting (SC) term. We find that the SC term introduces the intrinsic coupling between the different general Brillouin zones (GBZs) of the particle and hole subbands in our model, which drives the total GBZ to the BZ. Furthermore, when the SC term is finite, we show a possibility that the GBZ is the same as the BZ. We also find that the Bloch winding number of the quasi-one-dimensional model can predict the number of Majorana zero states faithfully in this case, which implies that the conventional BBC is reconstructed in our model. The localization strengths of the Majorana zero states are also calculated to show the reconstruction of conventional BBC. The finite-size effect, related experimental regimes in the electric circuit, and the fragility of this reconstructed conventional BBC are also discussed in this work.

Double Majorana vortex zero modes in superconducting topological crystalline insulators with surface rotation anomaly

The interplay of time-reversal and $n$-fold rotation symmetries ($n=2,4,6$) is known to bring a new class of topological crystalline insulators (TCIs) having $n$ surface Dirac cones due to surface rotation anomaly. We show that the proximity-induced $s$-wave superconductivity on the surface of these TCIs yields a topological superconducting phase in which two Majorana zero modes are bound to a vortex, and that $n$-fold rotation symmetry ($n=2,4,6$) enriches the topological classification of a superconducting vortex from $\mathbb{Z}_2$ to $\mathbb{Z}_2\times\mathbb{Z}_2$. Using a model of a three-dimensional high-spin topological insulator with $s$-wave superconductivity and two-fold rotation symmetry, we show that, with increasing chemical potential, the number of Majorana zero modes at one end of a vortex changes as $2\to1\to0$ through two topological vortex phase transitions. In addition, we show that additional magnetic-mirror symmetry further enhances the topological classification to $\mathbb{Z} \times \mathbb{Z}$

Topological phase transitions, Majorana modes, and quantum simulation of the Su–Schrieffer–Heeger model with nearest-neighbor interactions

The Su--Schrieffer--Heeger (SSH) model is one of the simplest topological lattice models and is a promising candidate for quantum simulations of topological systems. In this study we investigate an interacting SSH model with nearest-neighbor interactions using mean field theory and an exact diagonalization method after the Jordan-Wigner transformation. We obtain a rich phase diagram with six different phases. Majorana bound states are found and confirmed by analyzing wave functions and Majorana number. In addition, we propose to use superconducting devices to construct a one-dimensional quantum circuit, where Cooper pair boxes and Josephson junctions are alternately connected in series, to theoretically simulate the interacting SSH model. This approach also provides us a toolkit to simulate correlated topological systems.

Topological phase diagram of the disordered 2XY model in presence of generalized Dzyaloshinskii–Moriya interaction

The topological index of a system determines its edge physics. However, in situations such as strong disorder where due to level repulsion the spectral gap closes, the topological indices are not well-defined. In this paper, we show that the localization length of zero modes determined by the transfer matrix method reveals much more information than the topological index. The localization length can provide not only information about the topological index of the Hamiltonian itself, but it can also provide information about the topological indices of the ‘relative’ Hamiltonians. As a case study, we study a generalized XY model (2XY model) further augmented by a generalized Dziyaloshinskii–Moriya-like (DM) interaction parameterized by that after fermionization breaks the time-reversal invariance. The parent Hamiltonian at which belongs to the BDI class is indexed by an integer winding number while the daughter Hamiltonian which belongs to class D is specified by a Z2 index . We show that the localization length, in addition to determining Z2, can count the number of Majorana zero modes leftover at the boundary of the daughter Hamiltonian—which are not protected by the winding number anymore. Therefore the localization length outperforms the standard topological indices in two respects: (i) it is much faster and more accurate to calculate and (ii) it can count the winding number of the parent Hamiltonian by looking into the edges of the daughter Hamiltonian.

LSZ-reduction, resonances and non-diagonal propagators: Fermions and scalars

We analyze in details the effects associated with mixing of fermionic fields. In a system with an arbitrary number of Majorana or Dirac particles, a simple proof of factorizability of residues of non-diagonal propagators at the complex poles is given, together with a prescription for finding the “square-rooted” residues to all orders of perturbation theory, in an arbitrary renormalization scheme. Corresponding prescription for the scalar case is provided as well.

Proposal for direct measuring of the Majorana number in a topological superconductor using a quantum dot

We propose to directly measure the Majorana number for one-dimensional topological superconductors using a quantum dot. The setup consists of two topological superconducting wires with four Majorana zero modes, which are coupled to an external quantum dot. The measurement is achieved by utilizing the definition of the Majorana number, which is the charge-parity flipping when changing the boundary condition for the topological superconductor. We consider a control of the boundary condition with voltage gates. When the voltage on the gate are modulated sequentially, the boundary conditions changes and the parity of the superconducting state flips. We demonstrate that this parity flipping will change the electron occupation probability of the quantum dot, which reflects the value of the Majorana number.

Central Charge of Periodically Driven Critical Kitaev Chains.

Periodically driven Kitaev chains show a rich phase diagram as the amplitude and frequency of the drive is varied, with topological phase transitions separating regions with different number of Majorana zero and π modes. We explore whether the critical point separating different phases of the periodically driven chain may be characterized by a universal central charge. We affirmatively answer this question by studying the entanglement entropy (EE) numerically and analytically for the lowest entangled many particle eigenstate at arbitrary nonstroboscopic and stroboscopic times. We find that the EE at the critical point scales logarithmically with a time-independent central charge, and that the Floquet micromotion gives only subleading corrections to the EE. This result also generalizes to multicritical points where the EE is found to have a central charge that is the sum of the central charges of the intersecting critical lines.

Magnetic field driven nodal topological superconductivity in monolayer transition metal dichalcogenides

Recently, Ising superconductors that possess in-plane upper critical fields Hc2 much larger than the Pauli limit field are under intense experimental study. Many monolayer or few layer transition metal dichalcogenides are shown to be Ising superconductors. Here we show that in a wide range of experimentally accessible regimes where the in-plane magnetic field is higher than the Pauli limit field but lower than Hc2, a 2H-structure monolayer NbSe2 or similarly TaS2 becomes a nodal topological superconductor. The bulk nodal points appear on the Γ−M lines of the Brillouin zone where the Ising SOC vanishes. The nodal points are connected by Majorana flat bands, and the flat bands are associated with a large number of Majorana zero energy edge modes that induce spin-triplet Cooper pairs. This work demonstrates an experimentally feasible way to realize Majorana fermions in nodal topological superconductor, without any fine-tuning of experimental parameters.

Supersymmetry in closed chains of coupled Majorana modes

We consider a closed chain of even number of Majorana zero modes with nearest-neighbour couplings which are different site by site generically, thus no any crystal symmetry. Instead, we demonstrate the possibility of an emergent supersymmetry (SUSY), which is accompanied by gapless Fermionic excitations. In particular, the condition can be easily satisfied by tuning only one coupling, regardless of how many other couplings are there. Such a system can be realized by four Majorana modes on two parallel Majorana nanowires with their ends connected by Josephson junctions and bodies connected by an external superconducting ring. By tuning the Josephson couplings with a magnetic flux $\Phi$ through the ring, we get the gapless excitations at $\Phi_{SUSY}=\pm f\Phi_0$ with $\Phi_0= hc/2e$, which is signaled by a zero-bias conductance peak in tunneling conductance. We find this $f$ generally a fractional number and oscillating with increasing Zeeman fields that parallel to the nanowires, which provide a unique experimental signature for the existence of Majorana modes.

Majorana fermions in finite-size strips with in-plane magnetic fields

We study the Majorana bound states arising in quasi-one-dimensional systems with Rashba spin-orbit coupling in the presence of an in-plane Zeeman magnetic field. Using two different methods, first, the numerical diagonalization of the tight-binding Hamiltonian, and second, finding the singular points of the Hamiltonian (see Refs. [1-4]), we obtain the topological phase diagram for these systems as a function of the chemical potential and the magnetic field, and we demonstrate the consistency of these two methods. By introducing disorder into these systems we confirm that the states with even number of Majorana pairs are not topologically protected. Finally, we show that a formal calculation of the $\mathbb{Z}_2$ topological invariants recovers correctly the parity of the number of Majorana bound states pairs, and it is thus fully consistent with the phase diagrams of the disordered systems.

Majorana zero modes in a ladder of density-modulated Kitaev superconductor chains

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 Z2 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.

Exactly solvable spin chain models corresponding to BDI class of topological superconductors.

We present an exactly solvable extension of the quantum XY chain with longer range multi-spin interactions. Topological phase transitions of the model are classified in terms of the number of Majorana zero modes, nM which are in turn related to an integer winding number, nW. The present class of exactly solvable models belong to the BDI class in the Altland-Zirnbauer classification of topological superconductors. We show that time reversal symmetry of the spin variables translates into a sliding particle-hole (PH) transformation in the language of Jordan-Wigner fermions – a PH transformation followed by a π shift in the wave vector which we call it the πPH. Presence of πPH symmetry restricts the nW (nM) of time-reversal symmetric extensions of XY to odd (even) integers. The πPH operator may serve in further detailed classification of topological superconductors in higher dimensions as well.

Topological phase transition in a ladder of the dimerized Kitaev superconductor chains**Project supported by the National Natural Science Foundation of China (Grant No. 11274102), the Program for New Century Excellent Talents in University of Ministry of Education of China (Grant No. NCET-11-0960), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20134208110001).

We investigate the topological properties of a ladder model of the dimerized Kitaev superconductor chains. The topological class of the system is determined by the relative phase θ between the inter- and intra-chain superconducting pairing. One topological class is the class BDI characterized by the ℤ index, and the other is the class D characterized by the ℤ2 index. For the two different topological classes, the topological phase diagrams of the system are presented by calculating two different topological numbers, i.e., the ℤ index winding number W and the ℤ2 index Majorana number ℳ, respectively. In the case of θ =0, the topological class belongs to the class BDI, multiple topological phase transitions accompanying the variation of the number of Majorana zero modes are observed. In the case of θ =π/2 it belongs to the class D. Our results show that for the given value of dimerization, the topologically nontrivial and trivial phases alternate with the variation of chemical potential.

Exceptional point description of one-dimensional chiral topological superconductors/superfluids in BDI class

We show that certain singularities of the Hamiltonian in the complex wave vector space can be used to identify topological quantum phase transitions for 1D chiral topological superconductors/superfluids in the BDI class. These singularities fall into the category of the so-called exceptional points (EP's) studied in the context of non-Hermitian Hamiltonians describing open quantum systems. We also propose a generic formula in terms of the properties of the EP's to quantify the exact number of Majorana zero modes in a particular chiral topological superconducting phase, given the values of the parameters appearing in the Hamiltonian. This formula serves as an alternative to the familiar integer (Z) winding number invariant characterizing topological superconductor/superfluid phases in the chiral BDI class.

Majoranas with and without a ‘character’: hybridization, braiding and chiral Majorana number

In this paper we demonstrate under what conditions a pseudo-spin degree of freedom or character can be ascribed to Majorana bound states (MBS). These exotic states can be created at the boundaries of non-interacting systems, corresponding to D, DIII and BDI in the usual classification scheme, and we focus on one dimension. We have found that such a character is directly related to the class of the topological superconductor and its description by a , rather than a , invariant which corresponds to the BDI class. We have also found that the DIII case with mirror symmetry, which supports multiple MBS, is in fact equivalent to the BDI class with an additional time-reversal symmetry. In all cases where a character can be given to the Majorana states we show how to construct the appropriate local operator explicitly with various examples. We also examine the consequences of the Majorana character by considering possible hybridization of MBS brought into proximity and find that two MBS with the same character do not hybridize. Finally, we show that having this character or not has no consequence on the braiding properties of MBS.

Detecting topological phase transition in 1D superconducting systems with next nearest neighbor hopping

We investigate the effect of next-nearest-neighbor hopping on topological quantum phase transitions (QPTs) that are characterized by the number of Majorana zero modes in one-dimensional (1D) p-wave superconducting systems. We also numerically analyze the scaling behavior and the universality of the Berry phase (BP) of the ground state close to the critical point. For critical line (I), the derivative of the ground-state BP is nonanalytic at the phase boundaries. For the phase boundary (II), a noncontractible BP of the ground state itself is also a witness of quantum phase transition.

Quasiclassical theory of disordered multi-channel Majorana quantum wires

Multi-channel spin–orbit quantum wires, when subjected to a magnetic field and proximity coupled to an s-wave superconductor, may support Majorana states. We study what happens to these systems in the presence of disorder. Inspired by the widely established theoretical methods of mesoscopic superconductivity, we develop á la Eilenberger a quasiclassical approach to topological nanowires valid in the limit of strong spin–orbit coupling. We find that the ‘Majorana number’ , distinguishing between the state with Majorana fermions (symmetry class B) and no Majorana fermions (class D), is given by the product of two Pfaffians of gapped quasiclassical Green's functions fixed by the right and left terminals connected to the wire. A numerical solution of the Eilenberger equations reveals that the class D disordered quantum wires are prone to the formation of the zero-energy anomaly (class D impurity spectral peak) in the local density of states that shares the key features of the Majorana peak. In this way, we confirm the robustness of our previous conclusions (Bagrets and Altland 2012 Phys. Rev. Lett.109 227005) on a more restrictive system setup. Generally speaking, we find that the quasiclassical approach provides a highly efficient means to address disordered class D superconductors both in the presence and in the absence of topological structures.

Stability of Majorana fermions in proximity-coupled topological insulator nanowires

It has been shown previously that a finite-length topological insulator nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and subject to a longitudinal applied magnetic field, realizes a one-dimensional topological superconductor with an unpaired Majorana fermion (MF) localized at each end of the nanowire. Here, we study the stability of these MFs with respect to various perturbations that are likely to occur in a physical realization of the proposed device. We show that the unpaired Majorana fermions persist in this system for any value of the chemical potential inside the bulk band gap of order 300 meV in Bi$_2$Se$_3$ by computing the Majorana number. From this calculation, we also show that the unpaired Majorana fermions persist when the magnetic flux through the nanowire cross-section deviates significantly from half flux quantum. Lastly, we demonstrate that the unpaired Majorana fermions persist in strongly disordered wires with fluctuations in the on-site potential ranging in magnitude up to several times the size of the bulk band gap. These results suggest this solid-state system should exhibit unpaired Majorana fermions under accessible conditions likely important for experimental study or future applications.