Majorana Fermion Modes Research Articles

Disorder upon disorder: Localization effects in the Kitaev spin liquid

In recent years, several magnetic Mott insulators with strong spin–orbit couplings were suggested to be proximate to the Kitaev quantum spin liquid (QSL) whose one of the most exciting features is the fractionalization of spin excitations into itinerant Majorana fermions and static Z2 fluxes. Motivated by the emergence of this plethora of 4d and 5d transition metal Kitaev materials and by the fact that some level of disorder is inevitable in real materials, here we study how the Kitaev QSL responds to various forms of disorder, such as vacancies, impurities, and bond randomness. First, we argue that the presence of the quenched disorder in the Kitaev QSL can lead to the Anderson localization of Majorana fermions and the appearance of Lifshitz tails. We point out that the Anderson localization of low-energy states is particularly strong in the extended Kitaev model with the time reversal symmetry breaking term. Second, we show that the disorder effects on the low-energy Majorana fermion modes can be detected in thermal transport. Third, we show that at finite temperatures the Z2 fluxes become thermally excited and give rise to an additional disorder for the Majorana fermions. This source of the disorder dominates at high temperatures. Fourth, we demonstrate that both the structure of the energy spectrum and the thermal transport properties of the disordered Kitaev QSL depend strongly on the character of disorder. While we find that both the site disorder and the bond randomness suppress the longitudinal thermal conductivity, the low-energy localization is stronger in the case of the site disorder.

Pair density waves in spinless media

We reveal that in a magnetically polarized medium, a specific commensurate triplet pair density wave (TPDW) superconducting (SC) state, the staggered d-wave $\Pi$-triplet state, may coexist with homogeneous triplet SC states and even dominate eliminating them under generic conditions. When only this TPDW SC state is present, we have the remarkable phenomenon of gapless superconductivity. This may explain part of the difficulties in the realization of the engineered localized Majorana fermion modes for topological quantum computation. We point out qualitative characteristics of the tunneling density of states, specific heat and charge susceptibility that identify the accessible triplet SC regimes in a spinless medium.

Certified Quantum Measurement of Majorana Fermions.

We present a quantum self-testing protocol to certify measurements of fermion parity involving Majorana fermion modes. We show that observing a set of ideal measurement statistics implies anti-commutativity of the implemented Majorana fermion parity operators, a necessary prerequisite for Majorana detection. Our protocol is robust to experimental errors. We obtain lower bounds on the fidelities of the state and measurement operators that are linear in the errors. We propose to analyze experimental outcomes in terms of a contextuality witness W, which satisfies 〈W〉 ≤ 3 for any classical probabilistic model of the data. A violation of the inequality witnesses quantum contextuality, and the closeness to the maximum ideal value 〈W〉 = 5 indicates the degree of confidence in the detection of Majorana fermions.

Zero-Energy Modes, Fractional Fermion Numbers and The Index Theorem in a Vortex-Dirac Fermion System

Physics of topological materials has attracted much attention from both physicists and mathematicians recently. The index and the fermion number of Dirac fermions play an important role in topological insulators and topological superconductors. A zero-energy mode exists when Dirac fermions couple to objects with soliton-like structure such as kinks, vortices, monopoles, strings, and branes. We discuss a system of Dirac fermions interacting with a vortex and a kink. This kind of systems will be realized on the surface of topological insulators where Dirac fermions exist. The fermion number is fractionalized and this is related to the presence of fermion zero-energy excitation modes. A zero-energy mode can be regarded as a Majorana fermion mode when the chemical potential vanishes. Our discussion includes the case where there is a half-flux quantum vortex associated with a kink in a magnetic field in a bilayer superconductor. A normalizable wave function of fermion zero-energy mode does not exist in the core of the half-flux quantum vortex. The index of Dirac operator and the fermion number have additional contributions when a soliton scalar field has a singularity.

Robust Zero Energy Modes on Superconducting Bismuth Islands Deposited on Fe(Te,Se).

Topological superconductivity is one of the frontier research directions in condensed matter physics. One of the unique elementary excitations in topological superconducting state is the Majorana Fermion (mode) which is its own antiparticle and obeys the non-Abelian statistics and is thus useful for constructing the fault-tolerant quantum computation. The evidence for Majorana Fermions (mode) in condensed matter is now quickly accumulated. We deposit Bi islands on the iron-based superconductors Fe(Te,Se) and find the easily achievable zero energy modes on the tunneling spectra on some Bi islands. The zero energy mode is robust and appears everywhere on the island. Temperature and magnetic field dependence of the zero energy mode are also investigated. We attribute these zero energy modes to the Majorana modes due to the proximity effect-induced topological superconductivity on the Bi islands with strong spin-orbit coupling effect.

Distribution of conductances in chiral topological superconductor junctions

We study the electronic transport of heterojunctions made of chiral topological superconductors (TSCs) with $N$ chiral Majorana fermion modes and quantum Hall insulators (QHIs) with integer Chern number $C$. In the weak disorder regime, we show the two-terminal conductance $\sigma_{12}$ of a QHI-TSC-QHI junction is generically non-quantized, but obeys a certain distribution determined by $C$ and $N$, which is induced by random SO($N$) rotations of chiral Majorana fermion mode basis on the TSC edges. Oppositely, in the strong disorder regime, $\sigma_{12}$ tends to be a quantized value. We conclude with a brief discussion on the fractionally quantized thermal conductances of the junction.

Multiple Chiral Majorana Fermion Modes and Quantum Transport.

The chiral Majorana fermion is a massless self-conjugate fermionic particle that could arise as the quasiparticle edge state of a two-dimensonal topological state of matter. Here we propose a new platform for a chiral topological superconductor (TSC) in two dimensions with multiple N chiral Majorana fermions from a quantized anomalous Hall insulator in proximity to an s-wave superconductor with nontrivial band topology. A concrete example is that a N=3 chiral TSC is realized by coupling a magnetic topological insulator to the ion-based superconductor such as FeTe_{0.55}Se_{0.45}. We further propose the electrical and thermal transport experiments to detect the Majorana nature of three chiral edge fermions. A smoking gun signature is that the two-terminal electrical conductance of a quantized anomalous Hall-TSC junction obeys a unique distribution averaged to (2/3)e^{2}/h, which is due to the random edge mode mixing of chiral Majorana fermions and is distinguished from possible trivial explanations. The homogenous system proposed here provides an ideal platform for studying the exotic physics of chiral Majorana fermions.

Chiral Majorana fermion modes on the surface of superconducting topological insulators

The surface of superconducting topological insulators (STIs) has been recognized as an effective superconductivity platform for realizing elusive Majorana fermions. Chiral Majorana modes (CMMs), which are different from Majorana bound states localized at point defects, possess many exotic properties. Here we predict that CMMs can be achieved in experiments by depositing a ferromagnetic insulator overlayer on top of the STI surface. We simulate this heterostructure by employing a realistic tight-binding model and show that the CMM appears on the edge of the ferromagnetic islands and can be directly probed by STM only after the superconducting gap is inverted by the exchange coupling between the ferromagnet and the STI. In addition, multiple CMMs can be generated by tuning the chemical potential of the topological insulator. These results can be applied to both proximity-effect–induced superconductivity in topological insulators and intrinsic STI compounds such as PbTaSe2, BiPd and their chemical analogues, providing a route to engineering CMMs in those materials.

Noise signatures for determining chiral Majorana fermion modes

The conductance measurement of a half quantized plateau in a quantum anomalous Hall insulator-superconductor structure is reported by a recent experiment [Q. L. He \textit{et al.}, Science 357, 294-299 (2017)], which suggests the existence of the chiral Majorana fermion modes. However, such half quantized conductance plateau may also originates from a disorder-induced metallic phase. To identify the exact mechanism, we study the transport properties of such a system in the presence of strong disorders. Our results show that the local current density distributions of these two mechanisms are different. In particular, the current noises measurement can be used to distinguish them without any further fabrication of current experimental setup.

Chiral Majorana fermion modes regulated by a scanning tunneling microscope tip

The Majorana fermion can be described by a real wave function with only two phases (0 and {\pi}) which provide a controllable degree of freedom. We propose a strategy to regulate the phase of the chiral Majorana state by coupling with a scanning tunneling microscope tip in a system consisting of quantum anomalous Hall insulator coupled with a superconductor. With the change of the chemical potential, the chiral Majorana state can be tuned alternately between 0 and {\pi}, in which, correspondingly, the perfect normal tunneling and perfect crossed Andreev reflection appear. The perfect crossed Andreev reflection, by which a Cooper pair can be split into two electrons going into different terminals completely, leads to a pumping current and distinct quantized resistances. These findings may provide a signature of Majorana fermions and pave a feasible avenue to regulate the phase of Majorana state.

Interface and phase transition between Moore-Read and Halperin 331 fractional quantum Hall states: Realization of chiral Majorana fermion

We consider an interface separating the Moore-Read state and Halperin 331 state in a half filled Landau level, which can be realized in a double quantum well system with varying inter-well tunneling and/or interaction strength. We find in the presence of electron tunneling and strong Coulomb interaction across the interface, all charge modes localize and the only propagating mode left is a chiral Majorana fermion mode. Methods to probe this neutral mode are proposed. Quantum phase transition between the Moore-Read and Halperin 331 states is described by a network of such Majorana fermion modes. In addition to a direct transition, they may also be separated by a phase in which the Majorana fermions are delocalized, realizing an incompressible state which exhibits quantum Hall charge transport and bulk heat conduction.

Tunable odd-frequency triplet pairing states and skyrmion modes in chiral p-wave superconductor

Bogliubov-de Gennes equations are solved self-consistently to investigate the properties of bound states in chiral p-wave superconductive disks. It shows that either an s-wave or the mixed d- and s-wave state with odd-frequency and spin-triplet symmetry is induced at the vortex core, depending both on the chirality of the pairing states and on the vortex topology. It is also found that the odd-frequency triplet even parity (OTE) bound state can be manipulated with a local non-magnetic potential. Interestingly, with an appropriate potential amplitude, the zero-energy OTE bound state can be stabilized at a distance from the vortex core and from the local potential. Possible existences of the Majorana fermion modes are expected if the particle-hole symmetry property is applied to the zero-energy OTE bound state. Moreover, skyrmion modes with an integer topological charge have been found to exist.

Tunable quantum chaos in the Sachdev-Ye-Kitaev model coupled to a thermal bath

The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with N Majorana fermion modes coupled to another SYK model with N2 Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to temperature, it monotonically decreases from 2π/β (β = 1/(kBT), T is temperature) to zero as the coupling strength to the thermal bath increases. For a chain of SYK models, when they are uniformly coupled to the thermal bath, we show that the butterfly velocity displays a crossover from a sqrt{T} -dependence at relatively high temperature to a linear T-dependence at low temperature, with the crossover temperature also controlled by the coupling strength to the thermal bath. If only the end of the SYK chain is coupled to the thermal bath, the model can introduce a spatial dependence of both the Lyapunov exponent and the butterfly velocity. Our models provide canonical examples for the study of thermalization within chaotic models.

Giant Magnetic Band Gap in the Rashba-Split Surface State of Vanadium-Doped BiTeI: A Combined Photoemission and Ab Initio Study

One of the most promising platforms for spintronics and topological quantum computation is the two-dimensional electron gas (2DEG) with strong spin-orbit interaction and out-of-plane ferromagnetism. In proximity to an s-wave superconductor, such 2DEG may be driven into a topologically non-trivial superconducting phase, predicted to support zero-energy Majorana fermion modes. Using angle-resolved photoemission spectroscopy and ab initio calculations, we study the 2DEG at the surface of the vanadium-doped polar semiconductor with a giant Rashba-type splitting, BiTeI. We show that the vanadium-induced magnetization in the 2DEG breaks time-reversal symmetry, lifting Kramers degeneracy of the Rashba-split surface state at the Brillouin zone center via formation of a huge gap of about 90 meV. As a result, the constant energy contour inside the gap consists of only one circle with spin-momentum locking. These findings reveal a great potential of the magnetically-doped semiconductors with a giant Rashba-type splitting for realization of novel states of matter.

A Study of Topological Quantum Phase Transition and Majorana Localization Length for the Interacting Helical Liquid System

We consider a helical spin liquid system which shows majorana fermion modes at the edge. The interaction between the quasiparticles in this system induces phase transition, Majorana-Ising transition. We comply the density matrix renormalization group method to study this phase transition for the entire regime of the parameter space. We observe the presence of topological quantum phase transition for repulsive interaction, however this phase is more stable for the attractive interaction. The length scale dependent study shows many new and important results and we show explicitly that the major contribution to the excitation comes from the edge of the system when the system is in the topological state. We also show the dependence of Majorana localization length for various values of chemical potential.

Topological Quantum Phase Transition and Local Topological Order in a Strongly Interacting Light-Matter System

An attempt is made to understand the topological quantum phase transition, emergence of relativistic modes and local topological order of light in a strongly interacting light-matter system. We study this system, in a one dimensional array of nonlinear cavities. Topological quantum phase transition occurs with massless excitation only for the finite detuning process. We present a few results based on the exact analytical calculations along with the physical explanations. We observe the emergence of massive Majorana fermion mode at the topological state, massless Majorana-Weyl fermion mode during the topological quantum phase transition and Dirac fermion mode for the non-topological state. Finally, we study the quantized Berry phase (topological order) and its connection to the topological number (winding number).

Equivalence of topological mirror superconductivity and chiral superconductivity in one dimension

Recently it has been proposed that a unitary topological mirror symmetry can stabilize multiple zero energy Majorana fermion modes in one-dimensional (1D) time-reversal (TR) invariant topological superconductors. Here we establish an exact equivalence between 1D ``topological mirror superconductivity'' and chiral topological superconductivity in the BDI class which can also stabilize multiple Majorana-Kramers pairs in 1D TR invariant topological superconductors. The equivalence proves that topological mirror superconductivity can be understood as chiral superconductivity in the BDI symmetry class coexisting with time-reversal symmetry. Furthermore, we show that the mirror Berry phase coincides with the chiral winding invariant of the BDI symmetry class, which is independent of the presence of the time-reversal symmetry. Thus, the time-reversal invariant topological mirror superconducting state may be viewed as a special case of the BDI symmetry class in the well-known Altland-Zirnbauer periodic table of free fermionic phases. We illustrate the results with the examples of 1D spin-orbit coupled quantum wires in the presence of nodeless ${s}_{\ifmmode\pm\else\textpm\fi{}}$ superconductivity and the recently discussed experimental system of ferromagnetic atom (Fe) chains embedded on a lead (Pb) superconductor.

Existence of Majorana fermion mode and Dirac equation in cavity quantum electrodynamics

We present the results of low lying collective mode of coupled optical cavity arrays. We derive the Dirac equation for this system and explain the existence of Majorana fermion mode in the system. We present quite a few analytical relations between the Rabi frequency oscillation and the atom–photon coupling strength to explain the different physical situation of our study and also the condition for massless collective mode in the system. We present several analytical relations between the Dirac spinor field, order and disorder operators for our systems. We also show that the Luttinger liquid physics is one of the intrinsic concepts in our system.

Quantum simulation of Dirac fermion mode, Majorana fermion mode and Majorana-Weyl fermion mode in cavity QED lattice

Quantum simulation aims to simulate a quantum system using a controllable laboratory system that underline the same mathematical model. The cavity QED lattice system is the prescribed system to simulate the relativistic quantum effect. We quantum simulate the Dirac fermion mode, the Majorana fermion mode and also the Majorana-Weyl fermion mode and a crossover between them in cavity QED lattice. We present the different analytical relations between the field operators for different mode excitations. We also present an analysis of correlation function and the effect of dissipation at the zero Majorana fermion mode of cavity QED lattice.

Soliton-induced Majorana fermions in a one-dimensional atomic topological superfluid

We theoretically investigate the behavior of dark solitons in a one-dimensional spin-orbit coupled atomic Fermi gas in harmonic traps by solving self-consistently the Bogoliubov--de Gennes equations. The dark soliton---to be created by phase imprinting in future experiments---is characterized by a real order parameter, which changes sign at a point node and hosts localized Andreev bound states near the node. By considering both cases of a single soliton and multiple solitons, we find that the energy of these bound states decreases to zero when the system is tuned to enter the topological superfluid phase by increasing an external Zeeman field. As a result, two Majorana fermions emerge in the vicinity of each soliton, in addition to the well-known Majorana fermions at the trap edges associated with the nontrivial topology of the superfluid. We propose that the soliton-induced Majorana fermions can be directly observed by using spatially resolved radio-frequency spectroscopy or indirectly probed by measuring the density profile at the point node. For the latter, the deep minimum in the density profile will disappear due to the occupation of the soliton-induced zero-energy Majorana fermion modes. Our prediction could be tested in a resonantly interacting spin-orbit coupled ${}^{40}\text{K}$ Fermi gas confined in a two-dimensional optical lattice.