Majorana Excitations Research Articles

Probing Majorana Wave Functions in Kitaev Honeycomb Spin Liquids with Second-Order Two-Dimensional Spectroscopy.

Two-dimensional coherent terahertz spectroscopy (2DCS) emerges as a valuable tool to probe the nature, couplings, and lifetimes of excitations in quantum materials. It thus promises to identify unique signatures of spin liquid states in quantum magnets by directly probing properties of their exotic fractionalized excitations. Here, we calculate the second-order 2DCS of the Kitaev honeycomb model and demonstrate that distinct spin liquid fingerprints appear already in this lowest-order nonlinear response χ_{yzx}^{(2)}(ω_{1},ω_{2}) when using crossed light polarizations. We further relate the off-diagonal 2DCS peaks to the localized nature of the matter Majorana excitations trapped by Z_{2} flux excitations and show that 2DCS thus directly probes the inverse participation ratio of Majorana wave functions. By providing experimentally observable features of spin liquid states in the 2D spectrum, our Letter can guide future 2DCS experiments on Kitaev magnets.

Defect-Induced Low-Energy Majorana Excitations in the Kitaev Magnet α−RuCl3

The excitations in the Kitaev spin liquid (KSL) can be described by Majorana fermions, which have characteristic field dependence of bulk gap and topological edge modes. In the high-field state of layered honeycomb magnet α−RuCl3, experimental results supporting these Majorana features have been reported recently. However, there are challenges due to sample dependence, and the impact of inevitable disorder on the KSL is poorly understood. Here, we study how low-energy excitations are modified by introducing point defects in α−RuCl3 using electron irradiation, which induces site vacancies and exchange randomness. High-resolution measurements of the temperature dependence of specific heat C(T) under in-plane fields H reveal that, while the field-dependent Majorana gap is almost intact, additional low-energy states with C/T=A(H)T are induced by introduced defects. At low temperatures, we obtain the data collapse of C/T∼H−γ(T/H) expected for a disordered quantum spin system but with an anomalously large exponent γ. This leads us to find a power-law relationship between the coefficient A(H) and the field-sensitive Majorana gap. These results are consistent with the picture that the disorder induces low-energy linear Majorana excitations, which may be considered as a weak localization effect of Majorana fermions in the KSL. Published by the American Physical Society 2024

Evidence for field induced quantum spin liquid behavior in a spin-1/2 honeycomb magnet

<p>One of the most important issues in modern condensed matter physics is the realization of fractionalized excitations, such as the Majorana excitations in the Kitaev quantum spin liquid. To this aim, the 3<i>d</i>-based Kitaev material Na<sub>2</sub>Co<sub>2</sub>TeO<sub>6</sub> is a promising candidate whose magnetic phase diagram of <b>B</b> // <b>a*</b> contains a field-induced intermediate magnetically disordered phase within 7.5 T < |B| < 10 T. The experimental observations, including the restoration of the crystalline point group symmetry in the angle-dependent torque and the coexisting magnon excitations and spinon-continuum in the inelastic neutron scattering spectrum, provide strong evidence that this disordered phase is a field induced quantum spin liquid with partially polarized spins. Our variational Monte Carlo simulation with the effective K-J<sub>1</sub>-Γ-Γ'-J<sub>3</sub> model reproduces the experimental data and further supports this conclusion.</p>

Identifying Chern numbers of superconductors from local measurements

Fascination in topological materials originates from their remarkable response properties and exotic quasiparticles which can be utilized in quantum technologies. In particular, large-scale efforts are currently focused on realizing topological superconductors and their Majorana excitations. However, determining the topological nature of superconductors with current experimental probes is an outstanding challenge. This shortcoming has become increasingly pressing due to rapidly developing designer platforms which are theorized to display very rich topology and are better accessed by local probes rather than transport experiments. We introduce a robust machine learning protocol for classifying the topological states of two-dimensional (2D) chiral superconductors and insulators from local density of states (LDOS) data. Since the LDOS can be measured with standard experimental techniques, our protocol contributes to overcoming the almost three decades standing problem of identifying the topological phase of 2D superconductors with broken time-reversal symmetry.

Evidence of charge transfer in three-dimensional topological insulator/antiferromagnetic Mott insulator Bi1Sb1Te1.5Se1.5/α−RuCl3 heterostructures

We report magnetotransport measurements on ${\mathrm{Bi}}_{1}{\mathrm{Sb}}_{1}{\mathrm{Te}}_{1.5}{\mathrm{Se}}_{1.5}/{\mathrm{RuCl}}_{3}$ heterostructure nanodevices. ${\mathrm{Bi}}_{1}{\mathrm{Sb}}_{1}{\mathrm{Te}}_{1.5}{\mathrm{Se}}_{1.5}$ (BSTS) is a strong three-dimensional topological insulator (3D-TI) that hosts conducting topological surface states (TSS) enclosing an insulating bulk. $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ (namely, ${\mathrm{RuCl}}_{3}$) is an antiferromagnet that is predicted to behave as a Kitaev-like quantum spin liquid carrying chargeless Majorana excitations. Temperature ($T$)-dependent resistivity measurements on BSTS show the interplay between parallel bulk and surface transport channels. For $T<150$ K, surface state transport dominates over bulk transport. Multichannel weak antilocalization (WAL) is observed as a sharp cusp in the low-field magnetoconductivity, indicating strong spin-orbit coupling. The presence of top and bottom topological surface states (TSS) including a pair of electrically coupled Rashba surface states (RSS) is indicated. The nonlinear Hall effect, explained by a two-band model, further supports this interpretation. We observed that ${\mathrm{RuCl}}_{3}$ has an electrostatic effect on the TSS of BSTS, that is, it changes bandbending at the interface as a result of charge transfer. In this regard, the change in low-$T$ logarithmic resistivity upturn in the presence of out-of-plane magnetic field is analyzed employing the Lu-Shen model, supporting the presence of gapless surface states (TSS and RSS) with a $\ensuremath{\pi}$ Berry phase.

Majorana gap formation in the anisotropic Kitaev model with ordered flux configuration

We study the Kitaev model with direction dependent interactions to investigate how the flux configuration and/or the anisotropy in the exchanges affect the Majorana excitations. Systematic numerical calculations demonstrate how the anisotropy of the exchange couplings and flux configuration make the Majorana excitation gapped. The induced gapped quantum spin liquid states are distinct from the gapped one realized in the large anisotropic limit. The nature of gapped states can be explained by the superlattice potential due to flux configuration.

Quantum Phase Diagram and Spontaneously Emergent Topological Chiral Superconductivity in Doped Triangular-Lattice Mott Insulators.

The topological superconducting state is a highly sought-after quantum state hosting topological order and Majorana excitations. In this Letter, we explore the mechanism to realize the topological superconductivity (TSC) in the doped Mott insulators with time-reversal symmetry (TRS). Through large-scale density matrix renormalization group study of an extended triangular-lattice t-J model on the six- and eight-leg cylinders, we identify a d+id-wave chiral TSC with spontaneous TRS breaking, which is characterized by a Chern number C=2 and quasi-long-range superconducting order. We map out the quantum phase diagram with by tuning the next-nearest-neighbor (NNN) electron hopping and spin interaction. In the weaker NNN-coupling regime, we identify a pseudogaplike phase with a charge stripe order coexisting with fluctuating superconductivity, which can be tuned into d-wave superconductivity by increasing the doping level and system width. The TSC emerges in the intermediate-coupling regime, which has a transition to a d-wave superconducting phase with larger NNN couplings. The emergence of the TSC is driven by geometrical frustrations and hole dynamics which suppress spin correlation and charge order, leading to a topological quantum phase transition.

Edge states, Majorana fermions, and topological order in superconducting wires with generalized boundary conditions

We study the properties of one-dimensional topological superconductors under the influence of generic boundary conditions mimicking the coupling with external environments. We identify a general four-parameters classification of the boundary effects and show that particle-hole and reflection symmetries can be broken or preserved by appropriately fixing the boundary parameters. When the particle-hole symmetry is broken, the topological protection of the edge modes is lost due to the hybridization with the external degrees of freedom (quasiparticle poisoning). We assess the robustness of the edge modes in the various regimes by considering different quantifiers of topological properties. In particular, we investigate the resilience of the long-distance, edge-to-edge quantum mutual information and squashed entanglement, measuring the nonlocal correlations of the Majorana excitations. Besides their relevance for the open dynamics of topological systems, these results may provide a useful guide to the appropriate embedding of low-dimensional topological systems on nanodevices in realistic conditions.

Order fractionalization in a Kitaev-Kondo model

We describe a mechanism for order fractionalization in a two-dimensional Kondo lattice model, in which electrons interact with a gapless spin liquid of Majorana fermions described by the Yao-Lee (YL) model. When the Kondo coupling to the conduction electrons exceeds a critical value, the model develops a superconducting instability into a state where the spinor order parameter carries charge $e$ and spin $S=1/2$. The broken symmetry state develops a gapless Majorana Dirac cone in the bulk. By including an appropriate gauge string, we can show that the charge $e$, spinorial order develops off-diagonal long range order that allows electrons to coherently tunnel arbitrarily long distances through the spin liquid.

Topological squashed entanglement: Nonlocal order parameter for one-dimensional topological superconductors

Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that the end-to-end, long-distance, bipartite squashed entanglement between the edges of a many-body system, defined in terms of the edge-to-edge quantum conditional mutual information, is the natural nonlocal order parameter for topological superconductors in one dimension as well as in quasi one-dimensional geometries. For the Kitaev chain in the entire topological phase, the edge squashed entanglement is quantized to log(2)/2, half the maximal Bell-state entanglement, and vanishes in the trivial phase. Such topological squashed entanglement exhibits the correct scaling at the quantum phase transition, is stable in the presence of interactions, and is robust against disorder and local perturbations. Edge quantum conditional mutual information and edge squashed entanglement defined with respect to different multipartitions discriminate topological superconductors from symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For systems featuring multiple topological phases with different numbers of edge modes, like the quasi 1D Kitaev ladder, topological squashed entanglement counts the number of Majorana excitations and distinguishes the different topological phases of the system. In fact, we show that the edge quantum conditional mutual information and the edge squashed entanglement remain valid detectors of topological superconductivity even for systems, like the Kitaev tie with long-range hopping, featuring geometrical frustration and a suppressed bulk-edge correspondence.

Dulmage-Mendelsohn Percolation: Geometry of Maximally Packed Dimer Models and Topologically Protected Zero Modes on Site-Diluted Bipartite Lattices

A new kind of quantum percolation phenomenon---of particles hopping on a lattice with missing sites---has implications for the effect of nonmagnetic impurities in quantum antiferromagnets and could aid the search for Majorana excitations in solid-state devices.

Majorana excitations in the anisotropic Kitaev model with an ordered-flux structure

We investigate the anisotropic S = 1/2 Kitaev model on the honeycomb lattice with the ordered-flux structure. By diagonalizing the Majorana Hamiltonian for the flux configuration, we find two distinct gapped quantum spin liquids. One of them is the gapped state realized in the large anisotropic case, where low energy properties are described by the toric code. On the other hand, when the system has small anisotropy, the other gapped quantum spin liquid is stabilized by the ordered-flux configuration. Since these two gapped quantum spin liquids are separated by the gapless region, these are not adiabatically connected to each other.

Effect of strong intersite Coulomb interaction on the topological properties of a superconducting nanowire

For superconducting nanowire with the pairing of extended s-type symmetry, Rashba spin-orbit interaction in a magnetic field, the influence of strong intersite charge correlations on single-particle Majorana excitations is analyzed. This problem is investigated on the basis of the density matrix renormalization group numerical method. It is shown that with an increase in the repulsion intensity of electrons located at the neighboring sites, two subbands emerge in the lower Hubbard band of the open system. Based on calculations of the Majorana polarization and degeneracy of the entanglement spectrum, it was found that a topologically nontrivial phase with one edge state survives at the edge of each of the subbands where the concentration of electrons or holes is minimal. Keywords: superconducting nanowires, Majorana zero modes, strong electron correlations.

Effect of strong intersite Coulomb interaction on the topological properties of a superconducting nanowire

For superconducting nanowire with the pairing of extended s-type symmetry, Rashba spin-orbit interaction in a magnetic field, the influence of strong intersite charge correlations on single-particle Majorana excitations is analyzed. This problem is investigated on the basis of the density matrix renormalization group numerical method. It is shown that with an increase in the repulsion intensity of electrons located at the neighboring sites, two subbands emerge in the lower Hubbard band of the open system. Based on calculations of the Majorana polarization and degeneracy of the entanglement spectrum, it was found that a topologically nontrivial phase with one edge state survives at the edge of each of the subbands where the concentration of electrons or holes is minimal. Keywords: superconducting nanowires, Majorana zero modes, strong electron correlations.

Coexistence of topological and nontopological Fermi-superfluid phases

The two-dimensional spin-imbalanced Fermi gas subject to s-wave pairing and spin-orbit coupling is considered a promising platform for realizing a topological chiral-p-wave superfluid. In the BCS limit of s-wave pairing, i.e., when Cooper pairs are only weakly bound, the system enters the topological phase via a second-order transition driven by increasing the Zeeman spin-splitting energy. Stronger attractive two-particle interactions cause the system to undergo the BCS-BEC crossover, in the course of which the topological transition becomes first-order. As a result, topological and nontopological superfluids coexist in spatially separated domains in an extended region of phase space spanned by the strength of s-wave interactions and the Zeeman energy. Here we investigate this phase-coexistence region theoretically using a zero-temperature mean-field approach. Exact numerical results are presented to illustrate basic physical characteristics of the coexisting phases and to validate an approximate analytical description derived for weak spin-orbit coupling. Besides extending our current understanding of spin-imbalanced superfluid Fermi systems, the present approach also provides a platform for future studies of unconventional Majorana excitations that, according to topology, should be present at the internal interface between coexisting topological and nontopological superfluid parts of the system.

Majorana edge and corner states in square and kagome quantum spin- 32 liquids

Quantum spin liquids hosting Majorana excitations have recently experienced renewed interest for potential applications to topological quantum computation. Performing logical operations with reduced poisoning requires to localize such quasiparticles at specific point of the system, with energies that are well defined and inside the bulk energy gap. These are two defining features of second order topological insulators (SOTIs). Here, we show two spin models that support quantum spin liquid phases characterised by Majorana excitations and that behave as SOTIs, one of which is analytically solvable thanks to a theorem by Lieb. We show that, depending on the values of spin couplings, it is possible to localize either fermions or Majorana particles at their corners.

Microwave response of a chiral Majorana interferometer

We consider an interferometer based on artificially induced topological superconductivity and chiral 1D Majorana fermions. The (non-topological) superconducting island inducing the superconducting correlations in the topological substrate is assumed to be floating. This allows probing the physics of interfering Majorana modes via microwave response, i.e., the frequency dependent impedance between the island and the earth. Namely, charging and discharging of the island is controlled by the time-delayed interference of chiral Majorana excitations in both normal and Andreev channels. We argue that microwave measurements provide a direct way to observe the physics of 1D chiral Majorana modes.

Spin dynamics in the Kitaev model with disorder: Quantum Monte Carlo study of dynamical spin structure factor, magnetic susceptibility, and NMR relaxation rate

The Kitaev quantum spin liquid provides an ideal playground for the spin fractionalization to Majorana and vison excitations. The fingerprints manifest themselves in spin dynamics, which are accessible via many experimental techniques. However, extrinsic disorder effects are inevitable in real materials. Here, the authors compute finite-temperature spin dynamics in disordered Kitaev spin liquids and clarify that disorder seriously affects the vison excitations rather than Majorana quasiparticles. Furthermore, the softening of the vison excitations and Majorana zero modes emerge due to disorder, which will help identify Kitaev magnets.

Vacancy-Induced Low-Energy Density of States in the Kitaev Spin Liquid

The Kitaev honeycomb model has attracted significant attention due to its exactly solvable spin-liquid ground state with fractionalized Majorana excitations and its possible materialization in magnetic Mott insulators with strong spin-orbit couplings. Recently, the 5d-electron compound H$_{3}$LiIr$_{2}$O$_{6}$ has shown to be a strong candidate for Kitaev physics considering the absence of any signs of a long-range ordered magnetic state. In this work, we demonstrate that a finite density of random vacancies in the Kitaev model gives rise to a striking pileup of low-energy Majorana eigenmodes and reproduces the apparent power-law upturn in the specific heat measurements of H$_{3}$LiIr$_{2}$O$_{6}$. Physically, the vacancies can originate from various sources such as missing magnetic moments or the presence of non-magnetic impurities (true vacancies), or from local weak couplings of magnetic moments due to strong but rare bond randomness (quasivacancies). We show numerically that the vacancy effect is readily detectable even at low vacancy concentrations and that it is not very sensitive neither to nature of vacancies nor to different flux backgrounds. We also study the response of the site-diluted Kitaev spin liquid to the three-spin interaction term, which breaks time-reversal symmetry and imitates an external magnetic field. We propose a field-induced flux-sector transition where the ground state becomes flux free for larger fields, resulting in a clear suppression of the low temperature specific heat. Finally, we discuss the effect of dangling Majorana fermions in the case of true vacancies and show that their coupling to an applied magnetic field via the Zeeman interaction can also account for the scaling behavior in the high-field limit observed in H$_{3}$LiIr$_{2}$O$_{6}$.

Влияние сильного межузельного кулоновского взаимодействия на топологические свойства сверхпроводящей нанопроволоки

For superconducting nanowire with the pairing of extended s-type symmetry, Rashba spin-orbit interaction in a magnetic field, the influence of strong intersite charge correlations on single-particle Majorana excitations is analyzed. This problem is investigated on the basis of the density matrix renormalization group numerical method. It is shown that with an increase in the repulsion intensity of electrons located at the neighboring sites, two subbands emerge in the lower Hubbard band of the open system. Based on calculations of the Majorana polarization and degeneracy of the entanglement spectrum, it was found that a topologically nontrivial phase with one edge state survives at the edge of each of the subbands where the concentration of electrons or holes is minimal.