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The spatial structure of in-gap Yu-Shiba-Rusinov (YSR) bound states induced by a magnetic impurity in a superconductor is the essential ingredient for the possibility of engineering collective impurity states. Recently, a saddle-point approximation [Phys. Rev. B 105, 144503] revealed how the spatial form of a YSR state is controlled by an anisotropic exponential decay length, and an anisotropic prefactor, which depends on the Fermi velocity and Fermi-surface curvature. Here we analyze STM data on YSR states in NbSe$_2$, focusing on a key issue that the exponential decay length predicted theoretically from the small superconducting gap is much larger than the observed extent of YSR states. We confirm that the exponential decay can be neglected in the analysis of the anisotropy. Instead, we extract the anisotropic prefactor directly from the data, matching it to the theoretical prediction, and we establish that the theoretical expression for the prefactor alone captures the characteristic flower-like shape of the YSR state. Surprisingly, we find that up to linear order in the superconducting gap the anisotropic prefactor that determines the shape of YSR states is the same as the anisotropic response to the impurity in the underlying normal metal. Our work points out the correct way to analyze STM data on impurities in small-gap superconductors, and reveals the importance of the normal band structure's curvature and Fermi velocity in designing multi-impurity in-gap states in superconductors., Comment: 5+5 pages, 3+4 figures

Antiferromagnetic spin-1 chains host the celebrated symmetry protected topological Haldane phase, whose spin-1/2 edge states were evidenced in bulk by, e.g., Electron Spin Resonance (ESR). Recent success in assembling effective spin-1 antiferromagnetic chains from nanographene and porphyrin molecules opens the possibility of local, site-by-site, characterization. The nascent technique of combined ESR-STM is able to measure the spin dynamics with atomic real-space resolution, and could fully reveal and manipulate the spin-1/2 degree of freedom. In this work, we combine exact diagonalization and DMRG to investigate the local dynamic spin structure factor of the different phases of the bilinear-biquadratic Hamiltonian with single-ion anisotropy in presence of an external magnetic field. We find that the signature of the Haldane phase is a low-energy peak created by singlet-triplet transitions in the edge-state manifold. We predict that the signature peak is experimentally observable, although for chains of length above N = 30 its energy should be first tuned by application of external magnetic field. We fully characterize the peak in real-space and energy, and further show its robustness to weak anisotropy and a relevant range of temperatures., Comment: 7 pages, 3 figures

We investigate the quantum transport of the heat and the charge through a quantum dot coupled to fermionic contacts under the influence of time modulation of temperatures. We derive, within the nonequilibrium Keldysh Green's function formalism, generic formulas for the charge and heat currents by extending the concept of gravitational field introduced by Luttinger to the dynamically driven system and by identifying the correct form of dynamical contact energy. In linear response regime our formalism is validated from satisfying the Onsager reciprocity relations and demonstrates its utility to reveal nontrivial dynamical effects of the Coulomb interaction on charge and energy relaxations., Comment: 36 pages, 6 figures

Matrix Product States (MPS) and Operators (MPO) have been proven to be a powerful tool to study quantum many-body systems but are restricted to moderately entangled states as the number of parameters scales exponentially with the entanglement entropy. While MPS can efficiently find ground states of 1D systems, their capacities are limited when simulating their dynamics, where the entanglement can increase ballistically with time. On the other hand, quantum devices appear as a natural platform to encode and perform the time evolution of correlated many-body states. However, accessing the regime of long-time dynamics is hampered by quantum noise. In this study we use the best of worlds: the short-time dynamics is efficiently performed by MPSs, compiled into short-depth quantum circuits, and is performed further in time on a quantum computer thanks to efficient MPO-optimized quantum circuits. We quantify the capacities of this hybrid classical-quantum scheme in terms of fidelities taking into account a noise model. We show that using classical knowledge in the form of tensor networks provides a way to better use limited quantum resources and lowers drastically the noise requirements to reach a practical quantum advantage. Finally we successfully demonstrate our approach with an experimental realization of the technique. Combined with efficient circuit transpilation we simulate a 10-qubit system on an actual quantum device over a longer time scale than low-bond-dimension MPSs and purely quantum Trotter evolution.

In one-dimensional disordered systems with a chiral symmetry it is well-known that electrons at energy $E = 0$ avoid localization and simultaneously exhibit a diverging density of states (DOS). For $N$ coupled chains with zero-correlation-length disorder, the diverging DOS remains for odd $N$, but a vanishing DOS is found for even $N$. We use a thin spinless graphene nanotube with disordered Semenoff mass and disordered Haldane coupling to construct $N = 2$ chiral chain models which at low energy have two linear band crossings at different momenta $\pm K$ (two valleys) and disorder with an arbitrary correlation length $\xi$ in units of lattice constant $a$. We find that the finite momentum $\pm K$ forces the disorder in one valley to depend on the disorder in the other valley, thus departing from known analytical results which assume having $N$ independent disorders (whatever their spatial correlation lengths). Our main numerical results show that for this inter-dependent mass disorder the DOS is also suppressed in the limit of strongly coupled valleys (lattice-white noise limit, $\xi/a = 0$) and exhibits a non-trivial crossover as the valleys decouple ($\xi/a\gtrsim 5$) into the DOS shapes of the $N = 1$ continuum model with finite correlation length $\xi$. We also show that changing the intra-unit-cell geometry of the disordered Haldane coupling can tune the amount of inter-valley scattering yet at lowest energies it produces the decoupled-valley behavior ($N = 1$) all the way down to lattice white noise.

A chain of magnetic impurities deposited on the surface of a superconductor can form a topological Shiba band that supports Majorana zero modes and holds a promise for topological quantum computing. Yet, most experiments scrutinizing these zero modes rely on transport measurements, which only capture local properties. Here we propose to leverage the intrinsic dynamics of the magnetic impurities to access their non-local character. We use linear response theory to determine the dynamics of the uniform magnonic mode in the presence of external $ac$ magnetic fields and coupling the Shiba electrons. We demonstrate that this mode, which spreads over the entire chain of atoms, becomes imprinted with the parity of the ground state and, moreover, can discriminate between Majorana and trivial zero modes located at the ends of the chain. Our approach offers a non-invasive alternative to the scanning tunneling microscopy techniques used to probe Majorana zero modes. Conversely, the magnons could facilitate the manipulation of Majorana zero modes in topological Shiba chains., Comment: 5+9 pages, 8 figures, the final published version

We construct a 1D model Hamiltonian of spinless fermions for which the spinless analogue of $\eta$-pairing states are quantum many-body scars of the model. These states are excited states and display subvolume entanglement entropy scaling; they form a tower of states that are equally spaced in energy (resulting in periodic oscillations in the Loschmidt echo and in the time evolution of observables for initial states prepared in a superposition of them) and are atypical in the sense that they weakly break the eigenstate thermalization hypothesis, while the other excited states are thermal. We extend the approach by presenting models with a tower of scar states generated by multimers located at the edge of the Brillouin zone., Comment: 10+10 pages, 5 figures. Extended version with additional results on multimers

We combine density functional theory and scanning tunneling microscopy to study the proximity effects between a bulk Ru(0001) superconductor and an atomically thin overlayer of Co. We have identified that the Co monolayer can grow in two different stackings: the hcp and a reconstructed \textepsilon-like stacking. We analyze their electronic structure from both experiments and density functional theory. While the magnetic hcp stacking shows a weak proximity effect in combination with Shiba states and with almost no suppression of superconductivity of the substrate, the more complex \textepsilon-like stacking becomes almost fully superconducting and displays an edge state at the island rim. We identify this edge state as a trivial state caused by a local hcp rim around the \textepsilon-core. We explain the weak proximity effect between Ru and the magnetic hcp islands by a low transparency of the interface, while the large chemical unit cell of the non-magnetic \textepsilon-like stacking lifts the momentum conservation at the interface making it transparent and causing a clear proximity effect.

A magnetic impurity on a superconducting substrate induces in-gap Yu-Shiba-Rusinov (YSR) bound states, whose intricate spatial structure crucially influences the possibilities of engineering collective impurity states. By means of a saddle-point approximation we study the scattering processes giving rise to YSR states in gapped, two-dimensional superconductors. Further, we develop a theory which relates through a simple analytical expression an arbitrary energy dispersion of normal electrons in a two-dimensional host to the spatial features of the YSR states. Namely, we find that flatter segments of the Fermi surface with large Fermi velocity enhance the local density of states (LDOS) around the impurity. Our analytical approximation is quantitatively accurate against tight-binding calculations on various lattices with different Fermi surfaces, and it allows to predict the shape and orientation of YSR states observed in scanning tunneling spectroscopy experiments. We illustrate our results with a model of $\mathrm{NbSe}_2$., Comment: 14 pages, 5 figures

Hybrid quantum-classical algorithms have been proposed to circumvent noise limitations in quantum computers. Such algorithms delegate only a calculation of the expectation value to the quantum computer. Among them, the Variational Quantum Eigensolver (VQE) has been implemented to study molecules and condensed matter systems on small size quantum computers. Condensed matter systems described by the Hubbard model exhibit a rich phase diagram alongside exotic states of matter. In this manuscript, we try to answer the question: how much of the underlying physics of a 1D Hubbard chain is described by a problem-inspired Variational Hamiltonian Ansatz (VHA) in a broad range of parameter values ? We start by probing how much does the solution increases fidelity with increasing ansatz complexity. Our findings suggest that even low fidelity solutions capture energy and number of doubly occupied sites well, while spin-spin correlations are not well captured even when the solution is of high fidelity. Our powerful simulation platform allows us to incorporate a realistic noise model and shows a successful implementation of noise-mitigation strategies - post-selection and the Richardson extrapolation. Finally, we compare our results with an experimental realization of the algorithm on IBM Quantum's ibmq_quito device.

We show the stabilization of two trimer phases in a chain of spinless fermions with a correlated hopping term. A trimer fluid forms due to a gain in trimer kinetic energy and competes with a fluid of unbound fermions. Furthermore, we observe two intermediate phases where these two fluids coexist and do not spatially separate. Depending on the way trimers are created out of the Fermi sea, hybridization can occur, in which case the onset of correlations between the two fluids is well captured by a generalized BCS ansatz. These results are finally extended to the formation of larger multimers, which highlights the peculiarities of pair and trimer formation., Comment: 5+9 pages, 5+7 figures

The pair breaking potential of individual magnetic impurities in s-wave superconductors generates localized states inside the superconducting gap commonly referred to as Yu- Shiba-Rusinov (YSR) states whose isolated nature makes them ideal building blocks for artificial structures that may host Majorana fermions. One of the challenges in this endeavor is to understand their intrinsic lifetime, $\hbar/\Lambda$, which is expected to be limited by the inelastic coupling with the continuum thus leading to decoherence. Here we use shot-noise scanning tunneling microscopy to reveal that electron tunnelling into superconducting 2H-NbSe$_2$ mediated by YSR states is ordered as function of time, as evidenced by a reduction of the noise. Moreover, our data show the concomitant transfer of charges e and 2e, indicating that incoherent single particle and coherent Andreev processes operate simultaneously. From the quantitative agreement between experiment and theory we obtain $\Lambda$ = 1 $\mu$eV $\ll$ $k_BT$ demonstrating that shot-noise can probe energy- and time scales inaccessible by conventional spectroscopy whose resolution is thermally limited., Comment: 6 pages, 4 figures, Supplementary Information included as ancillary file

We compute the phase diagram of a one-dimensional model of spinless fermions with pair-hopping and nearest-neighbor interaction, first introduced by Ruhman and Altman, using the density-matrix renormalization group combined with various analytical approaches. Although the main phases are a Luttinger liquid of fermions and a Luttinger liquid of pairs, we also find remarkable phases in which only a fraction of the fermions are paired. In such case, two situations arise: either fermions and pairs coexist spatially in a two-fluid mixture, or they are spatially segregated leading to phase separation. These results are supported by several analytical models that describe in an accurate way various relevant cuts of the phase diagram. Last, we identify relevant microscopic observables that capture the presence of these two fluids: while originally introduced in a phenomenological way, they support a wider application of two-fluid models for describing pairing phenomena., Comment: 18 pages, 13 figures

Recently, great experimental efforts towards designing topological electronic states have been invested in layered incommensurate heterostructures which form various nano- and meso-scale domains. In particular, it has become clear that a delicate interplay of different spin-orbit terms is induced in graphene on transition metal dichalcogenide substrates. We therefore theoretically study various types of domain walls in spin-orbit coupling in graphene looking for robust one-dimensional propagating electronic states. To do so, we use an interface Chern number and a spectral flow analysis in the low-energy theory and contrast our results to the standard arguments based on valley-Chern numbers or Chern numbers in continuum models. Surprisingly, we find that a sign-changing domain wall in valley-Zeeman spin-orbit coupling binds two robust Kramers pairs, within the bulk gap opened due to a simultaneous presence of Rashba coupling. We also study the robustness to symmetry breaking and lattice backscattering effects in tight-binding models. We show an explicit mapping of our valley-Zeeman domain wall to a domain wall in gated spinless bilayer graphene. We discuss the possible spectroscopic and transport signatures of various types of spin-orbit coupling domain walls in heterostructures., Comment: 14 pages, 4 figures

We study the presence of higher-order van Hove singularities in mirror-symmetric twisted trilayer graphene. This geometry has recently emerged experimentally as a fascinating playground for studying correlated and exotic superconducting phases. We find that the trilayer hosts a zero-energy higher-order van Hove singularity with an exponent -1/3. The singularity is protected by the threefold rotation symmetry and a combined mirror-particle-hole symmetry and it can be tuned with only the twist angle and a perpendicular electric field. It arises from the combined merging of van Hove singularities and Dirac cones at zero energy, beyond the recent classifications of van Hove singularities. Moreover, we find that varying a third parameter such as corrugation brings the system to a topological Lifshitz transition, with anomalous exponent -2/5, separating regions of locally open and closed semiclassical orbits., Comment: 18 pages, 13 figures

Magnetic impurities in $s$-wave superconductors lead to spin-polarized Yu-Shiba-Rusinov (YSR) in-gap states. Chains of magnetic impurities offer one of the most viable routes for the realization of Majorana bound states which hold a promise for topological quantum computing. However, this ambitious goal looks distant since no quantum coherent degrees of freedom have yet been identified in these systems. To fill this gap we propose an effective two-level system, a YSR qubit, stemming from two nearby impurities. Using a time-dependent wave-function approach, we derive an effective Hamiltonian describing the YSR qubit evolution as a function of distance between the impurity spins, their relative orientations, and their dynamics. We show that the YSR qubit can be controlled and read out using the state-of-the-art experimental techniques for manipulation of the spins. Finally, we address the effect of the spin noises on the coherence properties of the YSR qubit, and show a robust behaviour for a wide range of experimentally relevant parameters. Looking forward, the YSR qubit could facilitate the implementation of a universal set of quantum gates in hybrid systems where they are coupled to topological Majorana qubits., Comment: 23 pages, 7 figures

Magnetic impurities in a superconductor induce Yu-Shiba-Rusinov (YSR) states inside the superconducting gap, whose energy depends on the strength of the coupling to the impurity and on the density of states (DOS) at the Fermi level. We consider DOS exhibiting a logarithmic or a power-law divergence at the Fermi level due to Van Hove singularities (VHS) and high-order Van Hove singularities (HOVHS), respectively. We find that the energy of the YSR states has the same functional form as in the constant DOS scenario, with the effect of the singularity being an enhancement of the effective coupling constants. In particular, the critical magnetic coupling strength at which the Shiba transition occurs is always lowered by a factor $1/\rho(\Delta/E_{\mathrm{c}}$), where $\Delta$ is the superconducting gap, $E_{\mathrm{c}}$ is the bandwidth, and $\rho(E)$ is the factor in DOS which diverges at $E=0$ for a VHS or HOVHS. Further, since the critical magnetic coupling is significantly reduced, a new regime becomes accessible where the transition point is controlled by the non-magnetic coupling constant. Interestingly, the slope of the Shiba energy curve at the Shiba transition is independent of impurity parameters and purely reflects the band structure. Additionally, we find that our main conclusions remain valid even when the Fermi level is not precisely tuned to the Van Hove singularity, but instead lies on an energy range of order the superconducting gap. Our results show that tuning a superconducting material towards a VHS or HOVHS enhances the possibilities for engineering YSR states, and for characterizing the superconductor itself., Comment: 10 pages, 5 figures

We analyze the orbital magnetic susceptibility from the band structure of twisted bilayer graphene. Close to charge neutrality, the out-of-plane susceptibility inherits the strong diamagnetic response from graphene. Increasing the doping, a crossover from diamagnetism to paramagnetism is obtained and a logarithmic divergence develops at the van Hove singularity of the Moir\'e lattice in the first band. The enhanced paramagnetism at the van Hove singularity is stronger for relatively large angle but gets suppressed by the flat spectrum towards the vicinity of the first magic angle. A diverging paramagnetic susceptibility indicates an instability towards orbital ferromagnetism with an orbital out-of-plane magnetization and a Landau level structure. The region of instability is however found to be practically very small, parametrically suppressed by the ratio of the electron velocity to the speed of light. We also discuss the in-plane orbital susceptibility at charge neutrality where we find a paramagnetic response and a logarithmic divergence at the magic angle. The paramagnetic response is associated with negative counterflow current in the two layers and does not admit a semiclassical description., Comment: 19 pages, 13 figures

We show that a simple one-dimensional model of spinless fermions with pair hopping displays a phase in which a Luttinger liquid of paired fermions coexists with a Luttinger liquid of unpaired fermions. Our results are based on extensive numerical density-matrix renormalisation group calculations and are supported by a two-fluid model that captures the essence of the coexistence region., Comment: 5+5 pages, 4+5 figures

Majorana fermions are promising building blocks of forthcoming technology in quantum computing. However, their non-ambiguous identification has remained a difficult issue because of the concomitant competition with other topologically trivial fermionic states, which poison their detection in most spectroscopic probes. By employing numerical and analytical methods, here we show that the Fano factor tomography is a key distinctive feature of a Majorana bound state, displaying a spatially constant Poissonian value equal to one. In contrast, the Fano factor of other trivial fermionic states, like the Yu-Shiba-Rusinov or Andreev ones, is strongly spatially dependent and exceeds one as a direct consequence of the local particle-hole symmetry breaking., Comment: 6 pages, 2 figures+ 11 pages supplementary material

The interplay between superconductivity, magnetic field and spin-orbit coupling can lead to the realization of non--trivial topological phases. Recent experiments have found signatures of such phases in magnetic nanoflakes formed by nanostructures coupled to a superconducting substrate. These heterostructures comprise a topologically non-trivial region surrounded by a trivial one due to the finite magnetic exchange field induced by the magnetic nanoflake. The analysis of the topological phase diagram of such a system shows that a similar phase separation occurs by tuning the chemical potential of the nanoflake. In this paper, we study such a possibility in detail, analyzing the spatial extent of the edge modes circulating around the nanoflake and discussing some practical implementations. We also show how the chirality of Majorana edge states can be probed using scanning tunneling spectroscopy with a double tip setup., Comment: 12 pages, 13 figures

Magnetic impurities inserted in a $s$-wave superconductor give rise to spin-polarized in-gap states called Shiba states. We study the back-action of these induced states on the dynamics of the classical moments. We show that the Shiba state pertains to both reactive and dissipative torques acting on the precessing classical spin that can be detected through ferromagnetic resonance measurements. Moreover, we highlight the influence of the bulk states as well as the effect of the finite linewidth of the Shiba state on the magnetization dynamics. Finally, we demonstrate that the torques are a direct measure of the even and odd frequency triplet pairings generated by the dynamics of the magnetic impurity. Our approach offers non-invasive alternative to the STM techniques used to probe the Shiba states., Comment: 14 pages, 3 figures

We investigate the phase diagrams of a one-dimensional lattice model of fermions and of a spin chain with interactions extending up to next-nearest neighbour range. In particular, we investigate the appearance of regions with dominant pairing physics in the presence of nearest-neighbour and next-nearest-neighbour interactions. Our analysis is based on analytical calculations in the classical limit, bosonization techniques and large-scale density-matrix renormalization group numerical simulations. The phase diagram, which is investigated in all relevant filling regimes, displays a remarkably rich collection of phases, including Luttinger liquids, phase separation, charge-density waves, bond-order phases, and exotic cluster Luttinger liquids with paired particles. In relation with recent studies, we show several emergent transition lines with a central charge $c = 3/2$ between the Luttinger-liquid and the cluster Luttinger liquid phases. These results could be experimentally investigated using highly-tunable quantum simulators., Comment: 15 pages, 18 figures

We show that dilute magnetic impurities in a conventional superconductor give origin to an odd-frequency component of superconductivity, manifesting itself in Yu-Shiba-Rusinov bands forming within the bulk superconducting gap. Our results are obtained in a general model solved within the dynamical mean field theory. By exploiting a disorder analysis and the limit to a single impurity, we are able to provide general expressions that can be used to extract explicitly the odd-frequency superconducting function from scanning tunneling measurements., Comment: 10 pages, 6 figures

We consider light propagation through a ballistic amplifying photonic honeycomb lattice below the lasing threshold. Two sublattices of the system are formed by the wave-guides with different complex dielectric permittivities, which results in the non-Hermitian Dirac equation for electromagnetic field. We reveal that there exists a critical length of the amplifying region for which the photonic lattice exhibits an amplifier to generator transition. The transmission and reflection probabilities at the normal angle of incidence are strongly enhanced at a critical length of the system. We also comment on the sensitivity of amplification to the direction of incident light and the thickness of the amplifying region., Comment: To appear in Physical Review A

We derive a general criterion for determining the onset of superradiant phase transition in electronic bands coupled to a cavity field, with possibly electron-electron interactions. For longitudinal superradiance in 2D or genuine 1D systems, we prove that it is always prevented, thereby extending existing no-go theorems. Instead, a superradiant phase transition can occur to a nonuniform transverse cavity field and we give specific examples in non-interacting models, either through Fermi surface nesting or parabolic band touching. Investigating the resulting time-reversal symmetry breaking superradiant states, we find in the former case Fermi surface lifting down to four Dirac points on a square lattice model, with topologically protected zero-modes, and in the latter case topological bands with non-zero Chern number on an hexagonal lattice., Comment: 6 pages, 2 figures and supplementary material

We study the in-gap states that appear at the boundaries of both 1D and 2D topological superconductors. While the massless Majorana quasiparticles are guaranteed to arise by the bulk-edge correspondence, we find that they could be accompanied by massive Volkov-Pankratov (VP) states which are present only when the interface is sufficiently smooth. These predictions can be tested in an s-wave superconductor with Rashba spin-orbit coupling placed on top of a magnetic domain wall. We calculate the spin-resolved local density of states of the VP states about the band inversion generated by a magnetic domain wall and find that they are oppositely spin-polarized on either side of the topological phase boundary. We also demonstrate that the spatial position, energy-level spacing, and spin polarization of the VP states can be modified by the introduction of in-plane electric fields., Comment: 10 pages, 8 figures

We study the unconventional superconducting correlations caused by a single isolated magnetic impurity in a conventional s-wave superconductor. Due to the local breaking of time-reversal symmetry, the impurity induces unconventional superconductivity which is even in both space and spin variables but odd under time inversion. We derive an exact proportionality relation between the even-frequency component of the local electron density of states and the imaginary part of the odd-frequency local pairing function. By applying this relation to scanning tunneling microscopy spectra taken on top of magnetic impurities immersed in a Pb/Si(111) monolayer, we show experimental evidence of the occurrence of the odd-frequency pairing in these systems and explicitly extract its superconducting function from the data., Comment: 6 pages, 3 figures, revised version

Proximitizing magnetic textures and $s$-wave superconductors is becoming a platform for engineering topological superconductivity and Majorana fermions by the means of exchange processes. However, the consequences of orbital effects have not yet been fully taken into account. In this work, we investigate the magnetic skyrmion texture-induced orbital effects using a Ginzburg-Landau approach and clarify the conditions under which they can induce superconducting vortices. These orbital effects are then included in Bogoliubov-De-Gennes theory containing the exchange interaction, as well as superconducting vortices (when induced). We find that the topological phase is largely stable to all investigated effects, increasing the realistic promise of skyrmion-superconductor hybrid structures for realization of topological superconductivity., Comment: 6 pages, 5 figures and 4 pages, 4 figures of supplementary material

We investigate the $T=0$ phase diagram of a variant of the one-dimensional extended Hubbard model where particles interact via a finite-range soft-shoulder potential. Using Density Matrix Renormalization Group (DMRG) simulations, we evidence the appearance of Cluster Luttinger Liquid (CLL) phases, similarly to what first predicted in a hard-core bosonic chain [M. Mattioli, M. Dalmonte, W. Lechner, and G. Pupillo, Phys. Rev. Lett. 111, 165302]. As the interaction strength parameters change, we find different types of clusters, that encode the order of the ground state in a semi-classical approximation and give rise to different types of CLLs. Interestingly, we find that the conventional Tomonaga Luttinger Liquid (TLL) is separated by a critical line with a central charge $c=5/2$, along which the two (spin and charge) bosonic degrees of freedom (corresponding to $c=1$ each) combine in a supersymmetric way with an emergent fermionic excitation ($c=1/2$). We also demonstrate that there are no significant spin correlations., Comment: Submission to SciPost

A trijunction made of three topological semiconducting wires, each supporting a Majorana bound state at its two extremities, appears as one of the simplest geometry in order to perform braiding of Majorana fermions. By embedding the trijunction into a microwave cavity allows to study the intricate dynamics of the low-energy Majorana bound states (MBSs) coupled to the cavity electric field under a braiding operation. Extending a previous work (Phys. Rev. Lett. 2019, 122, 236803), the full time evolution of the density matrix of the low-energy states, including various relaxation channels, is computed both in the adiabatic regime, as well as within the Floquet formalism in the case of periodic driving. It turns out that in the stationary state the observables of the system depend on both the parity of the ground state and on the non-Abelian Berry phase acquired during braiding. The average photon number and the second order photon coherence function $g^{(2)}(0)$ are explicitly evaluated and reveal the accumulated non-Abelian Berry phase during the braiding process., Comment: 9 pages, 1 figure

Magnetic skyrmions are nanoscale spin configurations that can be efficiently created and manipulated. They hold great promises for next-generation spintronics applications. In parallel to these developments, the interplay of magnetism, superconductivity and spin-orbit coupling has proved to be a versatile platform for engineering topological superconductivity predicted to host non-abelian excitations, Majorana zero modes. We show that topological superconductivity can be induced by proximitizing magnetic skyrmions and conventional superconductors, without need for additional ingredients. Apart from a previously reported Majorana zero mode in the core of the skyrmion, we find a more universal chiral band of Majorana modes on the edge of the skyrmion. We show that the chiral Majorana band is effectively flat in the physically relevant regime of parameters, leading to interesting robustness and scaling properties. In particular, the number of Majorana modes in the (nearly-)flat band scales with the perimeter length of a deformed skyrmion configuration, while being robust to local disorder., Comment: 16 + 3 pages, 3 figures + Supplementary Materials

We consider the effect of disorder on the spectrum of quasiparticles in the point-node and nodal-line superconductors. Due to the anisotropic dispersion of quasiparticles disorder scattering may render the Hamiltonian describing these excitations non-Hermitian. Depending on the dimensionality of the system, we show that the nodes in the spectrum are replaced by Fermi arcs or Fermi areas bounded by exceptional points or exceptional lines, respectively. These features are illustrated by first considering a model of a proximity-induced superconductor in an anisotropic two-dimensional (2D) Dirac semimetal, where a Fermi arc in the gap bounded by exceptional points can be realized. We next show that the interplay between disorder and supercurrents can give rise to a 2D Fermi surface bounded by exceptional lines in three-dimensional (3D) nodal superconductors., Comment: 10 pages, 5 figures

Majorana zero modes are fractional quantum excitations appearing in pairs, each pair being a building block for quantum computation . Some possible signatures of these excitations have been reported as zero bias peaks at endpoints of one-dimensional semiconducting wires and magnetic chains. However, 1D systems are by nature fragile to a small amount of disorder that induces low-energy excitations, hence obtaining Majorana zero modes well isolated in a hard gap requires extremely clean systems. Two-dimensional systems offer an alternative route to get robust Majorana zero modes. Indeed, it was shown recently that Pb/Co/Si(111) could be used as a platform for generating 2D topological superconductivity with a strong immunity to local disorder. While 2D systems exhibit dispersive chiral edge states, they can also host Majorana zero modes located on local topological defects. According to predictions, if an odd number of zero modes are located in a topological domain an additional zero mode should appear all around the domain's edge. Here we use scanning tunneling spectroscopy to characterize a disordered superconducting monolayer of Pb coupled to underlying Co-Si magnetic islands meant to induce a topological transition. We show that pairs of zero modes are stabilized: one zero mode positioned at a point in the middle of the magnetic domain and its zero mode partner extended all around the domain. The zero mode pair is remarkably robust, it is isolated within a hard superconducting energy gap and it appears totally immune to the strong disorder present in the Pb monolayer. Our theoretical scenario supports the protected Majorana nature of this zero mode pair, highlighting the role of magnetic or spin-orbit coupling textures. This robust pair of Majorana zero modes offers a new platform for theoretical and experimental study of quantum computing.

We study the dynamical process of braiding Majorana bound states in the presence of the coupling to photons in a microwave cavity. We show theoretically that the $\pi/4$ phase associated with the braiding of Majoranas, as well as the parity of the ground state are imprinted into the photonic field of the cavity, which can be detected by dispersive readouts techniques. These manifestations are purely dynamical, they occur in the absence of any splitting of the MBS that are exchanged, and they disappear in the static setups studied previously. Conversely, the cavity can affect the braiding phase, which in turn should allow for cavity controlled braiding., Comment: 4.5 pages+7 pages supplementary material, 4 figures

There is presently a tremendous activity around the field of topological superconductivity and Majorana fermions. Among the many questions raised, it has become increasingly important to establish the topological or non-topological origin of features associated with Majorana fermions such as zero-bias peaks. Here, we compare in-gap features associated either with isolated magnetic impurities or with magnetic clusters strongly coupled to the atomically thin superconductor Pb/Si(111). We study this system by means of scanning tunneling microscopy and spectroscopy (STM/STS). We take advantage of the fact that the Pb/Si(111) monolayer can exist either in a crystal-ordered phase or in an incommensurate disordered phase to compare the observed spectroscopic features in both phases. This allows us to demonstrate that the strongly resolved in-gap states we found around the magnetic clusters in the disordered phase of Pb have a clear topological origin., Comment: 11 pages, 5 figures. To be published in European Physical Journal Special Topics.dedicated to the conference FQMT'17

We theoretically investigate the tunneling spectroscopy of a system of two parallel one-dimensional helical conductors in the interacting, Luttinger liquid regime. We calculate the non-linear differential conductance as a function of the voltage bias between the conductors and the orbital momentum shift induced on tunneling electrons by an orthogonal magnetic field. We show that the conductance map exhibits an interference pattern which is characteristic to the interacting helical liquid. This can be contrasted with the different interference pattern from tunneling between regular Luttinger liquids which is governed by the spin-charge separation of the elementary collective excitations., Comment: 9 pages, 6 figures, final version

We show that bringing into proximity two topologically trivial systems can give rise to a topological phase. More specifically, we study a 1D metallic nanowire proximitized by a 2D superconducting substrate with a mixed s-wave and p-wave pairing, and we demonstrate both analytically and numerically that the phase diagram of such a setup can be richer than reported before. Thus, apart from the two "expected" well-known phases (i.e., where the substrate and the wire are both simultaneously trivial or topological), we show that there exist two peculiar phases in which the nanowire can be in a topological regime while the substrate is trivial, and vice versa., Comment: 8 pages, 6 figures (+ 6 pages, 3 figures in Appendixes)

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 topological and disorder-dominated phases. We confirm aforementioned results by considering a more microscopic approach which starts from the tunneling Hamiltonian between the three-dimensional s-wave superconductor and the nanowires. We find again that the interaction drives the system into the topological phase and, as the strength of the source term coming from the tunneling Hamiltonian increases, strong electron-electron interactions are required to reach the topological phase., Comment: 24 pages, 12 figures, published version

We consider a one-dimensional system combining local magnetic moments and a delocalized metallic band on top of a superconducting substrate. This system can describe a chain of magnetic impurities with both localized polarized orbitals and delocalized s-like orbitals or a conducting wire with embedded magnetic impurities. We study the interplay between the one-dimensional Shiba band physics arising from the interplay between magnetic moments and the substrate and the delocalized wire-like conduction band on top of the superconductor. We derive an effective low-energy Hamiltonian in terms of two coupled asymmetric Kitaev-like Hamiltonians and analyze its topological properties. We have found that this system can host multiple Majorana bound states at its extremities provided a magnetic mirror symmetry is present. We compute the phase diagram of the system depending on the magnetic exchange interactions, the impurity distance and especially the coupling between both bands. In presence of inhomogeneities which typically break this magnetic mirror symmetry, we show that the coexistence of a Shiba and wire delocalized topological band can drive the system into a non-topological regime with a splitting of Majorana bound states., Comment: 11 pages, 7 figures, final version

We theoretically study a Josephson junction based on a semiconducting nanowire subject to a time-dependent flux bias. We establish a general density matrix approach for the dynamical response of the Majorana junction and calculate the resulting flux-dependent susceptibility using both microscopic and effective low-energy descriptions for the nanowire. We find that the diagonal component of the susceptibility, associated with the dynamics of the Majorana states populations, dominates over the standard Kubo contribution for a wide range of experimentally relevant parameters. The diagonal term, thus far unexplored in the context of Majorana physics, allows to probe accurately the presence of Majorana bound states in the junction., Comment: 5 pages, 3 figures, 15 pages of supplemental material

We study theoretically a chain of precessing classical magnetic impurities in an $s$-wave superconductor. Utilizing a rotating wave description, we derive an effective Hamiltonian that describes the emergent Shiba band. We find that this Hamiltonian shows non-trivial topological properties, and we obtain the corresponding topological phase diagrams both numerically and analytically. We show that changing the precession frequency offers a control over topological phase transitions and the emergence of Majorana bound states. We propose driving the magnetic impurities or magnetic texture into precession by means of spin-transfer torque in a spin-Hall setup, and manipulate it using spin superfluidity in the case of planar magnetic order., Comment: 5 pages+9 pages supplementary material

Nambu-Goldstone theorem provides gapless modes to both relativistic and nonrelativistic systems. The Nambu-Goldstone bosons in insulating magnets are called magnons or spin-waves and play a key role in magnetization transport. We review here our past works on magnetization transport in insulating magnets and also add new insights, with a particular focus on magnon transport. We summarize in detail the magnon counterparts of electron transport, such as the Wiedemann-Franz law, the Onsager reciprocal relation between the Seebeck and Peltier coefficients, the Hall effects, the superconducting state, the Josephson effects, and the persistent quantized current in a ring to list a few. Focusing on the electromagnetism of moving magnons, i.e., magnetic dipoles, we theoretically propose a way to directly measure magnon currents. As a consequence of the Mermin-Wagner-Hohenberg theorem, spin transport is drastically altered in one-dimensional antiferromagnetic (AF) spin-1/2 chains; where the N\'eel order is destroyed by quantum fluctuations and a quasiparticle magnon-like picture breaks down. Instead, the low-energy collective excitations of the AF spin chain are described by a Tomonaga-Luttinger liquid (TLL) which provides the spin transport properties in such antiferromagnets some universal features at low enough temperature. Finally, we enumerate open issues and provide a platform to discuss the future directions of magnonics., Comment: 21 pages, 14 figures, 6 tables: Review article invited from J. Phys. D for special issue on magnonics

Just like insulators can host topological Dirac states at their edges, superconductors can also exhibit topological phases characterized by Majorana edge states. Remarkable zero-energy states have been recently observed at the two ends of proximity induced superconducting wires, and were interpreted as localized Majorana end states in one-dimensional (1D) topological superconductor. By contrast, propagating Majorana states should exist at the 1D edges of two-dimensional (2D) topological superconductors. Here we report the direct observation of dispersive in-gap states surrounding topological superconducting domains made of a single atomic layer of Pb covering magnetic islands of Co/Si(111). We interpret the observed continuous dispersion across the superconducting gap in terms of a spatial topological transition accompanied by a chiral edge mode and residual gaped helical edge states. Our experimental approach enables the engineering and control of a large variety of novel quantum phases. This opens new horizons in the field of quantum materials and quantum electronics where the magnetization of the domains could be used as a control parameter for the manipulation of topological states., Comment: 12 pages, 3 figures

We study theoretically the asymptotic behavior of the Shiba bound states associated with magnetic impurities embedded in both 2D and 1D anomalous superconductors. We calculate analytically the spatial dependence of the local density of states together with the spin polarization associated with the Shiba bound states. We show that the latter quantity exhibits drastic differences between s-wave and different types of p-wave superconductors. Such properties, which could be measured using spin-polarized STM, offer therefore a way to discriminate between singlet and triplet pairing in low-dimensional superconductors, as well as a way to estimate the amplitude of the triplet pairing in these systems., Comment: 18 pages, 5 figures

In this work we consider the influence of potential impurities deposited on top of two-dimensional chiral superconductors. As discovered recently, magnetic impurity lattices on an $s$-wave superconductor may give rise to a rich topological phase diagram. We show that similar mechanism takes place in chiral superconductors decorated by non-magnetic impurities, thus avoiding the delicate issue of magnetic ordering of adatoms. We illustrate the method by presenting the theory of potential impurity lattices embedded on chiral $p$-wave superconductors. While a prerequisite for the topological state engineering is a chiral superconductor, the proposed procedure results in vistas of nontrivial descendant phases with different Chern numbers., Comment: 4.5+5 pages

We theoretically study a one dimensional p-wave superconducting mesoscopic ring interrupted by a weak link and coupled inductively to a microwave cavity. We establish an input-output description for the cavity field in the presence of the ring, and identify the electronic contributions to the cavity response and their dependence on various parameters, such as the magnetic flux, chemical potential, and cavity frequency. We show that the cavity response is $4\pi$ periodic as a function of the magnetic flux in the topological region, stemming from the so called fractional Josephson current carried by the Majorana fermions, while it is $2\pi$ periodic in the non-topological phase, consistent with the normal Josephson effect. We find a strong dependence of the signal on the cavity frequency, as well as on the parity of the ground state. Our model takes into account fully the interplay between the low-energy Majorana modes and the gaped bulks states, which we show is crucial for visualizing the evolution of the Josephson effect during the transition from the topological to the trivial phase., Comment: 7 pages, 4 figures, extra references added

Cooper pairs in two-dimensional unconventional superconductors with broken inversion symmetry are in a mixture of an even-parity spin-singlet pairing state with an odd-parity spin-triplet pairing state. We study the magnetic properties of the impurity bound states in such superconductors and find striking signatures in their spin polarization which allow to unambiguously discriminate a non-topological superconducting phase from a topological one. Moreover, we show how these properties, which could be measured using spin-polarized scanning tunneling microscopy (STM), also enable to determine the direction of the spin-triplet pairing vector of the host material and thus to distinguish between different types of unconventional pairing., Comment: 11 pages + 9 pages of supplementary information, 9 figures + 8 figures (SI)