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Majorana-Kondo interplay in T-shaped double quantum dots
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Abstract
The transport behavior of a double quantum dot side-attached to a topological superconducting wire hosting Majorana zero-energy modes is studied theoretically in the strong correlation regime. It is shown that the Majorana zero energy mode can leak through the dot directly attached to topological superconductor to the side attached dot, giving rise to a subtle interplay between the two-stage Kondo screening and the half-fermionic nature of Majorana quasiparticles. In particular, the coupling to the topological wire is found to reduce the effective exchange interaction between the two quantum dots in the absence of normal leads. Interestingly, it also results in an enhancement of the second-stage Kondo temperature when the normal leads are attached. Moreover, it is shown that the second stage of the Kondo effect can become significantly modified in one of the spin channels due to the interference with the Majorana zero-energy mode, yielding the low-temperature conductance equal to $G=G_0/4$, where $G_0 = 2e^2/h$, instead of $G=0$ in the absence of the topological superconducting wire. Finally, in the case of a short wire, a finite overlap between the wave functions of the Majorana quasiparticles localized at the ends of the wire suppresses the quantum interference and a regular two-stage Kondo effect is restored.
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The prospect of using semiconductor quantum dots as an experimental tool to distinguish Majorana zero modes (MZMs) from other zero-energy excitations such as Kondo resonances has brought up the fundamental question of whether topological superconductivity and the Kondo effect can coexist in these systems. Here, we study the Kondo effect in a quantum dot coupled to a metallic contact and to a pair of MZMs. We consider a situation in which the MZMs are spin polarized in opposite directions. By using numerical renormalization group calculations and scaling analysis of the renormalization group equations, we show that the Kondo effect takes place at low temperatures, regardless of the coupling to the MZMs. Interestingly, we find that the Kondo singlet essentially decouples from the MZMs such that the residual impurity entropy can show local non-Fermi-liquid properties characteristic of the single Majorana excitations. This offers the possibility of tuning between Fermi-liquid and non-Fermi-liquid regimes simply by changing the quantum-dot-MZM couplings.
We study the low energy spectrum and transport properties of a correlated quantum dot coupled between normal and superconducting reservoirs and additionally hybridized with a topological superconducting nanowire, hosting the Majorana end-modes. In this setup the Majorana quasiparticle leaking into the quantum dot can be confronted simultaneously with the on-dot pairing and correlations. We study this interplay, focusing on the quantum phase transition from the spinless (BCS-type) to the spinful (singly occupied) configuration, where the subgap Kondo effect may arise. Using the selfconsistent perturbative treatment for correlations and the unbiased numerical renormalization group calculations we find that the Majorana mode has either constructive or destructive effect on the low-energy transport behavior of the quantum dot, depending on its spin. This spin-selective influence could be verified by means of the polarized STM spectroscopy.
We examine the influence of the superconducting proximity effect on the transport properties of a T-shaped double quantum dot strongly coupled to two normal, nonmagnetic or ferromagnetic leads. We show that the two-stage Kondo screening may be suppressed or enhanced by the presence of pairing correlations, depending on the specific geometric arrangement of the device. We explain our results by invoking effective decrease of Coulomb interactions by proximity effect and find qualitatively correct description in many cases, although spin-filtering effect stemming from spin-dependent Fano-Kondo interference occurs to be surprisingly fragile to the presence of induced pairing. The results are obtained within the numerical renormalization group framework in the limit of large superconducting gap, which allows for a reliable examination of the low-temperature sub-gap properties of the considered system. Nevertheless, finite temperature effects are also taken into account.
Majorana modes are zero-energy excitations of a topological superconductor that exhibit non-Abelian statistics1-3. Following proposals for their detection in a semiconductor nanowire coupled to an s-wave superconductor4,5, several tunnelling experiments reported characteristic Majorana signatures6-11. Reducing disorder has been a prime challenge for these experiments because disorder can mimic the zero-energy signatures of Majoranas12-16, and renders the topological properties inaccessible17-20. Here, we show characteristic Majorana signatures in InSb nanowire devices exhibiting clear ballistic transport properties. Application of a magnetic field and spatial control of carrier density using local gates generates a zero bias peak that is rigid over a large region in the parameter space of chemical potential, Zeeman energy and tunnel barrier potential. The reduction of disorder allows us to resolve separate regions in the parameter space with and without a zero bias peak, indicating topologically distinct phases. These observations are consistent with the Majorana theory in a ballistic system 21 , and exclude the known alternative explanations that invoke disorder12-16 or a nonuniform chemical potential22,23.
In the space of less than one decade, the search for Majorana quasiparticles in condensed matter has become one of the hottest topics in physics. The aim of this review is to provide a brief perspective of where we are with strong focus on artificial implementations of one-dimensional topological superconductivity. After a self-contained introduction and some technical parts, an overview of the current experimental status is given and some of the most successful experiments of the last few years are discussed in detail. These include the novel generation of ballistic InSb nanowire devices, epitaxial Al-InAs nanowires and Majorana boxes, high frequency experiments with proximitized quantum spin Hall insulators realised in HgTe quantum wells and recent experiments on ferromagnetic atomic chains on top of superconducting surfaces.
Majorana zero-modes hold great promise for topological quantum computing. Tunnelling spectroscopy in electrical transport is the primary tool to identify the presence of Majorana zero-modes, for instance as a zero-bias peak (ZBP) in differential-conductance. The Majorana ZBP-height is predicted to be quantized at the universal conductance value of 2e2/h at zero temperature. Interestingly, this quantization is a direct consequence of the famous Majorana symmetry, 'particle equals antiparticle'. The Majorana symmetry protects the quantization against disorder, interactions, and variations in the tunnel coupling. Previous experiments, however, have shown ZBPs much smaller than 2e2/h, with a recent observation of a peak-height close to 2e2/h. Here, we report a quantized conductance plateau at 2e2/h in the zero-bias conductance measured in InSb semiconductor nanowires covered with an Al superconducting shell. Our ZBP-height remains constant despite changing parameters such as the magnetic field and tunnel coupling, i.e. a quantized conductance plateau. We distinguish this quantized Majorana peak from possible non-Majorana origins, by investigating its robustness on electric and magnetic fields as well as its temperature dependence. The observation of a quantized conductance plateau strongly supports the existence of non-Abelian Majorana zero-modes in the system, consequently paving the way for future braiding experiments.
Recent experiments using the quantum dot coupled to the topological superconducting nanowire [M.T. Deng et al., Science 354, 1557 (2016)] revealed that zero-energy bound state coalesces from the Andreev bound states. Such quasi-particle states, present in the quantum dot, can be controlled by the magnetic and electrostatic means. We use microscopic model of the quantum-dot--nanowire structure to reproduce the experimental results, applying the Bogoliubov--de Gennes technique. This is done by studying the gate voltage dependence of the various types of bound states and mutual influence between them. We show that the zero energy bound states can emerge from the Andreev bound states in topologically trivial phase and can be controlled using various means. In non-trivial topological phase we show the possible resonance between this zero energy levels with Majorana bound states. We discuss and explain this phenomena as a result of dominant spin character of discussed bound states. Presented results can be applied in experimental studies by using the proposed nanodevice.
Robust zero bias transport anomalies in semiconducting nanowires with proximity-induced superconductivity have been convincingly demonstrated in various experiments. While these are compatible with the existence Majorana zero modes at the ends of the nanowire, a direct proof of their non-locality and topological protection is now needed. Here we show that a quantum dot at the end of the nanowire may be used as a powerful spectroscopic tool to quantify the degree of Majorana non-locality through a local transport measurement. Moreover, the spin polarization of dot sub-gap states at singlet-doublet transitions in the Coulomb blockade regime allows the dot to directly probe the spin structure of the Majorana wavefunction, and indirectly measure the spin-orbit coupling of the nanowire.
Watching Majorana bound states form
Majorana bound states (MBSs) are peculiar quasiparticles that may one day become the cornerstone of topological quantum computing. To engineer these states, physicists have used semiconductor nanowires in contact with a superconductor. Although many of the observed properties align with theoretical predictions, a closer look into the creation of MBSs is desirable. Deng et al. fabricated nanowires with a quantum dot at one end that served as a spectrometer for the states that formed inside the superconducting gap of the nanowire. Using this setup, topologically trivial bound states were seen to coalesce into MBSs as the magnetic field was varied.
Science , this issue p. 1557
The spin-resolved thermoelectric transport properties of a quantum dot coupled to ferromagnetic leads and side-coupled to a topological superconductor wire hosting Majorana zero-energy modes are studied theoretically. The calculations are performed in the linear response regime by using the numerical renormalization group method. It is shown that transport characteristics are determined by the interplay of Kondo correlations, exchange field due to the presence of ferromagnets and the strength of coupling to Majorana wire. These different energy scales are revealed in the behavior of the Seebeck and spin Seebeck coefficients, which exhibit an enhancement for temperatures of the order of the coupling strength to topological wire. Moreover, it is demonstrated that additional sign changes of the thermopower can occur due to the presence of Majorana zero-energy modes. These findings may provide additional fingerprints of the presence of Majorana fermions.
We have recently shown [Phys. Rev. B 89, 165314 (2014)] that a noninteracting quantum dot coupled to a one-dimensional topological superconductor and to normal leads can sustain a Majorana mode even when the dot is expected to be empty, i.e., when the dot energy level is far above the Fermi level of the leads. This is due to the Majorana bound state of the wire leaking into the quantum dot. Here we extend this previous work by investigating the low-temperature quantum transport through an interacting quantum dot connected to source and drain leads and side coupled to a topological wire. We explore the signatures of a Majorana zero mode leaking into the quantum dot for a wide range of dot parameters, using a recursive Green’s function approach. We then study the Kondo regime using numerical renormalization group calculations. We observe the interplay between the Majorana mode and the Kondo effect for different dot-wire coupling strengths, gate voltages, and Zeeman fields. Our results show that a “0.5” conductance signature appears in the dot despite the interplay between the leaked Majorana mode and the Kondo effect. This robust feature persists for a wide range of dot parameters, even when the Kondo correlations are suppressed by Zeeman fields and/or gate voltages. The Kondo effect, on the other hand, is suppressed by both Zeeman fields and gate voltages. We show that the zero-bias conductance as a function of the magnetic field follows a well-known universality curve. This can be measured experimentally, and we propose that the universal conductance drop followed by a persistent conductance of 0.5e2/h is evidence of the presence of Majorana-Kondo physics. These results confirm that this “0.5” Majorana signature in the dot remains even in the presence of the Kondo effect.
We study the spin-resolved transport properties of T-shaped double quantum
dots coupled to ferromagnetic leads. Using the numerical renormalization group
method, we calculate the linear conductance and the spin polarization of the
current for various model parameters and at different temperatures. We show
that an effective exchange field due to the presence of ferromagnets results in
different conditions for Fano destructive interference in each spin channel.
This spin dependence of the Fano effect leads to perfect spin polarization, the
sign of which can be changed by tuning the dots' levels. Large spin
polarization occurs due to Coulomb correlations in the dot, which is not
directly coupled to the leads, while finite correlations in the
directly-coupled dot can further enhance this effect. Moreover, we complement
accurate numerical results with a simple qualitative explanation based on
analytical expressions for the zero-temperature conductance. The proposed
device provides a prospective example of an electrically-controlled, fully
spin-polarized current source, which operates without an external magnetic
field.
Majorana fermions are the only fermionic particles that are expected to
be their own antiparticles. Although elementary particles of the
Majorana type have not been identified yet, quasi-particles with
Majorana-like properties, born from interacting electrons in the solid,
have been predicted to exist. Here, we present thorough experimental
studies, backed by numerical simulations, of a system composed of an
aluminium superconductor in proximity to an indium arsenide nanowire,
with the latter possessing strong spin-orbit coupling and Zeeman
splitting. An induced one-dimensional topological superconductor,
supporting Majorana fermions at both ends, is expected to form. We
concentrate on the characteristics of a distinct zero-bias conductance
peak and its splitting in energy--both appearing only with a small
magnetic field applied along the wire. The zero-bias conductance peak
was found to be robustly tied to the Fermi energy over a wide range of
system parameters. Although not providing definite proof of a Majorana
state, the presented data and the simulations support its existence.
We discuss the thermoelectrical properties of nanowires hosting Majorana edge
states. For a Majorana nanowire directly coupled to two normal reservoirs the
thermopower always vanishes regardeless of the value of the Majorana
hybridization. This situation changes drastically if we insert a quantum dot.
Then, the dot Majorana side coupled system exhibits a different behavior for
the thermopower depending on the Majorana hybridization parameter. Thermopower
reverses its sign when the half fermionic state is fully developed. As long as
the hybridization becomes finite the Seebeck coefficient behaves similarly to a
resonant level system. The sign change of the thermopower when Majorana physics
takes place and the fact that both, the electrical and thermal conductances
reach, their half fermionic value could serve as a proof of the existence of
Majorana edge states in nanowires. Finally, we perform some predictions about
the gate dependence of the Seebeck coefficient when Kondo correlations are
present in the dot.
We study properties of a quantum dot coupled to a one-dimensional topological
superconductor and a normal lead and discuss the interplay between Kondo and
Majorana-induced couplings in quantum dot. The latter appears due to the
presence of Majorana zero-energy modes localized at the ends of the
one-dimensional superconductor. We investigate the phase diagram of the system
as a function of Kondo and Majorana interactions and show, using both a
renormalization-group analysis and numerical simulations using the
density-matrix renormalization group method, that the Kondo fixed point is
generically unstable in such a system. Instead, the infrared fixed point is
controlled by the Majorana-induced coupling, and is characterized by the
non-trivial correlations between a localized spin on the dot and a fermion
parity of the topological superconductor and normal lead.
We investigate quantum transport through a quantum dot connected to source
and drain leads and side-coupled to a topological superconducting nanowire
(Kitaev chain) sustaining Majorana end modes. Using a recursive Green's
function approach, we determine the local density of states (LDOS) of the
system and find that the end Majorana mode of the wire leaks into the dot thus
emerging as a unique dot level {\it pinned} to the Fermi energy $\e_F$ of the
leads. Surprisingly, this resonance pinning, resembling in this sense a "Kondo
resonance", occurs even when the gate-controlled dot level $\e_\text{dot}(V_g)$
is far above or far below $\e_F$. The calculated conductance $G$ of the dot
exhibits an unambiguous signature for the Majorana end mode of the wire: in
essence, an off-resonance dot [$\e_\text{dot}(V_g)\neq \e_F$], which should
have G=0, shows instead a conductance $e^2/2h$ over a wide range of $V_g$, due
to this pinned dot mode. Interestingly, this pinning effect only occurs when
the dot level is coupled to a Majorana mode; ordinary fermionic modes in the
wire simply split and broaden (if a continuum) the dot level. We discuss three
experimental scenarios to probe Majorana modes in wires via these leaked/pinned
dot modes.
We study the limits of the energy resolution that can be achieved in the calculations of spectral functions of quantum impurity models using the numerical renormalization group (NRG) technique with interleaving ( z averaging). We show that overbroadening errors can be largely eliminated, that higher-moment spectral sum rules are satisfied to a good accuracy, and that positions, heights and widths of spectral features are well reproduced; the NRG approximates very well the spectral-weight distribution. We find, however, that the discretization of the conduction-band continuum nevertheless introduces artifacts. We present a modified discretization scheme which removes the band-edge discretization artifacts of the conventional approach and significantly improves the convergence to the continuum (Lambda-->1) limit. Sample calculations of spectral functions with high energy resolution are presented. We follow in detail the emergence of the Kondo resonance in the Anderson impurity model as the electron-electron repulsion is increased, and the emergence of the phononic side peaks and the crossover from the spin Kondo effect to the charge Kondo effect in the Anderson-Holstein impurity model as the electron-phonon coupling is increased. We also compute the spectral function of the Hubbard model within the dynamical mean-field theory, confirming the presence of fine structure in the Hubbard bands.
Conductance, on-site, and intersite charge fluctuations and spin correlations in the system of two side-coupled quantum dots are calculated using Wilson’s numerical renormalization group (NRG) technique. We also show the spectral density calculated using the density-matrix NRG, which for some parameter ranges remedies inconsistencies of the conventional approach. By changing the gate voltage and the interdot tunneling rate, the system can be tuned to a nonconducting spin-singlet state, the usual Kondo regime with an odd number of electrons occupying the dots, the two-stage Kondo regime with two electrons, or a valence-fluctuating state associated with a Fano resonance. Analytical expressions for the width of the Kondo regime and the Kondo temperature are given. We also study the effect of unequal gate voltages and the stability of the two-stage Kondo effect with respect to such perturbations.
We consider theoretically tunneling characteristic of a junction between a
normal metal and a chain of coupled Majorana bound states generated at
crossings between topological and non-topological superconducting sections, as
a result of, for example, disorder in nanowires. While an isolated Majorana
state supports a resonant Andreev process, yielding a zero bias differential
conductance peak of height 2e^2/h, the situation with more coupled Majorana
states is distinctively different with both zeros and 2e^2/h peaks in the
differential conductance. We derive a general expression for the current
between a normal metal and a network of coupled Majorana bound states and
describe the differential conductance spectra for a generic set of situations,
including regular, disordered, and infinite chains of bound states.
We study the effect of an external magnetic field in the Kondo regime of a
double-quantum-dot system in which a strongly correlated dot (the "hanging
dot") is coupled to a second, noninteracting dot that also bridges the gap
between two external leads. In zero field, the spectral function of the hanging
dot has previously been shown to exhibit a split-peak structure near the Fermi
level due to "Kondo resonance filtering" by the bridging dot. We show using the
numerical renormalization group that application of a magnetic field leads to a
subtle interplay between electronic interference, Kondo physics, and Zeeman
splitting with nontrivial consequences for the spectral and transport
properties. The value of the hanging-dot spectral function at the Fermi level
exhibits a nonuniversal field dependence that can be explained using a
generalized Friedel sum rule for a Kondo system with energy-dependent
hybridization. The magnetic field also accentuates the exchange-mediated
interdot coupling, which dominates the ground state at intermediate fields
leading to the formation of antiparallel magnetic moments on the dots. By
tuning gate voltages and the magnetic field, one can achieve complete spin
polarization of the linear conductance between the leads, raising the prospect
of applications of the device as a highly tunable spin filter. The system's
low-energy properties are qualitatively unchanged by the presence of weak
on-site Coulomb repulsion within the bridging dot.
We use the numerical renormalization group method to study the O(3)-symmetric version of the impurity Anderson model of Coleman and Schofield. This model is of general interest because it displays both Fermi liquid and non-Fermi liquid behaviour, and in the large UU limit can be related to the compactified two channel Kondo model of Coleman, Ioffe and Tsvelik. We calculate the thermodynamics for a parameter range which covers the full range of behaviour of the model. We find a non-Fermi liquid fixed point in the isotropic case which is unstable with respect to channel anisotropy.
We consider an experimentally feasible setup to demonstrate the existence and
coherent dynamics of Majorana fermion. The transport setup consists of a
quantum dot and a tunnel-coupled semiconductor nanowire which is anticipated to
generate Majorana excitations under some conditions. For transport under finite
bias voltage, we find that a subtraction of the source and drain currents can
expose the essential feature of the Majorana fermion, including the zero-energy
nature by gate-voltage modulating the dot level. Moreover, coherent oscillating
dynamics of the Majorana fermion between the nanowire and the quantum dot is
reflected in the shot noise via a spectral dip together with a pronounced
zero-frequency noise enhancement effect. Important parameters, e.g. for the
Majorana's mutual interaction and its coupling to the quantum dot, can be
extracted out in experiment using the derived analytic results.
We propose an experimental setup for detecting a Majorana zero mode
consisting of a spinless quantum dot coupled to the end of a p-wave
superconducting nanowire. The Majorana bound state at the end of the wire
strongly influences the conductance through the quantum dot: Driving the wire
through the topological phase transition causes a sharp jump in the conductance
by a factor of 1/2. In the topological phase, the zero temperature peak value
of the dot conductance (i.e. when the dot is on resonance and symmetrically
coupled to the leads) is e^2/2h. In contrast, if the wire is in its trivial
phase, the conductance peak value is e^2/h, or if a regular fermionic zero mode
occurs on the end of the wire, the conductance is 0. The system can also be
used to tune Flensberg's qubit system [PRL 106, 090503 (2011)] to the required
degeneracy point.
Electron transport through a normal-metal-quantum-dot-topological-superconductor junction is studied and reveals interlacing physics of Kondo correlations with two Majorana fermions bound states residing on the opposite ends of the topological superconductor. When the strength of the Majorana fermion coupling exceeds the temperature T, this combination of Kondo-Majorana fermion physics might be amenable for an experimental test: The usual peak of the temperature dependent zero bias conductance σ(V=0,T) splits and the conductance has a dip at T=0. The heights of the conductance side peaks decrease with magnetic field.
Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducted states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on HgTe/CdTe quantum wells are described that demonstrate the existence of the edge states predicted for teh quantum spin hall insulator. Experiments on Bi1-xSbx, Bi<2Se3, Bi2Te3 and Sb2Te3 are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.
Majorana zero modes (MZMs) emerging at the edges of topological superconducting wires have been proposed as the building blocks of novel, fault-tolerant quantum computation protocols. Coherent detection and manipulation of such states in scalable devices are, therefore, essential in these applications. Recent detection proposals include semiconductor quantum dots (QDs) coupled to the end of these wires, as changes in the QD electronic spectral density due to the MZM coupling could be detected in transport experiments. Here, we propose that multi-QD systems can also be used to manipulate MZMs through precise control over the QDs' parameters. The simplest case where Majorana manipulation is possible is in a double quantum dot (DQD) geometry. By using exact analytical methods and numerical renormalization-group calculations, we show that the QDs' spectral functions can be used to characterize the presence or not of MZMs “leaking” into the DQD. More importantly, we find that these signatures respond to changes in the DQD parameters such as gate voltages and couplings in a consistent fashion. Additionally, we show that different MZM-DQD coupling geometries (“symmetric,” “in-series,” and “T-shaped” junctions) offer distinct ways in which MZMs can be switched from dot to dot. These results highlight the interesting possibilities that DQDs offer for all-electrical MZM control in scalable devices.
We show that the Kondo screening in a correlated double-quantum-dot structure may be caused solely by the proximity of a superconductor which induces nonlocal pairing by Andreev reflection processes. This leads to an effective exchange interaction, which we estimate perturbatively, and we corroborate the analytical predictions by the numerical renormalization group calculations, using an effective model for the superconductor-proximized nanostructure. We determine the dependence of the relevant Kondo temperature on the coupling to the superconductor and predict a characteristic modification of conventional low-temperature transport behavior, which can be used to experimentally distinguish this phenomenon from other Kondo effects. The occurrence of nonlocal pairing exchange does not depend on the details of the proposed setup; therefore, it can also be relevant for the bulk materials, such as heavy-fermion compounds.
Spatial separation of Majorana zero modes distinguishes trivial from topological midgap states and is key to topological protection in quantum computing applications. Although signatures of Majorana zero modes in tunneling spectroscopy have been reported in numerous studies, a quantitative measure of the degree of separation, or nonlocality, of the emergent zero modes has not been reported. Here, we present results of an experimental study of nonlocality of emergent zero modes in superconductor-semiconductor hybrid nanowire devices. The approach takes advantage of recent theory showing that nonlocality can be measured from splitting due to hybridization of the zero mode in resonance with a quantum dot state at one end of the nanowire. From these splittings as well as anticrossing of the dot states, measured for even and odd occupied quantum dot states, we extract both the degree of nonlocality of the emergent zero mode, as well as the spin canting angles of the nonlocal zero mode. Depending on the device measured, we obtain either a moderate degree of nonlocality, suggesting a partially separated Andreev subgap state, or a highly nonlocal state consistent with a well-developed Majorana mode.
We study the low-energy physics of a one-dimensional array of superconducting quantum dots realized by proximity coupling a semiconductor nanowire to multiple superconducting islands separated by narrow uncovered regions. The effective electrostatic potential inside the quantum dots and the uncovered regions can be controlled using potential gates. By performing detailed numerical calculations based on effective tight-binding models, we find that multiple low-energy subgap states consisting of partially overlapping Majorana bound states emerge generically in the vicinity of the uncovered regions. Explicit differential conductance calculations show that a robust zero-bias conductance peak is not inconsistent with the presence of such states localized throughout the system, hence, the observation of such a peak does not demonstrate the realization of well-separated Majorana zero modes. However, we find that creating effective potential wells in the uncovered regions traps pairs of nearby partially overlapping Majorana bound states, which become less separated and acquire a finite gap that protects the pair of Majorana zero modes localized at the ends of the system. This behavior persists over a significant parameter range, suggesting that proximitized quantum dot arrays could provide a platform for highly controllable Majorana devices.
Topological states of quantum matter have been investigated intensively in recent years in materials science and condensed matter physics. The field developed explosively largely because of the precise theoretical predictions, well-controlled materials processing, and novel characterization techniques. In this Perspective, we review recent progress in topological insulators, the quantum anomalous Hall effect, chiral topological superconductors, helical topological superconductors and Weyl semimetals.
Realizing topological superconductivity and Majorana zero modes in the laboratory is one of the major goals in condensed matter physics. We review the current status of this rapidly-developing field, focusing on semiconductor-superconductor proposals for topological superconductivity. Material science progress and robust signatures of Majorana zero modes in recent experiments are discussed. After a brief introduction to the subject, we outline several next-generation experiments probing exotic properties of Majorana zero modes, including fusion rules and non-Abelian exchange statistics. Finally, we discuss prospects for implementing Majorana-based topological quantum computation in these systems.
Motivated by an important recent experiment [Deng et al., Science 354, 1557 (2016)], we theoretically consider the interplay between Andreev bound states(ABSs) and Majorana bound states(MBSs) in quantum dot-nanowire semiconductor systems with proximity-induced superconductivity(SC), spin-orbit coupling and Zeeman splitting. The dot induces ABSs in the SC nanowire which show complex behavior as a function of Zeeman splitting and chemical potential, and the specific question is whether two such ABSs can come together forming a topological MBS. We consider physical situations involving the dot being non-SC, SC, or partially SC. We find that the ABSs indeed tend to coalesce together producing near-zero-energy midgap states as Zeeman splitting and/or chemical potential are increased, but this mostly happens in the non-topological regime although there are situations where the ABSs could come together forming a topological MBS. The two scenarios(two ABSs forming a near-zero-energy non-topological ABS or a zero-energy topological MBS) are difficult to distinguish by tunneling conductance spectroscopy due to essentially the same signatures. Theoretically we distinguish them by knowing the critical Zeeman splitting for the topological quantum phase transition or by calculating the topological visibility. We find that the "sticking together" propensity of ABSs to produce a zero-energy midgap state is generic in class D systems, and by itself says nothing about the topological nature of the underlying SC nanowire. One must use caution in interpreting tunneling conductance measurements where the midgap sticking-together behavior of ABSs cannot be construed as definitive evidence for topological SC with non-Abelian MBSs. We also suggest some experimental techniques for distinguishing between trivial and topological ZBCPs.
The transport properties of a quantum dot coupled to ferromagnetic contacts and attached to a topological superconductor wire hosting Majorana zero-energy modes at its ends are studied theoretically in the Kondo regime. By using the numerical renormalization group method, the temperature and dot level dependence of the spectral function, the conductance, and its spin polarization are studied for different coupling strengths to a topological superconductor. It is shown that the transport characteristics are generally determined by the interplay of three competing energy scales, resulting from Kondo correlations, a ferromagnetic-lead-induced exchange field, and topological-wire-induced splitting. These two splittings are found to have opposite signs. Moreover, they can compensate for each other in an appropriate parameter regime.
Majorana zero modes are quasiparticle excitations in condensed matter systems that have been proposed as building blocks of fault-tolerant quantum computers. They are expected to exhibit non-Abelian particle statistics, in contrast to the usual statistics of fermions and bosons, enabling quantum operations to be performed by braiding isolated modes around one another. Quantum braiding operations are topologically protected insofar as these modes are pinned near zero energy, with the departure from zero expected to be exponentially small as the modes become spatially separated. Following theoretical proposals, several experiments have identified signatures of Majorana modes in nanowires with proximity-induced superconductivity and atomic chains, with small amounts of mode splitting potentially explained by hybridization of Majorana modes. Here, we use Coulomb-blockade spectroscopy in an InAs nanowire segment with epitaxial aluminium, which forms a proximity-induced superconducting Coulomb island (a â ∼ Majorana islandâ (tm)) that is isolated from normal-metal leads by tunnel barriers, to measure the splitting of near-zero-energy Majorana modes. We observe exponential suppression of energy splitting with increasing wire length. For short devices of a few hundred nanometres, sub-gap state energies oscillate as the magnetic field is varied, as is expected for hybridized Majorana modes. Splitting decreases by a factor of about ten for each half a micrometre of increased wire length. For devices longer than about one micrometre, transport in strong magnetic fields occurs through a zero-energy state that is energetically isolated from a continuum, yielding uniformly spaced Coulomb-blockade conductance peaks, consistent with teleportation via Majorana modes. Our results help to explain the trivial-to-topological transition in finite systems and to quantify the scaling of topological protection with end-mode separation.
A two-dimensional quantum system with anyonic excitations can be considered as a quantum
computer. Unitary transformations can be performed by moving the excitations around
each other. Measurements can be performed by joining excitations in pairs and observing the
result of fusion. Such computation is fault-tolerant by its physical nature.
The linear-response transport properties of a T-shaped double quantum dot strongly coupled to external ferromagnetic leads are studied theoretically by using the numerical renormalization group method. It is shown that when each dot is occupied by a single electron, for antiparallel alignment of leads' magnetizations, the system exhibits the two-stage Kondo effect. For parallel alignment, however, the second stage of the Kondo effect becomes suppressed due to the presence of ferromagnetic-contact-induced exchange field. The difference between the two magnetic configurations results in highly nontrivial behavior of the tunnel magnetoresistance, which for some parameters can take giant values. In addition, the dependence of the linear conductance and tunnel magnetoresistance on external magnetic field, the double-dot levels' position, and the spin polarization of the leads is thoroughly analyzed. It is shown that the second stage of the Kondo effect can be restored by fine-tuning of the magnetic field or the dots' levels. The effect of spin-dependent tunneling on the low-temperature transition from the high to low conducting state of the system, which occurs when changing the hopping between the dots, is also discussed.
We consider a quantum dot coupled to a topological superconductor and two
normal leads and study transport properties of the system. Using Keldysh
path-integral approach, we study current fluctuations (shot noise) within the
low-energy effective theory. We argue that the combination of the tunneling
conductance and the shot noise through a quantum dot allows one to distinguish
between the topological (Majorana) and non-topological (e.g., Kondo) origin of
the zero-bias conduction peak. Specifically, we show that, while the tunneling
conductance might exhibit zero-bias anomaly due to Majorana or Kondo physics,
the shot noise is qualitatively different in the presence of Majorana zero
modes.
Thermoelectric effects in transport through a quantum dot coupled to external ferromagnetic leads are investigated theoretically. The basic thermoelectric transport characteristics, such as thermopower, electronic contribution to the heat conductance, and the corresponding figure of merit, are calculated in the linear response regime by means of the density-matrix numerical renormalization group method. The case of a nonzero spin splitting of the electrochemical potential in the electrodes is also considered and the associated spin thermoelectric effects are analyzed. It is shown that the spin-dependent thermoelectric phenomena in the local moment regime depend generally on the exchange field induced by ferromagnetic contacts. In addition, the temperature dependence of the Seebeck coefficient is rather nontrivial, and depends on the spin polarization and spin relaxation in the leads. In the presence of ferromagnetic leads, the thermopower as a function of temperature may change sign more times than the thermopower for nonmagnetic leads. These changes can be thus used to determine the relevant Kondo behavior and Kondo energy scale in the system. Moreover, the effects of external magnetic field and different spin polarization of ferromagnetic leads are also analyzed.
We investigate the dynamical and transport features of a Kondo dot side coupled to a topological superconductor (TS). The Majorana fermion states (MFSs) formed at the ends of the TS are found to be able to alter the Kondo physics profoundly: For an infinitely long wire where the MFSs do not overlap (εm=0) a finite dot-MFS coupling (Γm) reduces the unitary-limit value of the linear conductance by exactly a factor 3/4 in the weak-coupling regime (Γm<TK), where TK is the Kondo temperature. In the strong-coupling regime (Γm>TK), on the other hand, the spin-split Kondo resonance takes place due to the MFS-induced Zeeman splitting, which is a genuine many-body effect of the strong Coulomb interaction and the topological superconductivity. We find that the original Kondo resonance is fully restored once the MFSs are strongly hybridized (εm>Γm). This unusual interaction between the Kondo effect and the MFS can thus serve to detect the Majorana fermions unambiguously and quantify the degree of overlap between the MFSs in the TS.
Electron transport through the T-shaped quantum-dot (QD) structure is
theoretically investigated, by considering a Majorana zero mode coupled to the
terminal QD. It is found that in the double-QD case, the presence of the
Majorana zero mode can efficiently dissolve the antiresonance point in the
conductance spectrum and induce a conductance peak to appear at the same energy
position whose value is equal to $e^2/2h$. This antiresonance-resonance change
will be suitable to detect the Majorana bound states. Next in the multi-QD
case, we observe that in the zero-bias limit, the conductances are always the
same as the double-QD result, independent of the parity of the QD number. We
believe that all these results can be helpful for understanding the properties
of Majorana bound states.
Sunto
Si dimostra la possibilità di pervenire a una piena simmetrizzazione formale della teoria quantistica dell’elettrone e del
positrone facendo uso di un nuovo processo di quantizzazione. Il significato delle equazioni di
Dirac
ne risulta alquanto modificato e non vi è più luogo a parlare di stati di energia negativa; nè a presumere per ogni altro
tipo di particelle, partieolarmente neutre, l’esistenza di « antiparticelle » corrispondenti ai « vuoti » cut energia negativa.
1. Models of magnetic impurities 2. Resistivity calculations and the resistance minimum 3. The Kondo problem 4. Renormalization group calculations 5. Fermi liquid theories 6. Exact solutions and the Bethe ansatz 7. N-fold degenerate models I 8. N-fold degenerate models II 9. Theory and experiment 10. Strongly correlated fermions Appendices.
Based on the s-d interaction model for dilute magnetic alloys we have calculated the scattering probability of the conduction electrons to
the second Born approximation. Because of the dynamical character of the localized spin system, the Pauli principle should
be taken into account in the intermediate states of the second order terms. Thus the effect of the Fermi sphere is involved
in the scattering probability and gives rise to a singular term in the resistivity which involves c log T as a factor, where c is the concentration of impurity atoms. When combined with the lattice resistivity, this gives rise to a resistance minimum,
provided the s-d exchange integral J is negative. The temperature at which the minimum ccurs is proportional to c1/5 and the depth of the minimum to c, as is observed. The predicted log T dependence is tested with available experiments and is confirmed. The value of J to have fit with experiments is about -0.2 ev, which is of reasonable magnitude. Our conclusion is that J should be negative
in alloys which show a resistance minimum. It is argued that the resistance minimum is a result of the sharp Fermi surface.
A logarithmic discretization procedure, alternative to the one traditionally employed in the Numerical renormalization-group computations of physical properties for impurity models, is introduced. While the traditional method neglects the coupling of the conduction states most localized around the impurity site to all other conduction states, this one constructs a nonorthogonal basis that diagonalizes the conduction-band Hamiltonian and neglects the overlap between basis states. Unlike the traditional procedure, which underestimates the spectral density of the coupling between the conduction band and the impurity, this one requires no ad hoc renormalization of coupling constants. Numerical examples covering the specific heat for the Kondo model and the impurity spectral densities for the uncorrelated Anderson model show that, for the same discretization parameters, this procedure is substantially more accurate than the traditional one.
Several topics involving renormalization group ideas are reviewed. The
solution of the s-wave Kondo Hamiltonian, describing a single magnetic impurity
in a nonmagnetic metal, is explained in detail. ''Block spin'' methods, applied
to the two dimensional Ising model, are explained. A relatively short review of
basic renormalization group ideas is given, mainly in the context of critical
phenomena. The relationship of the modern renormalization group to the older
problems of divergences in statistical mechanics and field theory and field
theoretic renormalization is discussed. The special case of ''marginal
variables'' is discussed in detail, along with the relationship of the modern
renormalization group to its original formulation by Gell-Mann and Low and
others.
Semiconductor InSb nanowires are expected to provide an excellent material platform for the study of Majorana fermions in solid state systems. Here, we report on the realization of a Nb-InSb nanowire-Nb hybrid quantum device and the observation of a zero-bias conductance peak structure in the device. An InSb nanowire quantum dot is formed in the device between the two Nb contacts. Due to the proximity effect, the InSb nanowire segments covered by the superconductor Nb contacts turn to superconductors with a superconducting energy gap $\Delta_{InSb}\sim 0.25$ meV. A tunable critical supercurrent is observed in the device in high back gate voltage regions in which the Fermi level in the InSb nanowire is located above the tunneling barriers of the quantum dot and the device is open to conduction. When a perpendicular magnetic field is applied to the devices, the critical supercurrent is seen to decrease as the magnetic field increases. However, at sufficiently low back gate voltages, the device shows the quasi-particle Coulomb blockade characteristics and the supercurrent is strongly suppressed even at zero magnetic field. This transport characteristic changes when a perpendicular magnetic field stronger than a critical value, at which the Zeeman energy in the InSb nanowire is $E_z\sim \Delta_{InSb}$, is applied to the device. In this case, the transport measurements show a conductance peak at the zero bias voltage and the entire InSb nanowire in the device behaves as in a topological superconductor phase. We also show that this zero-bias conductance peak structure can persist over a large range of applied magnetic fields and could be interpreted as a transport signature of Majorana fermions in the InSb nanowire.
The 1937 theoretical discovery of Majorana fermions-whose defining property is that they are their own anti-particles-has since impacted diverse problems ranging from neutrino physics and dark matter searches to the fractional quantum Hall effect and superconductivity. Despite this long history the unambiguous observation of Majorana fermions nevertheless remains an outstanding goal. This review paper highlights recent advances in the condensed matter search for Majorana that have led many in the field to believe that this quest may soon bear fruit. We begin by introducing in some detail exotic 'topological' one- and two-dimensional superconductors that support Majorana fermions at their boundaries and at vortices. We then turn to one of the key insights that arose during the past few years; namely, that it is possible to 'engineer' such exotic superconductors in the laboratory by forming appropriate heterostructures with ordinary s-wave superconductors. Numerous proposals of this type are discussed, based on diverse materials such as topological insulators, conventional semiconductors, ferromagnetic metals and many others. The all-important question of how one experimentally detects Majorana fermions in these setups is then addressed. We focus on three classes of measurements that provide smoking-gun Majorana signatures: tunneling, Josephson effects and interferometry. Finally, we discuss the most remarkable properties of condensed matter Majorana fermions-the non-Abelian exchange statistics that they generate and their associated potential for quantum computation.
Majorana fermions are particles identical to their own antiparticles. They have been theoretically predicted to exist in topological
superconductors. Here, we report electrical measurements on indium antimonide nanowires contacted with one normal (gold) and
one superconducting (niobium titanium nitride) electrode. Gate voltages vary electron density and define a tunnel barrier
between normal and superconducting contacts. In the presence of magnetic fields on the order of 100 millitesla, we observe
bound, midgap states at zero bias voltage. These bound states remain fixed to zero bias, even when magnetic fields and gate
voltages are changed over considerable ranges. Our observations support the hypothesis of Majorana fermions in nanowires coupled
to superconductors.
We propose a setup to measure the lifetime of the parity of a pair of
Majorana bound states. The proposed experiment has one edge Majorana state
tunnel coupled to a quantum dot, which in turn is coupled to a metallic
electrode. When the Majorana Fermions overlap, even a small relaxation rate
qualitatively changes the non-linear transport spectrum, and for strong overlap
the lifetime can be read off directly from the height of a current peak. This
is important for the usage of Majorana Fermions as a platform for topological
quantum computing, where the parity relaxation is a limiting factor.
Topological insulators are new states of quantum matter which can not be
adiabatically connected to conventional insulators and semiconductors. They are
characterized by a full insulating gap in the bulk and gapless edge or surface
states which are protected by time-reversal symmetry. These topological
materials have been theoretically predicted and experimentally observed in a
variety of systems, including HgTe quantum wells, BiSb alloys, and Bi$_2$Te$_3$
and Bi$_2$Se$_3$ crystals. We review theoretical models, materials properties
and experimental results on two-dimensional and three-dimensional topological
insulators, and discuss both the topological band theory and the topological
field theory. Topological superconductors have a full pairing gap in the bulk
and gapless surface states consisting of Majorana fermions. We review the
theory of topological superconductors in close analogy to the theory of
topological insulators.