Weyl point immersed in a continuous spectrum: an example from superconducting nanostructures
WWeyl point immersed in a continuous spectrum: an example from superconductingnanostructures
Y. Chen and Y. V. Nazarov Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
A Weyl point in a superconducting nanostructure is a generic minimum model of a topologicalsingularity at low energies. We connect the nanostructure to normal leads thereby immersing thetopological singularity in the continuous spectrum of the electron states in the leads. This setsanother simple and generic model useful to comprehend the modification of low-energy signularityin the presence of continuous spectrum. The tunnel coupling to the leads gives rise to new lowenergy scale Γ at which all topological features are smoothed. We investigate superconducting andnormal currents in the nanostructure at this scale. We show how the tunnel currents can be used fordetection of the Weyl point. Importantly, we find that the topological charge is not concentrated ina point but rather is spread over the parameter space in the vicinity of the point. We introduce andcompute the resulting topological charge density. We also reveal that the pumping to the normalleads helps to detect and investigate the topological effects in the vicinity of the point.
I. INTRODUCTION
The study of topological materials has been on thefront edge of the modern research in condensed matterphysics for the past decade . These materials are ap-pealing from fundamental point of view and for possi-ble applications , , , , including quantum informationprocessing ). The basis for applications is the topo-logical protection of quantum states, which makes thestates robust against small perturbations and leads tomany unusual phenomena, e.g. topologically protectededge states . The topological superconductors and Chern insulators are the classes of topologicalmaterials that are under active investigation.Most topological effects under consideration requirediscrete quantum states, for instance, electron, photon orphonon bands in a Brillouin zone of a periodic structure.Topological protection requires a gap in energy spectrum,that is, absence of continuous excitation spectrum at lowenergies. It is intuitively clear that immersing the dis-crete states in a continuous spectrum, and compromisingthe energy gaps in this way will lead to compromisingthe topology. One of the goals of the present paper topropose and investigate a simple model for this that canbe elaborated analytically to all details.We concentrate on Weyl points those are most gen-erally defined as topologically protected crossings of thediscrete energy levels in a parametric space. From gen-eral topological reasoning, such crossing requires tuningof three parameters, so it is natural to consider a three-dimensional parametric space.Recently, Weyl points - the topologically protectedcrossings in the spectrum of Andreev bound states - havebeen predicted in superconducting nanostructures .The specifics of superconductivity that these crossingsmay be pinned to Fermi level. This restricts the relevantphysics to low energies and the properties of the groundstate of the system. At a Weyl point, the energy of thelowest Andreev state crosses Fermi level, so it costs van-ishing energy to excite a quasiparticle in the vicinity of the point. A requirement of realization. This is why theWeyl points are usually considered in multi-terminal su-perconducting nanostructures where the parameters arethe superconducting phase differences of the terminals.Four terminals are thus needed to realize a Weyl point.This prediction gave rise to related experimental and the-oretical research A separate set of proposals aims torealization of Weyl points in devices combining Joseph-son effect and Coulomb blockade .It is important that weak spin-orbit interaction splitsthe energies of single-quasiparticle states.
Owing tothis, the ground state configuration is always a compo-nent of a spin doublet in a small finite region around thepoint and is spin-singlet otherwise.
The topologicalsingularity still remains since the energies of two singletstates still cross in a point owing to topological protec-tion.In we have noticed that continuous spectrum abovethe gap may modify the signatures of topology leadingto a non-quantized contribution to the transconductance.The continuous spectrum at low energies shall bring moredrastic modification. The most experimentally relevantway to bring a continuous spectrum into play is to cou-ple a system of discrete Andreev levels in the supercon-ducting nanostructure to normal leads. As we will seein detail in this Article, this brings new energy scale Γ,that is the rate of tunnelling to the leads from a discretestate. Since we are at a point of energy crossing, thissmall energy scale also implies a small scale in the pa-rameter space: the scale at which the energy splittingmatches Γ.We have studied tunnel coupling to discrete normalstates in where we propose a Spin-Weyl quantum unit.Importantly, we have found there that the tunnel cou-pling may break isotropy in the vicinity of the Weyl point.In the context of spintronics, we have recently studied thecharge and spin transport in normal leads tunnel-coupledto a Weyl-point superconducting nanostructure. This isessentially the same setup as we consider here. However,in we access the transport in the framework of master a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b gx a) E SSSS
W.P.A.B.S N V J N V J b) FIG. 1. a. In a four-terminal superconducting heterostruc-ture, the Andreev states may cross Fermi level in a point - aWeyl point - in 3D parameter space of superconducting phase.The resulting spectrum in the vicinity of the point is isotropicand conical for two singlet states ( x and g in the Figure) andflat for doublet states. The doublet states are split by spin-orbit interaction, and one doublet state is ground one in themere vicinity of the point. b. The setup under consideration.The Andreev bound states near Weyl point (A.B.S.W.P) aretunnel-coupled with the continuous spectrum of the electronstates in several normal-metal leads (two are shown in theFigure). The tunnel coupling results in an energy scale Γ atwhich the spectral singularities are smoothed. equation, that is, assuming that the energy differencesof Andreev states exceed much the tunnel energy scale.In this approximation the quantities characterizing thesetup retain singularities: the superconducting currenthas a jump at the point, the normal currents jump atvoltages corresponging to the energy levels, the Berrycurvature diverges upon approaching the point indicat-ing the point-like topological charge.In this Article, we investigate the setup at the en-ergy scale Γ revealing how the above-mentioned singular-ities are smoothed at this scale. We formulate a genericmodel of tunneling suitable for many leads that includesisotropy violation. Technically, the problem at hand is acase of non-equlibrium Green function technique fornon-interacting Fermions. However, we chose to presentan explicit derivation in terms of Heisenberg equation of motion for the operators of the superconducting cur-rent and those of the currents in the normal leads. Wecompute these quantities for equilibrium, stationary andadiabatic cases. Owing to simplicity of the generic setupunder consideration, all results are analytical.As expected, all singularities are smoothed. We findthe maximum derivative of the supercurrent with respectto the controlling phases, that is set by Γ, and the max-imum differential conductance in the tunneling currents.An experimentally relevant point is the sharp dependenceof tunneling currents in the vicinity of the point in thelimit of high voltages and temperatures. This can beused for detection of Weyl points at temperatures thatexceed the level splitting.We redefine Berry curvature in terms of the responsefunction in the limit of small frequencies. The divergenceof the redefined curvature gives the density of topologicalcharge, so we explicitly compute how the point-like topo-logical singularity is spread over the parameter space.In addition, we evaluate the tunneling currents gen-erated by an adiabatic change of the controlling phases.This is the case of parametric charge pumping : theresult of a change of the controlling phases along a closedcontour is a charge transferred to the leads that dependson the contour only. We show that this is a convenienttool for exploration of the vicinity of the Weyl point, in-cluding the smoothing of the singularities.The structure of the Article is as follows. We formulatethe model in Section II and perform necessary derivationsin Section III. We evaluate the superconducting currentsin equilibrium in Section IV. There are no tunneling cur-rents in equilibrium. They arise if the voltages are ap-plied to the leads, and we evaluate these currents forstationary voltages in Section V. Next, we turn to theadiabatic case computing the response functions in thelimit of low frequency. We redefine Berry curvature, eval-uate the response function and the density of topologicalcharge in Section VI. The Section VII concentrates oncharge pumping to the normal leads. We conclude inSection VIII. II. THE MODEL
We start with the effective Hamiltonian in the vicinityof a Weyl point following .Three independent superconducting phase differencescan be regarded as a 3D vector (cid:126)ϕ . Suppose the Weylpoints are situated at ± (cid:126)ϕ . In the vicinity of the pointat (cid:126)ϕ we expand (cid:126)ϕ = (cid:126)ϕ + δ (cid:126)ϕ , | δ (cid:126)ϕ | (cid:28) × H W = φ a ˆ τ a ; φ a = M ab δϕ b , (1)where ˆ τ a is a vector of Pauli matrices. This form suggestsconvenient coordinates (cid:126)φ for the vicinity of a Weyl pointthat are linearly related and thus equivalent to δ (cid:126)ϕ . Wewill make use of these coordinates through the paper. Inthese coordinates of dimension energy, the spectrum isisotropic and conical, E = ±| (cid:126)φ | . The coordinates arethus defined upon an orthogonal transform.Weak spin-orbit interaction within the nanostructuresplits the Andreev states in spin , resulting in the fol-lowing Hamiltonian,ˆ H W = φ a ˆ τ a + B a ˆ σ a , (2)ˆ σ a being a vector of Pauli matrices in spin space, and B a looks like an external magnetic field causing Zeemansplitting. However, (cid:126)B (cid:54) = 0 even in the absence of externalmagnetic field and represents the effect of the supercon-ducting phase differences on spin orientation. Owing toglobal time reversibility, the vectors (cid:126)B are opposite foropposite Weyl points, (cid:126)B ( − ϕ ) = − (cid:126)B ( ϕ ). The magni-tude of (cid:126)B can be estimated as the superconducting en-ergy gap ∆ times a dimensionless factor characterizingthe weakness of the spin-orbit interaction. For a con-crete number in mind, we can take B (cid:39) . (cid:39) . meV which corresponds to niobium. If there is an externalmagnetic field, it adds to (cid:126)B . We note however that ourestimation of B is about 3 T , so it requires a significantfield to change it.To represent the Hamiltonian in the second-quantization form, we introduce quasiparticle annihila-tion operators ˆ γ σ and associated 4-component Nambubispinors γ α , where α = ( i, σ ) combines spin and Nabmuindex i = e, h , ¯ γ i,σ ≡ (ˆ γ σ , − σ ˆ γ − σ ) to recast it to thestandard form, H W = 12 ¯ γ † α H W αβ ¯ γ β . (3)We note that γ † i,σ = − σγ − i, − σ This gives an isotropicspectrum which depends only on φ ≡ | (cid:126)φ | .(see Fig. Ia) The energies are E = ± φ for two spin-singlet states,ground one | g (cid:105) , and excited one | x (cid:105) , and E = ± B fortwo components of the spin doublet | ↑(cid:105) , | ↓(cid:105) . The ener-gies of the split doublet exhibit no singularity nor phasedependence in the vicinity of the Weyl point, while thespin-singlet states retain the conical spectrum.The ground state is magnetic ( | ↓(cid:105) ) in a narrow vicinityof the Weyl point, namely, at | φ | < B and spin-singletotherwise. (Fig. I a)We will need the current operators in 3 superconduct-ing leads. They are given by the derivatives of the Hamil-tonian with respect to the phases, I a = 2 e (cid:126) ∂ ˆ H W ∂ϕ a = 2 e (cid:126) M ab ˜ I b ; (4)˜ I a ≡ γ † α τ aαβ γ β (5)Since there is a trivial linear relation between I a and ˜ I a ,we will futher concentrate on the dimensionless quantities˜ I a . Let us bring in the coupling with the continuous spec-trum of electron states in several leads (Fig. I). We willdescribe the leads with a usual free-fermion Hamiltonianˆ H leads = (cid:88) k E k ˆ d † k,σ ˆ d ak,σ (6)where k labels the states of the quasi-continuous spec-trum in the leads, d k are the corresponding electron an-nihilation operators, E k are the corresponding energies.The states k are distributed over the leads, those are la-belled with a . We characterize a general non-equilibriumstate of the leads with the energy-dependent filling fac-tors f a ( E ) such that (cid:104) ˆ d † k,σ ˆ d k,σ (cid:105) = f a ( E k ) f or k ∈ a. (7)The crucial part of the Hamiltonian is the tunnellingbetween the electron states in the leads and the Andreevstate in the nanostructure. We will keep it in the mostgeneral form,ˆ H T = (cid:88) k,σ (cid:0) t k ˆ γ † σ − t (cid:48) k σ ˆ γ − σ (cid:1) ˆ d k,σ + h.c. (8)not specifying the spin-independent tunnel amplitudes t k , t (cid:48) k . In the course of the derivation, we will see whichcombinations of the amplitudes are the relevant parame-ters of the model. It is convenient to present the Hamil-tonian in the form of Nambu spinors2 ˆ H T = (cid:88) k γ † α T αβk d αk + h.c. (9)where the matrix T αβ depends on the Nambu index only, T k = (cid:18) t k t (cid:48)∗ k t (cid:48) k − t ∗ k (cid:19) (10)With this, we derive the operators of the current to anormal lead aJ a = e (cid:88) k ∈ a,σ i (cid:0) t k ˆ γ † σ − t (cid:48) k σ ˆ γ − σ (cid:1) ˆ d k,σ + h.c ; (11) J a = ie (cid:88) k ∈ a γ † α ( T αβk τ ) αβ d αk + h.c. (12) III. DERIVATION
The derivation of expressions for the currents in su-perconducting and normal leads can be accomplishedby standard methods of superconducting non-equilibriumKeldysh Green functions . However, for the sake ofcomprehensibility we give here an explicit derivation fromscratch. This is easy for the system under considerationand makes explicit the transition from quasi-continuousto continuous spectrum in the leads.Let us write down the Heisenberg evolution equationsfor the operators ˆ γ α , ˆ d αk,σ governed by the total Hamilto-nian ˆ H = ˆ H W + ˆ H leads + ˆ H T . We use bold-face notationsfor bispinors and ”check” for the corresponding 4 × i ˙ γ = ˇ H W γ + (cid:88) k ˇ T k d k (13) i ˙ d k = E k ˇ τ d k + ˇ T † k γ (14)Here, we implicitly assume a time-dependence of H W .Solving equations for each of ˆ d gives d k ( t ) = e − iE k ˇ τ t d k + (cid:90) dt (cid:48) ˇ g k ( t, t (cid:48) ) ˇ T † k γ ( t (cid:48) ) (15)where ˇ g k ( t, t (cid:48) ) = − ie − iE k ˇ τ ( t − t (cid:48) ) Θ( t − t (cid:48) ) . (16)Here, d describes the state of the leads. We substitudethis to Eq. 13 to obtain a closed equation for γ andexpress it in terms of d : γ ( t ) = (cid:90) dt (cid:48) ˇ G ( t, t (cid:48) ) (cid:88) k ˇ T k e − iE k ˇ τ t (cid:48) d k (17)where we have introduced the advanced Green functiondefined as[ i∂ t − ˇ H W ] ˇ G ( t, t (cid:48) ) − (cid:90) dt (cid:48)(cid:48) ˇΣ( t − t (cid:48)(cid:48) ) ˇ G ( t (cid:48)(cid:48) , t (cid:48) ) = δ ( t − t (cid:48) )(18)where the self-energy ˇΣ readsˇΣ( t, t (cid:48) ) = (cid:88) k ˇ T k ˇ g k ( t, t (cid:48) ) ˇ T † k (19)We substitute the expression (17) to the expressionsfor the current operators (5) and average over the non-equilibrium state of the leads using Eq. 7. This yields (cid:104) ˜ I a (cid:105) = 12 (cid:90) dt (cid:48) dt (cid:48)(cid:48) Tr[ˇ τ a ˇ G ( t, t (cid:48) ) ˇ F ( t (cid:48) , t (cid:48)(cid:48) ) ˇ¯ G ( t (cid:48)(cid:48) , t )] (20)where ˇ¯ G ( t, t (cid:48) ) ≡ ˇ G † ( t (cid:48) , t ) andˇ F = ˇ T k (cid:18) f k e iE k ( t (cid:48) − t )
00 ( ¯ f k ) e iE k ( t − t (cid:48) ) (cid:19) ˇ T † k (21)Here and further on, ¯ f k ≡ − f k . In a similar way, wederive the averages of the currents in the normal leads.They read: (cid:104) J a ( t ) (cid:105) = e (cid:90) dt dt dt Tr[ ˇ M a ( t, t ) ˇ G ( t , t )ˇ F ( t , t ) ˇ¯ G ( t , t )] + (cid:90) dt (cid:0) Tr[ ˇ D a ( t, t (cid:48) ) ˇ G ( t, t (cid:48) )] + h.c. (cid:1) . (22) Here, we defineˇ M a = − (cid:88) k ∈ a ˇ T k τ e − iE k τ ( t − t (cid:48) ) ˇ T † k ; (23)ˇ D a ( t, t (cid:48) ) = − i (cid:88) k ∈ a ˇ T k τ (cid:18) f k e iE k ( t (cid:48) − t ) f k e iE k ( t − t (cid:48) ) (cid:19) ˇ T † k . (24)So far, the expressions are valid for any spectrum inthe normal lead, either quasi-continuous or continuous.Let us now specify to continuous spectrum. For this, wedefine the following combinations of tunnel amplitudesin each lead:Γ a ( E ) = (cid:88) k ∈ a ( | t k | + | t (cid:48) k | ) δ ( E − E k ); (25) (cid:126) Γ a ( E ) = (cid:88) k ∈ a (2Re( t (cid:48) k t ∗ k ) , t (cid:48) k t ∗ k ) , | t k | − | t (cid:48) k | ) δ ( E − E k )(26)All the constituents of the expressions for the opera-tors can be expressed through Γ a ( E ) , (cid:126) Γ a ( E ). Those arethus the actual parameters of our model. The con-tinuous spectrum is implemented by assumption thatΓ a ( E ) , (cid:126) Γ a ( E ) are continuous and smooth functions of en-ergy. Moreover, a convenient and relevant assumption isthat these functions vary at an energy scale that exceedsby far that of the Weyl point. In this case, the energydependence can be disregarded and Γ a , (cid:126) Γ a are taken atzero energy.Let us see how ˇΣ, ˇ F , ˇ M a and ˇ D a are simplified underthese assumptions. In energy representation, the self-energy becomesˇΣ( (cid:15) ) = 14 π (cid:88) ± (cid:16) Γ( E ) ± (cid:126) Γ( E ) · (cid:126) ˇ τ (cid:17) (cid:15) ∓ E − i , (cid:126) Γ ≡ (cid:80) a Γ a , (cid:126) Γ a . The Hermitian part of ˇΣ in thelimit (cid:15) adds a constant term to H and therefore describesa shift, or renormalization of the Weyl point position inthe space of three pahses due to tunneling, δφ = − (cid:90) dE (cid:126) Γ( E ) E . (28)We will disregard this irrelevant redifinition of the Weylpoint position. The anti-Hermitian part of the self-energyis more imortant describing the decay of discrete statesinto the continuous spectrum,ˇΣ = 14 (cid:88) ± (cid:16) Γ( ± (cid:15) ) ± (cid:126) Γ( ± (cid:15) ) · (cid:126) ˇ τ (cid:17) ≈ Γ2 (29)where the limit of small (cid:15) has been implemented in thelast equality. The matrices ˇ
F , ˇ D a bring the informationabout the filling factors in the leads and are expressed asˇ F = (cid:88) a Γ a f + a + (cid:126) Γ a · (cid:126) ˇ τ f − a (30)ˇ D a = − i (cid:104) (cid:126) Γ a · (cid:126) ˇ τ f + a + Γ a f − a (cid:105) . (31) f ± ( (cid:15) ) ≡ f a ( (cid:15) ) ± ¯ f a ( − (cid:15) )2 (32)Finally, ˇ M a = − (cid:126) Γ a · (cid:126) ˇ τ /
2. With this, the terms with ˇ M a in Eq. 22 are related to superconducting currents, (cid:104) J a (cid:105) = − (cid:126) Γ a · (cid:126) ˜ I + (cid:90) dt (cid:0) Tr[ ˇ D a ( t, t (cid:48) ) ˇ G ( t, t (cid:48) )] + h.c. (cid:1) (33)From now on, we will denote the expectation values ofthe currents simply as J a , (cid:126) ˜ I . IV. CURRENTS IN EQUILIBRIUM
In equilibrium and stationary state, the Green func-tions are diagonal in energy representation,ˇ G, ˇ¯ G = 1 (cid:15) − ˇ H W ∓ i Γ2 . (34)There is also a convenient relation i ( ˇ G − − ˇ¯ G − ) = Γ (35)We note that in equilibrium f ( (cid:15) ) = ¯ f ( − (cid:15) ) and fillingfactors in all leads correspond to Fermi distribution atzero chemical potential, f a ( (cid:15) ) = f F ( (cid:15) ). With this, ˇ F =Γ f F . Invoking Eq. 35, we proveˇ G ˇ F ˇ¯ G = − if F ( ˇ G − ˇ¯ G ) (36)and the currents are expressed as (cid:126) ˜ I = − i (cid:90) d(cid:15) π Tr[ (cid:126) ˇ τ ( ˇ G − ˇ¯ G ) f F ( (cid:15) )] (37)Let us first recognize that the equilibrium super cur-rents are expressed from the derivatives of free energywith respect to (cid:126)φ . For an isolated superconductingnanostructure, that is, in the limit Γ (cid:28) B, φ , and at zerotemperature, the ground state energy is given throughthe positive energies of Andreev bound states, E g = − (cid:88) i E i Θ( E i ) (38)For the nanostructure under consideration, the Andreevbound states are E σ, ± = Bσ ± φ and the currents in thislimit read (cid:126) ˜ I = − (cid:126)n Θ( φ − B ) (39)The current has a cusp: that is, its derivative with re-spect to φ diverges in a point. This divergence may be in -1-0.8-0.6-0.4-0.20 0 2 4 6 8 10 a b FIG. 2. Smoothing of the superconducting current singularityat the scale of Γ. The curve a. corresponds to B (cid:28) Γ, whilethe curve b. to B = 5Γ. principle used for finding the Weyl point and is smoothedat the scale of Γ.At finite Γ, the Andreev energies correspond to thepoles of the Green functions. Their poles are shiftedby ± i Γ / ξ σ, ± ≡ arctan(2( Bσ ± φ ) / Γ), (cid:126) ˜ I = (cid:126)n π (cid:88) σ ( ξ σ, − − ξ σ, + ) (40)The cusps are smoothed by a finite Γ (see Fig. IV). Themaximum derivative with respect to φ is now finite andis of the order of Γ − : ∂ ˜ I∂φ = 2 π Γ for B (cid:28) Γ , π Γ for B (cid:29) Γ . (41)In equlibrium, we expect no currents to normal leads.Indeed, if there were currents, one could extract energyfrom the equilibrium system by applying voltages to thenormal leads. Technically, two terms in Eq. 33 canceleach other upon applying the relation (36). V. STATIONARY CURRENTS
Now we turn to the case of non-equilibrium filling fac-tors in the leads still assuming stationary Weyl pointHamiltonian. The currents are given by Eqs. 33, 20with energy-diagonal Green functions (34). To keep theformulas simple, we will specify to differential conduc-tances at vanishing temperature. The voltages in theleads only change the filling factors, at vanishing tem-perature ∂f a /∂ eV a = δ ( (cid:15) − eV a ), that is, the differen-tial conductances are contributed by the specific energies (cid:15) = ± eV a only.For the derivatives of supercurrents, we have2 π ∂(cid:126)I∂eV a = (cid:126)φ Γ a K o ( eV a )+(2 (cid:126)φ · (cid:126) Γ a ) (cid:126)φ + ( (cid:126)φ × (cid:126) Γ a )) K e ( eV a ) + (cid:126) Γ a K ( eV a ) (42) -0.4-0.3-0.2-0.100.10.20.30.40.50.60.7 0 2 4 6 8 10 a b c FIG. 3. The voltage derivative of the superconducting cur-rent. There is a single lead, (cid:126)
Γ = 0, we set φ = 3 .
0. The spinsplitting B is set to 0, Γ, 4Γ, for the curves a,b,c respectively. where the functions K o,e, are defined as ( K − σ ≡ (( (cid:15) − Bσ ) − Γ / − φ ) + Γ ( (cid:15) − Bσ ) ): K o = 2 (cid:88) σ ( (cid:15) − Bσ ) K σ ; K e = (cid:88) σ K σ ; (43) K = (cid:88) σ (( (cid:15) − Bσ ) + Γ / − φ ) K σ (44)We note that (cid:90) ∞ d(cid:15)K o = 2(arctan( φ + B ) + arctan( φ − B ))Γ φ ; (45) (cid:90) ∞ d(cid:15)K e = π Γ(Γ / φ ) ; (46) (cid:90) ∞ d(cid:15)K = π Γ2(Γ / φ ) (47)The derivatives are illustrated in Fig. V for a single leadand simple case (cid:126) Γ = 0. They peak at the positions ofresonant levels eV = φ + B , | φ − B | . The peak width isof the order of Γ. For singlet ground state (the curves a , b the finite current at zero voltage falls to zero in oneor two steps. For the doublet ground state, the currentthat is small at zero voltage rises at the first and dropsat the second resonant level.The differential conductances in the normal leads aregiven by: ∂J a ∂e V b = − (cid:126) Γ a · ∂(cid:126) ˜ I∂eV b +Γ δ ab π (cid:16) Γ a ( K ( eV a ) + 2 φ K e ( eV a )) + ( (cid:126) Γ a · (cid:126)φ ) K o (cid:17) (48)We plot in Fig. V an example of zero-voltage conduc-tances G , G , G for two leads. The diagonal con-ductances peak when the resonant levels are at zero en-ergy, | φ − B | = 0. The peak widths are of the order -0.200.20.40.60.811.21.41.6 -8-6-4-2 0 2 4 6 8-1.5-1-0.500.511.522.53-3 -2 -1 0 1 2 3 a b c x 10 c x 10 a b FIG. 4. An example of zero-voltage conductances. There aretwo leads, Γ = 0 . = 0 . (cid:126) Γ (cid:107) x , (cid:126) Γ (cid:107) y , the plotsare for (cid:126)φ in z-direction. The curves a,b,c, correspond two G , G , G . The transconductance is antisymmetric in thiscase, G = − G . Left pane: B (cid:28) Γ, right pane: B = 5Γ.The vertical scale of the curve c is increased by a factor of 10. of the conductance quantum G Q ≡ e / (cid:126) pi . An interest-ing feature is a Hall-like antisymmetric transconductance G = − G . It incorporates the effects of vector partsof Γ in two leads, G ∝ (cid:126)φ · ( (cid:126) Γ × (cid:126) Γ ) and changes sign if (cid:126)φ → − (cid:126)φ .For finite-voltage conductance, we restrict ourselves tothe case of a single lead. The example for | (cid:126) Γ | = Γ / eV = | φ ± B | , their width being of theorder of Γ. The peak values are of the order of G Q .The vector part of Γ brings anisotropy and asymmetryof conductances with respect to voltage and (cid:126)φ .At high voltages eV (cid:29) Γ , φ, B applied, the currentin the normal lead saturates at finite value J ∞ , as itis expected for the transport via resonant levels. Wenote a peculiar feature: this current retains the depen-dence on φ and its direction, this dependence is smoothedat the small scale of φ (cid:39) Γ only. Using the relations(42),(43),(45), we obtain J ∞ /e = Γ − ( (cid:126) Γ · (cid:126)φ ) + ( (cid:126) Γ) Γ / φ + Γ /
4) (49)This feature survives rather high temperatures φ (cid:28) k B T (cid:28) eV at which the thermal equilibration eventu-ally cancels the superconducting currents near the Weylpoint. This makes the feature highly proficient for ex-perimental detection of Weyl points in a practical situa-tion where the finite temperature prevents the detectionthrough the supercurrent. One would look at the vari-ation of the tunnel current under variation of φ to finda signal that is concentrated near the point and showsanisotropy defined by Eq. 49. The maximum derivativefor (cid:126)φ ⊥ Γ ∂J∂φ = e (cid:126) (cid:126) Γ Γ (50)does not depend on the strength of the tunnel coupling,this guarantees a big amplitude of the detection signal. a b c d e FIG. 5. Differential conductance for the case of a single lead. | (cid:126) Γ | = Γ / (cid:126)φ ⊥ (cid:126) Γ,the conductance is even in V and φ . The dashed curves cor-respond to (cid:126)φ (cid:107) (cid:126) Γ, and G ( V, φ ) = G ( − V, − φ ). The parametersare: a. B = 0, φ = 3Γ; b. B = 5Γ, φ = 3Γ; c. B = 3Γ, φ = 5Γ; d. B = 5Γ, φ = 0; e. B = 0, φ = 0. The perpendicu-lar and parallel conductances coincide for the last two plots,since φ = 0. VI. REDIFINITION OF BERRY CURVATUREAND DENSITY OF TOPOLOGICAL CHARGE
In this Section, we consider adiabatic case. We assumeequilibrium filling factor in the leads and concentrate onthe case of vanishing temperature. If we change the con-trol phases slowly,the superconducting currents acquire acorrection proportional to time derivatives of the phases:˜ I α ( t ) = ˜ I α ( (cid:126)φ ( t )) + B αβ ( (cid:126)φ ) ˙ φ β (51)Thereby we define a tensor response function B αβ . Thesymmetric part of this tensor defines the dissipation inthe course of the slow change of the phases, dEdt = ˙ φ α B αβ ( (cid:126)φ ) ˙ φ β . (52)If the system under consideration is gapped, the dissipa-tive part is absent, while the antisymmetric part of theresponse function gives the Berry curvature of the groundstate of the system (see e.g. ) B αβ = 2Im (cid:104) ∂ α Ψ | ∂ Ψ Ψ (cid:105) . (53)It is convenient to introduce a pseudovector of Berry cur-vature B α = e αβγ For the superconducting Weyl point,the Berry curvature has been evaluated in . For thesinglet ground state, and in the coordinates in use it as-sumes the standard expression (cid:126)B = (cid:126)φ/ (2 φ ). The fluxof (cid:126)B through a surface enclosing the origin is 2 π mani-festing a unit point-like topological charge at the origin.However, (cid:126)B = 0 at φ < B where the ground state isdoublet. The continuity of the ground state is broken at φ = B and topological consideration that guarantees adivergentless (cid:126)B cannot be applied anymore.We evalute B αβ for the setup under consideration mak-ing use of Eq. 20. Given a modulation of the Hamiltonianˇ δH oscillating at frequency ω , the response of the cur-rents oscillating at the same frequency can be representedas ˜ I αω = (cid:90) d(cid:15) π
12 Tr[ˇ τ α ( ˇ G (cid:15) + ω ˇ δH ˇ G (cid:15) ˇ F (cid:15) ˇ¯ G (cid:15) + (54)ˇ G (cid:15) ˇ F (cid:15) ˇ¯ G (cid:15) ˇ δH ˇ¯ G (cid:15) − ω )] . (55)We obtain B αβ by substituting ˇ H = δφ α ˇ τ α and takingthe limit ω →
0. This is valid for ω (cid:28) Γ. We assumevanishing temperature when integrating over the energy.To present the answers in a compact form, we intro-duce a convenient expression K ≡ ( φ − B + Γ / + B Γ . The dissipative part of the response functionreads: B αβ = Γ πK (cid:18) δ αβ + φ α φ β B K (cid:19) (56)It is plotted in Fig. VI for two values of magnetic field.We note that the dissipative part at small Γ is pro-portional to Γ except φ = B This is because the dis-sipation requires an excitation of an electron-hole pair a b x 10c x 10
FIG. 6. Dissipative part of the response function. We assume φ (cid:107) z and plot B zz , B xx = B yy . Curve a: B = 0, B zz = B xx .Curves b,c: B = 3 in the normal leads, which is a second-order tunnel-ing process . At the resonance threshold φ = B , and B (cid:29) Γ, the dissipative part of the response function isstrongly anisotropic: it is (cid:39) Γ − for the direction (cid:107) (cid:126)φ and (cid:39) B − otherwise.Following , we redefine Berry curvature as an asym-metric part of the response function. For any discretespectrum and zero temperature, this redefinition wouldbe exact retaining all topological properties of the cur-vature provided the limit ωto ω (cid:28) δ , δ beingthe level spacing in the spectrum. However, in our casethe spectrum is continuous, that is, δ = 0, and the limit ω → ω (cid:28) Gamma . Nevertheless, theredefined curvature coincides with the standard expres-sion at φ, B (cid:29)
Γ, that is, far from a close vicinity of thepoint or the resonance φ = B . General expression reads (cid:126)B = (cid:126)φ πφ (cid:34)(cid:88) ± arctan 2( φ ± B )Γ + φ − Γ / − B K (cid:35) (57)We plot it in Fig. VI for several B . At the origin, (cid:126)B ∝ (cid:126)φ ,the maximum at B = 0 is | (cid:126)B | ≈ . − and is achievedat φ ≈ . ρ ( φ ) = 12 π div (cid:126)B (58)This is the most important manifestation of embeddinga topological singularity into a continous spectrum. Thepoint-line unit charge is spead over the parameter spaceconcentrating either near the origin or, at B (cid:29) Γ at thesurface φ = B . We evaluate ρ ( φ, B ) = Γ π B + φ + Γ / K (59) a b c FIG. 7. Redefined Berry phase × φ . The curves a,b,c corre-spond to B = 0 , , φ (cid:29) Γ. a b x 100 c x 300 FIG. 8. The density of topological charge. Curves a, b, ccorrespond to B = 0 , , At small Γ, the density is proportional to Γ arising froma complex tunneling process. Its maximum value (cid:39) Γ − at B = 0 and (cid:39) B − Γ − at B (cid:29) Γ. We plot the densityat several values of B in Fig. VI VII. CURRENTS IN NORMAL LEADS:PUMPING
A slow change of control phases may lead to the cur-rents in the normal leads proportional to the time deriva-tives of the phases, J a = e (cid:32) (cid:126)A a ( (cid:126)φ ) · d(cid:126)φdt (cid:33) (60) (cid:126)A a being (cid:126)φ -dependent proportionality coefficients. Letus recognize this as a case of parametric pumping, a phe-nomenon that has been intensively discussed in quan-tum transport , also in the context of superconduct-ing nanostructures with normal leads . An ac modula-tion of φ is expected to result in an ac normal current,that is difficult to measure. However, it can also give riseto a dc current, that is, to pumping. If (cid:126)φ is changingperiodically along a closed contour, the charge per cycledepends on the contour only, and, by virtue of Stokestheorem, is given by a flux of the curl of (cid:126)A through thecontour, Q a = (cid:90) T dtJ a ( t ) = (cid:123) dS ( (cid:126)N · curl (cid:126)A ) . (61)We evaluate (cid:126)A making use of Eq. 33 and expanding theGreen functions up to first order in ˇ δH . We notice thatthe currents, since the filling factors are in equilibrium,are only due to the vector parts of Γ. Two groups ofterms in Eq. 33 that cancel each other in stationaryequilibrium case can be rewriten as J a = 12 Tr[( (cid:126) Γ a · ˇ (cid:126)τ [ ˇ f , ˇ G ] ˇ¯ G ] (62)The commutator in this expression in energy representa-tion can be rewritten as( f ( (cid:15) ) − f ( (cid:15) − ω )) ˇ G (cid:15),(cid:15) − ω (63)Since we are to expand to the first order in ω , this willgive a weight of ∂ (cid:15) f in the integration over (cid:15) , and we canneglect small ω in the Green functions. The quantitiesunder evaluation just sample Green functions in an en-ergy interval (cid:39) k B T near zero energy, this interval goingto zero at vanishing temperature. This is in contrast tothe response functions explored in the previous Section,those are determined by integration over all relevant en-ergies. Nevertheless, the expression of (cid:126)A has qualitativelysimilar features, the values being concentrated at φ (cid:39) Γif B (cid:28) Γ or at φ = B(cid:126)A a = − Γ πK (cid:18) (cid:126) Γ a Γ + (cid:126)φ ( (cid:126) Γ a · (cid:126)φ )Γ 4 B K + (Γ a × (cid:126)φ ) (cid:19) (64)Since we discuss the pumping, the curl of (cid:126)A — let us callit the effective field —is more relevant for us:curl (cid:126)A a = − Γ2 πK [( (cid:126) Γ a × (cid:126)φ )4Γ( φ + Γ / (cid:126) Γ a (( B + Γ / − φ )+ (cid:126)φ ( (cid:126)φ · (cid:126) Γ a )( φ + Γ − B )] . (65)The natural axis in (cid:126)φ space is set by the direction of (cid:126) Γ a . In the above expression, we have separated the ef-fective field into azimuthal, axial, and radial component.The dimension of effective field is E − . Far from theresonance, the azimuthal field is estimated as (cid:39) Γ φ − ,and axial/radial field as (cid:39) Γ φ − . Thus, the typical Q a /e for the contours that do not cross the resonance are small, (Γ /φ ) , (Γ /phi ) respectively. At the reso-nance φ = B (cid:29) Γ, the azimuthal field is estimated as B − Γ − , and axial/radial field as B − . At B (cid:39) Γ, andnear the origin, all field components are estimated as Γ .This implies that we can achieve Q a (cid:39) e for small con-tours with dimension Γ provided they are close to theorigin.We illustrate this with the following examples (Fig.VII). For pumping in the lead a , it is convenient to choosethe coordinate system such that z (cid:107) (cid:126) Γ a . We probe theaxial component of the effective field by taking a circularorbit with radius R in the plane z = 0, that is centeredat the origin.(Fig. VII a). The axial field is positive atthe origin, and changes sign at φ = (cid:112) B + Γ /
4. Thetotal flux in z = 0 plane is zero. The charge per cycle forthis orbit is given by Q a /e = 2 | (cid:126) Γ a | Γ R ( R + Γ / B ) − R B . (66)It reaches maximum that does not depend on magneticfield, Q a = 2 e | (cid:126) Γ a | Γ , (67)and gets back to zero for the contours of bigger radius. Toprobe the azimuthal field, one chooses a contour in e.g. x = 0 plane, that follows the axis at the scale max(B , Γ)to enclose the maximum positive flux. The charge percycle in this case does not depend on the contour detailsand equals Q a = − πe | (cid:126) Γ a |
4Γ Γ / B Γ / B . (68)The vector parts of Γ are generally different in differentleads, so that the same contour is oriented differentlyfor different leads. We conclude that the pumping to thenormal leads provides an interesting possibility to explorethe vicinity of the Weyl point. VIII. CONCLUSIONS
To conclude, we have investigated the properties of aWeyl point immersed to a continuous spectrum. We takea Weyl point in a superconducting nanostructure thatis tunnel-coupled to the electronic states in the normalleads. The tunnel coupling gives rise to a new energyscale Γ, that corresponds to a scale in parametric space.We investigate in detail how the topological and spec-tral singularities of the Weyl point are smoothed on thisscale. We evaluate the superconducting currents in equi-librium, the superconducting and normal-lead currentsat constant voltages applied to the leads. We find sharpfeatures in high-voltage tunnel currents that may be usedto detect the Weyl points in experiment.0 z=0 xy + -x=0 yz + - a bA.B. FIG. 9. Pumping to a normal lead, (cid:126) Γ a (cid:107) z . A. Probing the axial effective field. A circular contour with radius R in z = 0 planeis centered at the origin. The plot: dependence of the charge per cycle on R for B = 0 , x = 0 plane that goes along the axis at the scale max B, Γ encircles the whole flux in this direction. Thevalue of the charge per cycle does not depend on the contour details and is given as function of B in the plot. Importantly, we consider the adiabatic variation ofcontrol phases. This permits us to redefine Berry curva-ture and evaluate the density of topological charge that isnot point-like but rather spread around the origin as themanifestation of coupling to the continuous spectrum.We investigate the pumping to normal leads and findthat it witnesses the peculiarities of Weyl point at thescale of Γ and opens up new perspectives for experimental exploration of Weyl point singularities.
ACKNOWLEDGMENTS
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