Topological properties of multi-terminal superconducting nanostructures: effect of a continuous spectrum
Recently, it has been shown that multi-terminal superconducting nanostructures may possess topological properties that involve Berry curvatures in the parametric space of the superconducting phases of the terminals, and associated Chern numbers that are manifested in quantized transconductances of the nanostructure. In this Article, we investigate how the continuous spectrum that is intrinsically present in superconductors, affects these properties. We model the nanostructure within scattering formalism deriving the action and the response function that permits a re-definition of Berry curvature for continuous spectrum.
We have found that the re-defined Berry curvature may have a non-topological phase-independent contribution that adds a non-quantized part to the transconductances. This contribution vanishes for a time-reversible scattering matrix. We have found compact expressions for the redefined Berry curvature for the cases of weak energy dependence of the scattering matrix and investigated the vicinity of Weyl singularities in the spectrum.
The study of topological materials has been on the front edge of the modern research in condensed matter physics for the past decade Xu and Balents (2018); Yao and Wang (2018); Pacholski et al. (2018); Hossain et al. (2018); Tan et al. (2018). These materials are appealing from fundamental point of view and for possible applications (TI-based PhotodetectorBrems et al. (2018); Tang et al. (2017), spintronicsGötte et al. (2014), field-effect transistorMaciejko et al. (2010), catalystChen et al. (2011) and quantum computingNayak et al. (2008); Aasen et al. (2016)). The basis for applications is the topological protection of quantum states, which makes the states robust against small perturbations and leads to many unusual phenomena, e.g. topologically protected edge statesKane and Mele (2005); Wu et al. (2006); Fu et al. (2007). The topological superconductorsQi et al. (2009, 2010); Das et al. (2012); Fu and Berg (2010) and Chern insulatorsHaldane (1988); Regnault and Bernevig (2011); Zhang and Qi (2014); Thonhauser and Vanderbilt (2006) are the classes of topological materials that are relevant for the present paper. In the case of the Chern insulator the topological characteristic is an integer Chern numberQi and Zhang (2011); Moore and Balents (2007) computed with the Green’s function of electrons occupying the bands in a Brillouin zone of a material - WZW formWitten (1983); Wang et al. (2010a, b); Essin and Gurarie (2011). The first Chern number reduces to the sum of first Chern numbers of the filled bands. For each band, the first Chern number is defined as an integral of the Berry curvature over the Brillouin zoneNiu et al. (1985); Thouless et al. (1982). The Berry curvature is commonly definedM. V. Berry (1984) as with being the wavefunction in this band and being the parameters: in this case two components of a wavevector. If the Chern number of a crystal is not zero, the edge states necessarily appear at the interface between the crystal and the vacuum (since the Chern number of the vacuum is zero). The dimensionality of topological materials in real space is restricted by three from above, which significantly limits possible topological phases.
However, there is a way to circumvent this fundamental limitation. Recently, the multi-terminal superconducting nanostructures with conventional superconductors were proposed to realize the topological solids in higher dimensionsRiwar et al. (2016). Such nanostructures host discrete spectrum of so called Andreev bound statesAndreev (1964); de Gennes and Saint-James (1963); Beenakker and van Houten (1991). The energies and wavefunctions of these states depend periodically on the phases of superconducting terminals. This sets an analogy with a bandstructure that depends periodically on the wavevectors. The dimensionality of this bandstructure is the number of terminals minus one. Also, as it was notedRiwar et al. (2016), the multi-terminal superconducting nanostructures cannot be classified as the high-dimensional topological superconductors from the standard periodic table of topological phasesKitaev (2009). The authors of Riwar et al. (2016) have considered in detail 4-terminal superconducting nanostructures and proved the existence of Weyl singularitiesLu et al. (2015); Soluyanov et al. (2015) in the spectrum. The Weyl singularity is manifested as level crossing of Andreev bound states at a certain point in 3-dimensional phase space. Each Weyl singularity can be regarded as a point-like source of Berry curvature. Owing to this, a nonzero two-dimensional Chern number can be realized and is manifested as a quantized transconductance of the nanostructure. This transconductance is the response of the current in one of the terminals on the voltage applied to the other terminal in the limit of small voltage, this signifies an adiabatic regime.
The peculiarity of the system under consideration is the presence of a continuous spectrum next to the discrete one. These states are the extended states in the terminals with energies above the superconducting gap. Were a spectrum discrete, the adiabaticity condition would imply the level spacing being much larger than the driving frequency. The level-spacing is zero for a continuous spectrum, so this complicates the adiabaticity conditions. This has been pointed out already in Ref.Riwar et al. (2016) but was not investigated in detail. We note the generality of the situation: a generic gapped system might have a continuous spectrum above the certain threshold, and the adiabaticity condition required for the manifestations of topology needs to be revisited in this situation.
The aim of the present article is to investigate this question in detail for a generic model of a superconducting nanostructure. We have studied the linear response of currents on the changes of superconducting phases in the terminals. We model a multi-terminal superconducting nanostructure within the scattering approachNazarov and Blanter (2009). In this approach the terminals of the nanostructure are described with semiclassical Green’s functions and the scatterer coupled to the terminals is described by a unitary (in real time) S-matrix. Although it is not crucial, we made use of Matsubara formalism which conveniently allows us to concentrate on the ground state of the system and the limit of zero temperature is formally achieved by considering continuous Matsubara frequencies. So we do the calculations in imaginary time formalismSchÃ¶n and Zaikin (1990). At the first step, we obtain the general effective action describing the nanostructure in terms of the S-matrix and time-dependent semiclassical Green’s functions of the terminals. At the second step, we expand the action to the second power in time-dependent phases of the terminals. At the third step, we concentrate on the limit of small voltage and driving frequency, to obtain the response function relevant for topological properties.
We can use the properly anti-symmetrized response function as a generalized definition of the Berry curvature that is suitable for the systems with and without a continuous spectrum. The main result of the present article is that so-defined Berry curvature is contributed to by a continuous spectrum as well as discrete one even in the case of energy-independent S-matrix. We derive an explicit formula for it. This solves the paradox mentioned in Riwar et al. (2016): the Berry curvature associated with discrete Andreev bands is discontinuous when the highest Andreev bound state merges with the continuum, which indicates that the integral of the Berry curvature defined only for discrete spectrum will not reduce to an integer. The redefined Berry curvature that we find is continuous. It gives rise to integer Chern numbers if the S-matrix is time-reversible. If it does not we reveal a specific additional non-topological contribution that does not depend on the superconducting phases. We note the the importance of the energy scales much larger than superconducting gap in this context. This is why we also discuss in detail the case of an energy-dependent S-matrix the energy scale of variation of which may be in any relation with superconducting gap. We find that the non-topological contribution depends on the regularization of the S-matirx at large energies. In particular, it vanishes if the S-matrix is regularized as , this corresponds to no conduction between the terminals.
The paper is organized as follows. In Sec. II we introduce the details of a model of a multi-terminal superconducting nanostructure and review the main aspects of a scattering matrix approach formalism in this case. The derivation and discussion of the response function are given in Sec. IV. In Sec. VI we discuss the specific behaviour near the Weyl singularities, in the absence and presence of a weak spin-orbit coupling. In Sec. V we apply the general formulae to the case of a scattering matrix that varies only slightly on the scale of the superconducting gap . In Sec. VII we address the energy-dependent S-matrices at arbitrary energy scale for a specific model of an energy dependence. We conclude the paper with the discussion of our results (Sec. VIII). The technical details of the derivations are presented in Appendices.
Ii Multi-terminal superconducting nanostructure
Generally a multi-terminal superconducting nanostructure (Fig. 1) is a small conducting structure that connects superconducting leads. The leads are macroscopic and are characterized by the phases of the superconducting order parameter. Each lead labeled by has its own superconducting phase and one of the leads’ phase can be set to zero value , according to the overall gauge invariance. The nanostructure design and these phases determine the superconducting currents in each lead, that are the most relevant quantities to observe experimentally.
We aim to describe a general situation without specifying the nanostructure design. To this end, we opt to describe the system within the scattering approach pioneered by Beenakker Beenakker (1997). The superconducting leads are treated as terminals: they are regarded as reservoirs which contain macroscopic amount of electrons and are in thermal equilibrium. A common assumption that we also make in this article is that all terminals are made from the same material and thus have the same modulus of the superconducting order parameter . At sufficiently low temperatures and applied voltages one can disregard possible inelastic processes in the nanostructure and concentrate on elastic scattering only. Following the basics of the scattering approachNazarov and Blanter (2009), we assume spin-degenerate transport channels in terminal . The conducting structure connecting the terminals is a scattering region and is completely characterized by a scattering matrix which generally depends on energy and is a unitary matrix at any . In Matsubara formalism we use imaginary energy and the matrix satisfies the condition . All the details of the nanostructure design are incorporated into the scattering matrix.
The electrons and holes in the superconducting transport channels involved in the scattering process may be described as plane waves that scatter in the region of the nanostructure and then return to the corresponding terminals. Amplitudes of incoming and outgoing waves are linearly related by the -matrix. The numbers of transport channels in the terminal denoted as determines the dimension of the scattering matrix: , where and counts for the spin.
The electrons and holes experience Andreev reflection in the superconducting terminals: the electrons are converted into holes and turn back, the same happens to holes. The Andreev reflection is complete at the energies smaller than the superconducting gap . Therefore, electron-hole waves may be confined in the nanostructure giving rise to discrete energy levels called Andreev bound states (ABS). The amplitudes and phases of these confined states are determined by the scattering matrix and Andreev reflection phases that involve the superconducting phases of the corresponding terminals. One can find the energies of the ABS through Beenakker’s determinant equationBeenakker and van Houten (1991):
where is the S-matrix at the real energy , is a Pauli matrix acting in the spin space and is the diagonal matrix in channel space ascribing the stationary superconducting phases of the terminals to the corresponding channels, where label the channels and is the terminal corresponding to the channel . The ABS energies and the corresponding eigenvectors in the space of the channels depend parametrically on independent phases and thus can be viewed as a bandstructure defined in a ”Brilluoin zone” of phases. It was notedRiwar et al. (2016) that (without spin-orbit interaction) three independent parameters are needed to tune the dimensional band structure of energy levels of ABS to reach the Weyl singularity at zero energy. It was also notedRiwar et al. (2016) that only one parameter is required to satisfy the condition for the highest ABS to touch the continuum above the gap (). The ABS merges the continuum in this case and this implies that one cannot change this level adiabatically even for arbitrarily slow change of the parameters. When the incommensurate small voltages are applied to two terminals to sweep the phasesRiwar et al. (2016), the system passes the points where the highest level merges with the continuum. This makes it questionable to apply the adiabaticity reasoning in this case. This makes it necessary to consider the contribution of the continuous spectrum to the response function of the currents in the limit of slow change of the parameters.
The most general way to describe the nanostructure under consideration is to use an action method. This method has been pioneered in the context of a simple Josephson junction in SchÃ¶n and Zaikin (1990). In this method one deals with an action of the nanostructure that depends on the time-dependent superconducting phases . The transport properties of the nanostructure as well as quantum fluctuations of the phases in case the nanostructure is embedded in the external circuit SchÃ¶n and Zaikin (1990), can be derived from this action.
One of the advances of this Article is the derivation of such action for multi-terminal nanostructure and arbitrary S-matrix in Matsubara formalism. The details of the derivation are given in IX. Here we give the answer:
here and are matrices in a space that is a direct product of the space of channels, the imaginary-time space, spin and Nambu space. The matrix is diagonal in the corresponding energy representation, therefore it depends on the difference of the imaginary time indices only. Its Nambu structure is given by
where is the electron energy-dependent S-matrix (see App. IX). The matrix is composed of the matrices diagonal in energy and diagonal in time in the following way:
where is the 3rd Pauli matrix acting in Nambu space and the Nambu structure has been made explicit in . This form assumes that is the same in all the terminals. If it is not so, the matrix also acquires the dependence on the channel index. It is worth noting that so that are projectors. The matrix can be associated with the semiclassical Green’s function in a terminalEilenberger (1968); Nazarov and Blanter (2009): is the diagonal matrix in channel space ascribing the time-dependent superconducting phases of the terminals to the corresponding channels, where label the channels and is the terminal corresponding to the channel . We note the gauge invariance of the action: due to the invariance of the trace under unitary transformations, the superconducting phases can be ascribed to the terminal Green’s functions as well as to the scattering matrix. Let us assume that the matrix does not depend on spin. Then the trace over spin is trivial. It is convenient to apply the unitary transformation as in to all the matrices in . This transforms the matrix to . Then the projectors take a simple form and the matrix in reduces to the lower block-triangular form in Nambu space. The determinant is then equal to the determinant of the lower right block of the transformed matrix . Then the action takes the form
the S-matrix in Matsubara formalism is subject to the unitarity constraint,
In what follows we concentrate on the zero-temperature limit , so the summations over discrete frequencies are replaced with integrations .
iii.1 Stationary phases
In the stationary case with constant and the value of the action gives the stationary phase-dependent ground state energy of the nanostructure .
where Trace is now over the channel space and the Trace over spin space is taken explicitly as a factor of unless specifically addressed. The operator introduced here has the properties of the inverse of the Green’s function although it is not related to an operator average: its determinant as function of complex vanishes, , at imaginary values corresponding to the ABS energies (compare with ). In addition to these singularities the operator has two cuts in the plane of complex corresponding to the presence of a continuous spectrum in the terminals above the gap . We choose the cuts as shown in Fig. 2. The expression can be simplified in the case when the S-matrix does not depend on energy
the second (divergent) contribution here does not depend on the superconducting phases so we omit it. To compute the integral it is convenient to choose the basis in which the unitary matrix is diagonal. This is a unitary matrix, so the eigenvalues are unimodular complex numbers. The phases of the eigenvalues are related to the energies of ABS: . The eigenvalue is doubly degenerate and corresponds to the values . The eigenvalues come in complex conjugated pairs , where corresponds to the Nambu-counterpart of the th eigenvector. So only the eigenvalues correspond to the quasiparticle states with positive energies. We will label them with positive indices . In what follows we define a ”bar” operation that links these pairs where is some eigenvector of . We note, however, that this operation is not a convolution, since .
In this basis we can rewrite the integral as
Evaluation of the integral brings to the known result
where are the stationary phase-dependent ABS energies, as discussed above. The derivative of the ground state energy with respect to a stationary phase in terminal gives the stationary current in the corresponding terminal,
We expect this relation to hold in the adiabatic limit. In the following Section, we will access the time-dependent currents concentrating on the next order correction in the limit of small frequencies.
Iv Response function of the currents
To compute the response function of the currents we assume small nonstationary phase addition to the stationary phases , , and expand the action to the second order in (first order vanishes automatically since is nonstationary ). We give the details in Append. X. The total contribution to the action reads
being the Fourier transform of . The frequency-dependent response function of the current is given by
here the stationary phases are ascribed to the S-matrix. We use a shorthand notation and define a set of matrices that project channel space onto the space of the channels in the terminal , if is a channel in terminal and otherwise. The first term in vanishes at zero frequency and in the case of the energy-independent S-matrix. The second term does not depend on frequency . In the limit of zero frequency the second and the third terms reproduce the stationary response function of the currents
Let us consider the limit of small and concentrate on the first order correction to the adiabatic limit
We note that the response function is analytic in the vicinity of . This is guaranteed by the gap in the density of states, which is given by the energy of the lowest ABS. Away from the zero-energy Weyl singularity it can be estimated as with being the total number of ABS in the nanostructure. The vicinity of a Weyl singularity has to be treated more carefully as we discuss in Sec. VI. Let us note that for any system with a discrete spectrum the quantity can be related to the Berry curvatureM. V. Berry (1984); Niu et al. (1985); Thouless et al. (1982). For any state in the discrete spectrum the Berry curvature corresponding to this state is given by with labeling discrete states and being the wavefunction of the corresponding state. In our case we are interested in the total Berry curvature of the superconducting ground state defined as where labels the (spin-degenerate) wavefunctions of the BdG equation with positive eigenvaluesRiwar et al. (2016). However, the adiabaticity condition which justifies the expansion in for the case of discrete spectrum requires the frequency to be much smaller than the smallest energy spacing between the levels.
In our system, the continuous spectrum above the superconducting gap is present. In principle, any continuous spectrum can be approximated with a discrete spectrum with a vanishing level spacing . By doing this we can utilize the previous expression for the response function since it is valid for the discrete spectrum. However, the adiabaticity condition which is necessary for this expression to be valid would reduce to . This condition contains an artificially introduced and is by construction very restrictive in . On the other hand, the expansion in Eq. is valid under a physically meaningful and less restrictive condition . Taken all that into account, we conclude that the response function defined in Eq does not have to reduce to the expression for a total Berry curvature of a superconducting ground state of a system discussed above. The topological properties of this quantity also have to be investigated separately.
One may conjecture that the resulting response function in Eq. reduces to the sum of the Berry curvatures of the discrete ABS spectrum, so that it is not contributed to by the continuous spectrum. This conjecture relies on the analogy between the expressions for the total Berry curvature and the superconducting ground state energy. In the case when the S-matrix is energy-independent, only the discrete states contribute to the ground state energy. Thus motivated, in the following we investigate the response function defined by means of Eq in detail. We find that there is a contribution from the continuous spectrum to this quantity as well as from the discrete one. We also find that in general the integral of over the phases that would normally define an integer Chern number, is not integer. Therefore, contains a non-topological contribution. This non-topological part is contributed by the continuous as well as the discrete part of the spectrum.
The tensor defined in Eq. is antisymmetric (since ). The concrete expression for reads:
The first term here resembles the usual WZW formEssin and Gurarie (2011) for a Chern number. Usually, the form contains the matrix Green’s functionsEssin and Gurarie (2011), in our case the form utilizes the matrix defined by Eq. . We note however that in distinction from common applications of WZW forms here one cannot regard as a smooth function of parameters defined on a compact manifold without a boundary. This is because in general this matrix has different limits at positive and negative infinite energies for and for that also depend on the phases. Due to this reason the integral of the first term over a compact surface without a boundary in a space of phases does not have to reduce to an integer . The second term in Eq. has a form of a total derivative with respect to a phase of a periodic and smooth function, so the integral of this one over a compact surface will give zero.
In order to obtain the value of this integral let us consider first the variation of this value upon the small smooth variation of the matrix that comes from the small variation of the S-matrix , so . The value of the integral of the second contribution in Eq. does not contribute to the integral over a compact submanifold in phase space, so we needn’t consider its variation. It is known Qi et al. (2008) that the variation of the first contribution to reduces to the total derivatives
The value of the integral of second term in over a compact submanifold in phase space vanishes if the submanifold does not pass Weyl singularities corresponding to , because it has a form of a total derivative of a smooth function. Evaluation of the integral in yields the following contribution to the variation of
We note that this contribution is generally nonzero and does not depend on phases.
Let us turn to the evaluation of the topological charge that is proven to be very useful in the field Wang et al. (2010b). The value of the topological charge is defined in a usual way with the divergence of the topological field
To compute the topological charge we need to consider a special variation of the S-matrix that just corresponds to the stationary phase derivative . Since the expression under the trace in does not depend on phases, the topological charge vanishes at any point where the field is well-defined, or alternatively is finite. The Weyl singularities give rise to the point-like integer charges being the sources of the field . We consider this in detail in Sec. VI. This situation is in complete analogy with that of the standard Berry curvature of a discrete spectrum where Weyl singularities correspond to band crossings. However, we have computed the topological charge for the particular phase-dependence of the S-matrix on phases (). We have not considered the topological charge in the space of 2 phases and some other parameter characterizing the scattering matrix, this charge could be nonzero and have a continuous distribution. The investigation of the general parametric dependence of the S-matrix is beyond the scope of the present article.
We separate the field into three parts: a part produced by the point-like charges, divergenceless field that is zero in average, and a constant part . The value of the integral
is given by the flux of the topological field through the corresponding surface. This flux reduces to the integer for the first contribution to , vanishes for the second divergenceless contribution and may result in some value for the constant part of the field. We stress that the last contribution being present is the main distinction from the common case. The value of this constant field is then given by the integration of the variation :
This constant field can contribute to the flux through any plane in the phase space.
where is the normal vector to this plane. As it has been shown in Ref. Riwar et al. (2016) the value of is directly related to the observable transconductance between the leads and . Therefore, in contrast to the conclusions of Ref. Riwar et al. (2016) the value of transconductance does not always quantize although the change of transconductance with a phase can be quantized.
So, in principle a nonzero non-topological contribution to can be present. This contribution is nonzero if the S-matrix is not regularized at infinite energy such that . If the S-matrix is regularized in this way, then the matrix is defined on a compact space of parameters , so the first contribution to Eq. would reduce to an integer (with proper normalization). If it is not regularized this way, then this boundary term leads to the presence of a non-topological contribution to the response function, that comes due to the presence of a continuous spectrum and, formally, from the fact that the matrix is not defined on a compact space, as discussed above. In the limit of energy-independent S-matrix, this contribution reduces to the antisymmetric part of the Landauer conductanceNazarov and Blanter (2009); Landauer (1957). In this case, if the bare S-matrix (without the stationary phases of terminals ascribed) is non-symmetric (which means the breaking the time-reversibility condition) we obtain a nonzero value of . If the S-matrix is time-reversible, the non-topological contribution is zero and the integer quantization of transconductance is restored.
V Weak energy dependence of the matrix
In the description of the realistic nanostructure a reasonable approximation is to consider the S-matrix to be constant on the scale of . It corresponds to the case of a short nanostructure (smaller than the superconducting coherence length). So a logical approximation would be to describe the nanostructure with a constant S-matrix at all energies. The response function is given by an integral over energy in Eq. . Would this integral accumulate in the region , then the approximation of a constant S-matrix at all energies would be accurate. However, there can be a significant contribution from the energy scales to the integral yielding . In this case the energy dependence of the S-matrix at the large energies becomes important. To investigate this we consider the contributions from the small scales and from the large scales in the Subsections V.1 and V.2 respectively.
v.1 Energy-independent matrix:
In this Subsection we analyze the small-scale () contribution to . For this we approximate the S-matrix to be constant at all energies and extend the integration limits to infinity. The second term in vanishes since . The integral in the first term in converges on the scale . This statement only necessarily holds if the S-matrix is energy-independent. Otherwise, the contribution from the larger scales can be present and we investigate it in V.2. Similarly to , the result of integration under consideration can be expressed in terms of the eigenvalues and eigenvectors of the unitary matrix . We use the same notations and for the eigenvectors related to the complex conjugated eigenvalues pair and correspondingly as described after Eq.. We remind that the phase of the eigenvalue with is related to the energy of ABS as . We also remind that is degenerate and corresponds to the energy of one of the ABS . Upon crossing this point in phase space, this ABS state exchanges the wave function with its Nambu counterpart with the eigenvalue . Due to this we call such points gap touching singularities.
Evaluating the integral yields
where label the eigenvalues of , and the summation goes over indices with both signs. If the number of channels is odd, there is an eigenvector of corresponding precisely to the eigenvalue . Then the index corresponds to this state. If the number of channels is even, the indices in Eq. do not take the zero value. In the following we consider the number of channels to be even. The logarithm here has a branch cut along the real axis as (see Fig. 3) to avoid the gap touching singularity ambiguity . Let us consider the behaviour of in the vicinity of the gap touching singularity. Since the wave function corresponding to is discontinuous upon crossing this singularity, it is not obvious that is continuous. However, one can observe that the first term is a sum of Berry curvatures of individual levels multiplied by the eigenvalue-dependent prefactors . This prefactors vanish for the discontinuous wavefunctions at the gap touching degeneracy and guarantee the continuity of the first term. Also, one can show that the second term in Eq. is continuous. Consequently, is continuous at this point (see Fig. 4). The only possibility for to be ill-defined at some points in phase space is the zero-energy Weyl singularity where diverges (see Sec.VI).
The response function is expressed in terms of eigenvalues and eigenvectors of the matrix . So is the ABS contribution to the ground state Berry curvature, which was conjectured as a result for (see Sec. IV). It was shownRiwar et al. (2016) that this ABS contribution is given by , . Since one of the wavefunctions contributing to this sum is discontinuous at the gap touching singularity, we conclude that is discontinuous contrary to . One can understand the difference between and by considering the computation of the integral in the first term in Eq. by means of complex analysis (in the plane of complex ). By shifting the integration contour to the upper half-plane, one can see that the integral is contributed to by the poles, corresponding to ABS and the cut above the gap (see Fig. 2). The contribution from the poles results in , but the contribution from the cut, , is equally important (see Fig. 4).
For the integrated we obtain in accordance with Eq.
so the value of transconductance is not necessarily quantized in the approximation of the energy-independent S-matrix.
v.2 Contribution from the large scales
In the previous Section we have shown that the non-topological contribution to the transconductance comes from the boundary terms at (see Eq.). This means that, contrary to intuition, there is an essential contribution to coming from the energy scales much larger than the energy gap. In order to investigate the large energy contribution we assume the regularization of the S-matrix at large energies. So, in this Subsection we consider for a particular energy-dependence of the S-matrix. It is chosen such that the S-matrix is regularized at infinity such that it varies slowly on the scale of a superconducting gap and . This S-matrix corresponds to a complete isolation of the terminals at the largest energies. With this regularization, the matrix is defined on a compact parameter space and the first contribution in must reduce to an integer. Due to the scale separation, there are two contributions to . One comes from the scales and is given by the same result . Another one comes from the scales .
For negative energies, the large scale contribution with asymptotic accuracy equals
with the notation .
For positive ones:
So, the both contributions give the following addition to the response function
Both terms here do not depend on phases. The first one is exactly equal to the constant part of the topological field defined previously with an opposite sign (computed for an energy-independent S-matrix case). So after integration over two phases, it cancels the non-topological contribution from small scales in . Since we assume a regularization , the second term is zero (), so the total mean value of the transconductance is quantized in correspondence with the theory of characteristic classes.
The second contribution to in Eq. contains the energy-derivative of the S-matrix under the integral. Due to this the energy scale of its dependence drops out from the integral. So, one may expect that it contributes to the large scale contribution to . However, with asymptotic accuracy it vanishes in the limit when the S-matrix varies slowly on the scale . Indeed, in the limit
In this limit for , the integrand equals
with asymptotic accuracy, since the expression under the trace does not depend on phases. For the integrand vanishes since for .
Vi The vicinity of a Weyl point
In this Section, we investigate the Berry curvature in the vicinity of a Weyl singularity, that occurs at some point in the 3-dimensional phase space. Such Weyl points have been analyzed in Riwar et al. (2016) assuming spin symmetry, in Yokoyama and Nazarov (2015) the analysis has been extended to cover weak spin-orbit interaction. Without spin-orbit coupling, the Weyl points are situated at zero energy and diverges near the point. A conical spectrum of ABS is found in the vicinity of the point Riwar et al. (2016). A weak spin-orbit coupling splits the energy cones in spin and shifts the Weyl point to a finite energy Yokoyama and Nazarov (2015). Further, we discuss separately the cases of vanishing and weak spin-orbit coupling.
vi.1 Vanishing spin-orbit coupling
When the spin-orbit (SO) coupling is absent, the Weyl singularities are located at some points in the phase space and occur at zero energy . To consider the vicinity of the singularity, we assume a small phase deviation from the singularity point and assign it to each channel via the diagonal matrix . In the vicinity, defined by Eq. only has non-zero contributions from the first term of quasi-WZW term. The second term vanishes asymptotically when the energy approaches zero, as shown in Eq. 34. Conform to these approximations, we extend the domain of the integration over the phases to infinity since is concentrated near the singularity point.
To compute , we approximate the matrix near the Weyl point with the expression that keeps the first orders in and of the variation: , being the scattering matrix in the singularity point at . Conveniently, we can replace with in Eq.(21). We find the variation by expanding the S-matrix in :
We can contract the dimension of projecting it to two eigenvectors of that achieve singular values at the Weyl point. Following Riwar et al. (2016), we separate the singular part of and write in the basis of ABS eigenvectors and satisfying , :
where are the Pauli matrices in the space of these two eigenvectors, and the components of are proportional to the components of : , .
The form of is similar to the generic form of Green’s function of a two-level system. We expect that the two poles of should be positioned symmetrically on the imaginary axis due to BdG particle-hole symmetry. Indeed, we find these poles at . Using the trace relations of Pauli matrices, we reduce in the leading order to the Berry curvature of the corresponding levels :
We note that in this section all the matrices have the spin index. For an dimensional space of superconducting phases, the singularities are concentrated in the dimensions and the relevant space is reduced to a -dimensional subspace . For certainty, we set the indices , and consider the curvature defined in the plane at a fixed phase .
The dependence of the integral of the curvature with respect to superconducting phases , witnesses the change of first Chern number when the integration plane passes the singularity point. Since we only concentrate on the vicinity of the Weyl singularity, the integral under the approximations made can only indicate the change of the Chern number, rather than its total value that can be determined by integration over the regions far from the singularity point. To compute the integrated , we notice from Eq.(39) that the energy spectrum is linear in , and introduce a linear relation with being a real invertible matrix. The integrated is then obtained as:
determining the orientation of the deviation.
This implies that whenever the integration plane passes the Weyl point, the first Chern number is changed by . This manifest the the integer values of the topological charge. The integrated in Eq.(41) specifies the flux of the Berry field penetrating the plane which is either above or below the singularity point. This flux, owing to symmetry, is a half of the total flux, this explains the half-integer values. Therefore, the main contribution to Eq. in the vicinity the Weyl point is given by the Berry curvatures of the two levels that are close to zero energy, and can be presented as
vi.2 Weak Spin-Orbit Coupling
Let us turn on a weak spin-orbit interaction and take it into account perturbatively giving a small spin-dependent change to the scattering matrix that preserves its unitarity, as is done in Yokoyama and Nazarov (2015). The first order variation thus reads
where the last equality sign implies the commutation relation . Here, are the Pauli matrices in spin space and being the corresponding Hermitian matrix in the channel space characterizing the spin-orbit effects. Owing to the time reversibility, , yet in the vicinity of the singularity we may disregard its dependence on superconducting phases.
As in the previous Subsection, we project the matrix onto singular subspace that has now dimension of to account for spin, and replace it with the matrix . Writing the latter in the basis of eigenvectors :
. We can conveniently choose the spin quantization axis in the direction of replacing the operator with its eigenvalues for spin up and down, respectively.
The spin-orbit coupling lifts the spin degeneracy of the ABS in the vicinity of a Weyl point. The poles at imaginary energies become for spin up and for spin down. Contrary to the spin-degenerate case, the singularities at are no longer at zero energy. Instead, they are shifted to , see Fig. 7. The conical singularity of the spectrum remains and the topology is still protected, as we will explain below in detail.
The ABS energies cross zero energy when
is satisfied. (Here, we introduce a positively defined matrix . Eq.(46) defines an ellipsoidal surface in the 3D superconducting phase space that encloses the singularity at where . Outside the ellipsoid, two positive imaginary poles at