Transport properties of a Weyl superconductor in a vortex lattice

(1)

Transport properties of a Weyl

superconductor in a vortex lattice

Thesis

submitted in partial fulfillment of the requirements for the degree of

Master of Science in

Physics

Author : Gal Lemut

Student ID : s2108364

Supervisor : Prof. dr. C. W. J. Beenakker

2nd corrector : Dr. V. Cheianov

(2)

(3)

Transport properties of a Weyl

superconductor in a vortex lattice

Gal Lemut

Instituut-Lorentz for Theoretical Physics P.O. Box 9506, 2300 RA Leiden, The Netherlands

July 16, 2019

Abstract

Building on the discovery that a Weyl superconductor in a magnetic field supports chiral Landau level motion along the vortex lines, we investigate its transport properties out of equilibrium. We show that the vortex lattice

carries an electric current I = 1

2(Q 2

eff/h)(Φ/Φ0)V between two normal metal

contacts at voltage difference V , with Φ the magnetic flux through the

sys-tem, Φ0 the superconducting flux quantum, and Qeff < e the renormalized

charge of the Weyl fermions in the superconducting Landau level. Because the charge renormalization is energy dependent, a nonzero thermoelectric coefficient appears even in the absence of energy-dependent scattering pro-cesses.

(4)

(5)

Chapter

1

Introduction

1.1 Preface

In condensed matter physics we are often given a task to describe and pre-dict the behaviour of electrons in various forms of matter. Although this seems like a straightforward task in terms of a microscopic theory, the sheer variety of possible crystalline structures and symmetries gives a rise to a whole zoo of possible effective Hamiltonians. This can cause the generally well defined particles such as electrons to exhibit exotic behaviour that can strongly influence the properties of different materials. In some occasions electrons can even appear as massless, in which case they seem to behave as light particles/photons. Drawing inspiration from the particle physics we call the three-dimensional realisation of these particles Weyl fermions.

It has been theoretically predicted [1] that one can find such crystalline ma-terials called Weyl semimetal, where the effective behaviour of the electrons can in fact be described by such massless Weyl fermions. In combination with an external magnetic field this systems exhibit linearly dispersed chiral Landau levels that allow for directed transport channels. Such theoretical predictions have motivated many experiments that managed to demonstrate the existence of Weyl fermions in different proposed materials such as TaAs, TaP, etc. [2–7] If one would manage to construct a superconducting realiza-tion of such systems, these transport channels would then become localized in the Abrikosov vortex lines, through which the magnetic field penetrates the superconductor.

(8)

Figure 1.1: Experimental observations of the Weyl cones in TaAs. ARPES two

dimensional dispersion-relation measurements for different scans of kx, ky, kz

mo-menta compared to the theoretical predictions. Plot adapted from Ref. [2]

These mixed-charge channels of a Weyl superconductor will be the main topic of this master thesis, where we will focus on their thermal and electrical transport signatures. In the first chapter we will introduce the theoretical concepts of a Weyl semimetal, its superconducting counterpart and their cor-responding chiral Landau levels. The second chapter will then describe our

main work, published in Physical Review B.1 _{In this work we have}

analyti-cally calculated the transport properties using the scattering formalism and compared them with a numerically realised tight-binding. Finally we will conclude the thesis with a short summary of our findings.

1_{G. Lemut, M. J. Pacholski, İ. Adagideli and C. W. J. Beenakker, Effect of charge} renormalization on the electric and thermoelectric transport along the vortex lattice of a Weyl superconductor, Phys. Rev. B 100, 035417 (2019 ).

(9)

1.2 Weyl semimetal

1.2.1 Metals, insulators and semimetals

As a starting point it is important to introduce the Weyl semimetal. As all periodic crystalline structures which host electrons, it can be described by the band theory. In this approach we are calculating the electronic bands of our system, which are filled to a certain energy called the Fermi energy. Looking at the band structure we traditionally divide our crystalline mate-rials into two groups. If the Fermi energy lies within a band, our material will have states available at the Fermi level. This allows for charge-carrying excitations of arbitrarily small energies, causing these materials to conduct electric current. We call this band the conduction band and the materials with these properties are called metals. If on the other hand the Fermi level lies in a gap between two bands, below the conductance band and above the lower valence band, there will be no available states at the Fermi level making our material an insulator. There is however a third possibility when the conductance and the valence band overlap by a small amount. If then our Fermi level lies within this overlap our system will be gapless with a neg-ligible density of states at the Fermi level. This type of materials are called semimetals; see Fig. 1.2. This thesis will focus on a specific example of this class called the Weyl semimetals.

Density of states

Ener

gy

Metal Semimetal Insulator/ Semiconductor Conduction band Valence band Conduction band Conduction band Valence band

Figure 1.2: Illustration ofensity of states for metals, semimetals,

insula-tors/semiconductors (from left to right) . The dotted line represents the Fermi Energy of the systems.

(10)

1.2.2 From the Dirac equation to a Weyl fermion

Let’s start by addressing the question: “What is a Weyl fermion?”. If we want to find out the answer we need to turn to particle physics and have a look at the Dirac equation [8]:

i~γµ∂µΨ = mcΨ , (1.1)

which can be rewritten more specifically in the Weyl representation as:

0 i~∂t+ i~cσ · ∇

i~∂t− i~cσ · ∇ 0

Ψ = mc2₁_{Ψ ,} _(1.2)

where σ represents the vector of the two dimensional Pauli matrices and Ψ

is a four component spinor. In the massless (e.i. m = 0) regime we can split

this equation in the eigen-spaces of the chiral operatorγ5 = diag(−12, 12) for

two-dimensional spinorsΨ = (ΨL, ΨR)T whereL and R indicate the left and

right-handed eigenspaces. We can now write the two independent equations as:

i~∂tΨR,L= ∓i~cσ · ∇ΨR,L. (1.3)

This chiral solutions were first proposed by a German physicist and math-ematician Herman Weyl and can be written in the form of the Schrödinger equations with chiral Hamiltonians:

HR,L = ±vk · σ . (1.4)

This is now a Hamiltonian of a so-called Weyl fermion where k_{= −i~∇ is the}

momentum operator andv is the group velocity. It is a gapless Hamiltonian

with a linear dispersion EWeyl = ±v|k|, as can be seen in Fig. 1.3, and a

vanishing density of states at the Weyl point (kx = ky = kz = 0), making it a

semimetal. Scientists have long believed that neutrinos could be a candidate for such Weyl fermions until they were shown to not be completely massless. Today we actually know that none of the fundamental particles is a Weyl fermion, however it turns out that it’s Hamiltonian can effectively describe excitations of a class of materials in condensed matter [2–7].

(11)

Figure 1.3: Dispersion relation of a Weyl Hamiltonian of a set chirality described

by Eq. (1.4) with a unitless velocity set to v = 1. The thick line describes the

kx = ky = 0 case with a gap closing Weyl point at kx = ky = kz = 0 while the thin

lines represent subbands for different values ofkx andky.

1.2.3 Topological protection

The previously derived Weyl Hamiltonian can be thought of as a three di-mensional realisation of the massless two didi-mensional Dirac Hamiltonian, but despite their apparent similarities, the Weyl Hamiltonian features some quite unique properties. As we will see these features arise from the topolog-ical protection of the Weyl point. [15] To get a simple understanding of this protection we have to first turn back and investigate the conventional Dirac Hamiltonian:

HDirac = vxσxkx+ vyσyky, (1.5)

with a linear dispersion relation: EDirac = ±

q

v2

xk2x+ v2yky2. (1.6)

This Hamiltonian also describes a semimetal with a Dirac gap closing point

at kx = ky = 0. We can now look at the stability of the Dirac point under

generic small perturbations of the form P

i=x,y,zfi(k)σi. Around the Dirac

point we will be able to rewrite the perturbed Hamiltonian as:

e

HDirac = vxσxkx+ vyσyky+

X

i=x,y,z

(12)

with a perturbed dispersion relation: EDirac0 = ±

q

v2

x(kx+ mx)2 + v2y(ky + my)2+ m2z, (1.8)

Where we have now relabelled fi(k = 0) = mi for simplicity. It is now clear

that while mx, my perturbation terms will just shift the Dirac cone, from

k = (0, 0) to k = (−mx, −my), any perturbation including mz will destroy

the Dirac point and open a gap as shown in Fig. 1.4.

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 ky(a.u.) E (a .u .) (a) -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 ky(a.u.) E (a .u .) (b)

Figure 1.4: a) Gapples Dirac dispersion relation described by Eq. (1.7) forkx = 0.

Thick line represents thekx= 0 subband while other subbands are represented by

thin lines. b) The dispersion relation of the the same Dirac Hamiltonian with an

additional magnetization termmzσz that causes a gap opening.

We can now look back at the Weyl Hamiltonian for a chosen chirality:

HWeyl= vxσxkx+ vyσyky + vzσzkz, (1.9)

with a generalized three dimensional linear dispersion relation:

EWeyl = ±

q

v2

xk2x+ vy2ky2+ v2zkz2. (1.10)

If we now repeat the small perturbationsP

i=x,y,zfi(k)σianalysis and expand

it around the Weyl point we will get the perturbed dispersion relation: e

EWeyl = ±

q

v2

x(kx+ mx)2+ v2y(ky + my)2+ vz2(kz+ mz)2. (1.11)

It is now clear that, while all the possible small perturbations still shift

the Weyl point from k = (0, 0, 0) to k = (−mx, −my, −mz), they cannot

introduce a gap; see Fig. 1.5. This means that any small perturbations will not destroy the Weyl point and will keep the Hamiltonian gapless which is a

(13)

fundamental difference from the previously discussed two-dimensional Dirac Hamiltonian. We say that the Weyl cone is topologically protected. In fact, as we will see later, we can properly define a conserved topological invariant which protects the Weyl cone from opening.

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 k_z_(a.u.) E (a .u .) (a) -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 k_z_(a.u.) E (a .u .) (b)

Figure 1.5: Same as Fig. 1.4, but now for a three-dimensional Weyl Hamiltonian,

for which the term mzσz only shifts the cone instead of creating a gap opening.

Thick line represents the kx = ky = 0 subband while other subbands are

repre-sented by thin lines.

1.2.4 Weyl fermions in crystalline structures

We have so far focused on a infinite system Hamiltonian which can be thought of as an effective Hamiltonian around some Weyl point in a crystal model. In fact any generic two-band three-dimensional Hamiltonian expanded around a crossing of two bands has the form:

Heff =

X

α=x,y,z

vαδkασα, (1.12)

which is now the anisotropic generalization of the Weyl Hamiltonian with a set chirality. It actually turns out that in a condensed matter system a single Weyl fermion is not enough. We will show that they always need to come in pairs of opposite chiralities. To understand this we will first need to look at the Landau-levels in our system. These are the quantum analogues of the cyclotron orbits caused by an applied magnetic field. For calculation simplicity we can look at the Hamiltonian of right-handed isotropic Weyl

fermion and set the magnetic field in the z-direction B = ˆzBz, without the

loss of generality:

H = vhσzkz+ σx(kx− eAx) + σy(ky − eAy)

i

(14)

We can now define the operators A, A† _{using the momentum and vector} potential operators: A = r 1 2e~Bz (kx− eAx+ iky + ieAy) (1.14) , A†= r 1 2e~Bz (kx− eAx− iky− ieAy) , (1.15)

and rewrite the Hamiltonian as: H = vkz v √ 2e~BzA v√2e~BzA† −vkz . (1.16)

It can be easily shown that A, A† = 1, which in turn allows us to use the

harmonic-oscillator algebra. If we want to inspect the transport properties of our system we are interested at the states close to the zero energy. This

motivates us to look for the zero-energy solutions HΨ = 0. We can now

square the Hamiltonian and equivalently search for the solutions ofH2_{Ψ = 0.}

H2 _{= v}2_k2 zσ0+ 2ev2~Bz AA† ₀ 0 A†_A = v2_k2 zσ0+ 2ev2~BzN + 1 0₀ _N . (1.17) In accord with the harmonic oscillator-algebra we know that the we can find eigenstates of N as |ni with the lowest eigenstate N |0i = 0 |0i. Using this

we can find the zero-energy state withkz = 0 and a spinor structure:

N + 1 0 0 N 0 |0i = 0 0 |0i , (1.18)

This state is also an eigenstate of the Hamiltonian for finite values ofkz with

a linear dispersion in thez direction:

H 0 |0i = vkz 0_|0i . (1.19)

and a Landau degeneracy proportional to the magnetic fluxN = LxLyeBz/2π

for the other two directions. This degenerate chiral mode is called the zeroth Landau level. If we repeat the calculation for the left-handed Weyl fermion we will find a the same dispersion relation with an opposite sign. We can

(15)

1.2.5 Chiral anomaly

Although this linear-dispersion zeroth Landau level may seem completely normal we quickly run into an anomalous effect when we apply an external

electric field E. Let us consider the case where E k B = ˆzBz. The electrons

feel the Coloumb force caused by this electric field changing the momenta as

∂tk= eE . (1.20)

Since we are looking at a Landau level with a fixed chirality it has either left or right movers. Because of this the applied field shifts all the states towards higher momenta. This would mean that the applied electric field

would change the particle number as ∂thˆniR,L ∝ ±E · B which can be seen

in Fig. 1.6.

E

k

(a)

E

k

∂tk ∝ E (b)

Figure 1.6: a) Dispersion relation of a single chiral mode with states filled up to the Fermi energy. b) The same dispersion relation with an additional external electric field in the direction parallel to the velocity of the chiral mode, causing a

shift in momentum∂tk ∝ E changing the number of particles in the system.

This quantum phenomenon is called chiral anomaly. In particle physics this anomaly is a phenomenon of its own however for a solid-state system we know that new particles cannot be created by an applied electric field. This contradiction can easily be resolved if we have an additional cone of the opposite chirality, for which the electric field would seem to annihilate particles. Now the applied field simply converts the left-movers to right-moving particles; see Fig. 1.7, creating a current in the direction of the electric field. This resolution can also be described by the Nielsen-Ninomya [14] theorem which states that Weyl fermions have to always come in pairs of opposite chiralities (left, and right-handed).

(16)

E

k

(a)

E

k

∂tk ∝E (b)

Figure 1.7: Same as Fig. 1.6, but now for 2 effective chiral modes of opposite chiralities (R,L) for a periodic crystalline invariant system. Both chiral mode now

gets a momentum shift∂tk ∝ E while the left going particles get transferred into

the effectively right-chiral mode.

This can in fact be understood in an even simpler way. In a condensed matter we know that the energy dispersion relation has to be periodic. It is then easy to see that a single chiral Landau level cannot satisfy this condition requiring us to have pairs of Weyl cones of opposite chiralities.

1.2.6 Berry curvature

Most of the properties of the Weyl fermions can also be understood from the topological point of view. For this we will need to define an appropriate topological invariant. To understand this let’s first take a look at a generic

Weyl semi-metal as a n-band system. We know that the electrons in its nth

band are described by the eigenvectors |ki_n of the Weyl Hamiltonian:

HR,L= ±vfk · σ. (1.21)

We can now calculate its Berry connection A and curvature [9] F according to the:

A_n(k) = i hk|_n∇k|ki_n , (1.22)

Fn(k) = ∇k× An(k) , (1.23)

where we use ∇k to denote (∂kx, ∂ky, ∂kz). Using this definition it is now

possible to calculate the Berry curvature for our Weyl Hamiltonian which turns out to be:

FR,L(k) = ±

1 4π

k

(17)

with a corresponding divergence:

∇k· FR,L = ±δ(k) . (1.25)

From this expression we can recognize that each Weyl node represents either a monopole source or an anti monopole sink of the Berry curvature. We can now calculate the total field flux by integrating the Berry curvature over a Fermi sheet that surrounds the monopole:

CL,R=

I

FR,L· dS = ±1 . (1.26)

This integral of the Berry curvature gives us the charge of our monopole. By looking at the Berry curvature we can now see that a periodic Brillouin zone cannot support a single source of the Berry curvature, since it’s field lines need to be closed just as the usual magnetic lines. This is just another way of saying that in order to have closed Berry curvature lines, we need to have two Weyl cones of opposite chiralities, one as a source and the other as a sink of the Berry curvature field; see Fig. 1.8.

Figure 1.8: Berry curvature described by the Eq. (1.24) of a pair of Weyl fermions of opposite chiralities where one represents a monopole source and the other a sink of the Berry curvature.

It turns out that the charge of the monopole of the Berry monopole also defines a topological invariant of the two-dimensional Fermi surface manifold.

(18)

From Eq. (1.26) we can see that the Chern number of a right or left-handed

Weyl fermion is either 1 or −1 respectively. Since the topological invariant

cannot be changed with continuous transformations the Weyl point will be topologically protected for any small perturbation, and can only be opened when two Weyl cones of opposite chiralites, and thus opposite monopole charges, annihilate each other.

1.3 Weyl superconductor

1.3.1 Bogoliubov-de Gennes formalism

Now that we have introduced the Weyl semimetal we can think about the ad-dition of superconductivity [11]. In this work we will use the Bogoliubov-de Gennes [10] formalism to describe the elementary excitations of the super-conductor as a mixture of electrons and holes.

The use of this formalism is necessary because the charge inside a super-conductor is no longer a conserved quantity. For convenience we can now double the degrees of freedom and separate them into an electron and hole subsectors, and rewrite the Hamiltonian in the following form:

H = 1 2Ψ †_H BdGΨ, Ψ =           a1 .. . aN T    a†1 .. . a†_N              , (1.27)

where we have definedΨ as a vector of N annihilation and creation operators

and T is the time-reversal symmetry operator. The newly defined HBdG

is called the Bogoliubov-de Gennes Hamiltonian and it will be the central object of our further analysis. It can be written in a block structure where the diagonal blocks represent what we now call electron and hole Hamiltonians and the off-diagonal blocks represent the coupling between the two subsectors that arises from superconducting effects. A generic BdG Hamiltonian can

(19)

now be written in the block form as:

HBdG= H0 ∆ ∆∗ _{−T H} 0T−1, (1.28)

where ∆ represents the superconducting coupling. As a consequence of the

doubling of the degrees of freedom, the newly defined HBdGexhibits

particle-hole symmetry C = iνyτ , where ν is used to denote Pauli matrices acting in

the electron-hole subspace. This newly defined symmetry anticommutes with

the BdG Hamiltonian: CHBdGC−1 = −HBdG. By employing the particle-hole

symmetry it is easy to see that each eigenvector vi of the BdG Hamiltonian

will have a corresponding partner Cvi, which is also an eigenstate, with the

opposite eigenvalue:

HBdGvi = Eivi

CHBdGC−1Cvi = EiCvi

HBdGCvi = −EiCvi. (1.29)

In order to find the elementary excitations of our superconductor we will now diagonalize the BdG Hamiltonian with a unitary transformation U :

H = 1

2Ψ

†_U†_{U H}

BdGU†U Ψ . (1.30)

We can now introduce new operatorsbidefined as UΨ = (b1, . . . , b2N)T, which

arise from the diagonalization of the BdG Hamiltonian. It now seems that

we have constructed2N independent operators bi from the starting N

oper-ators ai. To resolve this we have to remember the newly arisen particle-hole

symmetry of the BdG Hamiltonian, which implies that the BdG eigenvectors

vi have a particle-hole partnervj = Cvi, j 6= i; see Eq. (1.29). Using this fact

we can find the relation:

bi = vi†Ψ = (v

† j)

∗

CΨ = vT_j(Ψ†)T = Ψ†vj = b†j (1.31)

This relation tells us that we can identify half of the excitations bi with

their particle-hole partners with the opposite eigenvalues and rewrite UΨ =

(b1, . . . , b2N)T = (b1, . . . , bN, b†1. . . b

†

N)T. We can now see that in the end we

still get N distinguishable operators since we have just artificially doubled

the degrees of freedom. Since U is a unitary transformation the newly defined

operators bi still satisfy the fermionic anticommutation relations: {bi, b†j} =

{ai, a

†

j} = δi,j, while their commutator with the Hamiltonian can be written

as: h

(20)

bi represent elementary excitations of our total Hamiltonian which can now be rewritten as: U HBdGU†=          E1 0 0 0 0 0 0 . .. 0 0 0 0 0 0 EN 0 0 0 0 0 0 −E1 0 0 0 0 0 0 . .. 0 0 0 0 0 0 −EN          , (1.32) H = 1 2Ψ †_U†_{U H} BdGU†U Ψ = N X i=1 Eib†ibi+ const . (1.33)

These new excitationsbi are called the Bogoliobov quasiparticles which can

be written as a linear combination of the annihilation and creation (particle and hole) operators:

bi = vi†Ψ = N X j=1 h (vi)∗jaj + (vi)∗j+Na † j i (1.34)

These excitations are now no longer eigenstates of the charge operator and come in pairs with opposite mixed charges. As a consequence of this particle-hole symmetry doubling, the BdG Hamiltonian now has a certain redundancy in the spectrum where creating a quasiparticle with momentum k and energy E is directly equivalent to annihilating its particle-hole counterpart with momentum −k at the energy −E.

Theoretically this formalism can be used to describe any system with an additional superconductor pairing, but in reality a Weyl superconductor has not yet been discovered. We can still construct an effective Weyl super-conductor by proximitizing our Weyl semimetal with a supersuper-conductor, so that the cooper pairs are allowed to tunnel into our material, creating an effective superconducting coupling. Since the Weyl semimetal is a three di-mensional material, we cannot proximitize it directly but we can imagine a layered structure, shown in Fig. 1.9, with interchanging Weyl semimetal and superconductor layers [17].

(21)

Figure 1.9: Layered structure of the proposed Weyl superconductor with lattice

constant a0 [17] and a perpendicular magnetization β in the z direction. The

applied magnetic fieldB0 enters the superconductor in terms of vortices that form

a lattice, with a lattice constant d0, featuring two vortices of unit fluxφ0 = h/2e

per unit cell. Plot adapted from Ref. [17]

This kind of structure will give us an effective Weyl superconductor with the following BdG Hamiltonian:

HBdG = HWSM(k) ∆ ∆∗ _{−T H} WSM(k)T−1 . (1.35)

For concreteness we will from now use the specific realization of the Weyl semimetal Hamiltonian featuring two Weyl fermions of opposite chiralities:

HWSM(k) = vfτz(k · σ) + τ0(βσz − µσ0) . (1.36)

In this notation we will use σ, τ to denote Pauli matrices, where σ acts on

the spin sector, andτ acts in the subspace that separates the two Weyl cones

of opposite chiralities. For generality we have added a magnetization termβ

which separates the two cones in momentum space and a chemical potential

term µ. The energy spectrum of this realization of the Weyl superconductor

(22)

-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 k_z(a.u.) E (a .u .) (a) -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 k_z(a.u.) E (a .u .) (b)

Figure 1.10: a) Dispersion relation of two Weyl fermions of opposite chiralities in an infinite system. b) The dispersion relation of the same system with an additional superconducting coupling described by the BdG Hamiltonian from Eq. (1.35).Thick

line represents thekx = ky = 0 subband while other subbands are represented by

thin lines.

1.3.2 Magnetic field in a superconductor

It is now possible to once again explore the effects of an additional magnetic field. It turns out this is now no longer a straight forward task since our system is now a superconductor, which generally expels the magnetic field. The only way it can enter our superconductor, is in the form of Abrikosov

vortices, each carrying one magnetic flux quantum Φ0 = h/2e . The

ap-pearance of these vortices usually destroys the dispersionless structure of the Landau levels. It was shown [17] that contrary to the usual example the Weyl superconductor does retain the zeroth Landau levels when we introduce the magnetic field because of the index theorem. This specific feature motivated our research, where we investigated the transport signatures of these Weyl superconductor Landau levels.

The superconducting pairing can be separated into the absolute value and

the phase as: ∆ = ∆0eiφ. The magnetic field in the form of the Abrikosov

vortices can now be introduced to the BdG Hamiltonian as a space dependant phase satisfying the relation:

∇ × ∇φ = 2π B0

|B0|

X

n

(23)

and the continuity equation:

∇ · ∇φ = 0 , (1.38)

where xnrepresents the positions of the vortices and _|BB0₀_| describes the

direc-tion of the external magnetic field; see Fig. 1.9. By the use of the Anderson

transformationV , we can absorb the space dependant superconducting order

parameter into the gauge potential:

HBdG = V†HWSM (k − eA) ∆ ∆∗ T HWSM(k − eA)T−1 V (1.39) =HWSM(k + a + mvs) ∆0 ∆0 T HWSM(k + a + mvs)T−1 , (1.40) V =e iφ ₀ 0 1 , (1.41)

where we have now defined the vector fields a, vs that now wind around the

vortex cores:

a= 1

2∇φ, mvs=

1

2 ∇φ − eA . (1.42)

1.3.3 Weyl superconductor Landau Levels

To find the Landau levels we can now turn back to the previously

de-fined Weyl Hamiltonian, see Eq. (1.36), with a superconductings-wave BdG

Hamiltonian: HBdG(k) =HWSM (k + mvs+ a) ∆0 ∆0 −σyHWSM∗ (−k + mvs− a)σy . (1.43) We can now block-diagonalize this Hamiltonian by preforming the

appro-priate unitary transformation: U = exp i1

2θνyτzσz. Where tan θ =

∆ vfkz,

θ ∈ (0, π) and ν again represents Pauli matrices acting in the electron-hole

sector . If we then project into ν = sign(β)τ = τ = ±1 subspace we can

obtain the effective low-energy Hamiltonian:

H±= vf(−i∇⊥+ a ± κmvs) · σ⊥+ (β − mkz)σz∓ κµσ0. (1.44)

We use ⊥ to denote x, y components and define mkz =

q

∆2_{+ v}2

fkz2, κ =

(24)

Landau levels. We can quickly see that the Hamiltonian now anticommutes

with σz which gives us a nice way of calculating the Landau levels.

We can again define an operator A± = −i∂x+ax−∂y−iay±κ(mvsx−imvsy)

and its corresponding adjoint operator A†±. We can now separate the diagonal

and off-diagonal parts of our Hamiltonian and define f (kz) = β − mkz:

H±= f (kz)σz+ vf 0 A†± A± 0 . (1.45)

We can again square the Hamiltonian and obtain:

H2 _{= f (k} z)2σ0+ vf2 A†_±A± 0 0 A±A†± . (1.46)

Since the second term is now a square of a Hermitian operator its eigenvalues have to be larger or equal to zero. Because of this we are again looking for

the zeroth Landau level states of the form |Ψi = u_v where A±u = 0 or

A†±v = 0. We will be focusing on the positive part of the kz spectrum and

τ = +1 projection. In that case as it was shown in [17] A†+v = 0 has no

normalizable solutions and A+u = 0 has NLandau solutions. Using this fact

we can now find the zero energy eigenstate atf (kz) = 0:

Hu

0

= 0 (1.47)

Since this state remains the eigenstate of the Hamiltonian also for finite values

of f (kz) we get a chiral band with a dispersion relation E(kz) = f (kz). We

can now write the Hamiltonian of the zeroth Landau level as:

HL = f (kz) = (mkz − β) =

q

∆2 _{+ v}2

fkz2− β . (1.48)

We can now expand this expression to the first order around the Weyl point.

Here we choose the point at positive momentum kz = K = _v1

fpβ

2_{− ∆}2 _to

get the energy dispersion:

E(kz) ≈ vfκ0kz− vfκ0K . (1.49)

We have now obtained the linearized dispersion relation of the Landau level

with positive velocity vz = _∂k∂E_z and momentum. Since our system has a

particle-hole symmetry we know that this chiral Landau level will have a

(25)

chiral mode located at kz = −K with the same sign of vz and a dispersion

relation

E(kz) = vfκ0(kz − K) → eE(kz) = vfκ0(kz+ K) . (1.50)

Since these two modes are independent in the linearised regime aroundE = 0

we can write them together in a matrix notation to produce the effective Hamiltonian:

Heff(kz) = vfκ0(kzσ0− Kσz) . (1.51)

It is easy now to and undo the projection and transformation V to get

in the basis of the original Hamiltonian and find the structure of our zeroth Landau levels. ΨS =     0 u sin θ/2 0 u cos θ/2     Ψ0S =     u∗_{cos θ/2} 0 −u∗_{sin θ/2} 0     |0i , , (1.52)

where u, u∗ _{represent the spatial structure of the zeroth Landau levels. As}

tan θ = ∆

vfkz we can see that cos θ/2 =

1 2 r (1 + _√ vfkz ∆2_+k2 zv2f = 1 2 √ 1 + κ and sin θ/2 = 1 2 √

1 − κ. Since these states describe Bogoliubov quasiparticles they are not necessarily eigenstates of the charge operator. We can still

calculate the expectation value of charge operatorC = νz which can be seen

in Fig. 1.11. hΨS| νz|ΨSi hΨS|ΨSi = −κ, Ψ 0 S νz Ψ0_S Ψ0 S Ψ 0 S = κ . (1.53)

(26)

Figure 1.11: a) The Dispersion relation of Landau levels in a Tight-binding

model, for kx = ky = 0, of a Weyl superconductor with a colormap representing

the charge expectation valuesκ = q/e. Plot adapted from Ref. [17]

We have now managed to effectively describe the zeroth Landau levels inside a Weyl superconductor. In the next chapter, which describes our main work, we will investigate the thermal and electrical transport signatures of these chiral modes.

(27)

Chapter

2

Effect of charge renormalization on

electric and thermo-electric transport

along the vortex lattice of a Weyl

superconductor

2.1 Introduction

Weyl superconductors are nodal superconductors with topological protection [11, 12]: They have nodal points of vanishing excitation gap, just like d -wave superconductors [13], but in contrast to those the gapless states are not restricted to high-symmetry points in the Brillouin zone and can appear for conventional s-wave pairing. The nodal points (Weyl points) at ±K in a Weyl superconductor are protected by the conservation of a topological invariant: the Berry flux of ±2π at Weyl points of opposite chirality [14, 15]. The distinction between symmetry and topology has a major consequence for the stability of Landau levels in a magnetic field. While in a d -wave su-perconductor the strong scattering of nodal fermions by vortices in the order parameter prevents the formation of Landau levels [16], in a Weyl supercon-ductor an index theorem for chiral fermions protects the zeroth Landau level from broadening [17]. The appearance of chiral Landau levels in a super-conducting vortex lattice produces a quantized thermal conductance parallel

(28)

to the magnetic field, in units of 1/2 times the thermal quantum per h/2e

vortex [17]. The factor of1/2 reminds us that Bogoliubov quasiparticles are

Majorana fermions, “half a Dirac fermion” [18, 19].

In this paper we turn from thermal transport to electrical transport, by studying the geometry of Fig. 2.1 and addressing the question “What is the charge transported along the vortices in a chiral Landau level?” It is known [20] that the charge of Weyl fermions in a superconductor (pair potential

∆0) is reduced by a factor κ = K(∆0)/K(0). We find a direct

manifesta-tion of this charge renormalizamanifesta-tion in the electrical conductance, which is

quantized at 1₂(eκ)2_{/h per vortex. Because the charge renormalization is}

en-ergy dependent, a coupling between thermal and electrical transport appears even without any energy-dependent scattering mechanism — resulting in a nonzero thermo-electric effect in a chiral Landau level.

<latexit sha1_base64="i+0I/ycIMW0ywkkWryTcWYplJ3Q=">AAAB8XicbVDLSgNBEOyNrxhfUY9eBoPgKezqQU8S8OIxgpsEkhBmJ73JkJndZaZXCCHf4NXcxKsfJP6Nk8cl0YKBoqqG7q4oU9KS7/94ha3tnd294n7p4PDo+KR8etawaW4EhiJVqWlF3KKSCYYkSWErM8h1pLAZjR7nfvMVjZVp8kLjDLuaDxIZS8HJSaFG4qpXrvhVfwH2lwQrUoEV6r3yd6efilxjQkJxa9uBn1F3wg1JoXBa6uQWMy5GfICTxYpTduWkPotT415CbKGu5bi2dqwjl9SchnbTm4v/ee2c4vvuRCZZTpiI5aA4V4xSNr+X9aVBQWrsCBdGug2ZGHLDBblW1qZEeuqaCDbv/ksaN9Xgtuo/B5Xaw6qTIlzAJVxDAHdQgyeoQwgCJLzBO8w86828D+9zGS14qz/nsAbv6xdPM5D7</latexit>

<latexit sha1_base64="YYrXQ4IHxe4CFnmnUljHAuh8DQU=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKezqQU8S8OIxAfOAJITZSW8yZHZnmekVQsgXeDU38eoniX/j5HFJYsFAUVVDd1eYKmnJ93+93M7u3v5B/rBwdHxyelY8v2hYnRmBdaGVNq2QW1QywTpJUthKDfI4VNgMR89zv/mGxkqdvNI4xW7MB4mMpODkpFqjVyz5ZX8Btk2CFSnBCtVe8afT1yKLMSGhuLXtwE+pO+GGpFA4LXQyiykXIz7AyWK9KbtxUp9F2riXEFuoazkeWzuOQ5eMOQ3tpjcX//PaGUWP3YlM0owwEctBUaYYaTa/lfWlQUFq7AgXRroNmRhywwW5RtamhPHUNRFs3r1NGnfl4L7s14JS5WnVSR6u4BpuIYAHqMALVKEOAhDe4QNmnvZm3qf3tYzmvNWfS1iD9/0HDAWPFg==</latexit>

<latexit sha1_base64="bkJRm2WxDhFyVsKpgaEV52DGSwA=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKezqQU8S8OJJEjAPSEKYnfQmQ2Z3lpleIYR8gVdzE69+kvg3Th6XRAsGiqoaurvCVElLvv/j5ba2d3b38vuFg8Oj45Pi6VnD6swIrAuttGmF3KKSCdZJksJWapDHocJmOHqc+81XNFbq5IXGKXZjPkhkJAUnJ9Wee8WSX/YXYH9JsCIlWKHaK353+lpkMSYkFLe2HfgpdSfckBQKp4VOZjHlYsQHOFmsN2VXTuqzSBv3EmILdS3HY2vHceiSMaeh3fTm4n9eO6PovjuRSZoRJmI5KMoUI83mt7K+NChIjR3hwki3IRNDbrgg18jalDCeuiaCzbv/ksZNObgt+7WgVHlYdZKHC7iEawjgDirwBFWogwCEN3iHmae9mffhfS6jOW/15xzW4H39Av+ujw4=</latexit>

<latexit sha1_base64="oQAQQfCuyaU8l9+3HZn0wALZCMk=">AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKezqQU8S8OIxQfOAJITZSW8yZHZnmekVQsgXeDU38eoniX/j5HFJtGCgqKqhuytMlbTk+z9ebmt7Z3cvv184ODw6PimenjWszozAutBKm1bILSqZYJ0kKWylBnkcKmyGo8e533xFY6VOXmicYjfmg0RGUnByUu25Vyz5ZX8B9pcEK1KCFaq94nenr0UWY0JCcWvbgZ9Sd8INSaFwWuhkFlMuRnyAk8V6U3blpD6LtHEvIbZQ13I8tnYchy4ZcxraTW8u/ue1M4ruuxOZpBlhIpaDokwx0mx+K+tLg4LU2BEujHQbMjHkhgtyjaxNCeOpayLYvPsvadyUg9uyXwtKlYdVJ3m4gEu4hgDuoAJPUIU6CEB4g3eYedqbeR/e5zKa81Z/zmEN3tcvB2qPEw==</latexit>

Figure 2.1: a) Vortex lattice in a Weyl superconductor sandwiched between metal electrodes; b) Circuit to measure the electrical transport along the vortex lines. The

nonlocal conductanceG12= dI2/dV1 gives the current carried through the vortex

lattice by nonequilibrium Weyl fermions in a chiral Landau level.

In the next section 2.2 we summarize the effective low-energy theory of the superconducting vortex lattice [17], on which we base our scattering theory in Sec. 2.3, followed by a calculation of electrical and thermo-electric transport properties in Sec. 2.4. These analytical results are compared with numerical simulations of a tight-binding model in Sec. 2.5. We conclude in Sec. 2.6.

(29)

2.2 Landau level Hamiltonian in the vortex lattice 23 x y dx 0 dy 0

zeroth Landau level (kx= ky= 0, kz= K)

0 1 2 probabilit y densit y (1 /d x 0d y)0

Figure 2.2: Left panel: The red solid curves show the dispersion of Landau levels

in thekx–ky plane perpendicular to the magnetic field (energyE normalized by the

energyE1of the first Landau level). The black dotted curves show the dispersion in

zero magnetic field, with a Weyl cone at theΓ point of the magnetic Brillouin zone.

Right panel: Particle density profile in the zeroth Landau level, in the x–y plane

perpendicular to the magnetic field, for a wave vector at the Weyl point (k= K ˆz).

The magnetic unit cell is indicated by a white dashed rectangle. Both panels are calculated numerically for a Weyl superconductor with a triangular vortex lattice. The vortex cores are located at the bright points in the density profile. Similar plots for a square vortex lattice are in Ref. [17].

2.2 Landau level Hamiltonian in the vortex

lat-tice

We summarize the findings of Ref. [17] for the Landau level Hamiltonian of Weyl fermions in a superconducting vortex lattice, which we will need to calculate the transport properties.

2.2.1 Dispersion relation

A Landau level is a dispersionless flat band in the plane perpendicular to the magnetic field. The lowest (zeroth) Landau level is protected by chiral symmetry from scattering by the vortices, see Fig. 2.2. This is the Landau level on which we focus our analysis. It is a celebrated result of Nielsen and Ninomiya [14] that Weyl fermions in the zeroth Landau level have a definite

(30)

or antiparallel to B. To account for the electron-hole degree of freedom the number of bands is doubled for each chirality, so that we have four bands in total. Electron-like and hole-like bands are related related by the

charge-conjugation symmetry relationEχ(kz) = −Eχ(−kz).

-<latexit sha1_base64="TvOQTgcDhsqe62M0qV/A22A9tzQ=">AAAB73icbVDLSgMxFL3xWeur6tJNsAiuhhlFdFlw47KifUBbSibNtKGZyZjcEUrpJ7i1O3HrF4l/Y9rOpq0HAodzTrj3njBV0qLv/5KNza3tnd3CXnH/4PDouHRyWrc6M1zUuFbaNENmhZKJqKFEJZqpESwOlWiEw4eZ33gTxkqdvOAoFZ2Y9RMZSc7QSc+Bd9stlX3Pn4OukyAnZchR7ZZ+2j3Ns1gkyBWzthX4KXbGzKDkSkyK7cyKlPEh64vxfMEJvXRSj0bauJcgnatLORZbO4pDl4wZDuyqNxP/81oZRvedsUzSDEXCF4OiTFHUdHYt7UkjOKqRI4wb6TakfMAM4+g6WZoSxhPXRLB69zqpX3vBjec/BeWKn3dSgHO4gCsI4A4q8AhVqAGHPrzDB0zJK5mST/K1iG6Q/M8ZLIF8/wGxYY9a</latexit> <latexit sha1_base64="YEZhBvUwGuYNc3QD9YDa0pQXVHM=">AAAB7XicbVDLSgMxFL1TX7W+qi7dBIvgqmR0ocuCG5ct2Ae0pWTSTBuaTIbkjlBKv8Ct3YlbP0n8G9N2Nq0eCBzOOeHee6JUSYeU/gSFnd29/YPiYeno+OT0rHx+0XIms1w0uVHGdiLmhJKJaKJEJTqpFUxHSrSjydPSb78K66RJXnCair5mo0TGkjP0UoMOyhVapSuQvyTMSQVy1Afl797Q8EyLBLliznVDmmJ/xixKrsS81MucSBmfsJGYrdabkxsvDUlsrH8JkpW6kWPauamOfFIzHLttbyn+53UzjB/7M5mkGYqErwfFmSJoyPJWMpRWcFRTTxi30m9I+JhZxtE3sjEl0nPfRLh991/SuquG91XaCCs1mndShCu4hlsI4QFq8Ax1aAIHAW/wDovABIvgI/hcRwtB/ucSNhB8/QLNao7i</latexit> <latexit sha1_base64="czZyqug9IeGiVCIDgV4/KfcqYgQ=">AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgxiGjgi4LblxWsA9oS8mkmTY08yC5I5TSX3Brd+LWHxL/xsx0Nm09EDicc8K99/iJkgYp/XVKW9s7u3vl/crB4dHxSfX0rGXiVHPR5LGKdcdnRigZiSZKVKKTaMFCX4m2P3nK/Pab0EbG0StOE9EP2SiSgeQMM+mGuveDao26NAfZJF5BalCgMaj+9IYxT0MRIVfMmK5HE+zPmEbJlZhXeqkRCeMTNhKzfMM5ubLSkASxti9CkqsrORYaMw19mwwZjs26l4n/ed0Ug8f+TEZJiiLiy0FBqgjGJDuXDKUWHNXUEsa1tBsSPmaacbSlrEzxw7ltwlu/e5O0bl3vzqUvXq1Oi07KcAGXcA0ePEAdnqEBTeAwhnf4gIWjnYXz6XwtoyWn+HMOK3C+/wAY5Y+P</latexit> <latexit sha1_base64="j9Foqa8E6xohHJxjUaKrofxhxHw=">AAAB73icbVDLSgMxFL3js9ZX1aWbYBFcDRkVdFlw47KifUBbSibNtKGZzJjcEUrpJ7i1O3HrF4l/Y9rOpq0HAodzTrj3njBV0iKlv97G5tb2zm5hr7h/cHh0XDo5rdskM1zUeKIS0wyZFUpqUUOJSjRTI1gcKtEIhw8zv/EmjJWJfsFRKjox62sZSc7QSc/Uv+2WytSnc5B1EuSkDDmq3dJPu5fwLBYauWLWtgKaYmfMDEquxKTYzqxIGR+yvhjPF5yQSyf1SJQY9zSSubqUY7G1ozh0yZjhwK56M/E/r5VhdN8ZS51mKDRfDIoyRTAhs2tJTxrBUY0cYdxItyHhA2YYR9fJ0pQwnrgmgtW710n92g9ufPoUlCs076QA53ABVxDAHVTgEapQAw59eIcPmHqv3tT79L4W0Q0v/3MGS/C+/wCuTY9Y</latexit> <latexit sha1_base64="uXLl9Br1LCSV9/3R8NcQ0xBP0fU=">AAACCHicbVDLSgMxFM3UV62vUXe6CbZC3ZQZXeiyIILLCvYBbRky6Z02NJMZkoxYS8G1H+LW7sStPyH+jel0Nm09EDg554R7c/yYM6Ud59fKra1vbG7ltws7u3v7B/bhUUNFiaRQpxGPZMsnCjgTUNdMc2jFEkjoc2j6w9uZ33wCqVgkHvUohm5I+oIFjBJtJM8+gWfKdHrBIED2R7h0Vx56Lxclzy46FScFXiVuRoooQ82zfzq9iCYhCE05UartOrHujonUjHKYFDqJgpjQIenDON18gs+N1MNBJM0RGqfqQo6ESo1C3yRDogdq2ZuJ/3ntRAc33TETcaJB0PmgIOFYR3hWA+4xCVTzkSGESmY2xHRAJKHalLUwxQ8npgl3+d+rpHFZca8qzoNbrDpZJ3l0is5QGbnoGlXRPaqhOqLoFb2jDzS13qyp9Wl9zaM5K3tzjBZgff8BIWOZAQ==</latexit> <latexit sha1_base64="9QWzFWvpfC+YRsy0NKlo50cJ030=">AAACDHicbVDLSgMxFM3UV62vUZeCBFuhbsqMLnRZEMFlBfuAtgyZ9E4bmmSGJCOOpTvXfohbuxO3/oL4N06n3bT1QODknBPuzfEjzrRxnF8rt7a+sbmV3y7s7O7tH9iHRw0dxopCnYY8VC2faOBMQt0ww6EVKSDC59D0h7dTv/kESrNQPpokgq4gfckCRolJJc8+hWfKTHbBIEH1E1y68zqRKA+9l4uSZxedipMBrxJ3Topojppn/3R6IY0FSEM50brtOpHpjogyjHIYFzqxhojQIenDKNt+jM9TqYeDUKVHGpypCzkitE6EnyYFMQO97E3F/7x2bIKb7ojJKDYg6WxQEHNsQjytAveYAmp4khJCFUs3xHRAFKEmLWxhii/GaRPu8r9XSeOy4l5VnAe3WHXmneTRCTpDZeSia1RF96iG6oiiV/SOPtDEerMm1qf1NYvmrPmbY7QA6/sPUl2awQ==</latexit> <latexit sha1_base64="RxbXFT3gTHk2y3JS5wMVFFT8YG0=">AAAB73icbVDLTgIxFL2DL8QX6tJNIzFxRWZ0oUsSNyZuMMojAUI6pQMN7XRs75gg4RPcys649YuMf2OB2QCepMnJOae5954wkcKi7/96uY3Nre2d/G5hb//g8Kh4fFK3OjWM15iW2jRDarkUMa+hQMmbieFUhZI3wuHdzG+8cmOFjp9xlPCOov1YRIJRdNLTQ/etWyz5ZX8Osk6CjJQgQ7Vb/Gn3NEsVj5FJam0r8BPsjKlBwSSfFNqp5QllQ9rn4/mCE3LhpB6JtHEvRjJXl3JUWTtSoUsqigO76s3E/7xWitFtZyziJEUes8WgKJUENZldS3rCcIZy5AhlRrgNCRtQQxm6TpamhGrimghW714n9atycF32H4NSxc86ycMZnMMlBHADFbiHKtSAQR/e4QOm3os39T69r0U052V/TmEJ3vcfjuWP6g==</latexit> <latexit sha1_base64="jkrp9y3+ik0YnxZ4knSyxkMqzRg=">AAAB8HicbVDLSgMxFL3xWeur6tJNsAhuLDO60GXBjeCmgn1AW0omzbShSWZI7gi19Bfc2p249YfEv3GmnU1bDwQO55xw7z1BrKRDz/slG5tb2zu7hb3i/sHh0XHp5LThosRyUeeRimwrYE4oaUQdJSrRiq1gOlCiGYweMr/5KqyTkXnBcSy6mg2MDCVnmEnXT723XqnsVbw56Drxc1KGHLVe6afTj3iihUGumHNt34uxO2EWJVdiWuwkTsSMj9hATOYbTullKvVpGNn0GaRzdSnHtHNjHaRJzXDoVr1M/M9rJxjedyfSxAkKwxeDwkRRjGh2Lu1LKziqcUoYtzLdkPIhs4xjWsrSlEBP0yb81bvXSeOm4t9WvGe/XPXyTgpwDhdwBT7cQRUeoQZ14DCEd/iAGbFkRj7J1yK6QfI/Z7AE8v0H+W6QIQ==</latexit> <latexit sha1_base64="42jf7+CIizGaxR4E8YY0P27wpNk=">AAAB73icbVDLSgMxFL1TX7W+qi7dBIsgCCWjC10WRHBZ0T6gLSWTZtrQZGZM7gil9BPc2p249YvEvzFtZ9PWA4HDOSfce0+QKGmR0l8vt7G5tb2T3y3s7R8cHhWPT+o2Tg0XNR6r2DQDZoWSkaihRCWaiRFMB0o0guH9zG+8CWNlHL3gKBEdzfqRDCVn6KTnh+5Vt1iiZToHWSd+RkqQodot/rR7MU+1iJArZm3Lpwl2xsyg5EpMCu3UioTxIeuL8XzBCblwUo+EsXEvQjJXl3JMWzvSgUtqhgO76s3E/7xWiuFdZyyjJEUR8cWgMFUEYzK7lvSkERzVyBHGjXQbEj5ghnF0nSxNCfTENeGv3r1O6tdl/6ZMn/xShWad5OEMzuESfLiFCjxCFWrAoQ/v8AFT79Wbep/e1yKa87I/p7AE7/sPDFyPlQ==</latexit> <latexit sha1_base64="wE82dCWNPrrU17RQumY5baQRnbw=">AAAB73icbVDLSgMxFL1TX7W+qi7dBIvgxpLRhS4LIrisaB/QlpJJM21oMjMmd4RS+glu7U7c+kXi35i2s2nrgcDhnBPuvSdIlLRI6a+X29jc2t7J7xb29g8Oj4rHJ3Ubp4aLGo9VbJoBs0LJSNRQohLNxAimAyUawfB+5jfehLEyjl5wlIiOZv1IhpIzdNLzQ/eqWyzRMp2DrBM/IyXIUO0Wf9q9mKdaRMgVs7bl0wQ7Y2ZQciUmhXZqRcL4kPXFeL7ghFw4qUfC2LgXIZmrSzmmrR3pwCU1w4Fd9Wbif14rxfCuM5ZRkqKI+GJQmCqCMZldS3rSCI5q5AjjRroNCR8wwzi6TpamBHrimvBX714n9euyf1OmT36pQrNO8nAG53AJPtxCBR6hCjXg0Id3+ICp9+pNvU/vaxHNedmfU1iC9/0HD26Plw==</latexit> <latexit sha1_base64="r+4kZb44F3oqRnPo7j/dTuMDI4g=">AAAB8nicbVDLSgNBEOyNrxhfUY9eBoPgxbCrBz0GvHiMYB6QhDA76U3GzD6Y6RVCyD94NTfx6v+If+Nks5ckFgwUVTV0d/mJkoZc99cpbG3v7O4V90sHh0fHJ+XTs6aJUy2wIWIV67bPDSoZYYMkKWwnGnnoK2z548eF33pDbWQcvdAkwV7Ih5EMpOBkpSbeKDnGfrniVt0MbJN4OalAjnq//NMdxCINMSKhuDEdz02oN+WapFA4K3VTgwkXYz7EabbjjF1ZacCCWNsXEcvUlRwPjZmEvk2GnEZm3VuI/3mdlIKH3lRGSUoYieWgIFWMYrY4mA2kRkFqYgkXWtoNmRhxzQXZWlam+OHMNuGt371JmrdV767qPnuVmpt3UoQLuIRr8OAeavAEdWiAgFd4hw+YO+TMnU/naxktOPmfc1iB8/0HqDSRGw==</latexit> <latexit sha1_base64="oOEi9PLxsQSulL9Absd0OF4B92Q=">AAAB8nicbVDLSgNBEOyNrxhfUY9eBoPgxbCrBz0GvHiMYB6QhDA76U3GzD6Y6RVCyD94NTfx6v+If+Nks5ckFgwUVTV0d/mJkoZc99cpbG3v7O4V90sHh0fHJ+XTs6aJUy2wIWIV67bPDSoZYYMkKWwnGnnoK2z548eF33pDbWQcvdAkwV7Ih5EMpOBkpeboRskx9ssVt+pmYJvEy0kFctT75Z/uIBZpiBEJxY3peG5CvSnXJIXCWambGky4GPMhTrMdZ+zKSgMWxNq+iFimruR4aMwk9G0y5DQy695C/M/rpBQ89KYySlLCSCwHBaliFLPFwWwgNQpSE0u40NJuyMSIay7I1rIyxQ9ntglv/e5N0rytendV99mr1Ny8kyJcwCVcgwf3UIMnqEMDBLzCO3zA3CFn7nw6X8towcn/nMMKnO8/rN6RHg==</latexit>

Figure 2.3: Dispersion relation of the zeroth Landau level in a superconducting

vortex lattice, plotted from Eq. (2.1) forµ = 0, ∆0 = 0.5, β = 1. Only the

depen-dence on the momentum kz along the magnetic field B is shown, the dispersion

is flat in thex–y plane (see Fig. 2.2). The four branches are distinguished by the

sign of the chirality (solid or dashed) and by the sign of the electric charge (red

or blue). The zero-field Weyl points at kz = ±K are indicated by arrows. Each

branch has a degeneracyNLandau= eΦ/h set by the enclosed flux Φ = BW2.

The effect of a superconducting vortex lattice on this four-band dispersion is given by [17] Eχ(kz) = −(sgn kz)χM (kz) − χµκ(kz), M (kz) = β − q ∆2 0+ kz2, κ(kz) = d dkz M (kz), (2.1)

plotted in Fig. 2.3. _{(We have set ~ and the Fermi velocity v}F equal to

unity, so κ is dimensionless.) The magnitude of the superconducting pair

potential outside of the vortex cores is denoted by ∆0 and β is an internal

magnetization along thez-direction that breaks time-reversal symmetry even

in the absence of any external magnetic field. In Eq. (2.1) we have assumed

(31)

2.2 Landau level Hamiltonian in the vortex lattice 25

Provided that ∆0 < β there is a pair of Landau levels for each chirality,

located in the magnetic Brillouin zone near the Weyl points at kz = K and

kz = −K, with [11]

K(∆0) =

q

β2_{− ∆}2

0. (2.2)

The charge expectation value

Qχ = −e ∂Eχ ∂µ = eχκ(kz) = − eχkz p∆2 0+ k2z (2.3)

for a given chirality has the opposite sign at the two Weyl points. (We say

that the chiral Landau levels near kz = ±K are charge-conjugate.) When

kz = ±K is at the Weyl point, the charge renormalization factor equals ∓κ0,

with

κ0 = K(∆0)/K(0) =

q

1 − ∆2

0/β2, (2.4)

while κ(kz) varies linearly with energy away from the Weyl point [20].

2.2.2 Effective Hamiltonian

The dispersion (2.1) follows from the effective low-energy Hamiltonian [17]

H = U     H+ 0 0 0 0 · · 0 0 · · 0 0 0 0 H−     U†, (2.5a) Hχ = (kx+ eAχ,x)σx+ (ky + eAχ,y)σy + M σz− χµκσ0, (2.5b) U = exp 1 2iθνyτzσz, θ = arccos κ. (2.5c)

The 2 × 2 Pauli matrices να, τα, and σα (with α = 0 the corresponding unit

matrix) act on, respectively, the electron-hole, orbital, and spin degrees of

freedom. The full Hamiltonian H is an8 × 8 matrix and the 2 × 2 matrices

H± act on the σ index in the ν = τ = ±1 sector.

The central block in Eq. (2.5a) indicated by dots refers to higher-lying bands that are approximately decoupled from the low-energy bands.

Vir-tual transitions to these higher bands contribute orderµ2 _{terms that remove}

the discontinuity in the derivative ∂E/∂kz at kz = 0 for µ 6= 0. No such

(32)

The gauge field A_χ(r), dependent on the position r = (x, y) in the x–y

plane, defines the effective magnetic field Bχ = ∇ × Aχin thez-direction felt

by the Weyl fermions in the lattice of vortices at positions Rn,

Bχ = (1 + χκ)Φ0

X

n

δ(r − Rn) − χκB. (2.6)

There are Nvortex = BW2/Φ0 vortices of flux Φ0 = h/2e in an area W2

per-pendicular to the applied magnetic fieldB, so the spatial averageR Bχdr = Φ

equals the total enclosed flux Φ = BW2 _{independent of} _{κ or of the lattice}

of vortices. (In the numerics that follows we will use a square lattice for definiteness.)

2.2.3 Zeroth Landau level wave functions

As shown in Ref. [17], the Aharonov-Casher index theorem [21–23], together with the requirement that the wave functions are square-integrable at a

vor-tex core, implies that the zeroth Landau level eigenstates ψχ of Hχ, which

are rank-two spinors, are also eigenstates |±iσ of σz,

σzψχ = (sgn Qχ)ψχ. (2.7)

The eigenvalue is determined by the sign of the effective quasiparticle charge (2.3).

It follows that the eigenstates Ψχ of the full Hamiltonian H, which are

rank-eight spinors, have the form

Ψχ= eikzzfχ(x, y)e

1

2iθνyτzσz|sgn χi_ν|sgn χi_τ|sgn Q_χi_σ

= eikzz_f

χ(x, y)

h

[cos(θ/2)|sgn χiν|sgn χiτ|sgn Qχiσ

− sin(θ/2)(sgn Qχ)|−sgn χiν|sgn χiτ|sgn Qχiσ

i

]. (2.8)

The spatial density profile fχ(x, y) is peaked at the vortex cores, with a

power law decay |fχ|2 ∝ δr−1+|Qχ|/e at a distance δr from the core [17]. The

renormalization of the quasiparticle charge does not affect the degeneracy of the zeroth Landau level: each of the four chiral modes in Fig. 2.3 has a degeneracy

NLandau = eΦ/h (2.9)

(33)

Although the spatial density profile of these chiral modes is nonuniform,

the wave functions extend over the entire x–y plane — they are not

expo-nentially confined to the vortex cores (see Fig. 2.2). This is a qualitative difference between the zeroth Landau level of a Weyl superconductor and zero-modes bound to vortices in topological superconductors [24, 25].

2.3 Transmission through the NSN junction

Refering to the geometry of Fig. 2.1, we seek the transmission matrixtNSN for

propagating modes of electrons and holes transmitted from the first metal

contact N1 in the region z < 0, through the Weyl superconductor in the

region 0 < z < L, into the second metal contact N2 in the regionz > L.

2.3.1 Renormalized charge transfer

We start by examining a single NS interface, to study how a chiral mode in the superconductor injects a renormalized charge into the normal metal.

On the superconducting side z < L of the NS interface at z = L the

incident modes have positive chirality χ = +1. There is a mode ΨS with

perpendicular momentum kz near K and a mode Ψ0S with kz0 near −K. We

do not specify the transverse momentum kk = (kx, ky), which gives each

mode a degeneracy of NLandau = eΦ/h, see Eq. (2.9).

According to Eq. (2.8), the spinor structure of the chiral modes is

ΨS ∝ cos(θ/2)|++−iντ σ + sin(θ/2)|−+−iντ σ,

Ψ0_S ∝ cos(θ0/2)|+++iντ σ− sin(θ0/2)|−++iντ σ.

(2.10)

We have abbreviated |±±±iντ σ = |±iν|±iτ|±iσ and denote θ = θ(kz), θ0 =

θ(k0

z).

For the normal metal we take the free-electron Hamiltonian

HN =

1

2m(k

2 _{− k}2

F)νzτ0σ0, (2.11)

isotropic in the spin and valley degrees of freedom, in the high Fermi-momentum

limit kFlm → ∞ when the effect of the magnetic field on the spectrum may

be neglected (lm =

p

(34)

Because of the large potential step experienced upon traversing the NS

interface, the perpendicular momentumkz is boosted to+kF for the electron

component of the state and to −kF for the hole component. A state in N

moving away from the NS interface of the form ΨN∝ eikF(z−L)cos(θ/2)|++−iντ σ

+ e−ikF(z−L)_{sin(θ/2)|−+−i}

ντ σ (2.12a)

can be matched to the incident state ΨS in S, while the state

Ψ0N∝ eikF(z−L)cos(θ 0

/2)|+++iντ σ

− e−ikF(z−L)_sin(θ0_/2)|−++i

ντ σ (2.12b)

can be matched to Ψ0

S.

The charge transferred through the interface when ΨS 7→ ΨN equals the

renormalized charge from Eq. (2.3),

QN = hΨN|eνz|ΨNi = e cos θ = eκ =

−ekz

p∆2

0+ kz2

, (2.13)

dependent on the perpendicular momentum kz in S, before the boost to kF

in N. Whenkz = K, this gives

QN = −e

q

1 − ∆2

0/β2 = −κ0e ≡ −Qeff. (2.14)

This is for the transmission ΨS 7→ ΨN . The other transmission Ψ0S 7→ Ψ

0 N

transfers fork0z = −K a charge Q0N= +Qeff.

Similarly, at the opposite NS interfacez = 0 the chiral Landau level modes

in S moving away from the interface are matched to incoming states in N of the form ΦN∝ eikFzcos(θ/2)|++−iντ σ + e−ikFz_{sin(θ/2)|−+−i} ντ σ, (2.15a) Φ0N∝ eikFzcos(θ 0 /2)|+++iντ σ

− e−ikFz_sin(θ0_/2)|−++i

ντ σ. (2.15b)

2.3.2 Transmission matrix

At a given energy E relative to the Fermi level the perpendicular momenta

(35)

determined by the dispersion relation (2.1) with χ = +1. For µ = 0 the

expressions are simple,

kz = K + (β/K)E, kz0 = −K + (β/K)E. (2.16)

For any µ, particle-hole symmetry ensures that

kz(E) = −k0z(−E). (2.17)

The Landau level ΨS propagating from z = 0 to z = L accumulates a

phasekzL, and similarly Ψ0S accumulates a phasekz0L. The full transmission

matrix of the NSN junction at energyE can thus be written as

tNSN(E) = eikzL|ΨNihΦN| + eik 0 zL|Ψ0 NihΦ 0 N|, (2.18)

with kz and kz0 determined by Eq. (2.16).

We can rewrite Eq. (2.18) in the basis of propagating electron modes in

the normal metal. In the region z < 0 one has the basis states

|Ψ↑i = |e ↑i |h ↑i , |Ψ↓i = |e ↓i |h ↓i , (2.19a) |e ↑i = eikFz_|+++i

ντ σ, |h ↑i = e−ikFz|−++iντ σ,

|e ↓i = eikFz_|++−i

ντ σ, |h ↓i = e−ikFz|−+−iντ σ, (2.19b)

and similarly for z > L with kFz replaced by kF(z − L).

The transmission matrix is block diagonal in the spin degree of freedom,

tNSN(E) = t↑ (E) 0 0 t↓(E) , (2.20a) t↑ = eik 0 zL

cos2_(θ0_/2) _{− cos(θ}0_{/2) sin(θ}0_/2)

− cos(θ0_{/2) sin(θ}0_/2) _sin2_(θ0_/2)

,

t↓ = eikzL

cos2_(θ/2) _{cos(θ/2) sin(θ/2)}

cos(θ/2) sin(θ/2) sin2_(θ/2)

. (2.20b)

The 2 × 2 matrix t↑ acts on the electron-hole spinor |Ψ↑i and t↓ acts on

|Ψ↓i. We may write this more compactly as

t↑ = 1₂eik 0 zL ν0+ νze−iθ 0_ν y , t↓ = 1₂eikzL ν0+ νzeiθνy . (2.21)

(36)

These are each rank-one matrices, one eigenvalue equals 0 and the other

equals 1 in absolute value. The unit transmission eigenvalue is NLandau-fold

degenerate in the transverse momentum kk.

At the Fermi levelE = 0 the particle-hole symmetry relation (2.17) implies

k0

z = −kz, θ0 = π − θ, hence

tNSN(0) = 1₂e−ikzLσz ν0− νzσzeiθνy . (2.22)

One verifies that

tNSN(0) = νyσyt∗NSN(0)νyσy, (2.23)

as required by particle-hole symmetry.

2.4 Transport properties

The transmission matrix allows us to calculate the transport properties of the NSN junction, under the assumption that there is no backscattering of the chiral modes in the Weyl superconductor. To simplify the notation, we

write t for the Fermi-level transmission matrix tNSN(0). The submatrices of

electron and hole components are denoted bytee,thh,the, andteh. We define

the combinations

T±= t†_eet_ee± t†_het_he, (2.24a)

T+= 1₂(ν0+ νz)t†t, T− = 1₂(ν0+ νz)t†νzt. (2.24b)

2.4.1 Thermal conductance

As a check, we first recover the result of Ref. [17] for the quantization of the thermal conductance.

The thermal conductance Gthermal = J12/δT gives the heat current J12

transported at temperatureT0from contactN1toN2via the superconductor,

in response to a small temperature difference δT between the contacts. It

follows from the total transmitted quasiparticle current, Gthermal= 1₂g0NLandauTr t†t = g0

eΦ

(37)

2.4 Transport properties 31

with NLandau = eΦ/h the Landau level degeneracy and g0 = 1₃(πkB)2(T0/h)

the thermal conductance quantum. The factor 1/2 in the first equation

appears because the quasiparticles in the Weyl superconductor are Majorana

fermions. It is cancelled by the factor of two from Tr tt†_{= 2, in view of Eq.}

(2.22).

2.4.2 Electrical conductance

Referring to the electrical circuit of Fig. 2.1b, we consider the electrical

con-ductance G12= dI2/dV1, given by G12= e2 hNLandauTr T− = e 2 hNLandau 1 2Tr (ν0+ νz)t † νzt. (2.26)

In the linear response limit V1 → 0 we substitute t from Eq. (2.22), which

gives G12(0) = cos2θ e2 hNLandau= (eκ)2 h eΦ h . (2.27)

The conductance quantum e2_{/h is renormalized by the effective charge e 7→}

eκ. At µ = 0, when kz = K, the renormalization factor is κ20 = (Qeff/e)2 =

1 − ∆2

0/β2 from Eq. (2.14). Note that the conductance per h/2e vortex is

1 2(eκ0)

2_{/h, with an additional factor 1/2 to signal the Majorana nature of}

the quasiparticles.

At finite E = eV1 we must use the energy-dependent transmission matrix

(2.20), which gives

G12(E) = 1₂

e2

hNLandau cos θ + cos θ

0

+ cos2θ + cos2θ0 . (2.28)

Substituting Eq. (2.13) for cos θ and cos θ0 _at_k

z and kz0, given as a function

of E by Eq. (2.16), we find G12(E) = G12(0) 1 − ∆ 2 0E (β2_{− ∆}2 0)3/2 + O(E2₎ . (2.29)

The energy dependence of the differential conductance comes entirely from

the energy dependence of the effective charge: AtE = 0 the electron-like and

hole-like chiral Landau levels have precisely opposite effective charge ±Qeff,

but for E 6= 0 the effective charges differ in absolute value by an amount

(38)

2.4.3 Shot noise

At temperatures small compared to the applied voltage V2, the time

depen-dent fluctuations in the currentI2 are due to shot noise. The formula for the

shot noise power is [26]

P12= e3_V 1 h Tr (T+− T 2 −). (2.30)

This can again be written in terms of the Pauli matrixτz and evaluated using

Eq. (2.22), P12= e3_V 1 h 1 − 1 2κ 2₋ 1 2κ 4_. (2.31)

The shot noise vanishes when κ → 1, it is fully due to the charge

renormal-ization.

The Fano factorF , the dimensionless ratio of shot noise power and average

current, results as F = P12 eV1G12 = 1 κ2 − 1 2(1 + κ 2 ). (2.32)

2.4.4 Thermo-electricity

Because of the energy dependence of the effective charge, a temperature

difference δT between contacts 1 and 2 will produce an electrical current

I12 = α12δT in addition to a heat current. The thermo-electric coefficient

α12 is given by [27] α12= π2 3ek 2 BT0 lim E→0 d dEG12(E). (2.33)

Substitution of Eq. (2.29) gives

α12 = − π2 3ek 2 BT0G12(0) ∆2 0 (β2_{− ∆}2 0)3/2 = −g0eκ20NLandau ∆2 0 (β2_{− ∆}2 0)3/2 = −g0eNLandau (∆0/β)2 (β2_{− ∆}2 0)1/2 . (2.34)

(39)

2.5 Numerical simulations

To test these analytical results, we have carried out numerical calculations in a tight-binding model of the Weyl superconductor with a vortex lattice.

2.5.1 Tight-binding Hamiltonian

The Bogoliubov-de Gennes Hamiltonian HS in the superconducting region

0 < z < L is HS= H0 (k + eA) ∆ ∆∗ _−σ yH0∗(−k + eA)σy , (2.35a) H0(k) = t0 X α=x,y,z [τzσαsin kαa0+ τxσ0(1 − cos kαa0)] + βτ0σz − µτ0σ0. (2.35b)

The cubic lattice constant of the tight-binding model is a0 and t0 is the

nearest-neigbor hopping energy. In what follows we will set a0 and t0 both

equal to unity.

In the strong-type-II limit the magnetic field B = B0z penetrates theˆ

superconductor uniformly, with vector potential A = (−B0y, 0, 0). The

ab-solute value ∆0 of the pair potential ∆ = ∆0eiφ can also be taken uniform,

assuming that the size ξ0 = ~vF/∆0 of the vortex core is small compared to

the magnetic length lm =

p

~/eB0. For the analytical calculations this is

the only requirement. For the numerics we also take ξ0 small compared to

the tight-binding discretization lengtha0, and then ensure that a vortex core

(where the phase field is undefined) does not coincide with a lattice point.

This implies that a0 is large compared to the atomic lattice constant (which

itself must be much smaller than ξ0).

The vortices are arranged on a square lattice in the x–y plane, lattice

constant d0 = N0a0, with two h/2e vortices in a unit cell. The number

N0 = (a20eB0/h)−1/2 (2.36)

is set at an integer value. The phaseφ(r) winds around the vortex cores Rn

according to

∇ × ∇φ = 2πˆzP

(40)

In the normal metal leads z < 0, z > L we have ∆0 ≡ 0 and a large

chemical potentialµN, so only modes with a large longitudinal momentumkz

couple to the superconductor. We effectuate theµN→ ∞ limit by removing

the transverse x, y couplings in the leads, resulting in the Hamiltonian [28]

HN= νzτzσzsin kz+ νzτxσ0(1 − cos kz). (2.38)

Figure 2.4: Data points: Electrical conductance (top panel) and Fano factor

(bottom panel) in the superconducting vortex lattice (lattice constant d0), as a

function of the pair potential ∆0 at fixed magnetization β = 1, calculated from

the tight-binding model (lattice constant a0) for different lattice constant ratios

N0 = d0/a0. The black curves are the analytical predictions from the charge

renormalization factorκ, both in the approximation of a linearized dispersion (black

dashed curve, κ = κ0 =p1 − ∆2₀/β2) and for the full nonlinear dispersion (black

(41)

The gauge-invariant discretization of the Hamiltonian (2.35) in the mag-netic Brillouin zone is detailed in Ref. [17]. The scattering matrix is calcu-lated using the Kwant code [29].

2.5.2 Results

Results for the conductance and shot noise are shown in Fig. 2.4, as a function

of∆0 forβ = 1, µ = 0. The analytical predictions (2.27) for the conductance

and (2.32) for the Fano factor are given by the black curves. As a check, for

these curves we have also calculated the charge renormalization factorκ from

the full sinusoidal dispersion, without making the small-k expansion of Eq.

(2.1) — the difference with κ0 =p1 − ∆20/β2 is small.

Figure 2.5: Dependence on ∆0 for β = 0.5 of the thermo-electric coefficient

(2.33), calculated from the infinite-system analytics (black solid curve) or obtained from finite-size numerics (colored data points).

To assess finite-size effects in the numerics we show results for different

values of the ratio N0 = d0/a0 of magnetic unit cell and tight-binding unit

cell. As expected, the agreement between numerics and analytics improves

with increasing N0, for ∆0/β not close to unity. (At ∆0 = β the spectrum

becomes gapless and the low-energy analytics breaks down.)

(42)

the conductance determines the thermo-electric coefficient (2.33). We show

numerical results for α12 ∝ dG12/dE in Fig. 2.5, for a smaller β = 0.5 to

reduce the oscillations that disappear only slowly with increasingN0.

2.5.3 Test for isotropy of the charge renormalization

Figure 2.6: Same as Fig. 2.4, but for a magnetization β that is perpendicular

rather than parallel to the magnetic fieldB.

So far we assumed that the internal magnetizationβ is parallel to the external

magnetic field in the z-direction. This assumption is needed for our

low-energy analytics, but numerically we can take an arbitrary angle between the

(43)

2.6 Conclusion 37

βτ0σzin the Hamiltonian (2.35b) withτ0β·σ. Results for β = (β, 0, 0), so for

a magnetization perpendicular to the magnetic field, are shown in Fig. 2.6. There is no qualitative difference with Fig. 2.4 for the parallel configuration, the quantitative difference is that the finite-size effects are smaller.

2.6 Conclusion

In summary, we have shown how the charge renormalization e 7→ κe of

Weyl fermions in a superconducting vortex lattice modifies the electrical and thermo-electrical transport properties.

In the electrical conductance, the current per vortex is reduced by a factor

1 2κ

2 _{— a prefactor} _{1/2 because of the Majorana nature of the quasiparticles}

and a factorκ2 _{because of the effective charge. At the Weyl point}_{κ → κ}

0 =

p1 − ∆2

0/β2 depends on the ratio of the superconducting gap ∆0 and the

separation 2β of the Weyl points of opposite chirality.

The charge-squared renormalization of the electrical conductance is a sim-ple result, but perhaps not what one might have guessed by analogy with

the fractional quantum Hall effect, where a 1/3 fractional charge reduces the

conductance by 1/3 rather than 1/9. The key difference is that here the

quasiparticles are not in an eigenstate of charge; the charge renormalization

is due to quantum fluctuations, which give uncorrelated reductions by κ × κ

at entrance and exit. These quantum fluctuations of the charge are also re-sponsible for the large shot noise power that we have found, with a diverging

Fano factor (2.32) in the limit κ → 0.

The energy dependence of the charge renormalization implies that charge

transport parallel to the magnetic fieldB goes hand-in-hand with heat

trans-port. As a result, a nonzero thermo-electric coefficientα12along the field lines

appears in a chiral Landau level — something that would not be possible in the normal state: The Landau level contributes an energy-independent

num-ber of propagating modes along B (one mode per flux quantum) and the

chirality suppresses backscattering, so the energy derivative in Eq. (2.33) would vanish in the normal state.

There is much recent interest in thermo-electricity of Weyl fermions in a

Landau level [30–33], but that refers to currents perpendicular to B. Our

(44)

a mechanism for a nonzero effect parallel to the field lines.

In our calculations we have assumed a clean system, without impurity scattering. However, we expect the transport properties to be robust against non-magnetic disorder, which in the effective low-energy Hamiltonian (2.5)

would enter as a term proportional toσz that does not couple Landau levels

(45)

Appendix

A

Calculation of transport properties

from the continuum limit of the

tight-binding model

In the tight-binding model of Sec. 2.5.1 the wave matching at the normal-superconductor (NS) interface is implemented by a nearest-neighbor coupling on a square lattice of the Hamiltonians (2.35) in S to (2.38) in N. Microscop-ically this results in different matching conditions on the wave function than the matching conditions (2.12) from the analytical treatment of Sec. 2.3. In this Appendix we check that the continuum limit of the tight-binding model still gives the same results for the transport properties as obtained in Sec.

2.4 from the main text. For simplicity, we set µ = 0 and restrict our

consid-erations to E = 0.

A.1 Matching condition

The linearized Hamiltonian for the normal metal reads

HN= νzτzσzkz (A.1)

and for the superconductor it reads

HS =

τzσ ·(k − eA) + βσz ∆0eiφ

∆0e−iφ −τzσ ·(k + eA) + βσz

Transport properties of a Weyl superconductor in a vortex lattice

Transport properties of a Weyl

superconductor in a vortex lattice

Transport properties of a Weyl

superconductor in a vortex lattice

Gal Lemut

Abstract

Contents

Chapter

1

Introduction

1.1

Preface

1.2

Weyl semimetal

1.2.1

Metals, insulators and semimetals

Density of states

Ener

gy

1.2.2

From the Dirac equation to a Weyl fermion

1.2.3

Topological protection

1.2.4

Weyl fermions in crystalline structures

1.2.5

Chiral anomaly

E

k

E

k

E

k

E

k

1.2.6

Berry curvature

1.3

Weyl superconductor

1.3.1

Bogoliubov-de Gennes formalism

1.3.2

Magnetic field in a superconductor

1.3.3

Weyl superconductor Landau Levels

Chapter

2

Effect of charge renormalization on

electric and thermo-electric transport

along the vortex lattice of a Weyl

superconductor

2.1

Introduction

2.2

Landau level Hamiltonian in the vortex

lat-tice

2.2.1

Dispersion relation

2.2.2

Effective Hamiltonian

2.2.3

Zeroth Landau level wave functions

2.3

Transmission through the NSN junction

2.3.1

Renormalized charge transfer

2.3.2

Transmission matrix

2.4

Transport properties

2.4.1

Thermal conductance

2.4.2

Electrical conductance

2.4.3

Shot noise

2.4.4

Thermo-electricity

2.5