Weyl points in systems of multiple semiconductor-superconductor quantum dots
WWeyl points in systems of multiple semiconductor-superconductor quantum dots
John P. T. Stenger, David Pekker
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USA (Dated: July 17, 2019)As an analogy to the Weyl point in k-space, we search for energy levels which close at a single pointas a function of a three dimensional parameter space. Such points are topologically protected in thesense that any perturbation which acts on the two level subsystem can be corrected by tuning thecontrol parameters. We find that parameter controlled Weyl points are ubiquitous in semiconductor-superconductor quantum dots and that they are deeply related to Majorana zero modes. In thispaper, we present several semiconductor-superconductor quantum dot devices which host parametercontrolled Weyl points. Further, we show how these points can be observed experimentally viaconductance measurements.
I. INTRODUCTION
Like Dirac and Majorana fermions, Weyl fermions area solution to the relativistic Dirac equation. Further-more, like Dirac and Majorana fermions, Weyl fermionsemerge in certain solid state systems as quasiparticlemodes. In particular, they emerge in Weyl semimetals which are characterized by a band degeneracy point in3-dimensional k-space. This point, known as the Weylpoint, is topologically protected from environmental per-turbations. Any perturbation, which acts only on thetwo degenerate bands can, at most, move the Weyl pointto a different location in k-space.Recently, it has been shown that systems of multi-terminal Josephson junctions can host Weyl points withthe analogy that k-space is replaced by the space of phasedifferences between the terminals.
Just like the tradi-tional Weyl points, these points are immune to perturba-tions which, instead of removing the point, simply moveit around the space of phase differences. As charge is theconjugate variable to flux, it is natural to wonder if Weylpoints can also be found by replacing k-space with chargespace.In this work, we take the analogy even further by look-ing for Weyl points in any three dimensional space of con-trol parameters. In other words, we will look for WeylHamiltonians H = (cid:126)k · (cid:126)σ with (cid:126)k replaced by a three di-mensional set of control parameters. In particular, wewill show how to search for these points in systems ofthree and four quantum dots like the one depicted inFig. 1. The control parameters of the system can beanything that influences the Hamiltonian, however, wewill attempt to use the dot potentials ( (cid:15) , (cid:15) , ... ) whenpossible. The idea being that the potentials arise fromcharging back gates and that charge is the conjugate vari-able to flux. Recently, other control parameters havebeen considered such as magnetic field . Besides beingof fundamental interest, these parameter controlled Weylpoints are a signature that the chosen parameter spacefully controls the Hamiltonian of a two level system. If aWeyl point is found in any 3-dimensional space of controlparameters then it is guaranteed that those parametersprovide access to the entire Hilbert space for those twolevels. Because the Hilbert space is fully controllable, any Superconductor
Lead ∆ ∆ ∆ 𝑡, 𝛼 𝑡, 𝛼 𝜏 𝜀 𝜀 𝜀 𝐵 𝑥 𝐵 𝑥 𝐵 𝑦 𝐵 𝑦 𝐵 𝑦 𝜀 𝜀 a) b) Figure 1. Schematic of a three quantum dot system. a) threedots (blue) are proximity coupled to a parent superconductorwith strength ∆ , the potential on each dot (labeled (cid:15) i ) iscontrolled by a back gate, the dots couple to their neighborwith a hopping strength t and spin orbit coupling α , and theleftmost dot couples to a lead with strength τ . There is amagnetic field gradient in the system. The magnetic field onthe end dots points in the opposite direction and must haveboth an ˆ x and ˆ y component. b) orientation of the ground statein the two level system which forms the Weyl point. The Weylpoint is controlled by the parameter space ( (cid:15) , (cid:15) , B y ). unwanted perturbation to the system can be corrected.To find these parameter controlled Weyl points, weemploy unconventional superconductivity. Generic levelcrossings in quantum dot systems occur on a sheet, ina three dimensional parameter space (e.g. potentials),instead of at a single point. However, there are specialpoints in certain unconventional superconductor devicesin which the level crossing is Weyl-like. In particular,we will look for these Weyl points near the vicinity ofseparated Majorana zero modes (MZMs) in systems ofquantum dots. Kitaev predicted that MZMs will emergefrom spinless p-wave superconducting chains. It hasbeen shown that the Kitaev chain can be simulated bysemiconductor nanowires with spin orbit coupling whichare proximity coupled to a normal s-wave superconductorand which are subject to a magnetic field which is in-linewith the semiconductor.
Signatures of the MZMsin these devices have been observed experimentally.
Another way to simulate the Kitaev chain is with chainsof semiconductor-superconductor quantum dots.
Foran infinite chain of quantum dots, the MZMs are sepa- a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l rated in a continuous range of control parameters. How-ever, for a chain of a finite number of dots, as in Fig. 1,the MZMs only separate at discrete points. Our modelis very similar to this finite Kitaev chain. As we argue insection IV, we expect to find Weyl points in the vicinityof those points where MZMs maximally separate. Thereare a number of multi-dot systems in which Weyl pointsemerge, however, we suggest that the system in Fig. 1 isthe most experimentally accessible.The remainder of the paper is organized as follows.In Sec. II we present two spinless models which hostWeyl points, Sec. II A a 3-dot chain and Sec. II B a 4-dottri-junction. These models are solved analytically. InSec. III we add the spin degree of freedom to the 3-dotdevice and discuss the similarities and differences fromthe spinless version. In Sec III A we present the modelfor the device, in Sec. III B we show how the Weyl pointemerges and discuss which control parameters are rele-vant, in Sec. III C we propose an experimental methodfor observing the Weyl point, in Sec. III D we define thecharge of the Weyl point and discuss its origins, and inSec III E we discuss alternative control parameters. InSec. IV we discuss the relationship between Weyl pointsand Majorana operators. In Sec. V we conclude. II. SPINLESS MODELS
We begin our search with spinless, Kitaev-like mod-els of superconducting quantum dots. Although, spin-less models are unphysical they can be approximate so-lution of spinfull models under magnetic fields and are,therefore, a useful starting point. In sec. III we will studya spinful model and show how it corresponds to the spin-less version.In Sec. IV we observe that the Weyl points often ac-company MZMs. Therefore, we will narrow our searchto the parameter regimes where the MZMs become max-imally separated. The concept of maximal separation isalso discussed in Sec. IV. A. 3-dot chain
The first configuration is a chain of three dots for whichwe will try to use the chemical potential of each dot as the3-dimensional control parameter. However, we will showthat the parameter space of potentials is not enough tofind Weyl points in this device. The Hamiltonian for thissystem is the following, H = (cid:88) i =1 (cid:15) i c † i c i + t (cid:88) i =1 ( c † i c i +1 + c † i +1 c i )+∆ (cid:88) i =1 ( e iφ i c † i c † i +1 + e − iφ i c i +1 c i ) (1)where (cid:15) i is the potential on dot i , t is the couplingbetween dots and ∆ is the p-wave superconducting strength. States with an odd number of electrons (oddparity) and an even number of electrons (even parity)are uncoupled so we can consider the two parity sectorsseparately. Here we will discuss even parity.We allow the superconducting coupling to have a dif-ferent phase φ i depending on which dots are coupled.Consider, for example, when the two segments have op-posite phase ( φ = − φ = − ) and the system is at theKitaev point t = ∆ . For such a system, the even parityeigenstates at (cid:15) = (cid:15) = (cid:15) = 0 are: | ψ (cid:105) = | (cid:105) + 1 √ | (cid:105) − | (cid:105) ) | ψ (cid:105) = | (cid:105) − √ | (cid:105) + | (cid:105) ) | ψ (cid:105) = | (cid:105) − √ | (cid:105) − | (cid:105) ) | ψ (cid:105) = | (cid:105) + 1 √ | (cid:105) + | (cid:105) ) (2)where the first two states are degenerate H | ψ , (cid:105) = −√ | ψ , (cid:105) and the second two states are degenerate H | ψ , (cid:105) = √ | ψ , (cid:105) . For large ∆ , we can ignore thehigh energy states by projecting onto the low energy sub-space. Turning the potentials back on we find, (cid:104) ψ | H | ψ (cid:105) = 12 ( (cid:15) + (cid:15) ) + (cid:15) − √ (cid:104) ψ | H | ψ (cid:105) = 12 ( (cid:15) − (cid:15) ) (cid:104) ψ | H | ψ (cid:105) = 12 ( (cid:15) − (cid:15) ) (cid:104) ψ | H | ψ (cid:105) = 12 (3 (cid:15) + 3 (cid:15) ) + (cid:15) − √ (3)Representing this low energy projection in terms of thePauli matrices, we have: H ≈ (cid:15) − √
2∆ + ( (cid:15) + (cid:15) )(1 − σ z ) + 12 ( (cid:15) − (cid:15) ) σ x . (4)We see that two of the Pauli matrices can be controlledby (cid:15) and (cid:15) , however, (cid:15) does not lift the degeneracy andthere is no way to control σ y using the potentials.On the other hand, we can pick up σ y by controlling thephase of the superconductor. Consider the perturbationHamiltonian; H δ = δc † c † + δ ∗ c c (5)For some δ (cid:28) ∆ . Projecting this onto the low energysubspace we get, (cid:104) ψ | H δ | ψ (cid:105) = 1 √ δ + δ ∗ ) (cid:104) ψ | H δ | ψ (cid:105) = − δ ∗ √ (cid:104) ψ | H δ | ψ (cid:105) = − δ √ (cid:104) ψ | H δ | ψ (cid:105) = 0 (6) 𝑡𝑡𝑡 𝑒 𝑖2𝜋3 Δ Δ 𝑐 † 𝑐 𝑐 𝑐 † 𝑒 𝑖 Δ Figure 2. A four dot device that hosts Weyl points which canbe fully controlled by the potentials of the outer dots. Eachouter dot is coupled to the inner dot by a hopping term ofstrength t and a p-wave superconducting term with a partic-ular phase and a magnitude of ∆ . In terms of the Pauli matrices, the entire Hamiltonianbecomes H + H δ ≈ (cid:15) + (cid:15) + (cid:15) + 1 √ δ ] − √ (7) + 12 ( √ δ ] − (cid:15) − (cid:15) ) σ z + 12 (cid:18) (cid:15) − (cid:15) − Re[ δ ] √ (cid:19) σ x + Im[ δ ] √ σ y (8)Therefore, we can control the full Weyl Hamiltonian byincluding phase control on the superconductors. B. 4-dot tri-junction
Given the results of the last section, it is alluring toask if the Weyl Hamiltonian can be controlled entirelyby potentials if a fourth dot is added to the system. Wewill show that the answer is yes if the dot is added in aspecific way.Consider the system of four dots depicted in Fig 2. TheHamiltonian for the system is H = (cid:88) i =1 (cid:15) i c † i c i + t (cid:88) i =2 ( c † c i + c † i c )+∆ (cid:88) i =2 ( e iφ i c † c † i + e − iφ i c i c ) (9)where we choose φ = 2 π/ , φ = 4 π/ , and φ = 0 .The eigenstates at t = ∆ and (cid:15) = (cid:15) = (cid:15) = (cid:15) = 0 break up into 4 sets of 4-fold degenerate states. Each sethas 2 even parity states and 2 odd parity states. Onceagain, the lowest energy states are separated from thenext set of states by ∆ . Therefore, we can project small perturbations onto the low energy states without worry-ing about the other states. The two lowest energy, evenparity states are: | ψ (cid:105) = i √ (cid:34)(cid:114) | (cid:105) + | (cid:105) ⊗ (cid:0) − | (cid:105) + χ | (cid:105) + ( χ ) ∗ | (cid:105) (cid:1) + | (cid:105) ⊗ ( − i | (cid:105) + χ ∗ | (cid:105) + χ | (cid:105) ) 1 √ (cid:21) | ψ (cid:105) = √ (cid:34)(cid:114) | (cid:105) + | (cid:105) ⊗ (cid:0) − | (cid:105) − ( χ ) ∗ | (cid:105) + χ | (cid:105) (cid:1) + | (cid:105) ⊗ ( − i | (cid:105) + χ | (cid:105) − χ ∗ | (cid:105) ) 1 √ (cid:21) . (10)where χ = exp( iπ/ . Using this basis, we can write thelow energy projection of the Hamiltonian in terms of thePauli matrices. Keeping (cid:15) = 0 but turning on the otherthree potentials, we have: H ≈ −
16 ( (cid:15) + (cid:15) + (cid:15) ) σ z − √ (cid:15) + (cid:15) − (cid:15) ) σ x − √ (cid:15) − (cid:15) ) σ y (11)where we have dropped the terms which multiply theidentity matrix. We see that each of the three Paulimatrices can be controlled by the potentials under thethree outer dots. Since the Hilbert space of two levelsystems is exhausted by the Pauli matrices, any smallperturbation to the system can be corrected by tuningthese three potentials. In other words, the Weyl pointcannot be removed form the three dimensional space ofpotentials by perturbing the system with energies lessthan ∆ . However, if any two superconducting phasesare the same then we lose control over one of the Paulimatrices. III. SPINFUL 3-DOT CHAIN
Kitaev-like spinless models can be obtained as the lowenergy limit of a spinfull model in the presence of spinorbit coupling and a magnetic field. In the last section,we saw that in order to control all three Pauli matricesin the spinless 3-dot chain we needed to control the su-perconducting phase. However, we will see that insteadof controlling the phase difference of superconductors, wecan instead control the orientation of either the spin or-bit coupling or the magnetic field. We suggest that thespinful 3-dot device (see Fig. 1) is the most easily accessi-ble system for experimental study and that the magneticfield orientation is the most easily accessible control pa-rameter.
A. Spinful 3-dot chain model
We study a three dot Hamiltonian with on site poten-tial, nearest neighbor hopping, spin orbit coupling, prox-imity induced Andreev reflection, a magnetic field, andinteractions (see Fig. 1). This is the most general modelwhich hosts MZMs. The interactions are not necessaryto observe a Weyl point but we include it to demonstratethe stability of the Weyl point. H = H (cid:15) + H ∆ + H t + H α + H B + H U (12)Here the onsite potential Hamiltonian is, H (cid:15) = (cid:88) i,σ (cid:15) i c † iσ c iσ (13)where (cid:15) i is the onsite potential, i ∈ { , , } runs over thethree dots, and σ ∈ {↑ , ↓} . The Andreev Hamiltonian is, H ∆ = (cid:88) i (cid:16) ∆ i c † i, ↑ c † i, ↓ + ∆ ∗ i c i, ↓ c ,i ↑ (cid:17) (14)where ∆ i is the induced Andreev reflection amplitude ondot i .The hopping Hamiltonian is, H t = (cid:88) i,σ t i (cid:16) c † i,σ c i +1 ,σ + c † i +1 ,σ c iσ (cid:17) (15)where t i is the nearest neighbor hopping strength be-tween dot i and dot i + 1 . The spin-orbit coupling termis broken up into its two component directions: H α = H α x + H α y (16)with H α x = (cid:88) i α i cos( ξ i ) (cid:16) c † i, ↑ c i +1 , ↓ − c † i, ↓ c i +1 , ↑ + h.c. (cid:17) H α y = i (cid:88) i α i sin( ξ i ) (cid:16) c † i, ↑ c i +1 , ↓ + c † i, ↓ c i +1 , ↑ + h.c. (cid:17) (17)where α i is the overall strength of the spin orbit couplingbetween dot i and i + 1 , and ξ i is the angle that a line be-tween the two dots makes with the x-axis. The magneticfield Hamiltonian is also broken up into its componentdirections, H B = H B x + H B y + H B z (18)with H B x = (cid:88) i B i sin( θ i ) cos( φ i ) (cid:16) c † i, ↑ c i, ↓ + c † i, ↓ c i, ↑ (cid:17) H B y = i (cid:88) i B i sin( θ i ) sin( φ i ) (cid:16) c † i, ↑ c i, ↓ − c † i, ↓ c i, ↑ (cid:17) H B z = (cid:88) i B i cos( θ i ) (cid:16) c † i, ↑ c i, ↑ + c † i, ↓ c i, ↓ (cid:17) (19) where B i is the magnitude of the magnetic field on dot i , θ i is the polar angle, and φ i is the azimuthal angle ofthe field on dot i . The interaction term is, H U = (cid:88) i U i c † i, ↑ c i, ↑ c † i, ↓ c i, ↓ (20)where U i is the interaction strength on dot i .In what follows, we will assume that all three dots areisomorphic except where otherwise specified and we willrefer to isomorphic parameters by dropping the site index(e.g. we will refer to ∆ when ∆ = ∆ = ∆ ). B. Energy levels for the spinfull 3-dot chain
In order to obtain Weyl points, we want to mimic thespinless model already discussed (Eq. 1). There we hadthat the two p-wave pairing terms had opposite sign. Wecan mimic this behaviour, without applying a phase dif-ference directly to the superconductors, by instead con-trolling the magnetic field orientation under each dot. Wetake this approach as we expect it to be experimentallyless challenging than controlling the phase differences. Insubsection III E we show that controlling the supercon-ducting phase and the spin-orbit angle are both viablealternatives to the magnetic field orientation.Let us take θ = 0 and φ = π while φ = φ = 0 .Figure 3 shows the energy difference between the firsttwo even parity states as a function of (cid:15) and (cid:15) . Justlike in the spinless case, (cid:15) does not open or close theWeyl point. We see that the Weyl point drops down andpersists for B > ∆ . This topological phase transitioncoincides with the appearance of maximally separatedMZMs.As we have mentioned, we want to use a magnetic fieldgradient as the third control parameter. We have al-ready used a magnetic field gradient in the ˆ x -direction todrive the topological phase transition. Therefore we usea gradient in the ˆ y -direction as the third control param-eter. Keeping θ = 0 , let us take φ = Φ + π , φ = 0 ,and φ = − Φ and let us define B x = B cos(Φ) and B y = B sin(Φ) . Keeping B x constant, we can controlthe third axis of the Weyl point with B y .Although the magnetic field has to rotate over a rathershort distance, the gradient should be experimentallyachievable. The strength of the gradient depends on themagnetic g-factor and the distance between the dots. Al-though there is no strict requirements for the distancebetween the dots, let us say that the dots are about µ m apart which is the a typical length for Majo-rana nanowires. Then from experimentally observed g-factors , we need a gradient in the x-direction of about0.5 T/ µ m. This is rather large but is predicted to beachievable using nanomagnets . On the other hand, themagnetic gradient in the y-direction can be much smallerand should be achievable using electromagnets .Figure 4 shows the energy difference between the firsttwo even parity states as a function of various control a)b)c) 𝜀 𝜀 E Figure 3. Emergence of the Weyl point. Each panel showsthe energy difference, E, between the first two even paritystates as a function of the potential under dot 1, (cid:15) , anddot 3, (cid:15) , for different values of the magnetic field, B x . Allenergy parameters are in units of the induced gap, . (a) B x = 0 . . (b) B x = 1 . . (c) B x = 2∆ . For magneticfields below the Andreev coupling B x < ∆ , as in panel (a),the energy levels are gaped in the (cid:15) - (cid:15) plane and there isno Weyl point. The Weyl point emerges above B X > ∆ ,however, near B x = ∆ , as in panel (b), the point is smeared.On the other hand, the Weyl point is sharp in the top rightcorner of panel (c). parameters. Panel (a) shows the Weyl cone as a functionof B y and (cid:15) while panel (b) shows the cone as a functionof (cid:15) and (cid:15) . We conclude that we have linear dispersionas a function of all three control parameters ( (cid:15) , (cid:15) , B y ).Panel (c) shows that the Weyl point vanishes from thetwo dimensional (cid:15) - (cid:15) space when B y is turned on. In a) .95.94.93.92.91 𝜀 𝐵 𝑦 𝜀 c) 𝜀 𝜀 d) 𝜀 = 𝜀 b) 𝜀 𝜀 E Figure 4. Mapping out the Weyl point. Each panel shows theenergy difference, E, between the first two even parity statesas a function of various control parameters which are definedin the main text. All parameters are in units of . (a) (cid:15) = 1 . , (b) B y = 0 , (c) B y = 0 . , (d) B y = 0 . Panels(a) and (b) show two dimensional slices of the Weyl cone.These show that all three parameters ( (cid:15) , (cid:15) , B y ) control theenergy level separation. However, to prove that there is aWeyl point we need panel (c) which shows that B y opens thegap everywhere in the 2D ( (cid:15) , (cid:15) ) space. Panel (d) shows that (cid:15) does not open the gap and therefore cannot be used as acontrol parameter. other words, there is no line B y = a(cid:15) + b(cid:15) (for arbitrary a and b ) on which the point stays closed. Panel (d) showsthat (cid:15) does not open or close the Weyl point. The Weylpoint’s immunity to (cid:15) means that this potential cannotbe used as a third control parameter.The immunity of the Weyl point to (cid:15) is also an exam-ple of the topological protection of the point. Just as thestandard Weyl point is a source of Berry curvature in k-space, the parameter controlled Weyl point is a source ofthe curvature in parameter space. Since we have controlover all three Pauli matrices, there are no small pertur-bations (compared to ∆ ) which removes the Weyl pointfrom the 3-dimensional parameter space. On the otherhand, perturbations larger than ∆ can be damaging intwo ways. First, they could simply move the Weyl pointto locations in parameter space which are out of the rangeof realistic tuning parameters. Second, they can closethe gap which could potentially destroy they Weyl point.Figure 5 shows the effect of several types of perturba-tions. In each panel, the yellow curve is the unperturbedenergy difference between the first and second even paritystate and the blue curve is the perturbed energy differ-ence. In fact, we see that even some perturbations onthe order of ∆ do not remove the Weyl point. In Fig. 5a,for example, the interaction strength is tuned to U = ∆ .In Fig. 5b we perturb the relative hopping strength be-tween dots. This type of perturbation, which breaks the a) b) c) 𝜀 𝜀 d) E 0.10.2 𝜀 𝜀 𝜀 𝜀 𝜀 𝜀 Figure 5. Protection of the Weyl point. Each panel shows theenergy difference, E, between the first two even parity statesunder various small perturbations overlaid with the unper-turbed case. All parameters are in units of . The yellowcurve in each panel is the unperturbed case. The blue curve isfor (a) U = ∆ , (b) t = 0 . t = 0 . t and α = 0 . α = 0 . α ,(c) B = 0 . B = 0 . B , (d) ∆ = 1 . and ∆ = 1 . isomorphism between dots, takes the Weyl point off ofthe diagonal (cid:15) = (cid:15) . This behaviour can also be seenin Fig 5c and Fig. 5d where we break the magnetic andsuperconducting isomorphisms respectively. C. Measuring the Weyl point
The Weyl point in the three dot chain can be observedexperimentally by tunnel coupling a metallic lead to oneof the dots in the system. In this setup, electrons tun-nel into the dots from the lead and then Andreev reflectoff of the superconductor contacts. We calculate the dif-ferential conductance using the well established masterequation formalism. ˙ P i = (cid:88) j (cid:0) Γ j → i P j − Γ i → j P i (cid:1) (21)where P i is the probability of the system being in state i . We use the steady state approximation ˙ P i = 0 andconnect the lead to the first dot. The rates Γ i → j aregiven by, Γ i → j = Γ i → jp + Γ i → jh (22)with Γ i → jp = τ f ( E j − E i − eV ) (cid:88) σ | (cid:104) j | c † σ | i (cid:105) | (23) Γ i → jh = τ (1 − f ( E j − E i − eV )) (cid:88) σ | (cid:104) j | c σ | i (cid:105) | where | i (cid:105) is the eigenvector with eigenvalue E i , f ( ω ) is theFermi distribution, and τ is the coupling between the lead 𝑉 𝛿 𝐵 𝑦 𝜀 𝜀 𝛿 −0.05 𝑉 𝛿 𝜀 𝜀 𝐵 𝑦 𝛿 𝑉 −0.05 𝐵 𝑦 𝜀 𝜀 𝛿 Figure 6. Measuring the Weyl point via tunnel conductance.Each scan is done in a different direction in the three dimen-sional parameter space. The top row shows the different paths δ i ( (cid:15) , (cid:15) , B y ) in the 3-dimensional parameter space that areused. The bottom row shows the differential conductance asa function of bias voltage V and the corresponding parameter δ i . All parameters are in units of 2 ∆ . and the first dot. Once we know all of the probabilities P i , we can calculate the current using, I = (cid:88) i,j P i (cid:16) Γ i → jp − Γ i → jh (cid:17) (24)and the differential conductance is simply dI/dV whichwe calculate numerically.Results of the transport calculation are shown in Fig. 6.The goal in experiment will be to check that the en-ergy gap, between the first and second even parity states,opens along all paths through the center of the three di-mensional parameter space. The data in Fig 6 is takenalong the paths δ i ( (cid:15) , (cid:15) , B y ) depicted on the top row. Ithappens that the ground state of this system is an oddparity state which is separated in energy from the firsteven parity state at the Weyl point. Therefore, we seetwo finite bias conductance curves (bottom row) whichcross at the Weyl point and reopen regardless of the pa-rameter path that is chosen.Note that the MZMs maximally separate when the oddparity ground state and one of the even parity statesare degenerate. This can be seen in Fig 6 at around δ ≈ − . in all three panels. As expected (see Sec. IV),the Weyl point appears in the vicinity of maximally sep-arated MZMs. D. Curvature of the energy degeneracies
Weyl points in k-space are characterized by a non-zero,integer Chern number. Similarly, we can define an anal-ogous integer for the parameter controlled Weyl point byintegrating over parameter space instead of k-space. Justlike the normal Chern number, we can define this integral 𝜀 𝜀 𝐵 𝑦 𝜀 𝜀 𝐵 𝑦 𝜀 𝜀 𝐵 𝑦 a) b) c) Figure 7. Surfaces in the three dimensional parameter spacewhere the first and second even parity states are degenerate.Going from right to left B x = 0 . , . , . , matching thevalues in Fig. 3. The global charge, calculated by integratingover the blue box, is always C = +2 . The charges enclosedby the green boxes are C = +1 for both panels (a) and (c). using a gauge field given by the Berry connection. (cid:126)A n ( (cid:126)r ) = i (cid:104) n ( (cid:126)r ) | (cid:126) ∇ r | n ( (cid:126)r ) (cid:105) (25)where | n (cid:105) is the n th energy level eigenstate and (cid:126)r = { (cid:15) , (cid:15) , B y } . The local curvature is then given by thecurl of this gauge field, (cid:126) Ω n ( (cid:126)r ) = (cid:126) ∇ r × (cid:126)A n ( (cid:126)r ) . (26)Integrating the curvature on a closed surface, we get aninteger which is zero if the surface does not enclose adegeneracies. We will call this integer the charge of adegeneracy surface or point, C n = 12 π (cid:73) d(cid:126)S · (cid:126) Ω n . (27)Let us apply this formalism to better understand the pa-rameter controlled Weyl point of our three dot device(seen in e.g. Fig 3c).In Fig. 7 we show the surfaces of degeneracy betweenthe first two even parity states. We see that the Weylpoint emerges when two positively charged degeneracysurfaces collide in the B y = 0 . plane. After the col-lision, the Weyl point breaks off from the surface andtakes half of the total charge with it. The global chargeis unchanged throughout the entire process. In principle,the positively charged Weyl point could be removed bycombining with a negatively charged degeneracy point.However, there are no negative charges within the pa-rameter regimes we have scanned. E. Using spin-orbit angle or phase differences asthe third control parameter
So far we have used the magnetic field gradient as thethird control parameter. However, there are at least twoother possible choices, namely a superconducting phasedifference between dots or a spin-orbit angle. These pa-rameters are likely more difficult to control experimen-tally than the magnetic field gradient but they are equallyvalid. 𝜋 𝜋2 𝜋 𝜉𝜙 𝑠𝑐 𝜀 𝜀 𝜀 𝜀 = 𝜀 𝜀 = 𝜀 𝜀 E a) b) c) d) Figure 8. Energy difference between the first two even paritystates. a) as a function of the spin-orbit angle ξ and thepotentials (cid:15) = (cid:15) . b) as a function of the dot potentials (cid:15) and (cid:15) for ξ = π/ . c) as a function of the phase of thethird superconducting contact φ sc = − i ln(∆ / | ∆ | ) while theother phases are zero, Im(∆ ) = Im(∆ ) = 0 . d) as a functionof the dot potentials for φ sc = π/ In Fig. 8 we plot the energy difference between the firsttwo even parity states showing that both the supercon-ducting phase and the spin-orbit angle can be used tocontrol the Weyl point. Figure 8a shows the Weyl pointin the parameter space ( (cid:15) , (cid:15) , ξ ) where ξ is the spin-orbit angle from Eq. 17. We see in Fig. 8b that ξ opensthe gap everywhere in the 2-dimensional ( (cid:15) , (cid:15) ) spacemeaning that it can truly be used as the third control pa-rameter. Similarly, in Fig. 8c we see the Weyl point in theparameter space ( (cid:15) , (cid:15) , φ sc ) where φ sc = − i ln(∆ / | ∆ | ) is the superconducting phase angle on the first dot. Weset the phase on the other dots to zero. In Fig. 8d wesee that φ sc also opens the gap everywhere in the 2-dimensional space ( (cid:15) , (cid:15) ) and is therefore another validcontrol parameter. IV. WEYL POINT FROM MAXIMALLYSEPARATED MAJORANA MODES
We have seen that Weyl points emerge in several p-wave superconducting devices. This is not an accident.In fact, Weyl Hamiltonians arise naturally from a com-parison of the algebra of Majorana operators and that ofthe Pauli matrices (i.e. the quaternion algebra). Indeed,even the Majorana operators themselves form a quater-nion algebra. Consider a pair of Majorana operators γ x and γ y which form the parity operator P = iγ x γ y . Anypair of these three operators γ x , γ y , and P multiplied to-gether gives the third in exactly the same way as the Paulimatrices. Therefore, the Pauli matrices are a represen-tation of a pair of Majorana operators and their parity.However, Hamiltonians come with pairs of Majorana op- 𝛾 𝑎 𝛾 𝑏 𝛾 𝑐 𝛾 𝑙 𝛾 𝑚 𝛾 𝑛 𝛾 𝑎 𝛾 𝑏 𝛾 𝑐 𝛾 𝑑 a) b) Figure 9. Topological superconducting nanowires (blue) withMajorana end modes (red and green) in configurations thathost Weyl points. The green Majoranas are active in the for-mation of the Weyl point while the red Majoranas are auxil-iary. erators, not single operators. Therefore, to reproduce theWeyl Hamiltonian, we need to find a quaternion algebrathat involves only even numbers of Majorana operators.One such algebra involves pairs of three different Ma-jorana operators. Consider for example the system inFig. 9a where there are six Majorana operators but onlythree of them are coupled ( γ a , γ b , γ c ). Now consider theoperators iγ a γ b , iγ a γ c , and iγ b γ c . Once again, these op-erators form a quaternion algebra and can be representedby the Pauli matrices. Therefore, the Hamiltonian H a = it ab γ a γ b + it ac γ a γ c + it bc γ b γ c (28)can be represented by the Pauli matrices, ¯ H a = (cid:126)t · (cid:126)σ (29)where (cid:126)t = ( t ab , t ac , t bc ) . Of course, the Hilbert spaceof H a is twice that of ¯ H a . However, the even and oddsubspaces are degenerate and are both described by ¯ H a .The same algebra can be found in other geometries aswell. Consider, for example, the system in Fig. 9b. TheHamiltonian that describes this system is, H b = it ad γ d γ a + it bd γ b γ d + it cd γ c γ d . (30)At first glance, these operators do not obey the quater-nion algebra, for example, ( iγ d γ a )( iγ b γ d ) (cid:54) = ± γ c γ d . How-ever, since parity is conserved, the total parity operator P = γ a γ b γ c γ d is simply a number (either ± ). Thus, wecan replace pairs of operators with the opposite pair (e.g. γ d γ a = P γ b γ c ). Then the Hamiltonian can be rewrittenas H b = iP ( t ad γ b γ c + t bd γ a γ c + t cd γ a γ b ) (31)which is simply the Hamiltonian in Eq. 28 and is thereforea representation of the Weyl Hamiltonian.Of course these are systems of separated MZMs. Insystems of quantum dots, we do not expect Majoranasto fully separate. However, Weyl points still emerge nearnear "maximally separated MZMs". As we study chainsof only a few quantum dots, the MZMs do not separateover a continuous range of parameters but only at dis-crete points. One such point can be seen in Fig. 6 where Dot 1 Dot 2 Dot 3 Dot 1 Dot 2 Dot 3 Dot 1 Dot 2 Dot 3 P r o b a b ili t y P r o b a b ili t y 𝜀 = 0.75 𝜀 = 0.85 𝜀 = 0.95𝜀 = 0.75 𝜀 = 0.85 𝜀 = 0.95 Figure 10. Probability distribution of the two Majorana de-compositions. The top row shows the probability distribution P x,iσ (yellow) and P y,iσ (blue) for a system with a magneticfield gradient as in the main text. Here the Majoranas cannotseparate to different dots yet they still separate into differentspin states at (cid:15) = (cid:15) = (cid:15) = (cid:15) = 0 . . For comparison, in thebottom row we show an identical system with the exceptionthat the magnetic field is uniform. Here the two Majoranadecompositions separate into the dots at the opposite ends ofthe chain. (cid:15) is given in units of . we claimed that the MZMs are maximally separated. Letus now clarify what we mean by maximal separation. Wedefine a pair of Majorana operators as ( γ x,iσ , γ y,iσ ) suchthat c iσ = 1 / γ x,iσ + iγ y,iσ ) destroys an electron on dot i with spin σ . Furthermore, let | ψ (cid:105) be the ground stateof our system and | ψ (cid:105) be the first excited state. Thenwe can decompose the excited state into the two typesof Majorana operators. We define the two probabilitydistributions, P x,iσ = | (cid:104) ψ | γ x,iσ | ψ (cid:105) | P y,iσ = | (cid:104) ψ | γ y,iσ | ψ (cid:105) | . (32)By maximal separation, we mean that we are at the pointin parameter space where these two distributions overlapthe least.In Fig. 10 we show the probability distributions fortwo different systems as a function of the potential (cid:15) = (cid:15) = (cid:15) = (cid:15) on the dots. The top row shows the prob-ability distributions for the spinful 3-dot system of themain text. Because there is a magnetic field gradient,the MZMs cannot separate to different dots, however, at (cid:15) = 1 . the MZMs maximally separate by occupyingdifferent spin sectors. The bottom row shows an iden-tical system with the exception that the magnetic fieldis uniform. We see that (cid:15) = 1 . is indeed the pointof maximal separation. As the MZMs have more roomto move in this case, they separate to either side of thechain. V. CONCLUSION
We have shown that Weyl points arise naturally inmulti-quantum dot systems with superconducting leadstuned to the vicinity of maximally separated MZMs.We propose a measurement scheme wherein current istunneled into the first dot in a three dot chain. TheWeyl point is visualized as the crossing of dI/dV peaksat a single point in the 3-dimensional parameter space.Generically, we expect dI/dV peaks to cross along 2-dimensional sheets in a 3-dimensional parameter space. If instead the peaks close at only a single point, then theparameter space can be used to control each Pauli ma-trix individually and, therefore, any perturbation to thesystem can be corrected by the control parameters.We thank Sergey Frolov for helpful discussions. Thiswork is supported by NSF PIRE-1743717. C. Herring, Phys. Rev. , 365 (1937). K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B ,075129 (2011). A. A. Burkov and L. Balents, Phys. Rev. Lett. , 127205(2011). S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian,C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang,A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, andM. Z. Hasan, Science , 613 (2015). B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao,J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen,Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X ,031013 (2015). L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos,and M. Soljačić, Science , 622 (2015). J. S. Meyer and M. Houzet, Phys. Rev. Lett. , 136807(2017). R.-P. Riwar, M. Houzet, J. S. Meyer, and Y. V. Nazarov,Nature Communictaions , 11167 (2016). H.-Y. Xie, M. G. Vavilov, and A. Levchenko, Phys. Rev.B , 161406 (2017). H.-Y. Xie, M. G. Vavilov, and A. Levchenko, Phys. Rev.B , 035443 (2018). T. Yokoyama and Y. V. Nazarov, Phys. Rev. B , 155437(2015). Z. Scherübl, A. Pályi, G. Frank, I. Lukács, G. Fülöp,B. Fülöp, J. Nygård, K. Watanabe, T. Taniguchi,G. Zaránd, and S. Csonka, arXiv:1804.06447 (2018). A. Y. Kitaev, Physics-Uspekhi , 131 (2001). C. Nayak, S. H. Simon, A. Stern, M. Freedman, andS. Das Sarma, Rev. Mod. Phys. , 1083 (2008). C. Beenakker, Annual Review of Condensed MatterPhysics , 113 (2013). J. Alicea, Reports on Progress in Physics , 076501(2012). J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. , 040502 (2010). J. Alicea, Phys. Rev. B , 125318 (2010). R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev.Lett. , 077001 (2010). Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. , 177002 (2010). J. D. Sau, S. Tewari, R. M. Lutchyn, T. D. Stanescu, andS. Das Sarma, Phys. Rev. B , 214509 (2010). A. Ptok, A. Kobiałka, and T. Domański, Phys. Rev. B , 195430 (2017). V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P.A. M. Bakkers, and L. P. Kouwenhoven, Science ,1003 (2012). J. Chen, P. Yu, J. Stenger, M. Hocevar, D. Car, S. R.Plissard, E. P. A. M. Bakkers, T. D. Stanescu, andS. M. Frolov, Science Advances (2017), 10.1126/sci-adv.1701476. M. T. Deng, S. Vaitiekenas, E. B. Hansen, J. Danon,M. Leijnse, K. Flensberg, J. Nygård, P. Krogstrup, andC. M. Marcus, Science , 1557 (2016). S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon,J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani,Science , 602 (2014). I. C. Fulga, A. Haim, A. R. Akhmerov, and Y. Oreg, NewJournal of Physics , 045020 (2013). J. P. T. Stenger, B. D. Woods, S. M. Frolov, and T. D.Stanescu, Phys. Rev. B , 085407 (2018). P. Zhang and F. Nori, New Journal of Physics , 043033(2016). Z. Su, A. B. Tacla, M. Hocevar, D. Car, S. R. Plissard,E. P. A. M. Bakkers, A. J. Daley, D. Pekker, and S. M.Frolov, Nature Communications , 585 (2017). L. Maurer, J. Gamble, L. Tracy, S. Eley, and T. Lu, Phys.Rev. Applied , 054071 (2018). M. Drndić, K. S. Johnson, J. H. Thywissen, M. Prentiss,and R. M. Westervelt, Applied Physics Letters72