Abstract
Recent years have brought an explosion of activities in the research of topological aspects of condensedmatter systems. Topological phases of matter are accompanied by protected surface states or exotic degenerate excitations such as Majorana modes or Haldane’s localized spinons. Topologically protected degeneracies can, however, also appear in the bulk. An intriguing example is provided by Weyl semimetals, where topologically protected electronic band degeneracies and exotic surface states emerge even in the absence of interactions. Here we demonstrate experimentally and theoretically that Weyl degeneracies appear naturally in an interacting quantum dot system, for specific values of the external magnetic field. These magnetic Weyl points are robust against spin–orbit coupling unavoidably present in most quantum dot devices. Our transport experiments through an InAs doubledot device placed in magnetic field reveal the presence of a pair of Weyl points, exhibiting a robust groundstate degeneracy and a corresponding protected Kondo effect.
Introduction
Mathematical tools borrowed from topology find more and more applications in contemporary condensedmatter physics^{1,2,3}. In Weyl semimetals^{4,5}, e.g., the electronic band structure exhibits isolated degeneracy points^{6}, where two bands touch; this effect is also accompanied by exotic surface states^{5,7}. In threedimensional systems, these degeneracy points can be protected by topology—and classified by a suitably chosen Chern number: continuous perturbations may displace these Weyl points in momentum space, but cannot break their degeneracy. Weyl pointrelated degeneracies of electronic states in molecules termed conical intersections are also thought to play a fundamental role in various phenomena in photochemistry^{8}. They have also been predicted to appear in the context of multiterminal Josephson junctions^{9}, and that of photonics^{10}, and have also been engineered and demonstrated in coupled superconducting qubits^{11,12}.
The simplest example of a Weyl point arises when a spin1/2 electron is placed in a homogeneous magnetic field (see Fig. 1a–c). In this example, the parameter space is spanned by the magneticfield vector B = (B_{x}, B_{y}, B_{z}) and the two energy eigenstates are degenerate at B = 0. We can associate a nonzero topological charge to this degeneracy point: the groundstate Chern number \(C({\cal{S}}) = 1\) evaluated on an arbitrary closed surface \({\cal{S}}\) surrounding the degeneracy point (see Methods for details). This nonzero Chern number promotes this B = 0 degeneracy point to a Weyl point and underlines the robustness of its (Kramers) degeneracy against perturbations.
Let us now turn to the case of two coupled interacting spins and investigate the fate of Weyl points in the presence of—possibly strong—spin–orbit interaction (SOI)^{13,14}. In the most general case, this system is described by the Hamiltonian H = H_{Z} + H_{int}, where \(H_{\mathrm{Z}} = \mathop {\sum}_{\alpha ,\beta } {\mu _{\mathrm{B}}} B_\alpha (\hat g_{\mathrm{L}}^{\alpha \beta }S_{\mathrm{L}}^\beta + \hat g_{\mathrm{R}}^{\alpha \beta }S_{\mathrm{R}}^\beta )\) describes the Zeeman coupling and \(H_{{\mathrm{int}}} = \mathop {\sum}_{\alpha ,\beta } {\hat J^{\alpha \beta }} {\mkern 1mu} S_{\mathrm{L}}^\alpha S_{\mathrm{R}}^\beta\) is just the exchange interaction. The SOI appears here through the anisotropic and dot dependent gtensors, \(\widehat {\mathbf{g}}_{{\mathrm{L}}/{\mathrm{R}}}\), and the anisotropic exchange coupling tensor, \(\widehat {\mathbf{J}}\).
In the absence of SOI (Fig. 1d–f), the gtensors as well as the exchange coupling are just scalars, \(\widehat {\mathbf{g}}_{{\mathrm{L}}/{\mathrm{R}}} \to g_{{\mathrm{L}}/{\mathrm{R}}}\) and \(\widehat {\mathbf{J}} \to J\). The energy spectrum (Fig. 1e) is therefore isotropic as a function of the magnetic field. For an antiferromagnetic coupling, the ground state becomes degenerate at a sphere of radius B = J/(μ_{B}g), where a singlet to triplet transition occurs (Fig. 1f).
According to naive expectations, a small SOI should mix the singlet and triplet states close to the sphere of degeneracies, and thereby remove the degeneracy immediately (Fig. 1g–i). Simple models, however, indicate that this may not be the case. For example, quasitwo and quasionedimensional quantum dots are often described in terms of a Rashba field, B_{so} \(\Vert\)E × p, with E a substrate or gateinduced (usually) vertical electric field, and p the momentum of the carriers^{15}. In such models, singlet–triplet mixing is found to be absent if the external field is aligned with the effective Rashba fields, whereas the degeneracy is lifted in other external field directions.
Unfortunately, the somewhat simplistic Rashbafield model is not quite appropriate for threedimensional or disordered quantum dots, similar to ours: it fails to account, e.g., for the strongly distorted, dotdependent gtensors measured earlier in InAs nanowire double quantum dots (DQDs)^{16,17}, as well as in our device, which are neither aligned with each other nor correlated with the geometry of the sample.
In this work, we consider two interacting spins confined in a spin–orbitcoupled DQD. Theoretically, we study a generic Hamiltonian with spindependent tunneling and distorted gtensors, and use topological considerations to provide the conditions for the existence of groundstate degeneracy points in the threedimensional magneticfield parameter space. Experimentally, we demonstrate the presence of two such magnetic Weyl points by doing transport spectroscopy in an InAs DQD as we explore the magneticfield parameter space. Furthermore, we show that these degeneracies lead to a twoelectron Kondo effect. Our results establish generic, robust, topologically protected degeneracy points, insensitive to microscopic details, in spin–orbitcoupled interacting spin systems.
Results
Magnetic Weyl points are topologically protected
We show now that the presence of degeneracy points is not a consequence of an oversimplified description or the simple dot geometries, but is rooted in topology. To show this, we consider first the most generic spin–orbitcoupled twospin Hamiltonian H above. Consider a sphere in the magnetic parameter space, centered at the origin, with a radius approaching infinity, and calculate the corresponding groundstate Chern number, C_{∞}. In this limit, each spin follows just the external field, yielding a Chern number, \(C_\infty = {\mathrm{sign}}\,{\mathrm{det}}(\widehat {\mathbf{g}}_{\mathrm{L}}) + {\mathrm{sign}}\,{\mathrm{det}}(\widehat {\mathbf{g}}_{\mathrm{R}}) \equiv C_\infty ^{\mathrm{L}} + C_\infty ^{\mathrm{R}}\). As long as both gtensor determinants are positive, we simply obtain C_{∞} = 2. As, by definition, C_{∞} counts the total topological charge carried by the degeneracy points in the entire magnetic field space, the finiteness of C_{∞} signals the existence of groundstate degeneracies with nonzero topological charge, typically located at single points, which we call magnetic Weyl points. Time reversal constrains the locations of these points (see Methods): a degeneracy point at B_{0} must have a partner at −B_{0}, carrying the same topological charge.
By these topological considerations, we expect that two Weyl points at ±B_{0} carry the total topological charge C_{∞} = 2. Using random spin Hamiltonians, we have numerically verified that this scenario of two magnetic Weyl points is generic and is indeed realized in cca. 98% of randomly generated twospin Hamiltonians. In the remaining cca. 2% of cases, the number of Weyl points is six, but the sum of their topological charges remains two.
Experimental observation of magnetic Weyl points
To demonstrate experimentally the existence of magnetic Weyl points in a spin–orbitcoupled interacting twospin system, we carried out lowtemperature electric transport measurements through a serial InAs nanowire DQD^{16,18,19,20,21} in the temperature range 60–300 mK. The setup is sketched in Fig. 2a (for sample fabrication and characterization see Methods and Supplementary Note 1). Alternative experimental techniques to explore these magnetic Weyl points are Landau–Zener^{22,23,24} or electrically driven spin resonance spectroscopy^{15}, as well as Pauliblockade spectroscopy^{21}, as applied to various twoelectron doubledot devices.
In the experiments, we focused on the (1,1) charge configuration of the device (see Fig. 2b), where the DQD contains two spatially separated and exchangecoupled spins. In this region, we expect that the ground state of the system is a singlet and the first excited state separated by ΔE ≈ J_{0} is a triplet (see Supplementary Note 2). The finite exchange splitting J_{0} ≈ 0.055 meV is demonstrated by the biasdependent differential conductance data presented in Fig. 2c. At the center of the (1,1) configuration, i.e., along the vertical dashed line, the conductance is suppressed at small biases, but increases once the bias is sufficiently high to induce inelastic cotunneling processes populating the triplet states. The differential conductance G = dI/dV_{bias} (white curve) has therefore two finitebias peaks (white lines) placed symmetrically at the first excited state of the DQD, at eV_{bias} = ±ΔE ≈ ± J_{0}. The asymmetry G(V_{bias}) ≠ G(−V_{bias}) can be attributed to asymmetric coupling to the leads.
We now switch on the magnetic field to tune the relative energies of the ground and excited states, and explore by the cotunneling spectroscopy outlined above, how the energy gap ΔE = ΔE(B) between the ground and first excited states varies with the field (Fig. 1h)^{25,26,27}. Two examples are shown in Fig. 2d, e, where we present the conductance G(B, V_{bias}) for magnetic fields B = B(sinθ cosϕ, sinθ sinϕ, cosθ) oriented along two different directions (see reference frame in Fig. 2a).
In Fig. 2d, e, the magneticfield dependence of the gap ΔE(B) is traced by the largeconductance features close to zero bias, also indicated by solid lines. The observed behavior is markedly different in the two cases: Fig. 2d displays a behavior in line with the naive singlet–triplet mixing argument and the gap remains open for all values of B. In Fig. 2e, however, the gap closes at around B_{0} = 70 mT, where a zerobias conductance peak develops (white continuous line), suggesting that this magneticfield vector corresponds to a magnetic Weyl point.
The scenario of the two magnetic Weyl points at opposite magnetic fields ±B_{0}, fits perfectly our experimental observations. To demonstrate this, we display the conductance G(θ, V_{bias}) in Fig. 2f for a fixed magneticfield length B = 75 mT and ϕ = 90°, while varying the polar angle θ over a range of 360° (see also Supplementary Note 3 for additional data). Our data indicate groundstate degeneracies at two opposite isolated points, θ ≈ 130° and θ ≈ 310°, but a finite gap otherwise. The solid lines in Fig. 2d–f, indicating the gap, are not only guides to the eye: they were computed from a twosite Hubbard model (see Methods), with parameters adjusted to yield a good overall match to experimental observations. Figure 2g visualizes the Berry curvature fields (see Methods) and the associated topological charges at the two Weyl points, as computed numerically from this twosite Hubbard model.
We support further the scenario of the two magnetic Weyl points by showing a more complete scan of the zerobias magnetoconductance ΔG(B) = G(B) − G(B = 0) in Fig. 2h. The four panels of Fig. 2h correspond to four azimuthal angles, ϕ ∈ {−45°, 0°, 45°, 90°} of the magnetic field, as depicted in the sketch on the left side of panel h. Each panel of Fig. 2h displays the zerobias magnetoconductance ΔG(θ, B) as the function of the polar angle θ and strength B of the magnetic field. The most prominent local maximum in the bottom right panel of Fig. 2h indicates that the twoelectron double dot has a magnetic Weyl point close to that region, ϕ ≈ 90°, θ ≈ 130°, and B ≈ 70 mT (also seen in Fig. 2e).
Twoelectron Kondo effect at the Weyl points
In our device, the observed groundstate degeneracy is accompanied by an increased zerobias conductance in the vicinity of the magnetic Weyl points (see white curve on Fig. 2e). This increased conductance is due to a twoelectron Kondo effect^{25,26,27,28,29}, as clearly revealed by the temperature and voltage dependence of our transport data in Fig. 3, complying with the Kondo behavior seen in other experiments^{30,31,32}. The differential conductance at the Weyl point exhibits, in particular, a pronounced zerobias Kondo peak with a height increasing upon decreasing temperature (see Fig. 3a). This increased lowtemperature conductance appears to be characteristic of the whole charge (1,1) domain, as demonstrated in Fig. 3b, presenting the temperature dependence of the zerobias conductance along the diagonal dashed line in Fig. 2b. In contrast, in the regions corresponding to (2,0) and (0,2) charge configurations, the ground state is unique; there the conductance shows thermal activation and is suppressed with decreasing temperatures.
Discussion
So far, we have argued that for two interacting spins, the appearance of groundstate degeneracies at a pair of timereversed magnetic Weyl points is generic and robust due to topological protection. The two Weyl points can change their positions in the magneticfield space as the twospin Hamiltonian is modified, but they cannot disappear, i.e., the corresponding degeneracy cannot split. Of course, this protection carries over to the observed Kondo effect.
In our experiments, we have tested the robustness of the Weyl points in two different ways. (1) We have also investigated other doubledot charge states and verified that the signatures of two Weyl points are indeed present there too, as predicted, although the quantum dot parameters as well as the location of the Weyl points changed considerably. (2) We have modified the microscopic parameters within the same charge state by varying the gate potentials. The Weyl points were displaced but have never disappeared (for further details, see Supplementary Notes 4 and 5).
Interestingly, in our simulations, we can find cases where four additional Weyl points emerge. In such cases, four out of the six Weyl points have a topological charge +1, whereas two of them have charge −1, adding up to a total topological charge +2, in agreement with our sum rule. This is shown in Fig. 4a, where the red (blue) spheres represent Weyl points with Chern number +1 (−1). As discussed below, this case—never realized in the simple Rashbafield model—may be relevant, e.g., in Si double dots^{33,34}.
The numerical simulation in Fig. 4a illustrates further topological protection: Weyl points can move around upon continuous deformation of the Hamiltonian and degeneracies of opposite topological charge can annihilate each other, but the total topological charge remains unchanged and assures the presence of degeneracies. A particular example of an annihilation process is presented in Fig. 4a, where arrows indicate the motion of degeneracy points, while their colors refer to their charge (see Supplementary Note 6 for a description of the deformation protocol).
The creation and annihilation of magnetic Weyl points should be observable in doubledot devices. Although using completely random parameters the probability of finding six Weyl points is only cca. 2% as illustrated of Fig. 4b, engineering the parameters allows for increasing ratio of six Weyl points. An analysis of our InAs doubledot setup using the experimentally determined gtensors and the ratio of spinconserving and spinrotating tunnel amplitudes (see Supplementary Note 6) yields that the probability of creating additional Weyl points is still low, cca. 10–20%. However, in other experimental systems with lower spinrotating tunneling amplitude, the likelihood of finding six Weyl points can increase significantly (50–100%) (see Supplementary Note 6). Silicon devices^{33,34} or bent carbon nanotubes could be promising candidates in this regard.
Remarkably, the argument applied to two coupled spins can be generalized to interacting multispin systems, such as magnetic trimers^{35}, atomic clusters, or multidot arrangements. In fact, the total topological charge for N noninteracting spins with size 1/2 and isotropic gtensors^{36} is C_{∞} = N. This suggests that for a generic Nspin system there are N magneticfield values where Weyl points of topological charge +1 appear and the ground state is degenerate. For even values of N, these degeneracies must appear as timereversed pairs, whereas for an odd number of spins a Weyl point must appear at B = 0, as also implied by Kramers’ theorem. These arguments can be readily extended to systems of spin S impurities as well, where the total topological charge adds up to C_{∞} = 2S N.
These general considerations have direct experimental implications. To demonstrate this, we generated an ensemble of random Hamiltonians for a spin–orbitcoupled threespin quantum dot system (see Supplementary Note 6) and analyzed the statistics of the number of magnetic Weyl points (Fig. 4c). We dominantly observe three Weyl points of charge +1 (~75%), but with a 25% likelihood, additional pairs of degeneracy points appear. From the trend observed in Fig. 4b, c, we anticipate that the more complex the interacting spin system, the easier it is (1) to find more Weyl points than the number of spins and (2) to measure the controlled creation and annihilation of Weyl points.
We thus established that magnetic Weyl degeneracies are generic in interacting quantum dot devices in the presence of SOI. The precise location and structure of these degeneracies may depend on microscopic details, but their presence is assured by topology. Their robustness has important physical implications such as the corresponding topologically protected Kondo effect observed.
Methods
Sample fabrication and measurement details
An array of Cr/Au (with 5/25 nm thickness) bottom gates (see Fig. 2a) with a width of 40 nm and a period of 100 nm was prepared by ebeam lithography and ebeam evaporation on a Si/SiO_{2} substrate. Exfoliated hexagonal boronnitride (hBN) flakes with a thickness of 20 nm were positioned on top of the bottom gates by a transfer microscope to electrically isolate the bottomgate electrodes from the nanowire. The 80 nm diameter InAs nanowire was placed on the hBN by a micromanipulator setup. The nanowire and the bottom gates were contacted by Ti/Au electrodes (10/80 nm), defined in a second ebeam lithography and ebeam evaporation step.
The sample was measured in Leiden Cryogenics CF400 cryofree dilution refrigerator, equipped with a twodimensional vector magnet. To vary the magnetic field in three dimensions, the sample holder probe was rotated manually to four different orientations ϕ ∈ {−45°, 0°, 45°, 90°}. After each rotation, the base temperature was different due to the different thermal contact between the probe and the cryostat. The differential conductance of the DQD was measured in a twopoint geometry by lockin technique at 237 Hz with 10 μV ac excitation with a homebuilt I/V converter. The conduction band was not fully depleted by the gates: chargeconfiguration labels in Fig. 2b therefore correspond to the number of electrons above closed shells in each quantum dot holding an unknown, large number of electron pairs.
Berry curvature and Chern number
Consider the groundstate manifold ψ_{0}(B) of a family of Hamiltonians H(B) parametrized by the magnetic field B. Assuming that ψ_{0} is differentiable in the vicinity of B, we define the Berry connection vector field \({\cal{A}} = ({\cal{A}}_x,{\cal{A}}_y,{\cal{A}}_z)\) as
The Berry curvature vector field \({\cal{B}} = ({\cal{B}}_x,{\cal{B}}_y,{\cal{B}}_z)\) is defined as the curl of the Berry connection,
It is noteworthy that although the Berry connection \({\cal{A}}\) is gauge dependent, the Berry curvature \({\cal{B}}\) is not.
Consider now a closed surface \({\cal{S}}\) in the magneticfield space, such that the ground state is nondegenerate at any point of \({\cal{S}}\). The (ground state) Chern number associated with this surface is then
For details, see Supplementary Note 7.
Magnetic Weyl points form timereversed pairs
If there is a magnetic Weyl point at B_{0}, then—by time reversal—there is also one at −B_{0}. This follows from the properties of time reversal, τ. (i) τ is an antiunitary operator, i.e., 〈τφτψ〉 = 〈φψ〉^{*} for any φ and ψ. (ii) τ changes the sign of each spin operator; hence, \(\tau H({\mathbf{B}})\tau ^\dagger = H(  {\mathbf{B}})\). (iii) From (ii) it follows that if H(B)ψ〉 = Eψ〉, then H(−B)τψ〉 = Eτψ〉. Thus, apart from an overall phase, τψ(B)〉 = ψ(−B)〉. Thus a degeneracy at B_{0} implies a degeneracy at −B_{0}, and at nondegenerate points \({\cal{B}}({\mathbf{B}}) =  {\cal{B}}(  {\mathbf{B}})\).
Twosite Hubbard model of the double quantum dot
Theoretical results in Fig. 2d–f, h were produced by a spin–orbitcoupled twosite Hubbard model, with Hamiltonian H = H_{0} + H_{Z}. Here, the Hamiltonian in the absence of magnetic field is
with U_{L/R} the strength of the Coulomb interaction on the left/right dot, n_{L/R} the occupation numbers, ε_{L/R} the gatecontrolled onsite energies, and \(t^{ss\prime } = t_0\delta _{ss\prime }  i\mathop {\sum}\nolimits_{\alpha = (x,y,z)} {t_\alpha } \sigma _{ss\prime }^\alpha\) a spinrotating hopping term^{37}, with realvalued hopping amplitudes t_{0}, t_{x}, t_{y}, t_{z}. The σ^{α} here denote Pauli matrices and are related to the spin operators in the usual way, e.g., \(S_{\mathrm{L}}^x = \mathop {\sum}\nolimits_{ss\prime } {c_{{\mathrm{L}}s}^\dagger } \sigma _{ss\prime }^xc_{{\mathrm{L}}s\prime }\). In an external magnetic field, we also add the Zeeman terms \(H_{\mathrm{Z}} = \mu _{\mathrm{B}}{\mathbf{B}} \cdot \left( {\widehat {\mathbf{g}}_{\mathrm{L}}{\mathbf{S}}_{\mathrm{L}} + \widehat {\mathbf{g}}_{\mathrm{R}}{\mathbf{S}}_{\mathrm{R}}} \right)\). The spinrotating interdot hopping as well as the nontrivial gtensors can be attributed to strong SOI in the InAs nanowires^{16,18,19,21}.
We have determined the values of the model parameters to provide a good overall agreement with the experimentally observed. For the methodology, see Supplementary Note 2. These parameters were then used to derive the theoretical results in Fig. 2d–f, h. The gtensors used were as follows:
Hoppings were set to t_{0} = 0.0525 meV, t_{x} = −0.0151 meV, t_{y} = 0.0565 meV, t_{z} = −0.0697 meV, and Coulomb energies to U_{L} = 1 meV, U_{R} = 0.6 meV. The onsite energies corresponding to the center of the (1, 1) hexagon of the charge stability diagram in Fig. 2b read ε_{L} = −U_{L}/2 and ε_{R} = −U_{R}/2.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge J. Asbóth, Á. Butykai, W. Coish, K. GroveRasmussen, B. Hensen, R. Maurand, P. Nagy, J. Paaske, F. Pollmann, and T. Tanttu for valuable discussions, and M. H. Madsen and C. B. Sørensen for technical support. This work was supported by the National Research Development and Innovation Office of Hungary within the Quantum Technology National Excellence Program (Project Number 20171.2.1NKP201700001), under OTKA Grants 124723 and 127900, the New National Excellence Program of the Ministry of Human Capacities, CA16218 nanocohybri COST Action, QuantERA SuperTop 127900, FET Open AndQC, the Danish National Research Foundation, and the Elemental Strategy Initiative conducted by the MEXT, Japan, and the CREST (JPMJCR15F3), JST.
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Experiments were designed by S.C. Wires were developed by J.N. and hBN crystals by K.W. and T.T. Devices were developed and fabricated by G. Frank, I.L., B.F., G. Fülöp, and S.C. Measurements were carried out and analyzed by Z.S., G. Frank, and S.C. Theoretical analysis was given by A.P., Z.S., G. Frank, and G.Z. A.P., Z.S. and G.Z. prepared the manuscript.
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Scherübl, Z., Pályi, A., Frank, G. et al. Observation of spin–orbit coupling induced Weyl points in a twoelectron double quantum dot. Commun Phys 2, 108 (2019). https://doi.org/10.1038/s4200501902002
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DOI: https://doi.org/10.1038/s4200501902002
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