Emulating heavy fermions in twisted trilayer graphene
EEmulating heavy fermions in twisted trilayer graphene
Aline Ramires and Jose L. Lado Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland Department of Applied Physics, Aalto University, 00076 Aalto, Espoo, Finland (Dated: February 23, 2021)Twisted van der Waals materials have been shown to host a variety of tunable electronic structures.Here we put forward twisted trilayer graphene (TTG) as a platform to emulate heavy fermionphysics. We demonstrate that TTG hosts extended and localized modes with an electronic structurethat can be controlled by interlayer bias. In the presence of interactions, the existence of localizedmodes leads to the development of local moments, which are Kondo coupled to coexisting extendedstates. By electrically controlling the effective exchange between local moments, the system canbe driven from a magnetic into a heavy fermion regime, passing through a quantum critical point.Our results put forward twisted graphene multilayers as a platform for the realization of stronglycorrelated heavy fermion physics in a purely carbon-based platform.
Twisted van der Waals multilayers have recently risenas a new family of systems allowing for the realization ofmultiple exotic quantum states of matter [1–10]. Theirversatility stems from the control of the strength of elec-tronic correlations and emergent topological propertiesby means of the twist angle between the different layers[11–18]. As paradigmatic examples, twisted graphene bi-layers have realized unconventional superconductivity [3],topological networks [6], strange metals [19], and Cherninsulators [9]. More complex twisted graphene multilay-ers, such as twisted bi-bilayers [7, 10] and trilayers [20–22], have provided additional platforms to realize similarphysics as twisted graphene bilayers. The possibility oftuning several twist angles in multilayer graphene [23–27]suggests that these systems may realize correlated statesof matter beyond the ones already observed in twistedbilayers.Heavy fermions are a family of strongly correlated ma-terials hosting a variety of fascinating quantum many-body states [28, 29] and unusual quantum critical phe-nomena [30–33]. Two main ingredients constitute theelectronic degrees of freedom in heavy fermions: local-ized states, usually associated with f-orbitals, and a seaof delocalized electrons. In these materials, strong inter-actions in the localized degrees of freedom generate lo-cal magnetic moments, which, through the Kondo effect,couple to the surrounding conduction electrons and dra-matically renormalize its mass. Remarkably, graphenemultilayers are known to host both dispersive [6, 34] andlocalized electronic states [2, 11, 35]. However, the coex-istence of dispersive and localized modes does not occurin the most studied instance of twisted graphene bilayersand bi-bilayers.Here we establish that twisted graphene trilayers canhost electronic states that mimic heavy fermion systemsin a purely carbon-based material. We show that themain ingredients, flat bands, and highly dispersive states,appear simultaneously and provide the starting point toengineer and control electronic phases analogous to theones observed in heavy fermions. We discuss how, in
FIG. 1. Top (a) and side (b) view of the twisted trilayergraphene system. Panel (c) and (d) display the electronicstructure in the absence and presence of interlayer bias V =0 . t ⊥ , respectively. The color code in (d) corresponds to thevalley quantum number V z . Panel (e) shows how the depen-dence of the flat band width, defined in the inset of (c), onthe interlayer bias. Panel (f) schematically represents the lowenergy sector with localized and itinerant modes. the presence of interactions, the highly degenerate flatbands lead to the formation of local spin and valley mo-ments, which become exchange coupled to dispersive he-lical modes, leading to the emergence of heavy fermions.We start by introducing the atomistic Hamiltonian forthe twisted trilayer, demonstrating that these systemshost the fundamental ingredients to realize heavy fermion a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b physics. We take a single orbital per carbon atom, yield-ing a tight-binding Hamiltonian of the form [11, 15] H = − t (cid:88) (cid:104) i,j (cid:105) ,s c † i,s c j,s − (cid:88) i,j,s ¯ t ⊥ ( r i , r j ) c † i,s c j,s (1)+ V d (cid:88) i,s z i c † i,s c i,s , where c † i,s ( c i,s ) creates (annihilates) electrons at site i with spin s . In the first term, t is the intralayer hoppingamplitude, and (cid:104) i, j (cid:105) restricts the sum to first neighbors.The second term accounts for interlayer hopping definedas ¯ t ⊥ ( r i , r j ) = t ⊥ ( z i − z j ) | r i − r j | e − β ( | r i − r j |− d ) [11, 15], where d is the interlayer distance and β controls the spatial decayof the hopping amplitudes. As a reference, for twistedgraphene multilayers t ≈ t ⊥ ≈ . t [36]. Thethird term corresponds to interlayer bias with magnitude V , with z i = ± d for the upper/lower layer and z i = 0for the middle layer. Here we consider a twisted trilayerstructure with the top and bottom layers aligned, rotatedby an angle θ with respect to the middle layer, as shownin Fig. 1 (a) and (b).The electronic structure for θ = 1 . ◦ is shown in Fig. 1(c) for V = 0. We highlight the emergence of flat bands,similar to the ones observed in twisted bilayer grapheneat the magic angle, coexisting with dispersive modes [24–27]. Interestingly, when applying an interlayer bias, thistwisted system develops even flatter bands, as shown inFig. 1 (d), in stark contrast with the case of magic-anglebilayers, for which an interlayer bias does not impact theelectronic structure strongly [5]. This can be quantita-tively assessed by computing the splitting of the nearlyflat bands at the Γ point as a function of the interlayerbias, as shown in Fig. 1 (e). In particular, it is observedthat for interlayer biases of V ≈ . t ⊥ the bandwidthbecomes drastically reduced. Moreover, in the presenceof interlayer bias, the electronic states can still be as-sociated with a well-defined valley quantum number V z (computed in the real-space basis using the valley oper-ator [37–41]). In summary, we have the coexistence ofvalley polarized and spin degenerate ultra flat states anddispersive modes, as shown schematically in Fig. 1 (f).We now focus on the flattest regime in the presenceof a bias, as shown in Fig. 2. By computing the layerpolarization, L , shown in Fig. 2 (a), we conclude thatboth the flat bands and dispersive modes are delocalizedbetween the three layers, with dispersive modes showinga small layer polarization. Moreover, the localization ofthe states in the moir´e unit cell can be characterized bymeans of the inverse participation ratio Ξ = (cid:80) i | Ψ n ( i ) | ,Fig. 2 (b). It is observed that the flat bands are stronglylocalized in the moir´e unit cell, whereas the dispersivemodes become more delocalized. The different localiza-tion can be directly observed from the local density of FIG. 2. (a,b) Band structure of the biased ( V = 0 . t ⊥ )twisted trilayer, highlighting the layer polarization of thestates, L , in (a), and their localization in the unit cell, quan-tified by the inverse participation ratio Ξ, in (b). Panels (c,d)show the local density of states for the flat modes at ω = 0(c) and for the dispersive modes at ω = 0 . t ⊥ (d). states (LDOS) at the energy of the localized and disper-sive modes, shown in Fig. 2 (c) and (d), respectively.To understand how the electronic structure is modi-fied in presence of interactions, we first study the prob-lem at the mean-field level. This is equivalent to first-principles electronic structure calculations performed forheavy fermion compounds [42–48]. Electronic interac-tions for the atomistic Hamiltonian take the form H int = (cid:80) ij v ( r i , r j ) ρ i ρ j , where ρ i = (cid:80) s c † i,s c i,s , and v ( r i , r j ) is ascreened Coulomb interaction. A mean-field decouplingallows to transform the previous density-density oper-ator into a single-particle operator of the form H MF = (cid:80) ijss (cid:48) χ ss (cid:48) ij c † i,s c j,s (cid:48) , where χ ss (cid:48) ij are the mean-field field pa-rameters. In the following, we will consider two mean-field ansatzes, one leading to a mean-field with spin polar-ization in the localized modes, and one leading to valleypolarization in the localized modes. It must be notedthat, although such mean-field electronic structure doesnot capture quantum fluctuations, it can be taken as anestimate of the backaction of the localized modes ontothe dispersive states. As a reference, in our calculations,we take an electronic repulsion equivalent to a chargingenergy of 50 meV, comparable to the one of twisted bi-layer graphene [49–53]. It is finally worth noting that thestrength of the Coulomb interactions can be externallycontrolled by means of screening engineering through asubstrate [50, 52]. FIG. 3. (a) Band-structure in the presence of finite spin(a) and valley (b) polarization of the localized modes. Pan-els (c) and (d) show a sketch of the order associated to thebandstructures in (a) and (b), respectively.
The mean-field band structures of the atomistic modelfor the case of a finite spin and valley polarization areshown in Fig. 3 (a) and (b), respectively, highlightingthat the polarization of the localized modes leads tostrong and qualitatively distinct band reconstructions.This phenomenology is analogous to the one found intwisted bilayers [49, 51, 53], with the key difference thattwisted bilayers do not have dispersive modes coexistingat the Fermi energy. In the proposed twisted trilayergraphene system, these results highlight that the disper-sive modes are strongly affected by the localized ones,signaling the existence of exchange coupling between ex-tended and localized modes, what can be understood interms of a Kondo coupling [54–57].The phenomenology obtained from the microscopicatomistic model within a mean-field approach shows thatbiased twisted trilayers have the fundamental ingredientsto display heavy fermion physics. In order to proceedwith the discussion, we now focus on the low energy ef-fective model for twisted trilayer graphene, captured by: H = H Helical + H LM + H Hyb + H Int . (2)Here H Helical corresponds to the propagating helical dis-persive modes. Note that, in contrast to the usual con-cept of helicity locking the direction of propagation withthe spin projection, here the helical modes have the di-rection of propagation locked with the valley DOF, while they are spin-degenerate: H Helical ≈ v F (cid:88) k> ,σ kd † k ⇑ σ d k ⇑ σ + (3) v F (cid:88) k< ,σ kd † k ⇓ σ d k ⇓ σ , where σ = {↑ , ↓} and v = {⇑ , ⇓} correspond to the spinand valley degrees of freedom (DOF), respectively, k de-notes the wavevector difference with respect to the Fermisurface, v F stands for the Fermi velocity. The flat bandsare accounted as localized states at each moir´e site withinternal spin and valley DOFs: H LM = (cid:88) i,σ,v (cid:15) LM f † ivσ f ivσ . (4)From the atomistic calculations, Fig. 1 (b), we accessthat the hybridization between the localized and helicalmodes preserves spin and valley DOF, therefore we write: H Hyb = δ (cid:88) i,σ,v f † ivσ d ivσ + h.c., (5)where δ is the hybridization strength. As a reference,our calculations show an effective hybridization strengthof δ ≈ H Int = U (cid:88) i,v,v (cid:48) ,σ,σ (cid:48) n if,vσ n if,v (cid:48) σ (cid:48) , (6)where U is a generalized Hubbard interaction parameter.The the sum should exclude terms with both v = v (cid:48) and σ = σ (cid:48) . We note that it is possible to write more genericinter-orbital coupling, yet SU(4) models are known toeffectively account for the interactions of localized modesin graphene multilayers [58, 59].We can now find an effective Kondo model by inte-grating out the the high energy DOF of the generalizedAnderson model. This can be done through a Schrieffer-Wolf transformation. Note that if the assumption of aspin and valley conserving hybridization holds, we can as-sign a unique index α = ( v, σ ) summed over four flavours,such that the problem can be casted as a SU(4) Kondomodel [60–65]: H Kondo = H Helical + J K (cid:88) i S i · s i , (7)where J K ≈ δ /U is the Kondo coupling, S i = Ψ † fi Γ Ψ fi ,and s i = Ψ † di γ Ψ di . Here Γ and γ stand for the arrayof 15 generators of SU(4) plus the identity, and Ψ † fi =( f † i ⇑↑ , f † i ⇑↓ , f † i ⇓↑ , f † i ⇓↓ ) and Ψ † di = ( d † i ⇑↑ , d † i ⇑↓ , d † i ⇓↑ , d † i ⇓↓ ).Interestingly, the effect of enlarging the symmetry groupassociated with the degrees of freedom that can scatterthrough the Kondo coupling by generalizing the Kondoeffect from SU(2) to SU(N) enhances the Kondo temper-ature according to T SU ( N ) K ≈ D ( N ρ J K ) /N e / ( Nρ J K ) , FIG. 4. (a) Sketch of the exchange between local momentsof strength J , and the Kondo coupling between extended andlocalized modes of strength J K . Panel (b) shows the de-pendence of the exchange coupling with the interlayer bias,showing an electrical switch from ferromagnetic to antifer-romagnetic. Green region in (b) shows the regime in which | J/J K | <
1. Panel (c) shows the phase diagram in the pres-ence of ferromagnetic exchange and panel (d) in the presenceof antiferromagnetic exchange. Panel (e) shows how the threephases can be explored by applying an interlayer bias betweenthe layers, stemming from the exchange control shown in (b). where ρ is the flat density of states and D the bandwidthassociated with the helical modes [63, 64]. Note that theKondo temperature is exponentially enhanced by increas-ing N [66–70]. The fact that the delocalized states arehelical does not play a critical role in the Kondo effectand on the renormalization procedure as long as the DOSis finite [71, 72].The existence of the heavy-fermion regime requiresthat Kondo interaction, J K , dominates the exchange cou-pling between localized moments J . As a reference, ouratomistic calculations yield an effective Kondo couplingof the order of J K ≈ . H Exc = J (cid:80) (cid:104) ij (cid:105) S i · S j . Interestingly, exchange couplingsin twisted van der Waals materials have been shown to betunable all the way from ferromagnetic to antiferromag-netic [16, 41, 73]. We now compute the exchange J be-tween two neighbouring localized modes by means of the magnetic force theorem [54–57]. In absence of interlayerbias, we obtain a ferromagnetic coupling between localmoments of | J ( V = 0) | ≈ V ≈ . t ⊥ . In thisregime, the Kondo coupling J K is the dominant inter-action, driving the twisted trilayer to the heavy-fermionregime. Importantly, depending on the sign of J , thephase diagram of the twisted trilayer will take two differ-ent forms. For ferromagnetic coupling, J <
0, a quantumcritical point separates the magnetically ordered phasefrom the Kondo screened phase, Fig. 4 (c). In com-parison, for antiferromagnetic coupling,
J >
0, the geo-metric frustration of the superlattice potentially leads toa quantum critical phase for a finite range of couplings[31, 75–77], as shown in Fig. 4 (d).Interestingly, the electric tunability of the twisted tri-layer allows for the exploration of these two types ofphase diagram with a single control parameter, the inter-layer bias, as displayed in (Fig. 4 (e)). This finding pro-poses a new perspective on the global phase diagram forheavy fermions [32, 33]. While a precise first-principlesestimate of all the couplings is challenging [24], the exactsystem’s parameters can be inferred from experimentslooking at the evolution of the tunneling spectra as afunction of temperature and magnetic fields in differentdirections. In particular, below the coherence tempera-ture, Fano-like resonances appear and the characteristicparameters of the line-shape can be extracted and iden-tified with model parameters such as the hybridizationbetween localized and delocalized states [78–82].To summarize, we have shown that twisted graphenetrilayers provide a van der Waals platform to engineerheavy fermion physics. In particular, we demonstratedthat the existence of electrically controllable flat bandsand dispersive modes provides the single-particle start-ing point to simulate Kondo lattices. In the presence ofinteractions, the flat bands lead to local spin and val-ley moments. Interestingly, the electric tunability of theinter-moment exchange couplings allows for the explo-ration of both the conventional and frustrated Kondolattice regimes, such that twisted trilayers can realizethe global phase diagram for heavy fermions within asingle material platform. Our results show that twistedgraphene multilayers provide a carbon-only platform toemulate rare earth compounds, opening new possibilitiesin the field of correlated twisted van der Waals materials.
Acknowledgments:
A. R. acknowledges the financialsupport from SNSF Ambizione. J. L. L. acknowledges thecomputational resources provided by the Aalto Science-IT project, the financial support from the Academy ofFinland Projects No. 331342 and No. 336243. We thankM. Sigrist, T. Neupert, P. Liljeroth and P. Rickhaus foruseful discussions. [1] G. Li, A. Luican, J. M. B. L. dos Santos, A. H. C. Neto,A. Reina, J. Kong, and E. Y. Andrei, Nature Physics ,109 (2009).[2] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,T. Taniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-Herrero, Nature , 80 (2018).[3] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,E. Kaxiras, and P. Jarillo-Herrero, Nature , 43(2018).[4] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir,I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang,A. Bachtold, A. H. MacDonald, and D. K. Efetov, Na-ture , 653 (2019).[5] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan-abe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean,Science , 1059 (2019).[6] P. Rickhaus, J. Wallbank, S. Slizovskiy, R. Pisoni,H. Overweg, Y. Lee, M. Eich, M.-H. Liu, K. Watanabe,T. Taniguchi, T. Ihn, and K. Ensslin, Nano Letters ,6725 (2018).[7] Y. Cao, D. Rodan-Legrain, O. Rubies-Bigorda, J. M.Park, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero, Nature , 215 (2020).[8] P. Rickhaus, F. de Vries, J. Zhu, E. Portol´es, G. Zheng,M. Masseroni, A. Kurzmann, T. Taniguchi, K. Wantan-abe, A. H. MacDonald, T. Ihn, and K. Ensslin, arXive-prints , arXiv:2005.05373 (2020), arXiv:2005.05373[cond-mat.mes-hall].[9] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu,K. Watanabe, T. Taniguchi, L. Balents, and A. F.Young, Science , 900 (2019).[10] X. Liu, Z. Hao, E. Khalaf, J. Y. Lee, Y. Ronen, H. Yoo,D. H. Najafabadi, K. Watanabe, T. Taniguchi, A. Vish-wanath, and P. Kim, Nature , 221 (2020).[11] E. Su´arez Morell, J. D. Correa, P. Vargas, M. Pacheco,and Z. Barticevic, Phys. Rev. B , 121407 (2010).[12] E. Su´arez Morell, P. Vargas, L. Chico, and L. Brey, Phys.Rev. B , 195421 (2011).[13] J. M. B. Lopes dos Santos, N. M. R. Peres, and A. H.Castro Neto, Phys. Rev. Lett. , 256802 (2007).[14] R. Bistritzer and A. H. MacDonald, Proceedings of theNational Academy of Sciences , 12233 (2011).[15] A. O. Sboychakov, A. L. Rakhmanov, A. V. Rozhkov,and F. Nori, Phys. Rev. B , 075402 (2015).[16] L. A. Gonzalez-Arraga, J. L. Lado, F. Guinea, andP. San-Jose, Phys. Rev. Lett. , 107201 (2017).[17] A. O. Sboychakov, A. V. Rozhkov, A. L. Rakhmanov,and F. Nori, Phys. Rev. Lett. , 266402 (2018).[18] A. Ramires and J. L. Lado, Phys. Rev. Lett. , 146801(2018).[19] Y. Cao, D. Chowdhury, D. Rodan-Legrain, O. Rubies-Bigorda, K. Watanabe, T. Taniguchi, T. Senthil, andP. Jarillo-Herrero, Phys. Rev. Lett. , 076801 (2020).[20] S. Chen, M. He, Y.-H. Zhang, V. Hsieh, Z. Fei,K. Watanabe, T. Taniguchi, D. H. Cobden, X. Xu,C. R. Dean, and M. Yankowitz, Nature Physics (2020),10.1038/s41567-020-01062-6.[21] J. M. Park, Y. Cao, K. Watanabe, T. Taniguchi, andP. Jarillo-Herrero, Nature (2021), 10.1038/s41586-021-03192-0. [22] Z. Hao, A. M. Zimmerman, P. Ledwith, E. Khalaf,D. Haie Najafabadi, K. Watanabe, T. Taniguchi, A. Vish-wanath, and P. Kim, arXiv e-prints , arXiv:2012.02773(2020), arXiv:2012.02773 [cond-mat.supr-con].[23] J. Liu, Z. Ma, J. Gao, and X. Dai, Phys. Rev. X ,031021 (2019).[24] A. Lopez-Bezanilla and J. L. Lado, Phys. Rev. Research , 033357 (2020).[25] E. Khalaf, A. J. Kruchkov, G. Tarnopolsky, and A. Vish-wanath, Phys. Rev. B , 085109 (2019).[26] S. Carr, C. Li, Z. Zhu, E. Kaxiras, S. Sachdev, andA. Kruchkov, Nano Letters , 3030 (2020).[27] Z. Wu, Z. Zhan, and S. Yuan, arXiv e-prints, arXiv:2012.13741 (2020), arXiv:2012.13741 [cond-mat.mes-hall].[28] P. Coleman, Many-Body Physics: From Kondo to Hub-bard (eds E. Pavarini, E. Koch and P. Coleman) (2015).[29] S. Wirth and F. Steglich, Nature Reviews Materials ,16051 (2016).[30] Q. Si and F. Steglich, Science , 1161 (2010).[31] A. Ramires, Nature Physics , 1212 (2019).[32] Q. Si, Physica Status Solidi (b) , 476.[33] P. Coleman and A. H. Nevidomskyy, Journal of Low Tem-perature Physics , 182 (2010).[34] P. San-Jose and E. Prada, Phys. Rev. B , 121408(2013).[35] Y. Xie, B. Lian, B. J¨ack, X. Liu, C.-L. Chiu, K. Watan-abe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Na-ture , 101 (2019).[36] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109(2009).[37] E. Colom´es and M. Franz, Phys. Rev. Lett. , 086603(2018).[38] A. Ramires and J. L. Lado, Phys. Rev. Lett. , 146801(2018).[39] A. Ramires and J. L. Lado, Phys. Rev. B , 245118(2019).[40] T. M. R. Wolf, J. L. Lado, G. Blatter, and O. Zilberberg,Phys. Rev. Lett. , 096802 (2019).[41] T. M. R. Wolf, O. Zilberberg, G. Blatter, and J. L. Lado,Phys. Rev. Lett. , 056803 (2021).[42] Y. Wang, L. G. Hector, H. Zhang, S. L. Shang, L. Q.Chen, and Z. K. Liu, Phys. Rev. B , 104113 (2008).[43] A. Shick, W. Pickett, and A. Liechtenstein, Journalof Electron Spectroscopy and Related Phenomena , 753 (2001).[44] D. A. Andersson, S. I. Simak, B. Johansson, I. A.Abrikosov, and N. V. Skorodumova, Phys. Rev. B ,035109 (2007).[45] B. Dorado, B. Amadon, M. Freyss, and M. Bertolus,Phys. Rev. B , 235125 (2009).[46] O. Eriksson, L. Nordstr¨om, M. S. S. Brooks, and B. Jo-hansson, Phys. Rev. Lett. , 2523 (1988).[47] F. Cricchio, F. Bultmark, O. Gr˚an¨as, and L. Nordstr¨om,Phys. Rev. Lett. , 107202 (2009).[48] E. Mendive-Tapia and J. B. Staunton, Phys. Rev. Lett. , 197202 (2017).[49] F. Guinea and N. R. Walet, Proceedings of the NationalAcademy of Sciences , 13174 (2018).[50] J. M. Pizarro, M. R¨osner, R. Thomale, R. Valent´ı, andT. O. Wehling, Phys. Rev. B , 161102 (2019).[51] T. Cea, N. R. Walet, and F. Guinea, Phys. Rev. B ,205113 (2019). [52] P. Stepanov, I. Das, X. Lu, A. Fahimniya, K. Watanabe,T. Taniguchi, F. H. L. Koppens, J. Lischner, L. Levitov,and D. K. Efetov, Nature , 375 (2020).[53] P. A. Pantaleon, T. Cea, R. Brown, N. R. Walet, andF. Guinea, arXiv e-prints , arXiv:2003.05050 (2020),arXiv:2003.05050 [cond-mat.str-el].[54] A. Liechtenstein, M. Katsnelson, V. Antropov, andV. Gubanov, Journal of Magnetism and Magnetic Ma-terials , 65 (1987).[55] A. Oswald, R. Zeller, P. J. Braspenning, and P. H.Dederichs, Journal of Physics F: Metal Physics , 193(1985).[56] S. Lounis and P. H. Dederichs, Phys. Rev. B , 180404(2010).[57] A. Szilva, M. Costa, A. Bergman, L. Szunyogh, L. Nord-str¨om, and O. Eriksson, Phys. Rev. Lett. , 127204(2013).[58] J. Kang and O. Vafek, Phys. Rev. Lett. , 246401(2019).[59] K. Seo, V. N. Kotov, and B. Uchoa, Phys. Rev. Lett. , 246402 (2019).[60] L. Borda, G. Zar´and, W. Hofstetter, B. I. Halperin, andJ. von Delft, Phys. Rev. Lett. , 026602 (2003).[61] G. Zar´and, A. Brataas, and D. Goldhaber-Gordon, SolidState Communications , 463 (2003).[62] R. L´opez, D. S´anchez, M. Lee, M.-S. Choi, P. Simon, andK. Le Hur, Phys. Rev. B , 115312 (2005).[63] M.-S. Choi, R. L´opez, and R. Aguado, Phys. Rev. Lett. , 067204 (2005).[64] M. Filippone, C. P. Moca, G. Zar´and, and C. Mora,Phys. Rev. B , 121406 (2014).[65] E. J. K¨onig, P. Coleman, and A. M. Tsvelik, Phys. Rev.B , 104514 (2020).[66] P. Coleman, Phys. Rev. B , 5255 (1983).[67] A. Ramires and P. Coleman, Phys. Rev. B , 035120 (2016).[68] A. Auerbach and K. Levin, Phys. Rev. Lett. , 877(1986).[69] A. J. Millis and P. A. Lee, Phys. Rev. B , 3394 (1987).[70] D. P. Arovas and A. Auerbach, Phys. Rev. B , 316(1988).[71] A. K. Mitchell, D. Schuricht, M. Vojta, and L. Fritz,Phys. Rev. B , 075430 (2013).[72] R. ˇZitko, Phys. Rev. B , 241414 (2010).[73] M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi,K. Kuroki, and L. Fu, Phys. Rev. X , 031087 (2018).[74] The effective exchange couplings depends on the chargingenergy, that can be controlled via dielectric engineering.[75] H. Zhao, J. Zhang, M. Lyu, S. Bachus, Y. Tokiwa,P. Gegenwart, S. Zhang, J. Cheng, Y.-f. Yang, G. Chen,Y. Isikawa, Q. Si, F. Steglich, and P. Sun, Nature Physics , 1261 (2019).[76] J. Zhang, H. Zhao, M. Lv, S. Hu, Y. Isikawa, Y.-f. Yang,Q. Si, F. Steglich, and P. Sun, Phys. Rev. B , 235117(2018).[77] S. Friedemann, T. Westerkamp, M. Brando, N. Oeschler,S. Wirth, P. Gegenwart, C. Krellner, C. Geibel, andF. Steglich, Nature Physics , 465 (2009).[78] P. Aynajian, E. H. da Silva Neto, A. Gyenis, R. E. Baum-bach, J. D. Thompson, Z. Fisk, E. D. Bauer, and A. Yaz-dani, Nature , 201 (2012).[79] M. Maltseva, M. Dzero, and P. Coleman, Phys. Rev.Lett. , 206402 (2009).[80] J. Figgins and D. K. Morr, Phys. Rev. Lett. , 187202(2010).[81] A. R. Schmidt, M. H. Hamidian, P. Wahl, F. Meier, A. V.Balatsky, J. D. Garrett, T. J. Williams, G. M. Luke, andJ. C. Davis, Nature , 570 (2010).[82] S. Ernst, S. Kirchner, C. Krellner, C. Geibel, G. Zwick-nagl, F. Steglich, and S. Wirth, Nature474