Dec 13, 2017 - ... multiplet with even fermion parity. This situation is reversed when â â« TK. Using the. arXiv:1712.04920v1 [cond-mat.supr-con] 13 Dec 2017 ...
Anisotropy and temperature effects on the spectral properties of Shiba impurities J. A. Andrade and Alejandro M. Lobos
arXiv:1712.04920v1 [cond-mat.supr-con] 13 Dec 2017
Facultad de Ciencias Exactas y Naturales Universidad Nacional de Cuyo, 5500 Mendoza, Argentina and Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET), Argentina We theoretically consider a quantum magnetic impurity weakly coupled to a superconductor and obtain the spectrum of subgap Yu-Shiba-Rusinov (YSR) states taking into account the effect of quantum fluctuations, anisotropy and thermal excitations. We consider the experimentally relevant case of a quantum impurity with easy- or hard-axis uniaxial anisotropy term Hanis = −D (S z )2 , which typically arise in magnetic atoms deposited on the surface of clean metals due to the presence of strong spin-orbit coupling and lack of inversion symmetry. Quantum fluctuations are taken into account at second-order in the coupling JK in the impurity Green’s functions self-energy, and considerably modify the subgap spectrum compared to the classical-spin approximation. For an impurity with easy-axis anisotropy (parameter D > 0), when either the spin S → ∞ or the anisotropy parameter D → ∞, the classical results are recovered very slowly (i.e., as 1/S or 1/D, respectively), implying that this limit is very hard to reach experimentally. On the other hand, when D < 0 (hard-axis anisotropy) the impurity never reaches the classical limit, and the position of the YSR states differ considerably from the usual formulas. Importantly, at finite temperature a thermally-excited quasiparticle continuum induced by the anisotropy arises inside the T = 0 gap, and cannot be reproduced within the classical approximation. Our results point towards the importance of taking into account quantum fluctuations for the correct interpretation of STM experiments. PACS numbers: 85.25.-j, 74.55.+v, 75.30.Hx, 75.30.Gw
The competition between superconductivity and quantum magnetism at the nanoscale gives rise to interesting exotic quantum phenomena1 . The advent of lowtemperature scanning-tunneling microscopy (STM) have allowed to examine the local density of states of magnetic impurities adsorbed on superconductors, revealing the presence of subgap excitation peaks, predicted in seminal papers by Yu, Shiba and Rusinov (YSR)2–4 . These states appear symmetrically around the Fermi level at energies within the superconducting gap ∆, and localize around the impurity. Recent progress in STM techniques have revealed a surprisingly complex behavior of YSR states, as a result of the interplay between quantum fluctuations, Kondo screening, crystal-field anisotropy, etc.5–16 . Since YSR states are naturally protected from decoherence processes by the presence of the superconducting gap17 , they have become increasingly attractive from the point of view of quantum information technologies. Moreover, theoretical proposals18–21 have predicted that a chain of magnetic impurities adsorbed ontop of a conventional s-wave superconductor could support Majorana zero-modes and therefore could be used in topological quantum computation schemes22 . Subsequent STM experiments realized on Fe atomic chains deposited ontop of superconducting Pb(111) or Pb(110) surfaces have revealed intriguing zero-bias peaks in the dI/dV signal, consistent with the Majorana zero-mode scenario23,24 . However, these experimental results are still under debate, and more theoretical and experimental work is needed to elucidate the presence (or not) of Majorana excitations. When considering magnetic impurities on superconductors, it is customary to make the simplifying assumption that the spins are classical objects, which is physi-
cally expected in the large-spin limit S → ∞. However, recent theoretical works have pointed out that the classical spin limit might be hard to achieve in realistic experimental conditions, and that quantum fluctuations should be taken into account for the correct (quantitative and qualitative) interpretation of the experimental data25,26 . However, taking quantum fluctuations into account introduces many-body correlations which are intimately connected to the celebrated Kondo problem, a paradigmatic phenomenon in condensed matter physics27–29 , and this fact considerably complicates the analytical treatment. The Kondo effect arises due to the s − d exchange coupling between a localized magnetic moment and the electrons in the conduction band. In the case of a normal metal (i.e., in the absence of pairing correlations), below a certain temperature known as Kondo temperature TK , the conduction-electron spins and the magnetic impurity become entangled in a many-body singlet (i.e., the Kondo singlet), and the magnetic moment of the impurity is effectively screened. However, for a superconducting host, the lack of electrons around the Fermi level due to the presence of the superconducting gap can result in an incomplete screening of the magnetic impurity, and gives rise to a complex interplay between superconductivity and Kondo correlations30,31 . In fact, in the case of an impurity spin S coupled to a single-band superconductor, the ground state of the full system (superconductor plus impurity) depends on the ratio between TK and ∆12,30–33 . When TK ∆, the many-body ground state has total spin ST = S − 1/2 and odd-fermion parity, and describes a Kondo-screened impurity, while the first excited (many-body) state corresponds to an unscreened ST = S multiplet with even fermion parity. This situation is reversed when ∆ TK . Using the
2 Lehmann representation34 , the intra-gap YSR states observed via STM spectroscopy can therefore be interpreted as parity-changing transitions between the ground-state and first-excited many-body states35,36 . When TK ∼ 0.3∆, a quantum phase transition (QPT) between these two states occurs32,33,37,38 , signalled by the crossing of the YSR states at E = 012,14 . This transition is also known as the “0 − π” transition in the context of electronic transport through quantum dots attached to superconducting leads39 . From the experimental perspective, real quantumimpurity systems deposited on top of clean metallic surfaces usually present single-ion magneto-crystalline anisotropy, which arise due to the presence of strong spin-orbit coupling and lack of inversion symmetry at the surface40 . Magnetic anisotropy has been identified25 as an important mechanism which can give rise to a multiple YSR structure, which has been observed in several experiments9–11,14,15,41 . It is worth noting that the effect of anisotropy cannot be described within the classical approximation. In that case the only known mechanism leading to YSR multiplets is the presence of higher angular-momentum conduction-electron scattering channels1,4,9–11 . These recent experimental advances point to the inherent complexity of realistic magnetic impurities on superconductors, and demand theoretical approaches that can go beyond the classical-spin approximation in order to interpret correctly the STM results. The classical-spin approximation breaks the rotational SU(2) and timereversal symmetries, and cannot describe the effects of the ever-present single-ion anisotropy in realistic STM systems. In that respect, it is interesting to point out recent theoretical efforts which take into account the quantum nature of magnetic impurities, either using the exact, but numerically costly, numerical renormalization group (NRG) method25,26,36,42 , or perturbation approaches in the parameter U of the corresponding Anderson model43–45 , which limited their attention to the ideal case of zero single-ion anisotropy. The Anderson model in a superconducting host has also been studied by means of the quantum Monte Carlo method46 , but its intrinsic difficulty to perform the analytical continuation to real frequencies makes this technique very hard to use in this case, where the sharp YSR resonances need to resolved. In fact, there are actually few theoretical methods that can reliably account for experimentally relevant effects, such as anisotropy and temperature. Besides these experimental motivations, fundamental questions also arise when magnetic impurities interact with a superconducting host, e.g., is it possible to make sense of the classical-spin approximation when either S → ∞ or D → ∞? Which is the regime of validity of the classical approximation? Despite its aforementioned short-comings, this approximation was able to predict the emergence of YSR states, and it is interesting to explore its regime of validity within a broader, better suited theoretical framework.
Based on these experimental and theoretical motivations, in this paper we study a quantum spin S coupled to a superconducting host via an exchange coupling JK . Taking JK as a small parameter, we use Green’s function perturbation theory and the Dyson’s equation34 , and obtain the spectrum of subgap YSR states, including the experimentally relevant effects of quantum fluctuations, uniaxial anisotropy and temperature, treating them on equal footing and in an unbiased way. Quantum fluctuations are encoded in the self-energy of the impurity, which is computed at second order in JK , and considerably modify the subgap spectrum as compared to the classical spin case. In the case of quantum impurities in normal metals, we note that the perturbative approach in JK is known to fail since the perturbative series is plagued by logarithmic divergencies ∼ ln(T /TK ) at low temperatures28,29 . However, in our case the presence of the energy gap ∆ acts as a low-energy cut-off for these divergencies, and allows us to extend the perturbative results to the limit T → 0. For an impurity with easy-axis anisotropy (parameter D > 0), when either the spin S → ∞ or the anisotropy parameter D → ∞, the perturbation method allows to recover the classical results, although this occurs very slowly (i.e., as 1/S or 1/D, respectively), implying that this limit is very hard to reach experimentally. Moreover, when D < 0 (hard-axis anisotropy) the impurity never reaches the classical limit, and the position of the YSR states differ considerably from the usual formulas. Importantly, at finite temperature a thermally-excited quasiparticle continuum induced by the anisotropy arises inside the T = 0 gap. This is a novel effect that cannot be reproduced within the classical approximation. Our results point towards the importance of taking into account quantum fluctuations for the correct interpretation of STM experiments. The rest of the paper is organized as follows. In Sec. II we present the theoretical model. In Sec. III we present analytical results obtained within second-order perturbation theory in JK at T = 0, and in Sec. IV we present the results at finite temperature. Finally, in Sec. V we summarize the main results and present the conclusions. I.
THEORETICAL MODEL
We describe a magnetic impurity coupled to a superconductor by the following Hamiltonian H = HSC + HK + Hanis .
(1)
Here HSC is the BCS Hamiltonian describing a conventional single-band s-wave superconductor X X † † HSC = εk c†kσ ckσ + ∆ ck↑ c−k↓ + c−k↓ ck↑ , (2) kσ
k
where c†k,σ (ck,σ ) creates (annihilates) an electron with momentum k in the conduction band and spin projection
3 σ along the zˆ axis, εk is the dispersion relation of normal quasiparticles, and ∆ is the superconductor gap. We describe the microscopic coupling of a quantum magnetic impurity to a superconducting host by the single channel s-d exchange (or Kondo) Hamiltonian1 JK X 1 h † Sz ck↑ ck0 ↑ − c†k↓ ck0 ↓ HK = √ V k,k0 2 i +S + c†k↓ ck0 ↑ + S − c†k↑ ck0 ↓ , (3) characterized by the antiferromagnetic exchange coupling JK . We note here that the multiple YSR structure observed in certain experiments9–11,14,15,41 could be related to the combination of multiple screening channels and anisotropy25 , or to the presence of screening channels with higher angular-momentum1,4,9 . Therefore, our simple single channel model cannot describe a YSR multiplet structure. This feature will be studied in a subsequent work, and here we keep the description as simple as possible. Finally, the single-ion magnetic anisotropy term 2
Hanis = −D (Sz )
(4)
describes a spin-S magnetic impurity with uniaxial anisotropy along the zˆ direction. Physically, the case of easy-axis anisotropy (D > 0) favours the maximal S z projections, i.e., m = ±S, while the hard-axis case (D < 0) favours an impurity ground state with projection m = 0 for S even, or m = ±1/2 for S odd. Yu, Shiba and Rusinov studied the above Hamiltonian in the approximation S → S, which effectively yields a single-particle Hamiltonian describing a superconductor with a local Zeeman term2–4 . This type of approximation, in the limit of infinite band width W → ∞, generates subgap excitations located at Ecl 1 − α2 =± , ∆ 1 + α2
(5)
where α = JK Sρ0 π/2, with ρ0 the density of states at the Fermi energy, and where the ± sign means that the YSR states are symmetrically located around the Fermi energy due to the electron-hole symmetry of the BCS Hamiltonian (2) (here the subscript “cl” stands for classical approximation). A YSR state can be interpreted as a discrete fermionic transition between the ground state and the first excited state, which belong to subspaces with different fermionic parity35 . As JK increases, these states become closer in energy, and consequently the positive and negative YSR states approach the Fermi energy. Eventually, when they become exactly degenerate the system experiences a parity- and spin-changing QPT, which is signalled by the crossing of the YSR states at the Fermi level. In the classical approximation, the QPT occurs for α = 1 [see Eq. (5)], and in the phase with α > 1, the local Zeeman potential becomes strong enough to bind an extra electron to the impurity spin, thus changing the fermionic parity in the ground state35 . Later,
FIG. 1. (Color online) a) Diagram of the experimental setup. b) Level spectrum for easy-axis anisotropy (D > 0). This figure shows the spin fluctuations from the ground state Sz = ±S to the firts exited states Sm = ±(S − 1) at zero ¯ = D(2S − 1), see text. c) Diagrammatic extemperature. ∆ pansion of the Green’s functions G(11) and G(12) [elements (11) and (12) respectively of the matrix G in Eq. (14)], using the second order self-energy Eq. (15). Only one of the second order diagrams of the infinite Dyson’s series is shown. The dashed red line represents the spin correlator hTτ S − (τ )S + (0)i, and the unperturbed Green’s functions are labelled by g0 and f0 .
Zittartz and M¨ uller-Hartmann30,31 studied a Kondo impurity in a superconducting host, adapting Nagaoka’s decoupling scheme of equations of motion for the Green’s function47 to the superconducting case, and recovered the YSR states in the presence of quantum fluctuations. They realized that the relevant condition signalling the QPT is actually ∆ ≈ TK 30,31 . Finally, the development of the NRG technique enabled to obtain a detailed description of the full many-body problem, and to prec c cisely obtain the critical ratio TK /∆ (e.g., TK /∆ = 0.27 c for a S = 1/2 impurity). When ∆ > TK , the ground state is an unscreened many-body state with total spin c ST = S and even fermion parity. For ∆ < TK a Kondo screened ground state with ST = S − 1/2 and odd parity occurs32,33,37,38 . We now outline the second-order perturbation theory, which is a standard many-body perturbation method34,48 . The type of processes which are taken into account are schematically shown in the Feynman diagram of Fig. 1(c). In Fig. 1(b) we show the fluctuations within the internal states of the impurity, asumming easy-axis anisotropy D > 0. In this case, and at T = 0, the system can only fluctuate between m = −S (m = S) and m = −S + 1 (m = S − 1) states. Since the perturbative approach is valid only in the limit JK → 0, here we assume TK ∆, i.e., far from the QPT and where the impurity spin is in the unscreened phase. Under this assumption, perturbation theory is a reliable method as the many-body Kondo correlations, which are non-perturbative, are minimized. The unperturbed Matsubara propagators in the superconductor in the absence of the magnetic impurity can be compactly
4 written in Nambu space as X G0 (τ − τ 0 ) = G0,k (τ − τ 0 ) k
X g0,k (τ − τ 0 ) f0,k (τ − τ 0 ) , (6) = f¯0,k (τ − τ 0 ) g¯0,k (τ − τ 0 )
giving rise to the classical YSR states. The other term in square brackets corresponds the quantity Z β
− + S S anis (iωl ) = dτ eiωl τ hTτ S − (τ ) S + (0)ianis , 0
(16)
k
where τ is the imaginary time. Here D E g0,k (τ − τ 0 ) = − Tτ ck↑ (τ ) c†k↑ (τ 0 ) 0
(7) 0
0
f0,k (τ − τ ) = − hTτ ck↑ (τ ) c−k↓ (τ )i0 D E f¯0,k (τ − τ 0 ) = − Tτ c†−k↓ (τ ) c†k↑ (τ 0 ) 0 D E † 0 0 g¯0,k (τ − τ ) = − Tτ c−k↓ (τ ) c−k↓ (τ ) ,
(8) (9) (10)
0
are the unperturbed electron-electron, electron-hole, hole-electron, and hole-hole propagators, respectively, and Tτ is the imaginary time-ordering operator. In Matsubara-frequency representation, these propagators can be expressed as1,48,49 Vk2 Uk2 + (11) iνn − Ek iνn + Ek 1 1 ∆ − (12), f0,k (iνn ) = f¯0,k (iνn ) = 2Ek iνn − Ek iνn + Ek g0,k (iνn ) = −¯ g0,k (iνn ) =
where νn = π(2n + 1)/β are the fermionic Matsub εk 1 2 ara frequencies, and where Uk = 2 1 + Ek and Vk2 = 21 1 − Eεkk are, respectively, the electron and hole weights of the Bogoliubov quasiparticle with energy Ek =
q
ε2k + ∆2 .
(13)
Assuming SU(2) symmetry in the superconducting bath, we can drop the spin indices in what follows. The Dyson’s series can be derived for the Nambu Green’s function, assuming that the Kondo temperature is negligible, and we obtain −1
G (iνn ) = G0 (iνn ) [1 − Σ (iνn ) G0 (iνn )]
2 X
JK Sz2 anis (iωl ) + S − S + anis (iωl ) 4β l
×G0 (iνn − iωl ) ,
0 = det [1 − Σ (EYSR ) G0 (EYSR )] .
(17)
Fourier transforming Eq. (6) and performing the sum in k, allows to obtain the expression g0 (z) f0 (z) G0 (z) = , f¯0 (z) g¯0 (z) W −2ρ0 z ∆ tan−1 √ (18) . =√ ∆ z ∆2 − z 2 ∆2 − z 2 Replacing this expression into Eq. (17) yields 0 = 1 − [g0 (EYSR ) ± f0 (EYSR )] h i × Σ(11) (EYSR ) ± Σ(12) (EYSR ) ,
(19)
where the different signs correspond to the positive and negative YSR states, and where Σ(αβ) (z) are matrix elements of Σ(z), obtained from the analytical continuation of Eq. (15).
(14)
where Σ (iνn ) is the self-energy matrix, computed at second order in JK : Σ(iνn ) =
i.e., the dynamical spin correlator representing the quantum fluctuations in the impurity given by the Feynman diagram in Fig. 1(c). Since the operators S ± do not commute with Hanis , this correlator contains dynamical Matsubara components ωl 6= 0 encoding the information about the magneto-crystalline anisotropy. Note that in the case of positive D > 0, transitions to excited states become less and less likely as D increases, and eventually in the limit D → ∞, the correlator hS − S + ianis (iωl ) → 0, and the classical YSR results are recovered. Performing the analytical continuation of the Matsubara frequencies into the complex plane, iνn → z, the YSR states can be found as the real poles of the Green’s function G (z) within the gap ∆. Therefore, finding the poles of G (z) amounts to solving for the real solutions of the determinant (see Eq. 14)
(15)
where ωl = 2πl/β are the bosonic Matsubara frequencies, and hSz2 ianis (iωl ) = δl,0 hSz2 ianis is the thermal average of the operator Sz2 , computed with respect to the magnetic anisotropy Hamiltonian (4). Since the operator Sz commutes with Hanis , this term has only a static Matsubara component, i.e. ωl = 0. This term is the contribution
II.
RESULTS AT T = 0
A.
Isotropic case D = 0
It is instructive to study first the isotropic case D = 0. In that case, the Hamiltonian Hanis vanishes, and the magnetic impurity has no internal dynamics. Therefore, the averages become hSz2 (iωl )ianis = S 2 δl,0 and hS − S + (iωl )ianis = Sδl,0 , and Eq. (15) reduces to Σ(ω) →
2 JK S(S + 1) G0 (ω) . 4
(20)
From this expression, note that the classical YSR result can be obtained by removing the fluctuations and
5
1.0
ECl (α=0.1)
●
●
●
●
EYSR [Δ]
0.8 0.6 0.4 0.2 0.0 0.0
ECl (α=0.47) ECl (α=0.65)
■
■ ▲
▲
0.5
1.0
■
▲ 1.5 S
● ● ● α=0.1 ■ α=0.47 ▲ α=0.65
■
■
■
▲
▲
▲
2.0
2.5
3.0
FIG. 2. (Color online) YSR state energy EYSR as a function of the spin impurity S for α = 0.1 (black points), α = 0.47 (blue squares) and α = 0.65 (red triangles). Continuous lines are a guide to the eye. The classical Shiba energy independent of S are showed as orange dashed horizontal lines for the corresponding values of α. The arrow marks the putative quantum phase transition predicted by perturbative approach.
which occurs at the critical value Sc = α2 /(1 − α2 ) (see Fig. 2). This is similar to Ref. 26, where a critical value Sc is found with NRG. For comparison, using α = 0.47, ˇ in that reference Zitko found phase transition between 2 < S < 5/2, whereas in our case we find the transition for an unphysical value Sc ∼ 0.28 < 0.5 (see blue line in Fig. 2). This result puts in evidence the limitations of the method, which becomes unreliable near the QPT (i.e., in this case, when S ≈ Sc ). We recall here that the transition actually arises from the competition between the Kondo correlations, which are absent in the perturbative approach, and ∆, and therefore is not related to the vanishing of Eq. (21). In order to ensure the validity of the perturbative approach, we therefore must restrict the values of S to Sc S. As can be seen in Fig. 2, the range of values of S for which this requirement is fulfilled becomes parametrically larger when α → 0 (see black line, which correspond to the value α = 0.1, for which already the case of the lowest physical spin S = 1/2 is already much larger than Sc ).
B.
Anisotropic case D 6= 0
2
keeping only the static term S , which yields Σ (ω) → 2 2 S G0 (ω)2–4 . Then, in the limit of infinite bandwidth JK W → ∞, replacing this expression into Eq. (19), we recover the classical expression (5). On the other hand, when quantum fluctuations are considered, the YSR energies become (for the same value of α) EYSR 1 − α2 (1 + 1/S) =± . ∆ 1 + α2 (1 + 1/S)
(21)
Therefore, in the isotropic case the perturbative ap 2 is qualitatively similar to the proach at order O JK classical YSR result, provided we renormalize α → α ˜= p α 1 + 1/S (as will see in the next sections, this only occurs in the isotropic case. In the presence of anisotropy, the position of the YSR states differ considerably from the classical limit). Eq. (21) is physically appealing, as we intuitively expect to recover the classical spin approximation in the limit S → ∞. In Fig. 2 we show the position of the YSR energy as a function of spin S for α = 0.1, 0.47 and 0.65, and compare it with the classical limit. As expected, the classical limit is only recovered when S → ∞ EYSR Ecl α2 1 → − . 2 2 ∆ ∆ (1 + α ) S
(22)
We also note that, due to the slow dependence 1/S, the classical result is very hard to reach experimentally. This result is qualitatively similar to that obtained by ˇ Zitko using NRG, who studied the quantum-to-classical crossover in isotropic impurities in superconductors (see Ref. 26). Through a numerical fit of the NRG results, he obtained that the classical limit is reached as 1/S 1.1 , very close to our analytical result. Interestingly, perturbation theory also predicts a QPT as a function of S,
We now focus on the case of finite anisotropy at T = 0. As we mentioned in the introduction, the presence of anisotropy changes qualitatively the low-temperature properties of quantum impurities. In particular, while easy-axis anisotropy tends to favour the classical limit when |D| → ∞, this is not the case for the hard-axis anisotropy, where the classical limit, strictly speaking, does not exist. Moreover, in this case we still need to distinguish the cases of half-integer and integer spin S, as the ground state is qualitatively different (doubly- or non-degenerate). We therefore distinguish in what follows three different cases: 1) D > 0, 2) D < 0 and S half-integer, and 3) D < 0 and S integer, which we analyze separately in the following.
1.
Case D > 0
The results in this section are depicted in Fig. 3 for D > 0, both for S half-integer [Fig. 3(a)] and S integer [Fig. 3(b)]. In this case, the ground state of the isolated impurity takes the maximal Sz projection m = ±S, and when D is very large with respect to ∆, the spin fluctuations become negligible because the system needs an energy of the order of D(2S−1) to be able to flip to states m = ±(S − 1) [see Appendix A and Fig. 1(b)]. This energy scale depends on D and S, and evinces that when S is increased, the system converges more quickly to the classical limit when the anisotropy is increased. In fact, either for S → ∞ or D → ∞, the system converges to the same classical limit. This behavior can be explained replacing the expression of the self-energy into the equation Eq. (19), in order to obtain the corresponding YSR states. This self-energy can be written as (see Appendix
6 A) Σ(ω) = Σk (ω) + Σ⊥ (ω),
(23)
where we have separated the static component 2
J Σk (ω) = K S 2 G0 (ω) 4
(24)
from the dynamical component containing the quantum fluctuations h 2 n J 0 (ω−D(2S−1)) Σ⊥ (ω) = 4K S G0 (ω+D(2S−1))+G 2 +F (ω + D(2S − 1)) − F (ω − D(2S − 1))]} (25) where ρ0 F(z) = √ tan−1 ∆2 − z 2 z ∆ × . ∆ z
√
Wz √ W 2 + ∆ 2 ∆2 − z 2
(26)
In the limit W → ∞, replacing the self-energy Eq. (23) into Eq. (19), we obtain the equation for the YSR pole r ∆ + EYSR 2α2 EYSR 2 ∆ + EYSR 1−α − = 0. (27) ∆ − EYSR πSD(2S − 1) ∆ − EYSR The solution to this equation is reduced to the classical YSR state given by Eq. (5) when either S → ∞ or D →
Σ⊥ (ω) =
∞ (the explicit analytical solution is very complicated and it is not very illuminating, thus we have only written the equation rather than the solution). Note that while the classical component of (23) is unchanged as compared to that of Eq. (20), the term corresponding to quantum fluctuations has considerably changed. This is expected since, as we mentioned before, the anisotropy is revealed only through quantum fluctuation processes in the impurity.
2.
Case D < 0 and S half-integer
As we mentioned, when the anisotropy is negative (hard-axis anisotropy), the impurity favors the states with minimal projection of Sz . In addition, when the spin S is half-integer, the isolated impurity has a doublydegenerate ground-state spanned by the states m = ±1/2, which implies that quantum fluctuations subsist even in the limit |D| → ∞. This means that the classical YSR energy is never achieved in that limit [see the negative axis in Fig. 3(a)]. The corresponding expression for the components of the self-energy are (see Appendix A) J2 Σk (ω) = K 4
2 1 G0 (ω), 2
(28)
which corresponds to Eq. (24) with S replaced by the value 1/2, and
2 JK [S (S + 1) − 3/4] [S (S + 1) + 1/4] G0 (ω) + 4 2 2 G0 (ω + 2|D|) + G0 (ω − 2|D|) × + F (ω + 2|D|) − F (ω − 2|D|) . 2
(29)
In order to get more physical insight, we now take the limit D → −∞ in this expression. In that case, note that the second line vanishes since G0 (ω ± 2|D|) → 0 and F (ω + 2|D|) − F (ω − 2|D|) → 0. Physically, taking the limit D → −∞ amounts to projecting out the states |m| > 1/2: the fluctuations between states ±1/2 ±3/2 . . . become negligible, but the fluctuations between states −1/2 1/2 remain. Therefore, at low energy the anisotropic original spin S maps onto an effective spin Seff = 1/2, which is coupled to the superconducting host via p the anisotropic Kondo exchange z ⊥ J˜K ≡ JK and J˜K ≡ JK [S (S + 1) + 1/4]. For this last case, the self-energy can be easily derived and writes 2 1 ˜z 2 2 ⊥ ˜ Σ (ω) = JK Seff + JK Seff G0 (ω) . (30) 4
ing that the classical limit cannot be reached. Consistently, the position of the YSR states as a function of S becomes 2 1 − α2 1 + S1 + 4S3 2 EYSR , = (31) 2 ∆ 1 + α2 1 + S1 + 4S3 2
It is easy to see that in terms of this effective Seff = 1/2 impurity, the transverse contribution is actually more important than the parallel (classical) contribution, mean-
In Fig. 3 (b) we show the YSR energy for D < 0 and spin S integer. In this case the ground state of the isolated impurity corresponds to m = 0 and therefore
very different from the classical limit Ecl given by Eq. (5) (note that the case S = 3/2 is an exception, for which EYSR = Ecl ). This behavior is shown in Fig. 3(a) for D < 0, where we note the difference with respect to the case D > 0.
3.
Case D < 0 and S integer
7
0.70
3.0
0.65 0.60
Ecl [Δ] S=3/2 S=5/2 S=7/2 S=15/2
0.55 0.50
A[1/Δ]/ρ0
3.5
EYSR [Δ]
a) 0.75
2.5 2.0 1.5 1.0 0.5 0.0 3.5
0.45
b)
Ecl [Δ] S=1 S=2 S=3 S=4
0.9 0.8
A[1/Δ]/ρ0
3.0
b) 1.0
EYSR [Δ]
a)
2.5 2.0 1.5 1.0
0.7
0.5
0.6
0.0 3.5
0.5
c)
0.4 -4
-2
0 D[Δ]
2
4
A[1/Δ]/ρ0
3.0 2.5
δ-peak
2.0 1.5
Precursor
1.0
of classical YSR state
0.5
FIG. 3. (Color online) YSR subgap-state energy EYSR as a function of the anisotropy for a) S half-integer and b) S integer. Here has been used the value α = 0.47.
0.0 3.5
d)
only the dynamical term Σ⊥ (ω) contributes to the selfenergy. The system needs an energy of the order of |D| in order to flip the spin to the states m = ±1 at T = 0. Therefore, when D → −∞ these fluctuations become negiglible, and the self-energy vanishes. Therefore, the effective coupling between the superconductor and the impurity tends to zero and the YSR levels shifts towards the edge of superconductor gap: the system effectively behaves as an unperturbed superconductor. The analytical expression for the self-energy in this case is
Σ(ω) = Σ⊥ (ω) 2 JK G0 (ω + |D|) + G0 (ω − |D|) = [S (S + 1)] 4 2 +F (ω + |D|) − F (ω − |D|)]} , (32)
where the static contribution Σk (ω) arising from the term Sz2 is absent. As |D| → ∞, it is easy to check that Σ (ω) → 0.
A[1/Δ]/ρ0
3.0 2.5 2.0 classical
1.5
YSR state
1.0 0.5 0.0 0.85
0.90
0.95
1.00
ω[Δ] FIG. 4. (Color online) Local density of states (LDOS) at the site of the impurity for a) D = 0, b) D = 0.01∆, c) D = 0.03∆ and d) D = ∆/2, at temperature T = 0.25∆. α = 0.1 and S = 3/2. The orange filling part represents the contribution of the anisotropy-induced continuum of thermally-excited quasiparticles.
III.
FINITE-TEMPERATURE RESULTS
The most important results of this work arise from the interplay between anisotropy, quantum fluctuations and thermal effects, which put in evidence the complexity of quantum impurities in superconductors. An excited spin ¯ above the impurity ground state lying at an energy ∆ state will be thermally populated with Boltzmann prob¯ ability ∼ e−∆/T , and new scattering channels, which are
8 ¯
0.98
0.4
W1+W2
0.95
T=0
W2
0.3
0.96
W/100
EYSR [Δ]
0.97
T=0.25Δ 0.2
Δ-2D
0.1 W1 0.0 0.00
0.02
0.04
0.06
0.94
0.08
0.10
0.12
0.14
D[Δ]
0.00
0.05
0.10 D[Δ]
0.15
0.20
FIG. 5. (Color online) Evolution of the subgap spectrum of excitations as a function of the anisotropy for spin impurity S = 3/2 and α = 0.1. The orange dashed line marks the classical YSR limit labelled by Ecl , the dotted black line is the edge of the thermal continuum described by the function ∆ − 2D, and the blue solid line is the YSR energy for T = 0. In addition, we show the maxima of the LDOS at T = 0.25∆ (gray dots). As expected, the D = 0 the position of the YSR state at T = 0.25∆ coincides with the value at T = 0. At finite D, the δ-peak representing the YSR state shifts to lower energies, following closely the continuum edge, and at the same time a new maximum emerges within the thermal continuum for D ∼ 0.014∆, at approximately the crossing of the black and blue lines. Finally, in the limit D → ∞, the classical YSR state given by Eq. (5) is recovered. Inset: spectral weight for the δ-peaks (W1 ), for the thermal continuum (W2 ), and their sum.
forbidden at T = 0, will be allowed. Intuitively, a thermally populated excited state in the impurity can decay into the lowest spin state, providing an amount of energy ¯ to a Bogoliubov quasiparticle. As a result, the en∆ ergy barrier to reach the quasiparticle continuum is now ¯ instead of ∆ as in the isotropic case. This gives ∆ − ∆, rise to an intra-gap continuum of excitations, originated from the combination of anisotropy, quantum fluctuations and finite temperature. It is important to stress that these three ingredients are necessary for this to occur. In order to obtain more physical insight into these new processes, we now focus for concreteness on the case D > ¯ = 2D. From the Eqs. (A13) 0 and S = 3/2, where ∆ and (A14) we obtain the following expression for the selfenergy
+ 23
Σ(ω, T ) = h ¯
2 Z Σ(ω, 0)
¯ G0 (ω+∆))+G0 (ω−∆) 2
+
¯ 2 2e−∆/T JK Z 4
n
3 2 2
G0 (ω) (33) io ¯ +F ω−∆ ¯ −F ω+∆ ,
where Σ(ω, 0) is the zero-temperature self-energy Eq. ¯ (23) for S = 3/2, and where Z = 2 + 2e−∆/T is the partition function (here an unimportant common factor has been ignored). Note that the most relevant dependence
on temperature is encoded in the factor e−∆/T /Z multiplying the term in curly brackets. A finite imaginary part in the self-energy, induced by thermally-excited quasipar ¯ ¯ for ticles, arises from the sum G0 (ω + ∆)/2 −F ω +∆ ¯ and from the sum G0 (ω − ∆)/2+F ¯ ¯ ω ≥ |∆− ∆|, ω−∆ ¯ Therefore, the superconducting quasifor ω ≤ −|∆ − ∆|. ¯ ≤ particle gap effectively reduces to the region −(∆ − ∆) ¯ ω ≤ ∆ − ∆ under the simultaneous effect of temperature and anisotropy. Moreover, note that this gap completely ¯ ≥ ∆. However, in the limit D → ∞ the closes for ∆ Boltzmann factor in Eq. (34) becomes negligible, and so does the imaginary part. Therefore, in this limit, the usual superconductor gap ∆ is recovered. This phenomenology should be experimentally observable for instance, in the dI/dV signal in STM experiments. To better visualize the experimental consequences of our theoretical results, we analyze the particle component of the local density of states (LDOS) at the site of the impurity, A (ω, T ) = − π1 Im G(11) (ω, T ) , (34) where G(11) (ω, T ) is the (11) element of the dressed Nambu Green’s function Eq. (14). In Fig. 4 we show the LDOS for S = 3/2 and α = 0.1, for different values of the anisotropy parameter D = 0, 0.01∆, 0.03∆ and ∆/2. We fix the temperature to T = 0.25∆. Note that since we have considered a BCS model with a static pairing parameter ∆, which strictly speaking should be obtained self-consistently48 , we cannot use values of T too close to Tc . Using the particlehole symmetry of our theoretical model, only the region ω > 0 has been plotted. For D = 0, a δ-peak representing the YSR state given by Eq. (21) is located at 0.967∆. As the anisotropy increases to D = 0.01∆, two main features are observed: a) an intra-gap continuum of thermally-excited quasiparticles emerges in the interval ¯ < ω < ∆ (see orange-shaded region), and the su|∆ − ∆| perconductor gap is effectively reduced, and b) the YSR δ-peak shifts towards the Fermi energy, appearing just below the thermal continuum edge. As D is increased, spectral weight is transferred from the YSR δ-peak to the thermal continuum, and a new resonance emerges within the continuum. For D = ∆/2, the original superconductor gap at T = 0 closes up, completely filled with thermally-excited quasiparticles [see Fig. 4(d)], and the YSR δ-peak, which can only occur in the absence of a quasiparticle continuum, dissappears. Concomitantly, the spectral weight of this δ-peak vanishes. Eventually, for even larger values of D, the only surviving feature at frequencies |ω| < ∆ is the resonance inside the thermal ¯ continuum, whose width is proportional to ∼ e−∆/T . In the limit D → ∞, the width of this resonance vanishes and the classical YSR state is recovered. This complex behavior can be better understood in Fig. 5, where we show the position of the Shiba states at T = 0 (continuous blue line) and the edge of the thermal continuum described by the function ∆ − 2D (dotted black line), as functions of D. In addition, we show
9 the maxima of the LDOS at T = 0.25∆ (gray dots) for energies ω < ∆. At D = 0 the position of the YSR state at T = 0.25∆ coincides with the value at T = 0. This is expected, since in the absence of anisotropy the LDOS does not depend on temperature. However, as D is increased the δ-peak representing the YSR state shifts to lower energies following closely the continuum edge (dotted black line). Simultaneously, a new maximum of the LDOS emerges within the thermal continuum for D ∼ 0.014∆, at the crossing of the black and blue lines. This new maximum is a remnant of the zerotemperature YSR state given by the blue line which, since now it lies within the thermal continuum, it coexists with thermally-excited quasiparticles and becomes broadened. The crossing point at D ∼ 0.014∆ marks the point at which the spectral weight starts to be transferred from the δ-peaks to the continuum, labelled respectively as W1 and W2 , in the inset of Fig. 5. Note that the sum of the contributions to the spectral weights is a smooth ¯ function of D, which behaves as ∼ e−∆/T as expected ¯ → ∞, the spectral from Eq. (34). Eventually, for ∆ weight becomes exponentially small, and the probability to find a thermally excited quasiparticle in the region |ω| < ∆ vanishes. Consequently as mentioned above, thermal broadening effects become negligible and we recover the classical YSR δ-peak at position Ecl given by Eq. (5). In addition, the original superconductor gap ∆ is restored. This is consistent with the narrowing of the resonance shown in Fig. 4(d), which eventually becomes a δ-peak in the limit D → ∞. It is worth mentioning that for impurities with spin S > 3/2, the presence of higher excited states would give rise to more continuum-edges coming from different transitions (i.e. ±S ±(S − 1), ±(S − 1) ±(S − 2), ...), and therefore a much more complex subgap spectrum of excitations is obtained.
IV.
SUMMARY AND CONCLUSIONS
We have studied the subgap spectrum of magnetic impurities deposited ontop of clean superconductors by means of perturbation theory in the fermionic Green’s function, and we have focused on the simultaneous effects of single-ion anisotropy and temperature, which are relevant effects in, e.g., STM experiments. Our motivation is to obtain a simple theoretical description of the subgap excitations, which at the same time allows to go beyond the classical spin approximation, and that allows to include the effect of anisotropy. Our results point to the necessity of including the impurity spin quantum fluctuations in order to correctly interpret the experimental results, even for systems with relatively “large” spins. While many theoretical and experimental works make use of the classical spin approximation, we have shown that many interesting features of quantum impurities on superconductors cannot be described within the classical framework due to its tendency to oversimplify the in-
herent complexity of these systems. This is particularly important in the presence of anisotropy and at finite temperature. One important point in our work is that the presence of anisotropy changes the behavior of quantum impurities on superconductors in a fundamental way. Since anisotropy is ever-present at the surface of superconductors with large atomic number such as Pb, it cannot be disregarded in the theoretical description. Recently Pb has been used as platform of many STM experiments. Second-order perturbation theory predicts, in the presence of anisotropy, a shift of the Shiba states with respect to the fully isotropic value (D = 0). In particular, for easy-axis anisotropy D > 0, our results converge smoothly to the classical-spin case in the limit D → ∞. However, this limit is slowly reached [as ∼ 1/(DS)], and we expect that quantum fluctuations are important even for impurities with “large” spin. On the other hand, for the case of easy-plane anisotropy D < 0 the classical limit is never reached. From the technical perspective, the perturbative approach used here combines its simplicity of implementation with the ability to describe the simultaneous effects of anisotropy and temperature in an unbiased way. This is not an easy task to implement with other available methods. However, its drawback is that it cannot qualitatively describe the quantum phase transition taking place when TK ≈ ∆ since it does not include the many-body Kondo correlations, which are inherently non-perturbative. The regime of applicability therefore corresponds to systems which are deep in the unscreened phase TK ∆, and we expect the second-order perturbative approach to become quantitatively accurate only in the limit S → ∞. It would be interesting to have a better quantitative estimation of the performance of the perturbative method using, for instance, an exact impurity-solver, such as NRG. In addition, we note that in our theoretical description, we have used a single superconducting band. While this has the advantage of simplifying the analytical treatment, we note that in order to describe realistic experiments, this might not be enough. For instance, as we have mentioned in the main text, the presence of YSR multiplets and multiple scattering channels is beyond our single band theory. As a result we have found, at most, a single δ-peak excitation. However, the most important conclusions of this work does not concern the quantitative accuracy of the method, but are related to the qualitative features, in particular, those involving the simultaneous presence of temperature and anisotropy. We have pointed out that when both of these are present, novel features, such as the presence of an anisotropy induced thermal continuum inside the T = 0 superconducting gap, emerge. We believe that the presence of such a continuum is a robust feature which could be experimentally tested. To the best of our knowledge, such an effect has not been experimentally addressed. Finally, we think our results will be
10 useful for the correct interpretation of STM experiments, and to make theoretical predictions which take into account the true quantum nature of magnetic impurities in superconductors.
Σ(iνn ) =
2 X
JK Sz2 0 (iωl ) + S − S + 0 (iωl ) 4β l,k
×G0,k (iνn − iωl ) ,
(A1)
with
ACKNOWLEDGMENTS
Sz2
(iωl ) = 0
S X m2 e−βEm , Z
(A2)
Am , iωl − (Em+1 − Em )
(A3)
m=−S
A.M.L. acknowledges financial support from Proyecto PICT-0217 2015-2017 (ANPCyT - Argentina), Proyecto PIP-11220150100364 (CONICET - Argentina) and Relocation Grant RD1158 - 52368 (CONICET - Argentina).
S−1 X
− + S S 0 (iωl ) = m=−S
1 [S(S + 1) − m(m + 1)] (e−βEm+1 − e−βEm ). Z (A4) ωl = 2πn/β is the bosonic Matsubara’s frecuency and β = 1/kB T depend of temperature Em = −Dm2 , P T . −βE m and the partition function is Z = m e . From the Eq. A2, we can see that hSz i0 is independent of iωl and all the dinamic part is in the fluctuations hS − S + i0 . If we do a analitic extension for iνn → ω + iδ (for simplicity, every time that we refer to ω, this has implicit the infinitesimal term iδ added) and develope the sum in the Matsubara frecuency ωl , the matrix element Σ(11) (ω) = −Σ(22) (−ω) is Am =
Appendix A: Calculation of Self-energy
In this appendix, we describe the intermediate steps for obtain the Eqs. 23, 28, 29, 32. Developing order second perturbation theory, we find that the self-energy is
2
Σ(11) (ω) =
JK 4
P
m2 e−βEm g0,k (ω) m,k Z
+Am
2
JK 4
1 − Z [S(S + 1) − m(m + 1)] e−βEm+1 g0,k (ω − ∆m ) i Uk2 Vk2 f (−E ) + f (E ) , k k ω−∆m −Ek ω−∆m +Ek
−
P
m,k
where ∆m = Em+1 − Em = −D (2m + 1). For T ∆, f (−Ek ) → 1 y f (Ek ) → 0. Factoring e−βEg (Eg is the ground state energy for the isolated impurity) of Z and
2
Σ(11) (ω) =
JK 4
P
m,k
m2 e−β (Em −Eg ) g0,k (ω) Z
2
(A5)
2
the exponentials, and replacing Uk = Vk = 1/2, (since that the part that contains εk is null by parity), we have:
2
−
JK 4
P 1 −β(Em+1 −Eg ) g0,k (ω − ∆m ) m − Z [S(S + 1) − m(m + 1)] e i R + Am2 ρ0 ω−∆ −dε√ε2 +∆2 . (A6) m
The integral of the Eq. A6 is Z
dε √ = g0 (ωm ) ε2 + ∆ 2 # W ωm p arctan √ 2 W 2 + ∆2 ∆2 − ωm " # √ W + ∆2 + W 2 −2ρ0 ln , (A7) ∆ ρ0
ωm ρ0 −2 p 2 ∆2 − ωm
ωm − "
11
Z ρ0
# " √ W + ∆2 + W 2 dε √ = g0 (ωm ) − 2ρ0 ln ∆ ωm − ε2 + ∆2 " # ωm ρ0 W ωm p −2 p arctan √ , 2 2 ∆2 − ωm W 2 + ∆2 ∆2 − ωm (A8)
where ωm = ω − ∆m have implicit ω, and g0 (ω) =
ωρ0 2 − ω2 ∆ k W × arctan √ , ∆2 − ω 2 X
g0,k (ω) = −2 √
(A9)
where W P is the semi-wide of band. Taking into accound that m Am = 0, the Eq. A6 is 2
Σ(11) (ω) =
JK 4
P h m2 e−β(Em −Eg ) m
Z
+ Am √ωm2 ρ0 2 arctan
√
∆ −ωm
g0 (ω) + 12 Bm g0 (ωm ) Wω m √ ,(A10) 2 2 2 2
W +∆
f0 (ω) =
Σ(12) (ω) =
m
0 + Am √ ∆ρ 2
2 ∆ −ωm
Z
arctan
√
f0 (ω) + 12 Bm f0 (ωm ) Wω m √ ,(A11) 2 2 2 2
W +∆
∆ −ωm
where
2
b1 =
(A12)
∆ −ωm
P h m2 e−β(Em −Eg )
4
f0,k (ω) = −2 √
At T = 0 and D > 0 the ground state for the isolated impurity is ±S and the expresion for the matricial self¯ = ∆−S = −∆S−1 = energy is as the Eq. 23 where ∆ D(2S−1). For better understanding, the figure 1b) shows the level diagram with the quantum fluctuations of the ¯ is the energy need for that impurity for this case, where ∆ the spin flips between Sz = ±S and Sz = ±(2S − 1). For D < 0 the ground state for the isolated impurity with S odd is ±1/2, and the expresion for the self-energy is the Eq. 28 plus Eq. 29, where ∆1/2 = −∆−3/2 = 2|D|. For S even, the ground state is the zero spin projection and the z component of the self-energy is zero. Then only the quantum fluctuations contribute to self-energy, and so the self-energy is as the Eq. 32, where ∆0 = −∆−1 = |D|. At finite temperature, from the Eqs. A10 and A11 the self-energy Σ (ω, T ) take the following form
with Bm = [S(S + 1) − m(m + 1)] (e−βEm+1 + −βEm )/Z. Proceding of analog form as Σ(11) , we get e that the matrix element Σ(12) (ω) = Σ(21) (ω) is 2 JK
∆ρ0 ∆2 − ω 2 k W × arctan √ . ∆2 − ω 2 X
2 Σ (ω, T ) = Z
Σ (ω, 0) +
N X
! bi e
¯ i /T −∆
,
(A13)
i=1
where Σ (ω, 0) is like Σ (ω) of the subsection II B 1, II B 2 or II B 3 depending of case, N = (2S + 1)/2 for S odd and N = S for S even, bi is a factor that depends of S ¯ i is the excitation energy from ground state and D, and ∆ ¯ 2 we can truncate to i-th exited state. In the limit T ∆ ¯ the sum of the Eq. A13 to the orden ϑ e−∆2 /T . For the special case D > 0, we obtain that
¯ + G0 (ω − ∆) ¯ G0 (ω + ∆)) 2 ¯ +F ω−∆ ¯ (S − 1) G0 (ω) + S −F ω+∆ 2 G0 (ω + ∆−S+1 ) + G0 (ω − ∆−S+1 ) +(2S − 1)Θ[S − 3/2] + F (ω + ∆−S+1 ) − F (ω − ∆−S+1 ) , (A14) 2
JK 4
¯ =∆ ¯ 1 and ∆−S+1 = D(2S − 3) where ∆
1
2 3 4 5
A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Rev. Mod. Phys. 78, 373 (2006). L. Yu, Acta Phys. Sin. 21, 75 (1965). H. Shiba, Progress of Theoretical Physics 40, 435 (1968). A. I. Rusinov, Zh. Eksp. Teor. Fiz. 56, 2047 (1969). A. Yazdani, B. A. Jones, C. P. Lutz, M. F. Crommie, and D. M. Eigler, Science 275, 1767 (1997).
6
7
8
9
M. E. Flatt´e and J. M. Byers, Phys. Rev. Lett. 78, 3761 (1997). A. Yazdani, C. M. Howald, C. P. Lutz, A. Kapitulnik, and D. M. Eigler, Phys. Rev. Lett. 83, 176 (1999). O. Fischer, M. Kugler, I. Maggio-Aprile, C. Berthod, and C. Renner, Rev. Mod. Phys. 79, 353 (2007). S.-H. Ji, T. Zhang, Y.-S. Fu, X. Chen, X.-C. Ma, J. Li,
12
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
W.-H. Duan, J.-F. Jia, and Q.-K. Xue, Phys. Rev. Lett. 100, 226801 (2008). M. Iavarone, G. Karapetrov, J. Fedor, D. Rosenmann, T. Nishizaki, and N. Kobayashi, J. Phys.: Condens. Matter 22, 015501 (2010). S.-H. Ji, T. Zhang, Y.-S. Fu, X. Chen, J.-F. Jia, Q.-K. Xue, and X.-C. Ma, App. Phys. Lett. 96, 073113 (2010). K. J. Franke, G. Schulze, and J. I. Pascual, Science 332, 940 (2011). J. Bauer, J. I. Pascual, and K. J. Franke, Phys. Rev. B 87, 075125 (2013). N. Hatter, B. W. Heinrich, M. Ruby, J. I. Pascual, and K. J. Franke, Nature Communications 6, 8988 (2015). M. Ruby, Y. Peng, F. von Oppen, B. W. Heinrich, and K. J. Franke, Phys. Rev. Lett. 117, 186801 (2016). Choi, R.-V. Deung-Jang, d. B. Carmen, U. Joeri, L. Miguel M., P. Nicols, and J. Ignacio, Nature Communications 8, 15175 (2017). B. Heinrich, L. Braun, J. I. Pascual, and K. J. Franke, Nature Physics 9, 765 (2013). S. Nadj-Perge, I. K. Drozdov, B. A. Bernevig, and A. Yazdani, Phys. Rev. B 88, 020407 (2013). J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Phys. Rev. Lett. 111, 186805 (2013). B. Braunecker and P. Simon, Phys. Rev. Lett. 111, 147202 (2013). F. Pientka, L. I. Glazman, and F. von Oppen, Phys. Rev. B 88, 155420 (2013). C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). S. Nadj-Perge, I. K. Drozdov, J. Li, H. Chen, S. Jeon, J. Seo, A. H. MacDonald, B. A. Bernevig, and A. Yazdani, Science 346, 602 (2014). R. Pawlak, M. Kisiel, J. Klinovaja, T. Meier, S. Kawai, T. Glatzel, D. Loss, and E. Meyer, Npj Quantum Information 2, 16035 (2016). ˇ R. Zitko, O. Bodensiek, and T. Pruschke, Phys. Rev. B 83, 054512 (2011). ˇ R. Zitko, Physica B: Condensed Matter (2017), https://doi.org/10.1016/j.physb.2017.08.019. J. Franck, F. Manchester, and D. Martin, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 263, 494 (1961). J. Kondo, Progress of Theoretical Physics 32, 37 (1964).
29
30
31
32
33
34
35
36 37
38
39
40
41
42
43
44
45
46
47 48
49
A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, New York, 1993). J. Zittartz and E. M¨ uller-Hartmann, Zeitschrift f¨ ur Physik A Hadrons and nuclei 232, 11 (1970). J. Zittartz, Zeitschrift f¨ ur Physik A Hadrons and nuclei 237, 419 (1970). T. Yoshioka and Y. Ohashi, Journal Phys. Soc. Japan 67, 1332 (1998). T. Yoshioka and Y. Ohashi, Journal of the Physical Society of Japan 69, 1812 (2000). G. D. Mahan, Many particle physics (Plenum, New York, 1981). A. Sakurai, Progress of Theoretical Physics 44, 1472 (1970). ˇ R. Zitko, Phys. Rev. B 93, 195125 (2016). K. Satori, H. Shiba, O. Sakai, and Y. Shimizu, Journal of the Physical Society of Japan 61, 3239 (1992). O. Sakai, Y. Shimizu, H. Shiba, and K. Satori, Journal of the Physical Society of Japan 62, 3181 (1993). A. Martn-Rodero and A. Levy Yeyati, Advances in Physics 60, 899 (2011). C. F. Hirjibehedin, C.-Y. Lin, A. F. Otte, M. Ternes, C. P. Lutz, B. A. Jones, and A. J. Heinrich, Science 317, 1199 (2007). D.-J. Choi, C. Rubio-Verd´ u, J. de Bruijckere, M. M. Ugeda, N. Lorente, and J. I. Pascual, Nature Communications 8, 15175 (2017). J. Bauer, A. Oguri, and A. C. Hewson, Journal of Physics: Condensed Matter 19, 486211 (2007). ˇ M. Zonda, V. Pokorn´ y, V. Janiˇs, and T. Novotn´ y, Phys. Rev. B 93, 024523 (2016). ˇ V. Janiˇs, V. Pokorn´ y, and M. Zonda, Eur. Phys. J. B 89, 197 (2016). ˇ M. Zonda, V. Pokorn´ y, V. Janiˇs, and T. Novotn´ y, Scientific Reports 5, 8821 (2015). D. J. Luitz and F. F. Assaad, Phys. Rev. B 81, 024509 (2010). Y. Nagaoka, Phys. Rev. 138, A1112 (1965). A. L. Fetter and J. D. Walecka, Quantum theory of manyparticle systems (McGraw-Hill, New York, 1971). M. Tinkham, Introduction to Superconductivity, 2nd Edition (McGraw-Hill, Inc., 1996).
