Spin thermopower in the overscreened Kondo model

We study the spin thermopower in the two-channel spin-1/2 Kondo model which exhibits the phenomenon of impurity spin overscreening and non-Fermi-liquid properties. While magnetic field lower than the Kondo temperature does not strongly polarize the impurity spin, we show that it nevertheless strongly affects the low-energy part of the spectral function. In turn, this leads to characteristic saturation of the spin Seebeck coefficient at the value of 0.388kB/|e| at T ~ T*, where T* ~ B^2/TK is the scale of the crossover between the intermediate-temperature non-Fermi-liquid regime and the low-temperature Fermi-liquid regime. We show that measuring the spin thermopower at low magnetic fields would provide a sensitive test for distinguishing regular Fermi liquid, singular Fermi liguid, and non-Fermi liquid behaviour in nanodevices.

Experimentally, such behaviour was observed in a setup where a small quantum dot (the impurity) was coupled to conduction leads (first conduction band) and a big quantum dot (second conduction band) [23]. By careful tuning of system parameters, it has been shown that the transport properties scale with temperature and gate voltage in accordance with the two-channel Kondo (2CK) model.
The characterization of nanodevices through transport properties provides information about the spectral function of the system. The low-temperature differential conductance is a measure of the spectral density at the Fermi level. As it is relatively easy to measure, it remains the most commonly used experimental probe. Additional information can be gathered by measuring other transport properties, such as the thermopower (Seebeck coefficient), which is a sensitive probe of the particle-hole asymmetry of the spectral function [24,25]. In this work, we study a related property, the spin thermopower (spin Seebeck coefficient) [25,26]. It is defined as the induced spin voltage across the device at zero spin current in the presence of a temperature gradient. The spin Seebeck coefficient S s probes the particle-hole asymmetry of the spectral function in the presence of external magnetic field. Since the different kinds of the Kondo effect have very different response to an applied magnetic field, they can be clearly distinguished by different behaviour of S s as a function of magnetic field B and the temperature T , as will become evident in the following.
The spin thermopower in the conventional Kondo model (single-channel S = 1/2 model, which is a RFL) has been studied in [26]. Related results for the charge thermopower in the model with attractive interaction have been reported in [27]. For T T K , the spin thermopower as a function of B peaks at B ∼ T K (B is measured in the units of the Zeeman energy gµ B B, while the temperature is measured in the units of energy k B T ). For B T K , the spin thermopower as a function of T peaks at T ∼ 0.3T K . Finally, for B, T T K , the spin thermopower peaks at B ∼ 3 T . These features can be well understood within a simple model for the Kondo peak splitting in applied magnetic field, taking into account thermal broadening effects.
The spin thermopower in the underscreened Kondo model (single-channel S = 1 model, which is SFL [16,17]) has been studied in [25]. Due to the decoupled residual impurity moment, even a tiny magnetic field strongly polarizes the impurity and splits the Kondo peak. The main difference between the S = 1/2 and 1 models is thus in the B T K regime, where the maximum in the spin thermopower as a function of T is now found for T ∼ 0.2B, rather than at constant T ∼ 0.3T K ; the amplitude of the maximum is comparable in both the cases, being somewhat smaller in the underscreened case in a wide parameter range.
The spin thermopower in the overscreened Kondo model (two-channel S = 1/2 Kondo model, which is a true NFL system) is the subject of this work. The magnetic field is a relevant perturbation which drives the system away from the NFL fixed point, which governs the behaviour of the system in the temperature range T * T T K , to the stable low-temperature RFL fixed point at T T * . We will show that the 2CK model exhibits very peculiar low-field behaviour of the spin thermopower, which has a maximum at the crossover scale T * ∝ B 2 /T K with a height which saturates at small B (unlike in the fully screened and underscreened cases, where the height goes to zero). The saturated value at the maximum is 0.388k B /|e| ≈ 34µV K −1 . We will compare the very interesting contrasting behaviour of the underscreened and overscreened models: while in the underscreened model, a small magnetic field that strongly polarizes the impurity spin affects mainly the high-energy part of the spectral function, in the overscreened model the polarization effect is small but it strongly affects especially the  [3,11]. (c) Quantum dot coupled to a large quantum dot [23]. low-energy part of the spectral function, thus leading to a large spin-thermal response at T * for any non-vanishing field.

Models, spin thermopower and method
The physical systems discussed in this work are schematically represented in figure 1. The nanodevice is coupled to two contacts kept at different temperatures. The electrons in these two contacts form one screening channel. In addition, in the 2CK case, the role of the second screening channel is played by the electrons in the additional large quantum dot. The electrons in this dot are electrostatically confined by the large Coulomb repulsion in order to eliminate processes in which an electron would tunnel from the large quantum dot to the contacts; such processes would destroy the two-channel physics (i.e. they correspond to an additional relevant operator which drives the system away from the NFL fixed point).
At temperatures low enough, such nanosystems can all be modeled using some variant of the Kondo impurity model, where the impurity degrees of freedom are described using a quantum mechanical spin operator S which is locally coupled with the spin density of the conduction band electrons. The corresponding Hamiltonian is thus where c † k,σ,i are creation operators for the itinerant electrons in channel i with momentum k, spin σ and energy k . The spin density of the band i can be expressed as Here f † 0,σ,i = (1/ √ N ) k c † k,σ,i is the combination of states which couples to the impurity, while σ = {σ x , σ y , σ z } is a vector of Pauli matrices. The index i ranges over the number of channels n (one or two). Depending on the value of the impurity spin S and the number of channels n, the impurity spin will be either fully screened (n = 2S), underscreened (n < 2S) or overscreened (n > 2S) [18,[28][29][30][31][32]. We fix the Kondo exchange coupling J to a constant value J/D = 0.2. Here D is the half-width of the conduction band which is assumed to have a flat density of states ρ = 1/2D. The half-bandwidth is also used as the energy unit, D = 1.
The Kondo temperature is the same (up to non-exponential prefactors) in all the three models, T K ∝ exp(−1/ρ J ). Numerically, we find where T K,W is the Kondo temperature as defined by Wilson through the impurity magnetic susceptibility, T K,W χ imp (T K,W ) = 0.07 [33,34]. The magnetic field is measured in units of the Zeeman energy (i.e. we absorb the factor gµ B into B).
In this work, we focus on the spin Seebeck coefficient S s , defined as [26] The spin thermopower is the ability of the device to convert the temperature difference T to spin voltage V s , assuming no spin current I s flows through the impurity. In units of k B /|e|, to be used in what follows, the spin Seebeck coefficient is determined by computing the transport integrals where f (ω) is the Fermi-Dirac function at temperature T . At the particle-hole symmetric point (the case considered in this work), one has I 0↑ = I 0↓ ≡ I 0 and I 1↑ = −I 1↓ ≡ I 1 . From the definition it is manifest that the spin thermopower probes the difference in the particle-hole asymmetry of the two spin species. The quantity T σ in equation (5) is the transmission function of electrons with spin σ in the first channel (i.e. the channel which corresponds to the source and drain leads). It is proportional to the imaginary part of the T matrix for the Kondo model, defined as G = G 0 + G 0 T G 0 , where G is the interacting Green's function for electrons, while G 0 is the non-interacting one. For a given model, it may be computed using the equation of motion approach. One finds where O σ is defined as the commutator between the annihilation operator on the first site of the Wilson chain (which is the same as the combination of conduction-band states which couples to the impurity, f 0,σ,1 ) and the Kondo coupling term, The result is a composite fermion operator [35,36] The transport integrals have been computed using the numerical renormalization group (NRG) method [33,34,37], using the approach described in [26]: the Lehmann representation of the spectral function is integrated over and the transport integrals I nσ are computed for all temperatures in a single NRG sweep. It has been shown that for the RFL case this leads to only a very small deviation (few per cent at most) from the results of the more accurate full-densitymatrix (FDM NRG) approach [38], and we have now verified that this is the case also for the SFL in magnetic field despite the strong magnetic polarizability of this model. We conclude this section by a remark about the role of the second quantum dot as regards the particle and heat flow. We have assumed that the voltage and the temperature in the leads are The inset shows how, for temperatures below the Kondo temperature (indicated by arrows), the curves for different fields from B = 10 −7 to 10 −6 collapse into a universal curve given by equation (7) and shown with green dotted line. The low temperature, T T * , asymptote of equation (9) is shown as red line. The intermediate temperature, T * T T K , asymptote of equation (10) is shown as violet line.
symmetrically shifted with respect to the equilibrium value. For this reason, neither charge nor spin current flow to the large quantum dot. The large dot thus only plays the role of an additional screening channel.

Spin Seebeck coefficient of the overscreened Kondo effect
The main result of this work is shown in figure 2. We plot the spin Seebeck coefficient of the 2CK model for a range of applied external magnetic fields B as a function of T . For any B, the spin thermopower curve has a single peak. For B T K , the peak occurs at the NFL-RFL crossover scale T * ∝ B 2 /T K . The B 2 scaling is a characteristic feature of the model and is not expected to be observed in single-channel models. In this regime, S s (T ) at the point of its maximum saturates at a value that is determined using analytical arguments in the following section. (A somewhat related saturation of the charge Seebeck coefficient has been noted in a different proposed physical realization of the 2CK model [39]. In that case, the symmetry breaking operator was the channel asymmetry, J 1 = J 2 .) The behaviour is universal and we find that the curves, when plotted as a function of T /T * , fully overlap at low temperatures; see the inset in figure 2.
For B T K , the spin thermopower peaks on the temperature scale of B. Such asymptotic behaviour is expected for any impurity model where the local degrees of freedom are described by a spin operator. The reason is that for large B, there will be a threshold for inelastic spin excitations given by the Zeeman energy. The spectral lineshapes, however, are expected to be model-dependent. (The small drop in the maximum of S s (T ) at the largest values of B in figure 2 is due to finite-bandwidth effects.)

Analysis of the asymptotic behaviour and the non-Fermi-liquid-regular Fermi-liquid crossover
The NFL state of the 2CK model is destabilized by a perturbation due to an external magnetic field acting on the impurity, forcing the system towards a RFL ground state. In [40] the authors exploited a connection to an exactly solvable classical boundary Ising model to obtain the exact finite-temperature transmission function, describing the crossover from low temperature, T T * , FL to intermediate temperature, Here is the Gamma function and 2 F 1 is the hypergeometric function. The crossover temperature scale, is proportional to the square of the magnetic field, which should be small enough, B T K , to ensure that the crossover temperature is well separated from the Kondo temperature. c B = O (1) is a fitting parameter; in our case c B = 0.436. This theory applies exactly in the B T K limit. We find that for the range of B considered in this work, the transmission function has a multiplicative field-dependent prefactor, which, however, largely cancels out when S s (T ) is computed as a ratio of two transport integrals; thus the universal scaling for S s (T ) applies well for B/T K 10 −1 .
To evaluate the spin Seebeck coefficient, the integrals over ω in equations (3) and (4) can be calculated exactly. The spin Seebeck coefficient, is a universal function of rescaled temperature T /T * . The inset to figure 1 shows how the NRG curves corresponding to different magnetic fields B T K collapse into a universal curve at temperatures well below T K . In this regime, the spin Seebeck coefficient reaches its maximum value of S s (T max ) = 0.388, independent of the field, at T max = 0.829T * . We were unable to find a closed-form expression for S s (T max ), but it can be computed numerically as the maximum value of the expression in equation (7).
At low temperatures, T T * , the transmission function is well approximated by its zero temperature form. For ω T * the transmission function deviates form its ω = 0 value, T σ (0, 0) = 1 2 , in a linear fashion The deviation T σ (ω, 0) − 1 2 reaches its maximal value of 0.13 at ω ≈ 4T * and drops to zero as ln ω √ ω in the high frequency limit. The low-temperature spin Seebeck coefficient, probing the lowfrequency part of the transmission function, equation (8), exhibits a typical Fermi liquid linear dependence on temperature. At higher temperatures, T * T T K , the transmission function takes a simpler form where K is the complete elliptic integral of the first kind. It can be cast in a scaling form where the scaling function F is linear in frequency for ω T , ∼ 0.082u, reaches a maximum value of 0.15 at ω ≈ 5T and drops to zero as ln u √ u in the high-frequency limit. Such a scaling causes the spin Seebeck coefficient to exhibit an anomalous NFL dependence on temperature To sum it up, the spin Seebeck coefficient of the 2CK model at low B is characterized by an asymmetric peak of a universal form with a maximum value of about 0.4 at the crossover temperature T * between RFL and NFL regimes. The low-temperature asymptote is proportional to T /T * , while the high-temperature one is proportional to √ T * /T .

Contrasting behaviour of spin thermopower in regular, singular and non-Fermi liquids
In figure 3, we compare the main characteristics of the spin thermopower in the three models considered. As a function of the applied magnetic field we plot the following quantities: the temperature T max of the maximum, of the spin Seebeck coefficient S s (T ), the value S s (T max ) of the spin Seebeck coefficient at this maximum, and the impurity magnetic moment We first note that the high-field behaviour is rather similar in all the three models, as has been already suggested in section 3. We find that the temperature of the maximum is almost the same in all the three cases and given approximately by T max ≈ 0.3B. The value of spin Seebeck coefficient at this maximum depends only on the impurity spin, not on the number of channels, and it saturates for high magnetic fields: for S = 1/2 we find S s (T max ) ≈ 1.32, while for S = 1 we obtain a smaller value S s (T max ) ≈ 0.9. The magnetizations at the maximum are also spin dependent: for S = 1/2 we find m(T max ) ≈ 0.45, while for S = 1 it is twice as large, m(T max ) ≈ 0.9.
A crude way to understand the high-field asymptotic behaviour is to postulate the following rough approximation for the transmission function in the large-B limit: i.e. a piecewise constant function with a jump at ω = σ B by a value proportional to the magnetization with c a constant of order 1. The increase in scattering is due to the opening of the inelastic scattering channel due to spin-flips, which is possible for energies above the Zeeman energy. This form is suggested by the more accurate renormalization group theory [36]. We find that the spin Seebeck coefficient obtained using equation (11) has a peak at T = 2.3B with a height of 0.89 (for c = 1 and taking the value of m from NRG). Considering the extreme crudeness of the model, the result is a fair approximation for the true values.
For all the three models, the crossover to the low-field behaviour occurs on the same scale of the Kondo temperature. The low-field asymptotic behaviour, however, is markedly different in the three models. In the RFL case, the maximum occurs at the constant temperature of T ∼ 0.3T K [26]. This is because the impurity spectral function is not significantly affected by magnetic fields lower than T K ; there is only a rigid shift of the two spin-projected components of the spectral function. The linear reduction of S s (T max ) is also well accounted for by simple Fermi liquid arguments [26]. RFLs exhibit Pauli paramagnetism with T K playing the role of the Fermi temperature, thus the magnetization of this model at low temperatures is given by In the SFL, the spin Seebeck coefficient peaks at a temperature on the scale where the system crosses over from the SFL regime at intermediate temperatures to the asymptotic RFL behaviour at low temperatures [10]. The crossover temperature, defined from the thermodynamic properties as the temperature where the impurity entropy decreases to one half of its value at the SFL fixed point, is found to be T td = 0.32B, with no logarithmic correction. The position of the peak in S s , T max , appears to have logarithmic corrections compared to T td . The maximum spin Seebeck coefficient S s (T max ) decreases to zero as a linear function, but again with logarithmic corrections. For completeness we also plot the impurity spin magnetization for the three models in the right panel of figure 3. We emphasize that strong spin polarization does not necessarily imply a large spin Seebeck coefficient. In fact, the underscreened model-with by far the largest polarization-has the smallest spin Seebeck coefficient in a wide interval of magnetic fields (in fact, only for B < 10 −2 T K is its spin Seebeck coefficient somewhat larger than that of the fully screened model).
To better understand the value of the spin Seebeck coefficient we study the impurity spectral functions (figure 4). At B = 0, the overall aspect of the spectral functions is the same: they all have a Kondo resonance with a width proportional to T K . The main differences are better visible in the close-up to the peak region, (figure 5). The resonance shape is analytical only in the RFL case. The approach to the asymptotic value in the SFL is logarithmic [17], while in the 2CK effect it has a square root non-analyticity [41,42]. Note also that the ω = 0 value of the spectral function is twice as big in RFL and SFL as compared to the NFL case [42]. For small fields, B T K ; the effect on the overall spectral function is the most significant in the SFL case where a strong spin polarization clearly manifests on all frequency scales (see figure 4). Figure 5, however, reveals that in the NFL case the relative change of the spectral function at low frequencies is very significant. The spin-up spectral function is much higher for ω > 0 compared to ω < 0 (by almost a factor of 2), and it peaks on the frequency scale of T * ∝ B 2 /T K . The peak height even exceeds the height of the zero-field spectral function at ω = 0. This low-frequency polarization effect is much stronger compared to that for (singular and regular) Fermi liquid systems. This explains the large spin Seebeck coefficient in the NFL case.

Conclusion
We have analyzed the spin thermopower of the two-channel Kondo model, which is a paradigmatic case of an impurity model with non-Fermi-liquid properties at low-energy scales. We have shown that the crossover to the Fermi-liquid ground state generated by applied magnetic field B < T K leads to a pronounced peak in the spin Seebeck coefficient as a function of the temperature. The peak position correspond to the crossover scale T * ∝ B 2 /T K , while its height saturates at a sizeable value of 0.388k B /|e|. This is to be opposed to the singlechannel Kondo models with S = 1/2 (RFL) and S = 1 (SFL), where for B < T K the peak height decreases at low B. These differences arise due to characteristically distinct ways in which the magnetic field affects the spectral function in different cases. In the 2CK model, the spectral function becomes strongly spin-dependent on the lowest energy scales, in stark contrast to the strongly spin polarizable underscreened S = 1 model, where the spin dependence is large on higher-energy scales, but is less pronounced in the region −5k B T < ω < 5k B T , which is relevant for transport. An analytical approach can be applied to the universal low-temperature regime of the two-channel Kondo model and we have shown that the saturation value of ∼ 0.388k B /|e| can be computed numerically to high precision.
The very different spin-thermoelectric responses in the three classes of quantum impurity models discussed here could thus be used to experimentally discriminate over-screened and underscreened behaviour from the regular Fermi liquid behaviour with confidence. This is related to the different responses of the spectral function on the magnetic field found in these three cases. We speculate that all models with S = n/2 (full screening) are characterized by a spin Seebeck coefficient peak at ∼ T K with decreasing height, all models with S > n/2 (underscreening) are characterized by a peak at ∼ B (perhaps with some logarithmic corrections) with decreasing height, while in models with S < n/2 (overscreening) the spin Seebeck coefficient peaks at the crossover scale and its height saturates. Here S is the impurity spin, while n is the number of screening channels.
On the applied side, the spin-thermoelectric effect might be of interest for spintronics as it enables generation of the spin current without any accompanying charge current. In this respect we note that the overscreened Kondo model outperforms the exactly screened and the underscreened ones, especially at magnetic fields below and around the Kondo temperature. The universal dependence of spin Seebeck coefficient on temperature with the peak position specified by magnetic field could be considered also for sensitive measurements of the magnetic field. It would be fascinating if the overscreened model that is playing such an important role in the fundamental studies of correlated electrons, ended up in a useful device.