The Kondo scaling for spin thermocurrent in strongly correlated electron systems

One of the most important feature in the Kondo physics is the universal scaling behavior. In this study, we analyze the transport behavior of the spin thermocurrent driven by a small temperature bias and under a weak magnetic field. We conclude that the spin thermocurrent exhibits a universal scaling behavior, similar to the spin susceptibility. The validity of our conclusion is also checked by using the numerically exact hierarchical equations of motion approach. From our fitting result of the scaling function, it is found that the Kondo temperature can be determined by the maximum of the spin thermocurrent at T/TK=0.383 .

In the Kondo systems, there exists a characteristic quantity, the Kondo temperature T K , which describes the ground energy of the many-body bounding singlet state [2,18].Since the low-energy properties are governed by the ground state, the systems, although with different physical parameters, manifest the same behavior if they are scaled by their respective Kondo temperatures [19][20][21].The single-parameter dependence with the same function form is so-called the Kondo scaling [22][23][24].The universality of such behavior is extensively explored [25][26][27][28] and serves as a smoking gun to identify the Kondo resonance [29][30][31].The scaling universality in the temperature dependence of physical quantities and their response to external fields at lower energies than the scaling energy, the Kondo temperature T K , is the central feature of the Kondo effect.
In contrast to voltage-bias-driven nonequilibrium conductance, recently, the experiments on temperature-bias-driven thermocurrent have been boosted due to the advanced thermocurrent spectroscopy to study the related universality in the Kondo physics [32][33][34].Notwithstanding the zero-bias thermocurrent (the applied voltage bias V → 0) in the middle of Coulomb valley changes its sign at some temperature T > T K [35] or some external magnetic field µ B B > k B T K [36], its slope, a Fermi liquid parameter [37,38] or the derivative with respect to the probing voltage bias, exhibits a universal sign transition at a fixed magnetic field µ B B = k B T K in the very low-temperature T ≪ T K [34].
In this work, we explore the universal Kondo scaling in the pure temperature-bias-driven system with increasing temperature.Due to the sign change of the thermocurrent with increasing temperature, enlightened by a lot of recent studies [36,39] and our experience [40,41], we move our focus from thermocurrent to spin thermocurrent.A concise conclusion arrives from our study that the spin thermocurrent shows a universal Kondo scaling as a function of T/T K .

Model Hamiltonian
Consider a quantum dot (QD) connects with two nonmagnetic metal electrodes and a weak magnetic field is locally applied on the dot.Its Hamiltonian writes In the Hamiltonian, σ labels the electron spin, n σ = d † σ d σ , and U is the onsite Coulomb interaction.When the external magnetic field is absent, ε ↑ = ε ↓ = ε d , and therefore the system possesses time reversal symmetry.Furthermore, at ε d = ε sym ≡ −U/2 the system also has the particle-hole (P-H) symmetry with ⟨n ↑ ⟩ = ⟨n ↓ ⟩ = 1 2 .At high temperatures, the energy level is randomly occupied by a single electron with either spin up or spin down.The P-H symmetry is removed under the magnetic field and the energy level shifts to The hybridization between dot and electrodes is described by with α = L, R denoting the left and right electrodes.The two electrodes are kept in different temperatures T L and T R .c † kασ and c kασ is the creation and annihilation operator of the k state in the electrode α, respectively.And the nonmagnetic electrodes have the Hamiltonian with ε kα independent on the spin.Such a general Anderson impurity Hamiltonian [42,43] has been successfully and broadly employing to study the Kondo effect [44,45].
When the energy level of the QD lies below the Fermi level of electrodes, the QD does not favor the double occupancy state due to the onsite interaction U and thus has a large proportion of the single occupancy state, with either spin up or spin down.Although the QD is connected to the two electrodes, the electron transport through the dot is impossible because of the extra energy cost resulting from adding or removing one electron.At low temperatures, the single electron on the dot together with the electrons in electrodes forms a many-body Kondo resonance state and the Kondo resonance state at the Femi level provides an electron transport channel.Under a temperature bias, the heater electrons, with high kinetic energy, flow from the hotter electrode to the colder electrode.The applied weak magnetic field breaks down the time-reversal symmetry and leads to an imbalance between spin-up and spin-down states, therefore the finite spin thermocurrent is also raised as well as the thermocurrent.The thermocurrent and spin thermocurrent are the response of the Kondo resonance state to both the magnetic field and the temperature bias, thus providing an indicator to investigate the properties of the Kondo resonance.

Similarity between spin susceptibility and spin thermocurrent
When the Kondo effect takes place, the local spin moment in the dot will be screened by the electrons of the electrodes whether or not the dot is at the P-H symmetry point [37,46].As a result, the dot behaves as a paramagnetic system and the impurity spin susceptibility has the Pauli-like form χ(T) ∼ χ 0 [1 − 0.41 , where χ 0 = χ(0) is the spin susceptibility at zero temperature at which the local moment is completely screened [48].By using the total magnetization 1 2 gµ B (n ↑ − n ↓ ) and the relationship n σ = ´dωf(ω)A σ (ω), the spin susceptibility at temperature T can evaluated as A σ (ω) is the density of states (DOS) and f(ω) is the Fermi-Dirac distribution of the dot at temperature T and chemical potential µ.In the present case, due to thermal contact with the electrodes, it is reasonable to let T = (T L + T R )/2 and µ = µ L = µ R , since no voltage bias is applied and the chemical potential is pinning to the Fermi level of the electrodes.It will be justified in the following part.We find that the spin paramagnetic susceptibility stems from the difference of the DOS of the two spin species under an external magnetic field.Further, we can approximate equation ( 4) to In the above equation, we have expanded Recall the other physical quantity that also relates to spin polarization, the spin thermocurrent, which is [49,50] Γ is the effective dot-electrode coupling and is fairly approximated to a constant.f ′ (ω) is the derivative of Fermi-Dirac function at the temperature T = (T L + T R )/2 and Fermi level µ = 0, due to the difference of Fermi-Dirac distribution coming from the small temperature bias ∆T = T L − T R between the two electrodes.Equation ( 6) also indicates that the spin thermocurrent is contributed by the different DOS of the two spins around the Fermi level.
As shown in equations ( 5) and ( 6), the spin susceptibility and the spin thermocurrent have a similar mathematical definition that either the right-hand side of both the equations is only dependent on the temperature T and the difference of the DOS between two spins in the dot.
In the absence of magnetic field, time-reversal symmetry makes n ↑ = n ↓ , as well as I ↑ = I ↓ , and therefore leads to that the spin polarization and the spin thermocurrent are zero, whatever if P-H symmetry is holding on.When a finite magnetic field is applied to the QD, although the time-reversal symmetry is broken down, from equation (1), it is found that the jointing charge-time reversal (CT) symmetry, i.e. d σ → d † σ , still holds whatever if it deviates from P-H symmetry.Under the CT symmetric transformation, it simultaneously changes both energy level ε d to 2ε sym − ε d and spin states, which leads to that hole occupancy and electron occupancy have the relationship n electron The similarity suggests that the spin susceptibility and the spin thermocurrent have some consistent characteristics, such as a similar scaling behavior.However, when T → 0, χ trends to χ 0 [46], the zero temperature spin susceptibility, while I sp → 0 [34].It results in the quantity χ − χ 0 approaching zero as T → 0. Hereafter, we indiscriminately call χ − χ 0 and I sp /( ∆T T Γµ B B) the spin susceptibility and the spin thermocurrent, respectively, as well as the original ones, χ and I sp .

Unchanged sign of spin thermocurrent
Either for thermocurrent or spin thermocurrent, they can be expressed by the sum of I electron .We find that if P-H transformation is applied, i.e.I electron σ ↔ I hole σ , corresponding to ε sym − δε shifting to ε sym + δε, it leads to that the sign of I th will change at ε sym , while the sign of I sp is not changed.From our numerical calculations, the statement is also confirmed.Therefore, I sp has the consistency sign when changing the energy level of QD in a wide range and can be directly used to explore potential universal characteristics.

Hierarchical equations of motion
To produce accurate spin susceptibility and spin thermocurrent at a broad temperature range, beyond the Fermi liquid theory [38], we resort to the hierarchical equations of motion (HEOM) approach [51][52][53].The method allows one to address the focused system, such as the QD in our present case, and treat the other part, here the electrodes, as the environment of the target system [54,55].The task in the method is to obtain the real-time reduced density operator ρ (0) (t) of the object system by solving the equations hierarchically constituted by the auxiliary density operators ρ . The j r in the sets of subscripts abbreviates the indices above α, σ, and the indices m, z of parameterized hybridization function rooting from equation (2), and other possible indices (not for the current case) to characterize the reduced system.The Liouville superoperator and In the action of superoperator C jr , commutation − is applied to even-tier auxiliary density operators and anticommutation + to odd-tier ones.The numbers γ jr = γ z mσ and η z mσ come from the parameterized hybridization function.Usually, in HEOM, we adopt a Lorentzian hybridization function with linewidth Γ and bandwidth of electrodes W. Since the current is mainly contributed by the DOS around Fermi level, we can approximate the effective dot-electrode coupling as Γ ασσ ′ (0) = Γ in our definition of spin thermocurrent I sp /( ∆T T Γµ B B) under a large bandwidth W. The details of the HEOM method can be further found in our previous publications [51][52][53][54][55].
In the implementation, by numerically solving equation ( 7), the reduced density matrix and auxiliary density matrices can be obtained at any time t, including that of after accessing the stable state.Other physical quantities are directly obtained from the reduced density matrix.Beside that, the spin-σ real-time current alternatively relates to first-tier auxiliary density matrix through The trace tr d means it being performed over the dot degrees of freedom.i is the imaginary number unit and e is the elemental charge of the electron.Thus, HEOM can smoothly cover a wide range of temperatures in the calculation of spin thermocurrent.HEOM can arrive at high precise results by systematically controlling the truncated tier to achieve the preset convergence criteria no matter the specific structure of the Hamiltonian of the reduced system.A lot of our studies have already shown that it is good at addressing the nonequilibrium or dynamical observables [56,57], including that of in the strongly correlated impurity systems.Because the computational resource rapidly increase with decreasing the temperature, we usually choose a set of parameters to have a larger T K so that the physical quantities can be very close to that of in zero temperature.In this work, in the unit of Γ, bandwidth W = 10, and the Coulomb repulsion U = 2.2 is adopted.Using HEOM, we have studied the thermocurrent in both single QD and double QD [40,41], and obtained consistent results with other work, in which the sign changing of thermocurrent is observed.To obtain high-quality data, we set the truncation tier level to n = 4 in equation (7), although convergence can be achieved with n = 3.From the numerical calculations, the highly accurate spin thermocurrent can be obtained at various temperatures, see figure 1.As per our previous analysis, the Hamiltonian possesses the CT symmetry.The spin thermocurrent produced by using HEOM is also perfectly subject to such symmetry.

Definition of the QD temperature at a finite temperature bias
When the QD contacts two electrodes at the same temperature T L = T R , it is natural to define the temperature of the QD using the temperature of the two electrodes.However, in our study, when the  temperature bias is applied on the two electrodes so as T L ̸ = T R , we define the temperature of the dot as The appropriateness of the definition of dot temperature can be demonstrated from the completely same spin susceptibilities, see figure 2, between the case of a finite temperature bias if we define T = 1 2 (T L + T R ) and the case without temperature bias.

Scaling behaviors
We concurrently calculate the spin susceptibility χ and the spin thermocurrent I sp in a single calculation for several energy levels ε d = −1.65,−1.1, −0.9, −0.7, −0.55 of the dot by using HEOM.In figure 3, we exhibit the unscaled raw data (χ − χ 0 )/χ 0 and I sp /( ∆T T Γµ B B).For ε d = −1.65 and ε d = −0.55,they are mutual under the P-H transformation, so that the spin-related quantities own the same values.At temperature T = 0, the two quantities, χ − χ 0 and I sp /( ∆T T Γµ B B), are zero for all the ε d .With increasing temperature, they separate from each other.We expect that after scaling them by their Kondo temperatures the different behaviors of both the spin susceptibility and the spin thermocurrent would simultaneously become the respective universal one.To investigate the universal Kondo scaling behaviors for the spin susceptibility and spin thermocurrent, we should first determine the Kondo temperature, as well as the case of deviating from the P-H symmetry point [58].The convenient way is to take Haldane's expression , where ε d = ε ↑ = ε ↓ as the weak magnetic field is absent, or use the Bethe ansatz relationship, k B T K = gµ B /4χ 0 [60], between the Kondo temperature and the zero temperature spin susceptibility.Due to the inaccessibility of zero temperature in our method, we use the spin susceptibility χ at T = 0.01 to replace χ 0 to calculate the Kondo temperatures.We compare the two determinations of the Kondo temperatures in figure 4. The two Kondo temperatures exhibit different values for the same energy level of QD.When the energy level deviates from the P-H symmetric point, they all increase exponentially.
As expected, for different ε d , by using the Kondo temperatures of Haldane's expression, all of the scaled curves of spin susceptibilities (χ − χ 0 )/χ 0 fall on the same line, see figure 5(a).Note that, the quantity (χ − χ 0 )/χ 0 having the universal scaling behavior is consistent with the conventional one χ/χ 0 having the universal scaling behavior but with different scaling functions.More interestingly, scaled by the same set of the Kondo temperatures, the spin thermocurrents I sp /( ∆T T Γµ B B) concurrently overlap each other as well, shown in figure 5(b).
Alternatively, we also scale the spin susceptibilities and the spin thermocurrents with the Kondo temperatures defined by the inverse of the spin susceptibility, k B T K = gµ B /4χ 0 [60], see figure 6.Here, we use the spin susceptibilities χ at T = 0.01 instead, which are almost saturated and thus are very close to the zero temperature spin susceptibilities.The alternative scaling behaviors are shown in figure 6.It is found that they fall on the same curve for the spin thermocurrent and for the spin susceptibility, respectively.All these data demonstrate that it exists the universal Kondo scaling behavior for the spin thermocurrent as well as the spin susceptibility.

The scaling function and the alternative extraction of the Kondo temperature
Although the system behaves the Fermi liquid characteristic at zero temperature limit, (χ − χ 0 )/χ 0 and I sp /( ∆T T Γµ B B) exhibit different dependence on T/T K , recognized from the nonconstant ratio of the two quantities.To distinguish their possible scaling functions, we first fit the χ/χ 0 with the function (1 + vT s (T/T K ) 2 ) −s with the Fermi liquid parameter v T [19].It recovers the Fermi liquid result χ/χ 0 = 1 + v T (T/T K ) 2 as T → 0. We obtain the fitting results s = 0.24 and v T = 2.15, where the Fermi liquid parameter v T is very close to the exact value of 2.26 [61].However, when the system towards the Fermi liquid domain, from the right-side hand of equation ( 6), the spin thermocurrent D ′ (0), which indicates I sp ∝ T [34].The scaling function of I sp /( ∆T T Γµ B B) should have a form of (T/T K ) f(T/T K ).The above form of scaling function for (χ − χ 0 )/χ 0 is no longer invalid for the spin thermocurrent.Instead, we fit the data of I sp /( ∆T T Γµ B B) with a rational function and the fitting parameters a = 21.72,b = 0.396, and c = 0.147, referring to figure 7. The other guideline besides the form of function for such a choice is the following, (i) f(x) → 0 when x → 0. (ii) as simple as possible.Note I sp /( ∆T T Γµ B B) progressively towards the constant, which implies I sp possessing a maximum at T/T K = √ c = 0.383 dependent on the Kondo temperature T K .It can give a clue to determine the Kondo temperature [62,63].As shown in figure 7, the maximum of I sp at T/T K = 0.383 is indicated by the red dashed arrow, which can be used to unambiguously determine the Kondo temperature, here, explicitly, the Kondo temperature defined by Haldane's expression.
As the temperature increases, the local moment grows the asymptotic freedom [64] and the spin susceptibility sets out to the Curie-like behavior, while the spin thermocurrent deviates from the universal scaling behavior due to the fading Kondo peak around Fermi level.Moreover, we add a value of spin thermocurrent I sp in figure 7 to show the deviation when T/T K approaches 1.As the data manifested in figure 7, our scaling function works well even when the temperature gets close to the Kondo temperature.When further increasing T larger than T K , the Kondo resonance is progressively weakened and the system enters the Coulomb block region.Thus I sp breaks away from the relationship with the Kondo temperature and ultimately decreases to their respective value which is determined by the position of the dot energy level ε d .

Discussion and summary
The validating of the universal scaling behavior of spin thermocurrent can be realized in a similar scheme of studying the spin Seebeck effect [65].The primary system with the Kondo resonance can resort to semiconductor QD [9] or organic radical molecule junctions [34,66].The local magnetic field can be generated by a delicately designed micromagnet [67], such as a Ti/Co micromagnet [68], and deposited next to the Kondo device.Usually, many fabricated devices are necessary in practice and the applied weak magnetic field is fixed during the measurement on one device.However, a very weak external magnetic field is always presented in the transport experiments of the Kondo effect [20], which can satisfy the demand of changing the strength of the local magnetic field by tuning the spin-orbit coupling with different orientations of the magnetization direction [68,69].As in the inverse spin Hall effect suggested by Hirsh, the spin current will induce a transverse charge imbalance [70], the spin thermocurrent can be detected with the assistance of charge current something like that of the spin Hall effect if it cannot be directly measured.
In summary, by way of the theoretical analysis and the accurate numerical calculation, we propose a universal scaling behavior for temperature-bias-driven spin thermocurrent I sp /( ∆T T Γµ B B) and also attempt to discuss the possible scaling function.We propose that the Kondo temperature can be determined by T K = T/0.383where a maximum spin thermocurrent appears at the temperature T. For spin thermocurrent, the value at zero temperature is zero and therefore we do not need to cool down too much to obtain scaling behavior.Moreover, in contrast to other physical quantities, the spin thermocurrent provides an unambiguous way to examine the universal scaling behavior so as to identify the Kondo resonance.All these features can be checked by future thermocurrent spectroscopy experiments.

Figure 1 .
Figure 1.The Iσ versus εσ curve manifests that the spin thermocurrent satisfies the CT symmetry.The numerical results are obtained by using the hierarchical equations of motion with the Coulomb repulsion U = 2.2, on-dot external magnetic field µBB = 0.02, and bandwidth of electrode W = 10, in the unit of the dot-electrode effective coupling Γ.

σ
= n hole σ , in which n electron σ is proportional to the contribution from state D(ω < 0) and n hole σ from D(ω > 0).Likewise, the CT symmetry requires I hole σ = I electron σ to hold on as well.Note that hole current has a positive contribution to the current, while the electron current has a negative contribution to the current.And the current of spin σ component I σ = I hole σ − I electron σ .Thus, CT symmetry transformation leads to that I ↑ = I hole ↑ − I electron ↑ changes to −I ↓ = I electron ↓ − I hole ↓ .Figure 1 exhibits our numerical confirmation of the conservation of CT symmetry transformation for I σ .The details of the numerical method will be introduced later.

Figure 2 .
Figure 2. The spin susceptibility, calculated under TL = TR and TL − TR = 0.002.If we definite T = 1 2 (TL + TR), the completely same spin susceptibility behavior can be obtained.The other parameters are Coulomb repulsion U = 2.2, on-dot external magnetic field µBB = 0.02, and bandwidth of electrode W = 10, in the unit of Γ.

Figure 3 .
Figure 3.The unscaled raw data for (a) the spin susceptibility and (b) the spin thermocurrent calculated under the same parameters, the Coulomb repulsion U = 2.2, temperature-bias ∆T = 0.002, on-dot external magnetic field µBB = 0.02, and bandwidth of electrode W = 10, for different energy levels ε d = −1.65,−1.1, −0.9, −0.7, −0.55 of the QD as the labels in the figure.

Figure 4 .
Figure 4. Comparison of the Kondo temperature TK from Haldane's expression and from the definition of zero temperature spin susceptibility.The spin susceptibility is in the unit of gµB.We use the spin susceptibility χ at T = 0.01 instead of that of T = 0, since the zero temperature is hard to reach in our method.

Figure 5 .
Figure 5.The universal Kondo scaling for (a) spin susceptibility and (b) spin thermocurrent by scaling the data of figure 4 with the Kondo temperatures of Haldane's expression.

Figure 6 .
Figure 6.The universal scaling behaviors for (a) spin susceptibility and (b) spin thermocurrent alternatively scaled by the Kondo temperature kBTK = gµB/4χ with spin susceptibility χ at T = 0.01 instead of T = 0.

Figure 7 .
Figure 7.The fitting result (the blue solid line) of the scaling function for the spin thermocurrent (the black squares).The red dashed arrow shows the maximum of spin thermocurrent I sp at T/T K = √ c = 0.383, which gives a convenient extraction of the Kondo temperature defined by Haldane's expression.