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Nonlinear Seebeck effect of SU($N$) Kondo impurity

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Pith's one-line read The nonlinear Seebeck coefficient of an SU(N) Kondo impurity becomes coupling-asymmetry dependent through the quadratic voltage response, which can enhance thermopower in beyond-half-filled systems.

arxiv 1908.00415 v1 pith:UEVIZ6ZG submitted 2019-08-01 cond-mat.mes-hall

D. B. Karki , Mikhail N. Kiselev This is my paper · ORCID
classification cond-mat.mes-hall
keywords kondoeffectsseebeckasymmetrycouplingimpuritysymmetricbeyond
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

The paper studies a single quantum dot with strong electron interactions, described by SU(N) symmetry, coupled to two metal leads. At low temperature the dot forms a Kondo singlet with the leads, and the standard theory for this regime is the local Fermi liquid picture, where the interacting system looks like a gas of weakly interacting quasiparticles. The authors add an important ingredient that is usually ignored: the two tunnel barriers connecting the dot to the left and right leads can have different strengths. This asymmetry is described by a parameter C. Using a nonequilibrium Keldysh formalism, they calculate the electric current when a voltage difference and a temperature difference are applied across the dot. From the condition of zero current they extract the Seebeck coefficient, which measures the voltage produced per unit temperature difference. In the linear response regime, the Seebeck coefficient turns out to be independent of C, as expected. In the nonlinear regime, however, the first correction to the Seebeck coefficient depends on C, and for quarter-filled SU(4) systems the correction grows with positive C, meaning a stronger left coupling can enhance the thermovoltage. The main caveat is that this nonlinear enhancement is sensitive to how the applied voltage is split between the two leads in the theory. The authors choose a split that depends on C in order to simplify the mathematics. If a different split is used, the quadratic-in-voltage term that drives the enhancement disappears. This does not invalidate the derivation, but it means the predicted enhancement is tied to a specific assumption about the electrostatics of the device.
Extended reading notes

Core claim

For a strongly coupled, asymmetrically coupled SU(N) Kondo impurity, the nonlinear Seebeck coefficient depends on the asymmetry parameter C through the transport coefficients L2_1 = (C/2) sin(2πm/N) and L11_12, so that beyond particle-hole symmetric variants (e.g., SU(4) with m=1) can have significantly enhanced thermopower at positive C, as expressed in Eqs. (36) and (41). The abstract states: 'beyond PH symmetric SU(N) Kondo variants are highly desirable ... to have significantly improved thermoelectric performance.'

Load-bearing premise

The ad hoc chemical potential partition μ_L = (eΔV/2)(1−C), μ_R = −(eΔV/2)(1+C) (Eqs. (12)-(13)) is chosen to enforce Eq. (11) and simplify the calculation, but every C-dependent nonlinear coefficient (L2_1, L11_12) that drives the claimed Seebeck enhancement is produced by this gauge. In the standard symmetric bias partition with μ_L = −μ_R = eΔV/2, the quadratic term L2_1 vanishes, so the central enhancement claim is contingent on this unjustified division of the voltage drop.

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Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation rests on standard Fermi liquid theory, Bethe ansatz results, and a specific voltage-division assumption. There are no data-fitted free parameters and no invented physical entities.

assumptions (5)
  • domain assumption The low-energy physics of the fully screened SU(N) Kondo impurity is described by the Nozieres local Fermi liquid Hamiltonian Eq. (7) including only up to four-fermion interactions.
    This is the standard strong-coupling fixed-point phenomenology, cited to Nozieres (1974) and Affleck-Ludwig (1993), and it underlies the entire calculation.
  • domain assumption The Bethe ansatz relation A = α2/α1² = (N−2)/(N−1) Γ(1/N) tan(π/N) / [√π Γ(1/2+1/N)] cot(mπ/N) in Eq. (8) holds for the SU(N) Kondo model with m electrons.
    This exact relation fixes the ratio of the second generation Fermi liquid coefficients and is imported from the integrability literature without derivation in this paper.
  • ad hoc to paper The chemical potentials satisfy μ_L cos²θ + μ_R sin²θ = ε_F = 0, yielding the specific partition μ_L = eΔV(1−C)/2 and μ_R = −eΔV(1+C)/2.
    This choice enforces the Fermi liquid condition Eq. (11) and simplifies the K integrals, but it fixes the voltage division between the leads in a way that is not experimentally derived and directly produces the C-dependent quadratic terms.
  • domain assumption Perturbative treatment of the interaction Hamiltonian Hint is valid for eΔV, T_L, T_R much smaller than the Kondo temperature T_K.
    The Keldysh calculation expands in these small parameters and retains terms up to quadratic response; this validity condition is stated in Section III.
  • domain assumption The left and right reservoirs remain in thermal equilibrium with Fermi functions f_L and f_R throughout the transport process.
    This is the standard assumption in quantum transport through a mesoscopic dot, used in the Keldysh Green's function expressions Eq. (24).

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Pith. "Pith review of Nonlinear Seebeck effect of SU($N$) Kondo impurity." pith.science (2026). https://pith.science/paper/UEVIZ6ZG

@misc{pith2026190800415,
  author       = {Pith},
  title        = {Pith review of: Nonlinear Seebeck effect of SU($N$) Kondo impurity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UEVIZ6ZG}},
  note         = {Machine review of arXiv:1908.00415}
}
abstract

We develop a theoretical framework to study the influences of coupling asymmetry on the thermoelectrics of a strongly coupled SU($N$) Kondo impurity based on a local Fermi liquid theory. Applying non-equilibrium Keldysh formalism, we investigate charge current driven by the voltage bias and temperature gradient in the strong coupling regime of an asymmetrically coupled SU($N$) quantum impurity. The thermoelectric characterizations are made via non-linear Seebeck effects. We demonstrate that the beyond particle-hole (PH) symmetric SU($N$) Kondo variants are highly desirable with respect to the corresponding PH symmetric setups in order to have significantly improved thermoelectric performance. The greatly enhanced Seebeck coefficients by tailoring the coupling asymmetry of beyond PH symmetric SU($N$) Kondo effects are explored. Apart from presenting the analytical expressions of asymmetry dependent transport coefficients for general SU($N$) Kondo effects, we make a close connection of our findings with the experimentally studied SU(2) and SU(4) Kondo effects in quantum dot nano structures. Seebeck effects associated with the theoretically proposed SU(3) Kondo effects are discussed in detail.

Figures

Figures reproduced from arXiv: 1908.00415 by the authors.

Figure 1
Figure 1. FIG. 1. Upper panel: Schematic representation of experi [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Linear (LR) and non linear (BLR) Seebeck coeffi [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Left panel: Plot of asymmetry dependent zero cur [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Left panel: Lines of zero charge currents in a single [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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    PH symmetric SU( N) Kondo effects As we anticipated earlier that for the half-filled SU(N) Kondo effects the transport coefficients satisfy the relation L 1 2 =L 2 1 =L 2 2 =0, therefore, corresponding 7 FIG. 2. Linear (LR) and non linear (BLR) Seebeck coeffi- cients with PH symmetric SU(2) and SU(4) Kondo effects for fixed value of the potential scattering δP. th...

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