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A Holographic Kondo Model

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Applications of the Gauge/Gravity Duality

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Abstract

This chapter presents our work on a holographic Kondo model. This model describes a magnetic impurity coupled to strongly correlated charge carriers, which are modelled by a holographic CFT. We extend the model by adding a second impurity and including an inter-impurity interaction. We obtain numerical evidence for a quantum phase transition in the two-impurity phase diagram. Computing correlation functions in the holographic single-impurity Kondo model, we observe Fano resonances in the corresponding spectrum and identify the Kondo resonance.

This chapter is based on A. O’Bannon, I. Papadimitriou and J. Probst, A Holographic Two-Impurity Kondo Model, JHEP 01 (2016) 103, [1510.08123] (Ref. [1]), on J. Erdmenger, C. Hoyos, A. O’Bannon, I. Papadimitriou, J. Probst and J. M. S. Wu, Holographic Kondo and Fano Resonances, Phys. Rev. D96 (2017) 021901, [1611.09368] (Ref. [2]), and on J. Erdmenger, C. Hoyos, A. O’Bannon, I. Papadimitriou, J. Probst and J. M. S. Wu, Two-point Functions in a Holographic Kondo Model, JHEP 03 (2017) 039, [1612.02005] (Ref. [3]).

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Notes

  1. 1.

    A LFL describes weakly interacting fermionic quasi-particles (‘dressed electrons’) whose excitations are in one-to-one correspondence with and have the same quantum numbers as their free counterparts (‘bare electrons’), but whose mass and other couplings assume different, renormalised values [5].

  2. 2.

    Intuitively, a conduction electron locks with the impurity into a spin singlet, which makes the impurity location impenetrable for other conduction electrons due to Pauli exclusion.

  3. 3.

    A different approach, using a delta-function source to describe a point-like impurity, is followed in Ref. [92].

  4. 4.

    Strictly speaking, the term ‘RKKY interaction’ only refers to a Heisenberg interaction that is induced by Friedel oscillations in the LFL at order \(\lambda _{\text {K}}^2\) in perturbation theory [4, 52]. In the large-N limit, however, the RKKY interaction is sub-leading in N and we need to add a Heisenberg interaction by hand [55]. In a (standard) abuse of terminology we will still call it RKKY coupling.

  5. 5.

    Surprisingly, numerical RG techniques reveal that the spin-spin correlator \(\langle S^A_{\text {I}} S^A_{\text {II}}\rangle \) decreases smoothly and monotonically as \(\lambda _{\text {RKKY}}/T_{\text {K}}\) increases from the FM limit, \(\lambda _{\text {RKKY}}/T_{\text {K}} \rightarrow -\infty \), where \(\langle S^A_{\text {I}} S^A_{\text {II}}\rangle =1/4\), the triplet value, to the AFM RKKY limit \(\lambda _{\text {RKKY}}/T_{\text {K}} \rightarrow + \infty \), where \(\langle S^A_{\text {I}} S^A_{\text {II}}\rangle =-3/4\), the singlet value [53, 54, 56, 58].

  6. 6.

    Our choices of spin and charge for \(\mathcal {O}\) indicate unambiguously that our model describes a Kondo rather than an Anderson model, and that the impurity spins are in totally anti-symmetric representations of SU(N). Totally symmetric representations, for example, would involve Schwinger bosons rather than Abrikosov pseudo-fermions [32, 98, 99], in which case \(\mathcal {O}\) would be fermionic. In the Anderson model, the impurity is a bi-linear of two physical f electrons which are charged under the U(1) of electromagnetism, in contrast to the pseudo-fermions which are neutral under that U(1) [4]. The Anderson model would thus require a complex scalar similar to our \(\mathcal {O}\), but built from a chiral fermion \(\psi \) and the f electron and hence neutral under the electric U(1).

  7. 7.

    Our choice \(K=1\) then guarantees, based on SU(N) representation theory arguments and Pauli exclusion alone, that overscreening cannot occur in our model [33].

  8. 8.

    We do not set \(M_+^2 = -1/4\), because then \(M^2_-\) would violate the BF bound, producing an instability.

  9. 9.

    Equations (5.32b) and (5.32c) are actually identical in form to the equations of motion in the holographic single-impurity Kondo model of Ref. [73], but where Ref. [73] had \(a_t^0\) and \(\phi \) we have \(\mathcal {A}_t^+\) and \(\phi \). However, we will see that the boundary conditions on \(\mathcal {A}_t^+\) and \(\phi \) in our two-impurity model are very different from those in Ref. [73] and that they will effectively couple \(\mathcal {A}_t^-\) to \(\mathcal {A}_t^+\) and \(\phi \).

  10. 10.

    References [117, 118] observed the same in holographic counterterms for irrelevant operators: they depend on sub-leading modes, but are well-defined for boundary conditions dual to multi-trace couplings.

  11. 11.

    In the holographic single-impurity Kondo model of Ref. [73], the finite boundary term involving the scalar field was different from our \(S_{\text {K}}\): it had the same form as our \(S_{\text {K}}\), but with \(\beta \rightarrow \alpha \) and \(\kappa \rightarrow 1/\kappa \). In that case, the linear combination of \(\alpha \) and \(\beta \) held fixed in the variational principle would not be \(\alpha - \kappa \beta \). Nevertheless, \(\alpha - \kappa \beta \) was held fixed in Ref. [73]. Those two wrongs made a right, in the following sense: the finite boundary term in Ref. [73], when evaluated on \(\alpha =\kappa \beta \), actually agrees with our \(S_{\text {K}}\), so the solutions for the scalar and the value of the on-shell action of Ref. [73] actually agree with those obtained using our \(S_{\text {K}}\). Similarly, the one-point function identified as \(\langle \mathcal {O} \rangle \propto N \alpha \) in Ref. [73] agrees with our \(\langle \mathcal {O} \rangle \propto N \beta \) when evaluated on \(\alpha = \kappa \beta \).

  12. 12.

    We show in the appendix of Ref. [1] that both Dirichlet and Neumann, and hence also mixed boundary conditions lead to normalisable solutions for a gauge field in \(AdS_2\).

  13. 13.

    The argument of the logarithm in Eq. (5.54) is made dimensionless by the unit \(AdS_3\) radius.

  14. 14.

    In contrast, if we had started with a Neumann boundary condition for \(\phi \) instead of Dirichlet, then \(\kappa \)’s RG transformation would be \(\kappa \longrightarrow \left(1+\kappa \ln (L)\right)/\kappa \), which is always marginally relevant since \(\kappa \) grows in the IR, \(L\rightarrow \infty \), for both \(\kappa <0\) and \(\kappa >0\) in the UV.

  15. 15.

    These two phases are also distinguished by their phase shifts. As in the holographic single-impurity Kondo model of Ref. [73], in our model the phase shift that accompanies Kondo screening appears holographically as a Wilson line of the Chern-Simons field in the x direction: if we compactify the x direction then the phase shift is \(\propto \oint _x A\). Equation (5.17b) shows that if \(\mathcal {A}_t^+\ne 0\), \(\phi \ne 0\) then \(A_x\ne 0\), while if \(\mathcal {A}_t^+=\phi =0\) then \(A_x=0\). Translating to the field theory, we find that a phase shift occurs in non-trivial states but not in the trivial state. In fact, the Chern-Simons field’s only role is to implement the phase shift and it will play no further role in the remainder of this section.

  16. 16.

    The appearance of non-trivial solutions for only one sign of a double-trace coupling is in fact generic in large-N field theory and in holography [107, 119, 120]. Adding a double-trace coupling shifts the quantum effective potential and generically will change the ground state only for one sign of the double-trace coupling constant, much the way a mass term added to a scalar field theory with quartic interaction will trigger scalar condensation only for negative mass-squared.

  17. 17.

    The explicit solutions can be found in Eqs. (3.23)–(3.27) in Ref. [3], where \(\delta \Phi \) and \(\delta \Phi ^\dagger \) are referred to as \(y_+\) and \(y_-\) respectively.

  18. 18.

    We discuss the positivity property satisfied by spectral functions of non-Hermitian operators such as our \(\mathcal{O}\) in Appendix A.

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    Jonas Probst

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Probst, J. (2018). A Holographic Kondo Model. In: Applications of the Gauge/Gravity Duality. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-93967-4_5

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