Abstract
We propose a model of a strongly-interacting two-impurity Kondo system based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, also known as holography. In a Landau Fermi Liquid, the single-impurity Kondo effect is the screening of an impurity spin at low temperature T . The two-impurity Kondo model then describes the competition between the Kondo interaction and the Heisenberg interaction between two impurity spins, also called the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. For spin-1/2 impurities, that competition leads to a quantum phase transition from a Kondo-screened phase to a phase in which the two impurity spins screen one another. Our holographic model is based on a (1 + 1)-dimensional CFT description of the two-impurity Kondo model, reliable for two impurities with negligible separation in space. We consider only impurity spins in a totally anti-symmetric representation of an SU(N ) spin symmetry. We employ a large-N limit, in which both Kondo and RKKY couplings are double-trace, and both Kondo and inter-impurity screening appear as condensation of single-trace operators at the impurities’ location. We perform the holographic renormalization of our model, which allows us to identify the Kondo and RKKY couplings as boundary conditions on fields in AdS. We numerically compute the phase diagram of our model in the plane of RKKY coupling versus T , finding evidence for a quantum phase transition from a trivial phase, with neither Kondo nor inter-impurity screening, to a non-trivial phase, with both Kondo and anti-ferromagnetic inter-impurity screening. More generally we show, just using SU(N ) representation theory, that ferromagnetic correlations must be absent at leading order in the large-N limit. Our holographic model may be useful for studying many open problems involving strongly-interacting quantum impurities, including for example the Kondo lattice, relevant for describing the heavy fermion compounds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
P. Coleman, Heavy fermions: electrons at the edge of magnetism, in Handbook of Magnetism and Advanced Magnetic Materials: fundamentals and Theory, vol. 1, H. Kronmuller and S. Parkin eds., John Wiley and Sons, U.S.A. (2007), pg. 95 [cond-mat/0612006].
P. Gegenwart, Q. Si and F. Steglich, Quantum criticality in heavy-fermion metals, Nature Phys. 4 (2008) 186 [arXiv:0712.2045].
Q. Si, Quantum criticality and global phase diagram of magnetic heavy fermions, Phys. Stat. Sol. B 247 (2010) 476 [arXiv:0912.0040].
Q. Si, Quantum criticality and the Kondo lattice, in Understanding Quantum Phase Transitions. Series: condensed matter physics, CRC Press, U.S.A. (2010), pg. 193 [arXiv:1012.5440] [INSPIRE].
Q. Si and F. Steglich, Heavy Fermions and quantum phase transitions, Science 329 (2010) 1161 [arXiv:1102.4896] [INSPIRE].
P. Coleman, Heavy Fermions and the Kondo lattice: a 21st century perspective, arXiv:1509.05769 [INSPIRE].
B. Keimer, S.A. Kivelson, M.R. Norman, S. Uchida and J. Zaanen, High temperature superconductivity in the cuprates, arXiv:1409.4673.
S. Doniach, The Kondo lattice and weak antiferromagnetism, Physica B+C 91 (1977) 231.
D. Goldhaber-Gordon et al., Kondo effect in a single-electron transistor, Nature 391 (1998) 156.
S. Cronenwett, T. Oosterkamp and L. Kouwenhoven, A tunable Kondo effect in quantum dots, Science 281 (1998) 540.
W.G. van der Wiel et al., The Kondo effect in the unitary limit, Science 289 (2000) 2105.
J. Kondo, Resistance minimum in dilute magnetic alloys, Prog. Theor. Phys. 32 (1964) 37 [INSPIRE].
C. Rizzuto, Formation of localized moments in metals: experimental bulk properties, Rept. Prog. Phys. 37 (1974) 147.
G. Grüner and A. Zawadowski, Low temperature properties of Kondo alloys, in Progress in Low Temperature Physics, vol. 7, part B, D. Brewer ed., Elsevier, The Netherlands (1978), pg. 591.
K.G. Wilson, The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. 47 (1975) 773 [INSPIRE].
H.R. Krishna-murthy, J.W. Wilkins and K.G. Wilson, Renormalization-group approach to the Anderson model of dilute magnetic alloys. 1. Static properties for the symmetric case, Phys. Rev. B 21 (1980) 1003 [INSPIRE].
H.R. Krishna-Murthy, J.W. Wilkins and K.G. Wilson, Renormalization-group approach to the Anderson model of dilute magnetic alloys. II. Static properties for the asymmetric case, Phys. Rev. B 21 (1980) 1044.
N. Andrei, Diagonalization of the Kondo Hamiltonian, Phys. Rev. Lett. 45 (1980) 379 [INSPIRE].
P. Wiegmann, Exact solution of s-d exchange model at T = 0, Sov. Phys. JETP Lett. 31 (1980) 364.
N. Andrei, K. Furuya and J.H. Lowenstein, Solution of the Kondo problem, Rev. Mod. Phys. 55 (1983) 331 [INSPIRE].
A. Tsvelick and P. Wiegmann, Exact results in the theory of magnetic alloys, Adv. Phys. 32 (1983) 453.
P. Coleman and N. Andrei, Diagonalisation of the generalised Anderson model, J. Phys. C 19 (1986) 3211 [INSPIRE].
N. Andrei, Integrable models in condensed matter physics, cond-mat/9408101.
P. Zinn-Justin and N. Andrei, The generalized multi-channel Kondo model: thermodynamics and fusion equations, Nucl. Phys. B 528 (1998) 648 [cond-mat/9801158].
A. Jerez, N. Andrei and G. Zaránd, Solution of the multichannel Coqblin-Schrieffer impurity model and application to multilevel systems, Phys. Rev. B 58 (1998) 3814 [cond-mat/9803137].
P. Coleman, Mixed valence as an almost broken symmetry, Phys. Rev. B 35 (1987) 5072.
N.E. Bickers, Review of techniques in the large-N expansion for dilute magnetic alloys, Rev. Mod. Phys. 59 (1987) 845 [INSPIRE].
O. Parcollet and A. Georges, Transition from overscreening to underscreening in the multichannel Kondo model: exact solution at large N , Phys. Rev. Lett. 79 (1997) 4665 [cond-mat/9707337].
O. Parcollet, A. Georges, G. Kotliar and A. Sengupta, Overscreened multi-channel SU(N ) Kondo model: large-N solution and conformal field theory, Phys. Rev. B 58 (1998) 3794 [cond-mat/9711192].
I. Affleck, A current algebra approach to the Kondo effect, Nucl. Phys. B 336 (1990) 517 [INSPIRE].
I. Affleck and A.W.W. Ludwig, The Kondo effect, conformal field theory and fusion rules, Nucl. Phys. B 352 (1991) 849 [INSPIRE].
I. Affleck and A.W.W. Ludwig, Critical theory of overscreened Kondo fixed points, Nucl. Phys. B 360 (1991) 641 [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
I. Affleck and A. Ludwig, Exact conformal-field-theory results on the multichannel Kondo effect: single-fermion Green’s function, self-energy, and resistivity, Phys. Rev. B 48 (1993) 7297.
I. Affleck, Conformal field theory approach to the Kondo effect, Acta Phys. Polon. B 26 (1995) 1869 [cond-mat/9512099] [INSPIRE].
A. Hewson, The Kondo model to heavy fermions, Cambridge University Press, Cambridge U.K. (1993).
D.L. Cox and A. Zawadowski, Exotic Kondo effects in metals: magnetic ions in a crystalline electric field and tunnelling centres, Adv. Phys. 47 (1998) 599 [cond-mat/9704103].
I. Affleck, The Kondo screening cloud: what it is and how to observe it, arXiv:0911.2209.
A. Georges, G. Kotliar, W. Krauth and M.J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68 (1996) 13 [INSPIRE].
C. Jayaprakash, H. Krishna-murthy and J. Wilkins, Two-impurity Kondo problem, Phys. Rev. Lett. 47 (1981) 737.
B. Jones and C. Varma, Study of two magnetic impurities in a Fermi gas, Phys. Rev. Lett. 58 (1987) 843.
R.M. Fye, J.E. Hirsch and D.J. Scalapino, Kondo effect versus indirect exchange in the two-impurity Anderson model: a Monte Carlo study, Phys. Rev. B 35 (1987) 4901.
B. Jones, C. Varma and J. Wilkins, Low-temperature properties of the two-impurity Kondo Hamiltonian, Phys. Rev. Lett. 61 (1988) 125.
B.A. Jones, B.G. Kotliar and A.J. Millis, Mean-field analysis of two antiferromagnetically coupled Anderson impurities, Phys. Rev. B 39 (1989) 3415.
B. Jones and C. Varma, Critical point in the solution of the two magnetic impurity problem, Phys. Rev. B 40 (1989) 324.
R.M. Fye and J.E. Hirsch, Quantum Monte Carlo study of the two-impurity Kondo Hamiltonian, Phys. Rev. B 40 (1989) 4780.
B. Jones, Antiferromagnetic phase instability in the two-impurity Kondo problem, in Field Theories in Condensed Matter Physics: a workshop, Z. Tesanovic ed., Addison-Wesley, U.S.A. (1990), pg. 87.
A. Millis, B. Kotliar and B. Jones, The two Kondo impurity problem: a large N biased review, in Field Theories in Condensed Matter Physics: a workshop, Z. Tesanovic ed., Addison-Wesley, U.S.A. (1990), pg. 159.
I. Affleck and A.W.W. Ludwig, Exact critical theory of the two impurity Kondo model, Phys. Rev. Lett. 68 (1992) 1046 [INSPIRE].
R.M. Fye, “Anomalous fixed point behavior” of two Kondo impurities: a reexamination, Phys. Rev. Lett. 72 (1994) 916.
I. Affleck, A. Ludwig and B. Jones, Conformal-field-theory approach to the two-impurity Kondo problem: comparison with numerical renormalization-group results, Phys. Rev. B 52 (1995) 9528 [cond-mat/9409100].
K. Ingersent and B.A. Jones, Low-temperature physics of the two-impurity, two-channel Kondo model, Physica B 199 (1994) 402.
J. Gan, Mapping the critical point of the two-impurity Kondo model to a two-channel problem, Phys. Rev. Lett. 74 (1995) 2583 [Erratum ibid. 74 (1995) 5287].
J. Gan, Solution of the two-impurity Kondo model: critical point, Fermi-liquid phase, and crossover, Phys. Rev. B 51 (1995) 8287.
A. Georges and A.M. Sengupta, Solution of the two-impurity, two-channel Kondo model, Phys. Rev. Lett. 74 (1995) 2808.
J.B. Silva et al., Particle-hole asymmetry in the two-impurity Kondo model, Phys. Rev. Lett. 76 (1996) 275.
B. Jones, The Kondo effect, in Handbook of Magnetism and Advanced Magnetic Materials: fundamentals and theory, vol. 1, H. Kronmuller and S. Parkin eds., John Wiley and Sons, U.S.A. (2007), pg. 149.
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M 2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
S. Kachru, A. Karch and S. Yaida, Holographic lattices, dimers and glasses, Phys. Rev. D 81 (2010) 026007 [arXiv:0909.2639] [INSPIRE].
S. Sachdev, Holographic metals and the fractionalized Fermi liquid, Phys. Rev. Lett. 105 (2010) 151602 [arXiv:1006.3794] [INSPIRE].
S. Kachru, A. Karch and S. Yaida, Adventures in holographic dimer models, New J. Phys. 13 (2011) 035004 [arXiv:1009.3268] [INSPIRE].
S. Sachdev, Strange metals and the AdS/CFT correspondence, J. Stat. Mech. 11 (2010) P11022 [arXiv:1010.0682] [INSPIRE].
W. Mück, The Polyakov loop of anti-symmetric representations as a quantum impurity model, Phys. Rev. D 83 (2011) 066006 [Erratum ibid. D 84 (2011) 129903] [arXiv:1012.1973] [INSPIRE].
A. Faraggi and L.A. Pando Zayas, The spectrum of excitations of holographic Wilson loops, JHEP 05 (2011) 018 [arXiv:1101.5145] [INSPIRE].
K. Jensen, S. Kachru, A. Karch, J. Polchinski and E. Silverstein, Towards a holographic marginal Fermi liquid, Phys. Rev. D 84 (2011) 126002 [arXiv:1105.1772] [INSPIRE].
N. Karaiskos, K. Sfetsos and E. Tsatis, Brane embeddings in sphere submanifolds, Class. Quant. Grav. 29 (2012) 025011 [arXiv:1106.1200] [INSPIRE].
S. Harrison, S. Kachru and G. Torroba, A maximally supersymmetric Kondo model, Class. Quant. Grav. 29 (2012) 194005 [arXiv:1110.5325] [INSPIRE].
P. Benincasa and A.V. Ramallo, Fermionic impurities in Chern-Simons-matter theories, JHEP 02 (2012) 076 [arXiv:1112.4669] [INSPIRE].
A. Faraggi, W. Mück and L.A. Pando Zayas, One-loop effective action of the holographic antisymmetric Wilson loop, Phys. Rev. D 85 (2012) 106015 [arXiv:1112.5028] [INSPIRE].
P. Benincasa and A.V. Ramallo, Holographic Kondo model in various dimensions, JHEP 06 (2012) 133 [arXiv:1204.6290] [INSPIRE].
H. Matsueda, Multiscale entanglement renormalization ansatz for Kondo problem, arXiv:1208.2872 [INSPIRE].
G. Itsios, K. Sfetsos and D. Zoakos, Fermionic impurities in the unquenched ABJM, JHEP 01 (2013) 038 [arXiv:1209.6617] [INSPIRE].
J. Erdmenger, C. Hoyos, A. O’Bannon and J. Wu, A holographic model of the Kondo effect, JHEP 12 (2013) 086 [arXiv:1310.3271] [INSPIRE].
J.M. Maldacena, Wilson loops in large-N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].
S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large-N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].
J.M. Camino, A. Paredes and A.V. Ramallo, Stable wrapped branes, JHEP 05 (2001) 011 [hep-th/0104082] [INSPIRE].
S. Yamaguchi, Wilson loops of anti-symmetric representation and D5-branes, JHEP 05 (2006) 037 [hep-th/0603208] [INSPIRE].
J. Gomis and F. Passerini, Holographic Wilson loops, JHEP 08 (2006) 074 [hep-th/0604007] [INSPIRE].
J. Gomis and F. Passerini, Wilson loops as D3-branes, JHEP 01 (2007) 097 [hep-th/0612022] [INSPIRE].
M. Blake, A. Donos and D. Tong, Holographic charge oscillations, JHEP 04 (2015) 019 [arXiv:1412.2003] [INSPIRE].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic entanglement entropy: an overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
P.M. Chesler and L.G. Yaffe, Numerical solution of gravitational dynamics in asymptotically anti-de Sitter spacetimes, JHEP 07 (2014) 086 [arXiv:1309.1439] [INSPIRE].
G.T. Horowitz, J.E. Santos and D. Tong, Optical conductivity with holographic lattices, JHEP 07 (2012) 168 [arXiv:1204.0519] [INSPIRE].
T. Senthil, S. Sachdev and M. Vojta, Fractionalized Fermi liquids, Phys. Rev. Lett. 90 (2003) 216403 [cond-mat/0209144].
T. Senthil, M. Vojta and S. Sachdev, Weak magnetism and non-Fermi liquids near heavy-fermion critical points, Phys. Rev. B 69 (2004) 035111 [cond-mat/0305193].
P. Kraus, Lectures on black holes and the AdS 3 /CFT 2 correspondence, Lect. Notes Phys. 755 (2008) 193 [hep-th/0609074] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
I. Papadimitriou, Holographic renormalization as a canonical transformation, JHEP 11 (2010) 014 [arXiv:1007.4592] [INSPIRE].
B.C. van Rees, Holographic renormalization for irrelevant operators and multi-trace counterterms, JHEP 08 (2011) 093 [arXiv:1102.2239] [INSPIRE].
B.C. van Rees, Irrelevant deformations and the holographic Callan-Symanzik equation, JHEP 10 (2011) 067 [arXiv:1105.5396] [INSPIRE].
D. Bensimon, A. Jerez and M. Lavagna, Intermediate coupling fixed point study in the overscreened regime of generalized multichannel SU(N ) Kondo models, Phys. Rev. B 73 (2006) 224445.
P. Nozières and A. Blandin, Kondo effect in real metals, J. Phys. France 41 (1980) 193.
A. Auerbach and D.P. Arovas, Schwinger bosons approaches to quantum antiferromagnetism, arXiv:0809.4836.
M. Mathur, I. Raychowdhury and R. Anishetty, SU(N ) irreducible Schwinger bosons, J. Math. Phys. 51 (2010) 093504 [arXiv:1003.5487] [INSPIRE].
E.I. Buchbinder, J. Gomis and F. Passerini, Holographic gauge theories in background fields and surface operators, JHEP 12 (2007) 101 [arXiv:0710.5170] [INSPIRE].
A. Castro, D. Grumiller, F. Larsen and R. McNees, Holographic description of AdS 2 black holes, JHEP 11 (2008) 052 [arXiv:0809.4264] [INSPIRE].
M. Fujita, S. Harrison, A. Karch, R. Meyer and N.M. Paquette, Towards a holographic Bose-Hubbard model, JHEP 04 (2015) 068 [arXiv:1411.7899] [INSPIRE].
I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].
D. Marolf and S.F. Ross, Boundary conditions and new dualities: vector fields in AdS/CFT, JHEP 11 (2006) 085 [hep-th/0606113] [INSPIRE].
C. Fefferman and C.R. Graham, Conformal invariants, in Elie Cartan et les Mathematiques d’aujourd’hui, Asterique, France (1985), pg. 95.
J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP 08 (2000) 003 [hep-th/9912012] [INSPIRE].
D. Martelli and W. Mück, Holographic renormalization and Ward identities with the Hamilton-Jacobi method, Nucl. Phys. B 654 (2003) 248 [hep-th/0205061] [INSPIRE].
I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, in AdS/CFT correspondence: Einstein metrics and their conformal boundaries. Proceedings, 73rd Meeting of Theoretical Physicists and Mathematicians, Strasbourg France September 11-13 2003, pg. 73 [hep-th/0404176] [INSPIRE].
W. Chemissany and I. Papadimitriou, Lifshitz holography: the whole shebang, JHEP 01 (2015) 052 [arXiv:1408.0795] [INSPIRE].
J. Erdmenger, C. Hoyos, A. O’Bannon, I. Papadimitriou, J. Probst and J. Wu, in preparation.
M. Bianchi, D.Z. Freedman and K. Skenderis, How to go with an RG flow, JHEP 08 (2001) 041 [hep-th/0105276] [INSPIRE].
I. Papadimitriou and K. Skenderis, Correlation functions in holographic RG flows, JHEP 10 (2004) 075 [hep-th/0407071] [INSPIRE].
M. Henningson and K. Skenderis, The holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
I. Papadimitriou, Multi-trace deformations in AdS/CFT: exploring the vacuum structure of the deformed CFT, JHEP 05 (2007) 075 [hep-th/0703152] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Diffeomorphisms and holographic anomalies, Class. Quant. Grav. 17 (2000) 1129 [hep-th/9910267] [INSPIRE].
A. Schwimmer and S. Theisen, Diffeomorphisms, anomalies and the Fefferman-Graham ambiguity, JHEP 08 (2000) 032 [hep-th/0008082] [INSPIRE].
A. Barut and R. Raczka, Theory of group representations and applications, World Scientific Publishing Company, Singapore (1986).
G.T. Horowitz and M.M. Roberts, Zero temperature limit of holographic superconductors, JHEP 11 (2009) 015 [arXiv:0908.3677] [INSPIRE].
J. Erdmenger, M. Flory and M.-N. Newrzella, Bending branes for DCFT in two dimensions, JHEP 01 (2015) 058 [arXiv:1410.7811] [INSPIRE].
R. Flint, M. Dzero and P. Coleman, Heavy electrons and the symplectic symmetry of spin, Nature Phys. 4 (2008) 643 [arXiv:0710.1126].
R. Flint, M. Dzero and P. Coleman, Supplementary material to heavy electrons and the symplectic symmetry of spin, arXiv:0710.1128.
O. Aharony and D. Kutasov, Holographic duals of long open strings, Phys. Rev. D 78 (2008) 026005 [arXiv:0803.3547] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1510.08123
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
O’Bannon, A., Papadimitriou, I. & Probst, J. A holographic two-impurity Kondo model. J. High Energ. Phys. 2016, 103 (2016). https://doi.org/10.1007/JHEP01(2016)103
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)103