Holographic Kondo and Fano resonances

Holographic Kondo and Fano resonances Johanna Erdmenger,1,* Carlos Hoyos,2,†Andy O’Bannon,3,‡Ioannis Papadimitriou,4,§ Jonas Probst,5,∥and Jackson M. S. Wu6,¶ 1Institut für Theoretische Physik und Astrophysik, Julius-Maximilians-Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany and Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 Munich, Germany 2Department of Physics, Universidad de Oviedo, Avda. Calvo Sotelo 18, 33007 Oviedo, Spain 3STAG Research Centre, Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom 4SISSA and INFN—Sezione di Trieste, Via Bonomea 265, I 34136 Trieste, Italy 5Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, United Kingdom 6Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35487, USA (Received 8 December 2016; published 12 July 2017) We use holography to study a (1þ1)-dimensional conformal field theory (CFT) coupled to an impurity. The CFT is an SUðNÞgauge theory at large N, with strong gauge interactions. The impurity is an SUðNÞ spin. We trigger an impurity renormalization group (RG) flow via a Kondo coupling. The Kondo effect occurs only below the critical temperature of a large-Nmean-field transition. We show that at all temperatures T, impurity spectral functions exhibit a Fano resonance, which in the low-Tphase is a large-N manifestation of the Kondo resonance. We thus provide an example in which the Kondo resonance survives strong correlations, and uncover a novel mechanism for generating Fano resonances, via RG flows between (0þ1)-dimensional fixed points. DOI: 10.1103/PhysRevD.96.021901 I. INTRODUCTION The Kondo effect is the screening of an impurity spin by a Landau Fermi liquid (LFL) at low T[1,2]. A variety of techniques, such as Wilson’s RG, large-N, CFT, and more [3], have captured many characteristic Kondo phenomena. Nevertheless, many questions resist solution, for example about inter-impurity interactions, subsystem entanglement entropy (EE), nonequilibrium processes like quantum quenches, and more. In particular, what happens when the LFL is replaced with strongly correlated electrons? For example, how does the Kondo effect change in a Luttinger liquid [4–8] or the Hubbard model [9,10]? In the latter case, experiments reveal dramatic effects of strong correlations, such as enhancement of the Kondo temperature, TK[53]. On the theory side, although special tools like bosonization [4–8] and uncontrolled mean-field approximations [9,11–17] have provided insight, in general, reliable techniques do not yet exist to answer questions about Kondo phenomena in strongly-correlated systems. To address all of the above, we have developed an alternative Kondo model, based on holographic duality [18–21]. Our model replaces the LFL by a (1þ1)- dimensional CFT in which spin SUð2Þis replaced by gauged SUðNÞ, with large Nand strong gauge interactions. Our model has already revealed novel strong-coupling phenomena in RG [18,19], interimpurity interactions [19] and EE [20]. Here we initiate the study of nonequilibrium phenomena in our model: we compute linear response (Green’s) functions of a charged bosonic impurity operator, O,in our model. We have two main results. First, we find a large-Nmanifestation of the Kondo resonance [2,22,23], a signature of the Kondo effect. As expected, our Kondo resonance appears only at Tbelow the critical temperature Tcof a mean-field transition that is common to large-NKondo models [23–27]:hOibecomes non-zero when T≤Tc. We thus prove unequivocally that our holographic model realizes a genuine Kondo effect, as opposed to some other impurity physics, and furthermore show that a large-NKondo resonance can survive strong correlations essentially intact. Second, at all T,O’s spectral function exhibits a Fano resonance [28,29], which occurs when a Lorentzian resonance is immersed in a continuum of states (in energy). A Fano resonance is characterized not only by its position, width, and height, like a Lorentzian, but also by an asymmetry parameter,q, which measures the relative strength of resonant versus non-resonant scattering. Our qincreases as T→Tþ c. When T≤Tc, the Fano lineshape arises from our Kondo resonance, which must be *[email protected]‑wuerzburg.de †[email protected] ‡[email protected] §[email protected] ∥[email protected]c.uk ¶[email protected] PHYSICAL REVIEW D 96, 021901(R) (2017) 2470-0010=2017=96(2)=021901(6) 021901-1 © 2017 American Physical Society RAPID COMMUNICATIONS antisymmetric due to particle-hole symmetry (PHS), and hence has the special value q¼1. Although Fano resonances have been observed in many impurity systems in one spatial dimension [29–32], ours arise from a qualitatively different mechanism. For instance, in side-coupled quantum dots (QDs) [29,31,32] the Lorentzian resonances are the discrete states on the QD, and the continuum comes from electronic scattering states in the leads. Coupling the two, for example by a Kondo coupling, can then produce Fano resonances. Our model also has an impurity coupled to a continuum in one spatial dimension, i.e. the CFT. However, our model has two couplings: the CFT’sSUðNÞgauge coupling and the Kondo coupling. The spectral function of Oinherits (0þ1)-dimensional scale invariance from the former, and so exhibits a continuum of states, in contrast to a QD’s discrete states. The Kondo coupling then triggers an RG flow from that (0þ1)-dimensional fixed point, and creates a resonance that cannot escape the continuum, hence producing a Fano line shape. To our knowledge, such a mechanism for producing Fano resonances is novel, and moreover is easy to generalize to any RG flow between (0þ1)-dimensional fixed points, as follows. Scale invariance implies that any spectral function will be a featureless continuum, which in (0þ1) dimensions means a power law (or logarithm) in frequency. A relevant deformation can then explicitly break scale invariance, trigger an RG flow to an IR fixed point—in which case we expect the continuum to survive—and may also produce resonances. In higher dimensions, the resonances would not have to be within the continuum, for example the two could be separated in momentum space. However, in (0þ1) dimensions the resonances have no place to escape the continuum, and hence must produce Fano line shapes. In fact, such a mechanism was at work in some previous cases, such as the large-NKondo model at sufficiently low T[33], and holographic duals of T¼0charged black branes [34–36]. However, the resulting Fano resonances went unidentified, leaving crucial physics overlooked, namely the relative strength of resonant versus nonresonant scattering, as measured by q. Our results not only provide a novel perspective on these cases, but also predict Fano resonances in RG flows between other (0þ1)- dimensional fixed points, such as Sachdev-Ye-Kitaev fixed points [36–43]. Further results of our model, including details of holographic renormalization useful for holographic impurity models in general, will appear in [44]. II. HOLOGRAPHIC KONDO MODEL We first briefly review some essential features of the CFT and large-Napproaches to the Kondo model, and how our model builds upon and extends them. The CFT approach [45] is based on s-wave reduction of LFL fermions about the impurity, plus linearization of the dispersion relation. In/out-going s-waves become relativistic left/right-moving fermions, ψLand ψR, in the radial direction, r. Reflecting ψRto r<0and relabeling ψR→ψLleads to ψLalone on the entire raxis, with impurity at r¼0. The ψLform a (1þ1)-dimensional chiral CFT with SUð2Þ1×Uð1Þspin and charge KacMoody currents, respectively. In the Hamiltonian, the Kondo interaction is δðrÞgKSAJA, with coupling constant gK, impurity spin SA, and spin current JA,A¼1,2,3. An antiferromagnetic coupling, gK>0, is marginally relevant, and triggers an RG flow to an IR chiral CFT characterized by a phase shift of ψLand impurity screening [45]. The large-Napproach begins by replacing spin SUð2Þ→SUðNÞ,followedbyN→∞with λK≡ NgKfixed [3,23,46,47]. We will only consider SAin totally antisymmetric SUðNÞrepresentations of rank Q, and introduce Abrikosov pseudofermions χvia SA¼ χ†TAχ, with SUðNÞgenerators TA,A¼1;…;N2−1. Doing so introduces an auxiliary Uð1Þacting only on χ, but with charge fixed by projecting onto states with χ†χ¼Q.AtlargeN,SAJA¼−O†O=2with O≡ψ† Lχ [19], which is charged under both the charge and auxiliary Uð1Þ’s. Our holographic model [18] begins by gauging SUðNÞ, thus introducing the ’t Hooft coupling, λ.We then add degrees of freedom to make the gauge theory a (1þ1)-dimensional CFT with sparse operator spectrum when Nand λboth →∞, but whose details otherwise are irrelevant. The theory is then holographically dual to Einstein-Hilbert gravity in (2þ1)-dimensional anti-de Sitter space, AdS3[48]. The charge Uð1ÞKac-Moody current is dual to a Uð1ÞChern-Simons gauge field, A, the auxiliary Uð1Þis dual to a Maxwell field aon an AdS2defect at r¼0, and Ois dual to a complex scalar field Φalso in AdS2, charged under both Aand a.As long as the stress-energy tensor is finite, at large Nwe can neglect backreaction of A,a,Φ(dual to fundamental fields) on the geometry (dual to adjoint fields). When T>0, the bulk metric is thus the BTZ black brane, ds2¼1 z2ðh−1ðzÞdz2−hðzÞdt2þdr2Þ; with hðzÞ¼1−z2=z2 Hwhere zH¼1=2πT, and unit AdS radius. The fields aand Φare localised to the asymptotically AdS2subspace at r¼0, with induced metric gmn (m; n ¼z,t). We describe the dynamics of A,a, and Φby the simple quadratic action [18], S¼−N 4πZBTZ A∧dAþSAdS2;ð1aÞ JOHANNA ERDMENGER et al. PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-2 RAPID COMMUNICATIONS SAdS2¼−NZx¼0 dzdtffiffiffiffiffiffi −g p1 4fmnfmn þðDmΦÞ†ðDmΦÞþM2Φ†Φ;ð1bÞ with field strength f¼da, covariant derivative DmΦ¼ ð∂mþiAm−iamÞΦ, and mass-squared M2. At the horizon z¼zHwe require regularity of all fields. At the boundary z¼0,a’s leading mode, a∼Q=z, is related to Q:Q≠0breaks χ’s PHS, so the PHS value Q¼0is dual to the PHS value Q¼N=2, and increasing jQj corresponds to increasing jQ−N=2j. The large-NKondo interaction −λKO†O=2is classically marginal, hence Ohas UV dimension Δ¼1=2, which fixes M2and hence Φ’s near-boundary expansion, Φ∼ffiffiffi z pðαlog zþβÞ. Introducing the Kondo interaction amounts to adding a boundary term ∝−λKO†O=2to S, which changes Φ’s boundary condition from α¼0to α¼ −λKβ[18,49,50]. For more details about the boundary terms, see [19,44]. A holographic scaling analysis reveals that λKruns logarithmically, λK¼1=log ðT=TKÞ, diverging at the dynamically-generated Kondo temperature, TK≡Λe−1=λK=ð2πÞ, with λKevaluated at the UV cutoff, Λ. A holographic antiferromagnetic UV Kondo coupling, λK>0, is thus marginally relevant, breaks conformal invariance, and triggers an RG flow. As mentioned above, our model has a mean-field phase transition [18]:hOi¼0(Φ¼0) when T>T cand hOi≠0 (Φ≠0) when T≤Tc. Condensate formation hOi≠0 breaks the charge and auxiliary Uð1Þ’s to the diagonal, and signals the Kondo effect, including a phase shift of ψL, dual to a Wilson line of A, and impurity screening, dual to reduction of fflux between z¼0and z¼zH. We refer to the T>T cand T≤Tcphases as “unscreened”and “screened,”respectively. In [18–21] we computed Tc numerically. Below we obtain an exact formula for Tc. III. FANO RESONANCES If a retarded Green’s function of complex frequency ω, GðωÞ, has a pole at ωp,GðωÞ∼Z ω−ωp, with complex residue Z¼ZRþiZI, then near the pole the spectral function ρðωÞ≡−2ImðGðωÞÞ will have a Fano resonance [28,29] (setting ImðωÞ¼0), ρFanoðωÞ¼ ðω−ω0þqΓ=2Þ2 ðω−ω0Þ2þðΓ=2Þ2;ð2Þ with position ω0¼ReðωpÞ, width Γ¼2jImðωpÞj, and asymmetry parameter q¼−ZR=ZIþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þZ2 R=Z2 I p.Fano resonances are antisymmetric when q¼1, meaning ρðωÞ is odd under PHS, and symmetric when q¼0(an antiresonance) or ∞(a Lorentzian), meaning ρðωÞis even. Fano resonances arise when a Lorentzian resonance is immersed in a continuum (in energy), due to interference between the two. The asymmetry parameter qcontains key dynamical information, specifically, q2∝the ratio of probabilities of resonant and non-resonant scattering. In our model, the AdS2subspace inherits scale invariance from AdS3, or in dual field theory language, the impurity inherits scale invariance from the CFT, so ρðωÞof impurity operators must be a featureless continuum. Our marginally-relevant Kondo coupling then breaks scale invariance and produces a resonance, while Q≠0breaks PHS. We will show that ρðωÞof Othen indeed generically exhibits asymmetric Fano resonances. IV. SPECTRAL FUNCTIONS We compute GðωÞholographically by solving for linearized fluctuations about solutions for the unscreened and screened phases [44,51,52]. At all T, we find that the Kac-Moody current’sGðωÞis unaffected by the impurity. In the unscreened phase, we find that all charged GðωÞ vanish, i.e. GOOðωÞ≡hOðωÞOð−ωÞi ¼ 0, while GO†OðωÞ≡hO†ðωÞOð−ωÞi ¼ N λK1−1 λKDðωÞ; DðωÞ≡H−1 2þiQ −iω 2πTþH−1 2−iQ þln2T TK; with Harmonic number H½x, and λKevaluated at Λ. The form of GOO†ðωÞis the same, but with Q→−Q. Scale invariance in (0þ1)-dimensions and Δ¼1=2imply a trivial UV continuum: limω→∞ρO†OðωÞ¼0. For given Qand T,GO†OðωÞhas poles in ωwhen DðωÞ¼0. Figure 1shows our numerical results for the FIG. 1. Positions of poles in the complex ω=ð2πTÞplane for Q¼1=2. Blue and purple denote lowest and next-lowest poles, respectively, of GO†OðωÞ(ReðωÞ>0) and GOO†ðωÞ (ReðωÞ<0), for T=Tcfrom 100 down to 1.001. Red and orange denote the same for GO†OðωÞfor T=Tcfrom 1 down to 0.2. Arrows indicate movement of poles as Tdecreases. HOLOGRAPHIC KONDO AND FANO RESONANCES PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-3 RAPID COMMUNICATIONS positions of the lowest (closest to ω¼0) and next-lowest poles of GO†OðωÞand GOO†ðωÞin the complex ω=ð2πTÞ plane, for Q¼1=2. Other Qgive similar results. As T→Tþ c, the lowest pole moves towards the origin, arrives there at Tc, and when T<T c, crosses into the ImðωÞ>0 region, signaling instability (not shown). We thus identify Tcas the Twhere Dðω¼0Þ¼0, Tc¼1 2TKexp −2ReH1 2þiQ: Figure 2shows the normalized spectral function ¯ ρO†OðωÞ≡−2λ2 K NImGO†OðωÞversus real ω=ð2πTÞfor Q¼ 1=2and T=Tc¼16, 8, 4, 2. We find a Fano resonance, as advertised, with asymmetric minimum and maximum. Numerically, ω0≈ReðωpÞand Γ≈2jImðωpÞj,asin(2), where ωpis GO†OðωÞ’s lowest pole. As T→Tþ c,qgrows: q≈1.7at T¼16Tcwhile q≈4at T¼2Tc. For Tjust above Tc,T≳Tc, expanding in Tabout Tc and in ωabout ω¼0gives, for GO†OðωÞ’s lowest pole, ωp 2πT¼−iT=Tc−1 ψ0½1 2þiQ;Z¼−iN λ2 K 2πTc ψ0½1 2þiQ;ð3Þ with digamma function ψ½x. The resonance height thus grows as ðT=Tc−1Þ−1and the width shrinks as T=Tc−1. It is therefore not related to a Kondo resonance, which grows logarithmically as T→Tþ K[22]. Indeed, at large N we expect the Kondo resonance only in the screened phase [23]. Our resonance is presumably a bound state of ψL and χ, heralding the nascent screened phase. The Zin (3) gives qthat depends only on Q, shown in Fig. 3. (Anti)symmetric values q¼1,0,∞occur when Q→0,∓∞, respectively. Indeed, Fig. 4shows that even for relatively modest Q¼1, the resonance is nearly Lorentzian, the minimum having practically vanished. In the screened phase, we have numerical results for GO†OðωÞ[18–21,44]. Figure 1shows our numerical results for the positions of the lowest and next-lowest poles in GO†OðωÞfor Q¼1=2. Other Qgive similar results. At T¼Tcthe poles are coincident with those of GO†OðωÞand GOO†ðωÞin the unscreened phase. As Tdecreases below Tc,GO†OðωÞ’s lowest pole, ωp, moves straight down the ImðωÞaxis. FIG. 2. The normalized spectral function ¯ ρO†OðωÞversus real ω=ð2πTÞfor Q¼1=2and, from shortest to tallest, T=Tc¼16 (red), 8 (green), 4 (orange), and 2 (blue). FIG. 3. Asymmetry parameter qversus Q, for T≳Tc. FIG. 4. The normalized spectral function, ¯ ρO†OðωÞ, versus real ω=ð2πTÞfor Q¼1and, from shortest to tallest, T=Tc¼1.04 (green), 1.02 (orange), and 1.01 (blue). FIG. 5. The normalized spectral function ¯ ρO†OðωÞversus real ω=ð2πTÞfor Q¼1=2and, from tallest to shortest, T=Tc¼0.998 (blue), 0.991 (orange), and 0.964 (green). JOHANNA ERDMENGER et al. PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-4 RAPID COMMUNICATIONS From our experience with the unscreened phase, we expect ωpto produce a Fano resonance in the normalized spectral function, ¯ ρO†OðωÞ. Crucially, ReðwpÞ¼0,soωp preserves PHS, ReðωÞ→−ReðωÞ, so we expect an antisymmetric Fano resonance at ReðωÞ¼0. Moreover, jImðωpÞjincreases as Tdecreases, and so should the width Γ. Figure 5confirms our expectations: ¯ ρO†OðωÞ’s only significant feature is a Fano resonance at ReðωÞ¼0 with q¼1, meaning perfectly antisymmetric minimum and maximum, and whose Γincreases as Tdecreases. Additionally, the height decreases, and indeed our numerics suggest limT→0¯ ρO†OðωÞ¼0. Figure 6shows our numerical results for the position of ωpversus small λ2 K N2hOi2=ð2πTÞ, or equivalently, Tjust below Tc,T≲Tc, for Q¼1=2. Figure 6also shows a linear fit demonstrating that.1 ωp∝−ihOi2:ð4Þ Our model’s mean-field behavior hOi∝ðTc−TÞ1=2then implies Γ∝Tc−Tfor T≲Tc. The behavior in (4) is in fact identical to that in a LFL at large N. In a LFL, the Kondo resonance is formally defined in the LFL fermion spectral function, and at large N appears only in the screened phase, with Γ∝hOi2[23].For T≲Tc, the mean-field behavior hOi∝ðTc−TÞ1=2then implies Γ∝Tc−T. Crucially, in the screened phase the Kondo resonance also appears in other spectral functions, due to operator mixing induced by the symmetry breaking [23]. In particular, a Kondo resonance should produce a pole in GO†OðωÞprecisely of the form in (4)2Our result (4) thus proves the existence of a Kondo resonance in our model when T≲Tc, with defining features essentially intact despite the strong interactions. V. CONCLUSION In a holographic model describing the interaction of a magnetic impurity with a strongly correlated CFT at large N, we discovered a novel mechanism for producing Fano resonances, namely via RG flows between (0þ1)-dimensional fixed points. The origin and consequences of such Fano resonances, in existing cases that have gone unidentified and in novel cases, deserve further study, particularly of the physics contained in the asymmetry parameter q. ACKNOWLEDGMENTS We would like to thank Ian Affleck, Nathan Andrei, Piers Coleman, Mario Flory, Henrik Johannesson, Andrew Mitchell, Max Newrzella, and Philip Phillips for helpful conversations and correspondence. C. H. is supported by the Ramon y Cajal fellowship RYC-2012-10370, the Asturian Grant No. FC-15-GRUPIN14-108 and the Spanish national Grant No. MINECO-16-FPA201563667-P. A. O’B. is a Royal Society University Research Fellow. J. P. is supported by the Clarendon Fund and St John’s College, Oxford, and by the European Research Council under the European Union’s Seventh Framework Programme (ERC Grant agreement 307955). [1] J. Kondo, Prog. Theor. Phys. 32, 37 (1964). [2] A. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, 1993). [3] D. L. Cox and A. Zawadowski, Adv. Phys. 47, 599 (1998). [4] D.-H. Lee and J. Toner, Phys. Rev. Lett. 69, 3378 (1992). [5] A. Furusaki and N. Nagaosa, Phys. Rev. Lett. 72, 892 (1994). [6] P. Fröjdh and H. Johannesson, Phys. Rev. Lett. 75, 300 (1995). [7] P. Fröjdh and H. Johannesson, Phys. Rev. B 53,3211(1996). [8] A. Furusaki, J. Phys. Soc. Jpn. 74, 73 (2005). [9] P. Fulde, V. Zevin, and G. Zwicknagl, Z. Phys. B 92, 133 (1993). [10] T. Schork and P. Fulde, Phys. Rev. B 50, 1345 (1994). [11] G. Khaliullin and P. Fulde, Phys. Rev. B 52, 9514 (1995). FIG. 6. Dots denote the position of GO†OðωÞ’s lowest pole versus λ2 K N2hOi2=ð2πTÞfor Q¼1=2. The solid line is a linear fit with slope ≈−17.6and intercept at the origin. 1In [44] we derive (4) without numerics, via a small-hOi expansion. 2For details, see for example chapter 18 of [23]. HOLOGRAPHIC KONDO AND FANO RESONANCES PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-5 RAPID COMMUNICATIONS [12] J. Igarashi, K. Murayama, and P. Fulde, Phys. Rev. B 52, 15966 (1995). [13] S. Tornow, V. Zevin, and G. Zwicknagl, arXiv:cond-mat/ 9701137. [14] T. Schork and S. Blawid, Phys. Rev. B 56, 6559 (1997). [15] B. Davidovich and V. Zevin, Phys. Rev. B 57, 7773 (1998). [16] H. T. Duc and N. T. Thang, Mod. Phys. Lett. B 13, 849 (1999). [17] W. Hofstetter, R. Bulla, and D. Vollhardt, Phys. Rev. Lett. 84, 4417 (2000). [18] J. Erdmenger, C. Hoyos, A. O’Bannon, and J. Wu, J. High Energy Phys. 12 (2013) 086. [19] A. O’Bannon, I. Papadimitriou, and J. Probst, J. High Energy Phys. 01 (2016) 103. [20] J. Erdmenger, M. Flory, C. Hoyos, M.-N. Newrzella, and J. M. S. Wu, Fortschr. Phys. 64, 109 (2016). [21] J. Erdmenger, M. Flory, C. Hoyos, M.-N. Newrzella, A. O’Bannon, and J. Wu, Fortschr. Phys. 64, 322 (2016). [22] P. Phillips, Advanced Solid State Physics (Cambridge University Press, 2012). [23] P. Coleman, Introduction to Many-Body Physics (Cambridge University Press, 2015). [24] P. Coleman and N. Andrei, J. Phys. C 19, 3211 (1986). [25] P. Coleman, Phys. Rev. B 35, 5072 (1987). [26] T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, 216403 (2003). [27] T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, 035111 (2004). [28] U. Fano, Phys. Rev. 124, 1866 (1961). [29] A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. Phys. 82, 2257 (2010). [30] V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie, and N. S. Wingreen, Science 280, 567 (1998). [31] V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie, and N. S. Wingreen, Phys. Rev. B 64, 165412 (2001). [32] J. Göres, D. Goldhaber-Gordon, S. Heemeyer, M. A. Kastner, H. Shtrikman, D. Mahalu, and U. Meirav, Phys. Rev. B 62, 2188 (2000). [33] O. Parcollet, A. Georges, G. Kotliar, and A. Sengupta, Phys. Rev. B 58, 3794 (1998). [34] T. Faulkner, H. Liu, J. McGreevy, and D. Vegh, Phys. Rev. D83, 125002 (2011). [35] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, and D. Vegh, Phil. Trans. R. Soc. A 369, 1640 (2011). [36] S. Sachdev, Phys. Rev. X 5, 041025 (2015). [37] S. Sachdev and J.-W. Ye, Phys. Rev. Lett. 70, 3339 (1993). [38] A. Kitaev, in KITP Strings seminar and Entanglement 2015 program, (2015). [39] J. Polchinski and V. Rosenhaus, J. High Energy Phys. 04 (2016) 001. [40] A. Jevicki, K. Suzuki, and J. Yoon, J. High Energy Phys. 07 (2016) 007. [41] J. Maldacena and D. Stanford, Phys. Rev. D 94, 106002 (2016). [42] A. Jevicki and K. Suzuki, J. High Energy Phys. 11 (2016) 046. [43] E. Witten, arXiv:1610.09758. [44] J. Erdmenger, C. Hoyos, A. O’Bannon, I. Papadimitriou, J. Probst, and J. M. S. Wu, J. High Energy Phys. 03 (2017) 039. [45] I. Affleck, Acta Phys. Pol. B 26, 1869 (1995). [46] N. Bickers, Rev. Mod. Phys. 59, 845 (1987). [47] P. Coleman, in Handbook of Magnetism and Advanced Magnetic Materials: Fundamentals and Theory, edited by H. Kronmuller and S. Parkin (John Wiley and Sons, New York, 2007), Vol. 1, p. 95. [48] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, and Y. Oz, Phys. Rep. 323, 183 (2000). [49] E. Witten, arXiv:hep-th/0112258. [50] I. Papadimitriou, J. High Energy Phys. 05 (2007) 075. [51] D. T. Son and A. O. Starinets, J. High Energy Phys. 09 (2002) 042. [52] P. K. Kovtun and A. O. Starinets, Phys. Rev. D 72, 086009 (2005). [53] T. Brugger, T. Schreiner, G. Roth, P. Adelmann, and G. Czjzek, Phys. Rev. Lett. 71, 2481 (1993). JOHANNA ERDMENGER et al. PHYSICAL REVIEW D 96, 021901(R) (2017) 021901-6 RAPID COMMUNICATIONS