Kondo model and solution techniques

model. This section is devoted to explaining the original Kondo model, its implications, di↵erent solution strategies and generalisations.

Here,J is the coupling constant at the Fermi temperatureT_F andN₀ denotes the electron density at the Fermi surface. ForJ >0, i.e. in the antiferromag-netic case, the coupling diverges at a finite temperature. This temperature is called the Kondo temperature, T_K, and is given by

TK =TF e^1/N0J. (3.5)

Apparently, the Kondo system becomes strongly coupled around the Kondo temperature, such that perturbation theory ceases to be reliable.

Following the renormalisation group flow in the opposite direction, the e↵ective coupling defined by (3.4) vanishes as T ! 1. So, the impurity decouples from the otherwise free electron gas in the bulk of the field theory.

This asymptotic freedom is, along with strong coupling at low energies, the reason why the Kondo model is often regarded as a toy model for quantum chromodynamics (QCD). It was hoped that a solution to the Kondo problem might shed some light on how to deal with strongly coupled QCD.

What happens forT .TK? Since perturbation theory is not reliable any-more, we must resort to di↵erent techniques, some of which will be presented in the following.

Wilson’s numerical renormalisation group

To begin with, we should state that the full resolution of the Kondo prob-lem as T ! 0 was performed first by Kenneth Wilson [63] with this nu-merical renormalisation group approach. He nunu-merically computed the full non-perturbative RG flow of the coupling down to zero temperature which actually shows that the divergence of J(T) appears not at the finite Kondo temperatureTK but only atT = 0. The divergence atTK in 3.4 is an artefact of cutting o↵ perturbation theory at finite order inJ. Going to higher orders changes the critical temperature at which the coupling diverges. Neverthe-less, we will go on to refer to (3.5) as the definition of TK.

Conformal field theory approach

In [87, 88], A✏eck and Ludwig proposed an elegant alternative way of deriv-ing the endpoints of the Kondo model’s RG flow by applyderiv-ing methods from conformal field theory. Essentially, they performed a partial wave decom-position centred around the impurity and retained only its s-wave. In this way, the system reduces to a 1+1-dimensional conformal field theory. The excitations of the host are given by left- and right-movers L(t+r), R(t r) away from the impurity atr = 0, wherer is a non-negative radial coordinate measuring the distance from the defect. By enhancing the range of the radial

coordinate to ( 1,1) and mirroring the right-movers defined on r > 0 to left-movers Ldefined onr <0, they mapped the Kondo model to and chiral model given by the Hamiltonian density

H = v_F 2⇡

†

Li@_{r L}+v_F (r)S~· ^†LT~ _L, (3.6) wherevF denotes the Fermi velocity which we set tovF =c= 1 in our setup.

At the impurity, matching conditions for the L-field must be provided and it turns out that at the RG fixed points, i.e. in the UV and the IR of the theory, the only di↵erence is due to these matching conditions which imprints itself in a phase shift. It is important to keep this approach in the back of our heads, as the holographic model presented in the next section will have similar properties.

Large-N ansatz

There are several ways to represent the impurityS~in the Kondo Hamiltonian (3.1). Above, we chose the totally antisymmetric representation⁹, but there are also others. In this case, as written down in equation (3.1), the impurity spin is modelled by Abrikosov pseudo-fermions, whose annihilation operator is given by . They are auxiliary degrees of freedom constrained to the impurity. In order to project our model back to physical degrees of freedom, the Abrikosov pseudo-fermions must obey a constraint, which is given by the quantisation condition

† =q . (3.7)

Here,q denotes the number of Abrikosov pseudo-fermions in any state which is also the number of boxes in the Young tableaux of the representation.

The vector symbol on S~ indicates that the original model was intended for a SU(2)-symmetry in three dimensions. However, it can obviously be extended toSU(N) by using symmetry generators T^a, a2{1, . . . N² 1}of the fundamental representation ofSU(N) instead. The completeness relation of the SU(N) generators is given by

T_↵^A T^A = 1 2

✓

↵

1 N ^↵

◆

. (3.8)

If we insert this into the CFT model (3.6), we find that the coupling to the impurity becomes

HK = ˆ 2

⇣OO^† q N ^L

L†

⌘ . (3.9)

9 This means that due to its composition, S~ = ^†T~ forms an antisymmetric repre-sentation ofSU(2) when acting on states.

where we defined ˆ = v_F and the scalar operator O = ^†_L , which has conformal dimension = 1/2. The ‘double trace’-term ⇠OO^† is the addi-tional marginally relevant term which is added in a localised fashion to the free chiral current in the holographic model of [60] and is therefore of central interest here.

Most interestingly, upon taking the large-N limit, N ! 1, it was shown that this variation of the Kondo model features a proper phase transition at a critical temperature Tc which is of the same order of magnitude as the Kondo temperature TK [89–94]. Below the critical temperature, T < Tc, the scalar operatorOcondenses in terms of mean-field theory, where close to the phase transition its behaviour is given by

h|O|i ⇠

✓T_c T Tc

◆1/2

, (3.10)

which is characteristic for order parameters in mean-field transitions. Due to its definition, O = ^† , the condensation of the scalar operator indicates the formation of a singlet state involving the impurity and the degrees of freedom in the host metal. This singlet state is often referred to as the Kondo screening cloud, as the e↵ects of the impurity on the host metal are screened at large distances due to the singlet formation.

This phase transition is a relict of the large-N limit and is not present in the original model. The transition at finite N is rather a cross-over than a phase transition, which is also consistent with the fact that there are no proper phase transitions possible in field theories of dimensionality d < 3 [95, 96]. Taking N to infinity is, however, a loop-hole to this theorem. On a computational level, the reason for the phase transition to occur at finite T ⇡ TK is given by equation (3.4). This is the one-loop approximation to the running, but as N ! 1, the higher order contributions actually vanish and the coupling diverges at a finite temperature [97].

We cannot expect that the large-N approach to the Kondo model captures all of its phenomenology. One of the most important features of the original Kondo model, namely the logarithmic contribution to the resistivity along with its associated minimum at finiteT, is absent in the large-N limit. How-ever, it is a very convenient starting point for holographic model building, which always utilises a large-N limit if we want to have a classical theory of gravity on the gravity side of the duality. Moreover, the gauge/gravity conjec-ture only works for conformal field theories, which may include (marginally) relevant operators, varying only the low energy behaviour of the theory. This is the case in the model at hand and in the next section, we will show how the authors of [60] used those features as a guideline in order to build a holographic model which shows strikingly similar phenomenology.