geodesic in the other half of the manifold, N , is simply given by changing the sign of the x-component of the geodesic we just solved for. The impor-tant part, its proper length, is the same as for the computed geodesic, so essentially we just need to take that twice in order to find the entanglement entropy via (2.91).
Note, that we do not need to renormalise the proper length of the geodesics as we subtract the divergent parts at asymptotic infinity automatically via the definition of the defect entropy (4.32). The divergence appears as the leading order term close to z = 0 in the proper length, which is both inde-pendent of the temperature and the asymptotic position `= +x(z = 0).
There is one more observation, that simplifies the analysis of the defect entropy dramatically. It is the fact, that while our embeddings X± are in principal arbitrary (numerical) solutions to (4.17), the background metrics in the ambient geometries, g±, are constant by construction and still given by the BTZ metric (3.19). This is due to having no propagating degrees of freedom in our 2+1-dimensional model with only topological fields in the 2+1-dimensional bulk.
Moreover, the geodesics in this metric are known analytically, see e.g. [109].
So instead of computing numerical solutions to the geodesic equation for each starting point on the defect D, we just need to fit the point and its normal with the known solutions, which is faster and exact. This way, the only source of numerical uncertainty entering our results of the defect entropy (4.32) is given by numerical errors in the embeddings themselves. To suppress those as much as feasible, we apply pseudospectral methods by computing the em-beddings and its derivatives on a Chebyshev-Gauss-Lobatto grid. For the details of the numerical approach, we refer the reader to appendix B.
of a d-form inD dimensions is in general defined by C =
Z
SD d 1 ⇤fd+1, (4.36)
where ⇤ denotes the Hodge star. According to the dictionary discussed in chapter 3, its asymptotic limit as z !0 is dual to the representation of the defect, which is hence given by
C ⌘ p ftz z!0 . (4.37)
If the scalar field vanishes and we are interested in static solutions, the equa-tion of moequa-tion for the gauge field reduces to
@z p fzt
| {z }
=C
= 0, (4.38)
which already tells us that the electric flux is constant along the hypersurface, i.e. p ftz = C everywhere. This is intuitive, since there is no charged field which could screen the flux. We have found a background solution for the gauge field, which still depends on on the induced metric and hence the embedding. In order to find out what this implies for the junction conditions, which are the equations of motion of the embedding, let’s have a look at the energy-stress due to the background solution of the gauge field. Following our analysis above, the energy-stress tensor for vanishing scalar is given by
Sµ⌫ = N
4 f↵ f↵ µ⌫ +Nfµ f⌫ = N
4 f↵ f↵ , (4.39) where in the last equality we used that fµ f⌫ = µ⌫f↵ f↵ /2 which holds in our case of a 1+1-dimensional, diagonal metric . Applying p ftz = C then yields
Sµ⌫ = N
2 C2 µ⌫. (4.40)
This is, remarkably, setting a constant tension on the hypersurface, since the energy-stress tensor is proportional to the induced metric, Sµ⌫ ⇠ µ⌫. In appendix A, we show that in such cases, there is a construction to generate solutions of the Israel junction conditions if we have a valid initial solution of constant tension.
To wrap it up, it works by following the geodesic normal flow of the hypersurface. To begin with, we can find a valid solution to the equations of motion of the field content on the hypersurface by imposing the gauge field to vanish, too. A possible solution for the embedding consistent with our
boundary conditions, X(z = 0) = 0, is then given by setting X(0)(z) = 0 in (4.12). In this case, all of our assumptions in appendix A hold true and we can apply the flow construction. To do this, we have to define a vector field which is normal to the hypersurface, normed, and for each starting point on the hypersurface, the integral curve of that point along the vector field satisfies the geodesic equation. We define new embeddingsX(s) :D ,!N as the set of points which we obtain by following the normal geodesic at each point on the initial hypersurface X(0) for a proper length s. After following the geodesic normal flow for an arc lengths, the embedding functionX+=X(s) is given by
X(s)(z) = zHartanh 0
@ sinh(s/L)
q
(z/zH)2+ sinh2(s/L) 1
A . (4.41)
Upon integrating the equation of motion (4.7) for the gauge field with van-ishing scalar field and an embedding given by (4.41), we find the analytic solution
a(s)t = CL2 zH
cosh(s/L)
✓
cosh(s/L) q
(z/zH)2+ sinh2(s/L)
◆
. (4.42) In order to eliminate the auxiliary variable s, we can plug equations (4.41) and (4.42) into the Israel junction conditions and find
tanh⇣s L
⌘
= L
4 NN C2. (4.43)
At this point, we should keep in mind thatscan have either sign. According to our convention, the normal vector field is pointing out of the bulk manifold N, such that its volume grows if s > 0, which is indeed the case as can be seen from (4.43). A sketch of the geodesic normal flow construction is shown in figure 4.2, where we see how the embedding changes as we increase the parameter s > 0, initially starting at the trivial and totally geodesic embedding withs = 0.
The extrinsic curvatureKs and the induced metric s as functions of the parameter s are readily derived to read
s =
✓Lcosh(s/L) zb
◆2
f(zb) dt2+f 1(zb) dz2 , (4.44) Ks = Lsinh(s/L) cosh(s/L)
zb2 f(zb) dt2+f 1(zb) dz2 , (4.45)
where zb denotes the point on the trivial embeddingX = 0, from which we start following the normal geodesic for an arc lengthsto find the embedding Xs. The induced metric and extrinsic curvature obviously satisfy
Ks= tanh(s/L) s/L , (4.46)
which is shown in appendix A to hold in more generality than in the context presented here.
From (4.43), we can already tell our first analytic result regarding the defect entropy. We remember from section 4.2, that the minimal surfaces whose area give us the defect entropy have to start normal to the defect hypersurface. As we just have seen, the embeddings for vanishing scalar fields are generated by using the same geodesics. Hence, the defect entropy SD as defined by (4.32) will simply be proportional to the arc length s = L artanh (LN N C2/4) used in our construction above. This will describe the overall o↵set of SD for any choice of C,N, and N.
Moreover, equation (4.43) puts a constraint on the matter content on the defect hypersurface. As the tanh function takes its values between 1 and +1, the allowed range of values for the parameters N, C, L, and N is restricted to
0LN N C2 4, (4.47)
where we already took into account that the appearing product of parameters is always positive. To the best of the authors knowledge, there is no such constraint on the Kondo model on the field theory side. Investigating its physical reason would be very interesting, but is left for future research.
This concludes the analysis of the normal phase in the static backreaction of the holographic Kondo model. In the next section, we will start from this background solution and describe the system’s behaviour in terms of the temperature T as it drops below the critical temperature Tc, at which point the scalar field becomes non-trivial.