At this point, it should be stressed that no holographic renormalisation is needed in our approach.
Secondly, we may compare the behaviour of the lowest lying quasinormal mode with what is expected from the Kondo model in condensed matter theory. According to [68, 116], the large-N Kondo resonance features a dis-tinct behaviour close to the critical temperature, which imprints itself in a quadratic relationship between the imaginary value of the lowest lying quasi-normal mode and the vacuum expectation value of the scalar operator, given by
|!ˆI|⇠2T hOi2. (5.25) To find out about this relationship, we can visualise our numerical results on a double logarithmic plot as shown in figure 5.12a. This reveals that the advertised relationship holds true close to T =Tc.
In figure 5.11, we see that ˆ!I/2⇡Tc also approaches zero as T ! 0. In fact, a log-linear plot shown in 5.12b shows that there is a similar relationship at very low temperatures, given by
ˆ
!I ⇠log (ThOi) , (5.26)
which displays strong deviation from mean-field theory. This is not unex-pected, as we discussed earlier: At very low temperatures, the system ap-proaches its ground state. Due to the fact that the scalar potential is not stabilised, the expectation value of the dual operator, O, is diverging. So, the behaviour (5.26) as shown in figure 5.12b is actually an artefact of our model and most likely would change if we stabilise the scalar potential.
T /Tc
ˆ
!I/2⇡Tc
0.2 0.4 0.6 0.8 1
0
0.04
0.06
0.08 0.02
Figure 5.11: Shown is the imaginary part of the lowest-lying quasinormal mode ˆ!I. The bullets • denote our data, while the diamonds ⇧ denote the data of [68], which shows excellent agreement. The real part is vanishing throughout the condensed phase.
2ThOi2 2⇡T
|ˆ!I| 2⇡T
10 5 10 4 10 3 10 2 10 1 1
10 1
10 2
10 3
10 4
(a)T !Tc
ˆ
!I 2⇡T
10 5 10 4 10 3 10 2 10 1 1
0.4
0.6
0.8 0.2
2ThOi2 2⇡T
(b)T!0
Figure 5.12: Behaviour of ˆ!I vs. 2T hOi2. Close to T = Tc (a), the re-lationship is linear, ˆ!I ⇠ 2T hOi2, while as T ! 0 (b), it is logarithmic, ˆ
!I ⇠log(ThOi).
In figure 5.11, we saw that in this case ˆ!I ! 0. Due to its definition (5.19), the thermalisation time scale diverges, ⌧th = 1/ˆ!I ! 1. So, does the system thermalise at all? The answer isyes, but without any time scale.
This is to be expected as at critical temperatures, field theories are usually well described by conformal field theories, which do not feature any scales.
Indeed, if we let the system evolve long enough, we eventually capture this conformally invariant behaviour by plotting the solution in dependence of logarithmic time. This is shown in figure 5.13. Note, that although the behaviour of hOi mimics a quasinormal mode by hOi ⇠ ei!log(t), it is not a quasinormal mode in the sense of the previous discussion in section 5.5.
Upon replacing t!log(t), we actually obtain polynomial decay of the form hOi ⇠t⇠, ⇠ ⌘i!, (5.27) where ⇠ is a dimensionless complex number. This clearly distinguishes this
‘frequency’ in logarithmic time from proper quasinormal modes as seen in section 5.5, which carry units of energy. The time coordinate t obviously should carry units of length and we must find a scale to compare it to in order to render the logarithm logt meaningful. In the numerics, we used the radius of the event horizon zH = (2⇡T) 1 for this purpose, so t is actually normalised w.r.t. the temperature which is T = Tc at the phase transition.
The overall polynomial form of the decay is, however, invariant under this rescaling.
It seems quite interesting to find a complex exponent, which might indi-cate the emergence a discrete scale invariance of the theory [117,118], i.e. the conformal invariance only remains intact for discrete time dilations t ! t, where takes discrete values, only. From figures 5.13a and 5.13b, we can fit the complex exponent⇠ to be given by
⇠= 0.502 + 1.501i⇡ 1 2 +3
2i , (5.28)
which seems reasonable. We will come back to the real part of this fallo↵
below.
As for the imaginary part, we need to investigate the topic of discrete scale invariance with caution: As we can see in figures 5.13c and 5.13d, the gauge-invariant quantities|hOi|andµ⇠ tdonotfeature oscillations in real time. These are the observables of the system, however, and thus it remains unclear, whether the imaginary part in (5.28) really has an implication like discrete scale invariance.
For holographic superconductors, it was found e.g. in [119–121] thatz = 2 and ⌫ = 1/2, independent of the dimensionality of the theory. This fits to
our analysis, since the holographic Kondo model of [60] essentially mimics a holographic superconductor in the AdS2 subspace of the defect hypersur-face. At this point, we need to be careful with the dimensions: The critical exponent ⌫ is not well-defined in our context, as is describes the spatial de-cay of correlation functions at criticality, which are obviously not present in the 0+1 dimensions of the dual defect to which the scalar operator O is constrained. Moreover, phase transitions as such are not possible at finite N in any theory for d 2 spatial dimensions due to the Mermin-Wagner-Coleman theorem [95, 96]. However, the large-N limit presents a loophole to this theorem as long-range fluctuations are suppressed. Thus, we take the results of [119–121] as strong hints that we can analytically continue⌫ = 1/2 to lower dimensions in the large-N limit.
The dynamical critical exponent of a theory close to criticality is defined via the scaling of its thermalisation time ⌧th by
1 ˆ
!I (5.19)
⌘ ⌧th ⇠⇠z ⇠
✓Tc T Tc
◆ z⌫
, (5.29)
where ⇠ denotes the correlation length of the system. So, we can extract the dynamical critical exponent z from the behaviour of the lowest lying quasinormal mode ˆ!I close to the critical temperature Tc. Its limit from the normal phase and the condensed phase need not be identical a priori.
Approaching Tc from the normal phase, we can extract the behaviour of ˆ
!I analytically if we linearise (3.45) around c. Indeed, as shown in [4], if we linearise T ⇡ c +(1)T ! +O(!2) and using T /Tc = exp(1/c 1/T), we find
⌧th ⌘ 1 ˆ
!I
= Im(1)T
c
✓T Tc
Tc
◆ 1
, (5.30)
so we deduce z⌫ = 1 orz = 2, as expected from [119–121].
On the other side, in figure 5.11 we see the linear behaviour of ˆ!I as we get close to the phase transition in the condensed phase. More quantitatively, a double logarithmic plot of |!ˆI| vs. 1 T /Tc close to T = Tc reveals a polynomial dependence of the form
|!ˆI|=a
✓T Tc Tc
◆b
, (5.31)
and we can fitb ⇡0.992, which is close enough toz⌫ = 1 so that we conclude z = 2 from both sides.
To conclude this discussion about critical dynamics in the holographic model of [60], we come back to figure 5.13 and the fitted decay constants
in equation (5.28). If we take |hOi| as the definite order parameter we can neglect the imaginary part in (5.28). If we can relate its power law fallo↵
at criticality directly to the dynamical critical exponent z, this would yield another powerful test for our numerics.
Indeed, the scaling behaviour of the order parameter can be generalised to the time dependent case and reads
|hO(t)i|T ⇠
✓T Tc Tc
◆
· f(t/⌧th), (5.32) with = 1/2 the critical exponent15 of the order parameter in equilibrium, see figure 3.3 and the corresponding discussion. From this, we can derive the limit forT !Tc, which is given by [122]
|hO(t)i|Tc ⇠t /z⌫, (5.33) where we used the relation ⌧th ⇠ ⇠z introduced above. Comparing the ex-ponent, /z⌫, to what we found numerically in (5.28) and using = 1/2 from the phenomenology in equilibrium, we once more findz⌫ = 1, orz = 2 if we assume⌫ = 1/2 in low dimensions.