AdS 5 /CFT 4 correspondence
4.4. Towards a Holographic Realization of Homes’ Law
4.4.1. Homes’ law in condensed matter
An interesting phenomenon to look for universal behavior in condensed matter systems, as ad-vertised in the introduction to this chapter, is the universal scaling law for superconductors empirically found by Homes et al. [42] by collecting experimental results. This so-calledHomes’
lawdescribes a relation between different quantities of conventional and unconventional super-conductors,i.e. the superfluid densityρsat zero temperature and the conductivityσDCtimes the critical temperatureTc,
ρs“CσDCpTcqTc, (4.220)
where Homes et al. report two different values of the constantCin unitsrcms´2for the different cases considered in [43]. The valueC“35is true for in-plane cuprates and elemental BCS su-perconductors, whereas for the cuprates along the c-axis and the dirty limit BCS superconductors they findC“65. The superfluid densityρsis a measure for the number of particles contributing to the superfluid phase. It can be thought of as the square of the plasma frequency19of the su-perconducting phaseωPs2, because the superconductor becomes “transparent” for electromagnetic waves with frequencies larger than
ρs”ω2Ps“ 4πnse2
m˚ . (4.221)
Here ns denotes the superconducting charge carrier density which describes the number of su-perconducting charge carriers per volume (and is very different from the superfluid densityρs), eis the elementary charge andm˚ is the effective mass of the charge carrier renormalized due to interactions. Another way to think about the superfluid densityρs is the London penetration depthλL which is basically the inverse of the superconducting plasma frequency, such that fre-quencies larger thanωPscorrespond to length scales smaller thanλL,i.e.ρs ”λ´L2. The critical temperature Tc is determined by the onset of superconductivity. The conductivityσDC and the superconducting plasma frequencyωPsare for instance obtained from reflectance measurements by extrapolation to theωÑ0limit of the complex optical conductivityσpωq,
σDC“ lim
ωÑ0Reσpωq, ωPs2 “ lim
ωÑ0
`´ω2Reǫpωq˘
, (4.222)
since the high frequency limit of the real part of the dielectric functionǫpωqis given by Reǫpωq “ǫ8´ω2Ps
ω2. (4.223)
whereǫ8is set by the screening due to interband transitions.
Alternatively, the superconducting plasma frequency may be obtained from the optical
conduc-19The definition of the plasma frequency and its relations to the dielectric function and superconductivity is discussed in detail in Section4.3.5.
Figure 4.6.
Schematic plots of the optical conductivity above the critical temperature Tc and near the absolute zeroT “0 for a dirty BCS su-perconductor. In the superconducting phase a gap develops for frequencies ω ă 2∆.
Note that in the dirty limit the quasi-particle scattering rate 1{τ is larger than 2∆. The missing area (red shaded region)i.e. the dif-ference between the area under the curve of Reσpω, T « Tcq and Reσpω, T ! Tcq which condenses into theδ-peak atω “ 0 and thus the superfluid strength (being the coefficient of the δpωq-function) is propor-tional to the missing area. Following the definitions given in (4.225) we see that the area under the Reσpω, T ! Tcq curve de-termines Ns whereas the red shaded area yieldsNn´Ns.
ω
Reσpωq
„δpωq
2∆ 1{τ
0
Reσpω, T «Tcq Reσpω, T !Tcq
tivity measured above and below the critical temperature with the help of the oscillator strength sum rule
ω2P 8 “
ż8
0
dωReσpωq, (4.224)
for the optical conductivity. This gives rise to an alternative definition of the superfluid density as compared to (4.222). We define the spectral weight in the normal and superconducting phase as follows,
Nn“ ż8
0
dωReσpωq ˇˇ ˇˇ
TąTc
“ω2Pn
8 , Ns“
ż8
0`
dωReσpωq ˇˇ ˇˇ
TăTc
. (4.225)
The superfluid densityρs describes the degrees of freedom in the superconducting phase which have condensed into a Diracδ-peak at zero frequency, whereρs can be viewed as the coefficient ofδpωq. Thisδ-peak in the real part of the conductivity gives rise to an infinite DC conductivity or zero resistivity. The superfluid density is equal to the difference between the integral over the optical conductivity (4.224) evaluated forT ăTcandT ąTcand generally yields identical values as compared to (4.222). Using the definitions of the spectral weight (4.225) we find
ρs“8pNn´Nsq. (4.226)
This is theFerrell-Glover-Tinkham sum rule. Note that in the definition ofNs we have excluded theδ-peak atω“0because the oscillator strength sum rule (4.224) requires that the area under the optical conductivity curve is identical above and belowTc,i.e. in the superconducting and normal state. Thus (4.226) determines the missing area of the spectral weight that condensed into theδ-peak atω“0, as illustrated in Figure4.6. Although the gap describes the creation of Cooper pairs, it is really the missing area which gives rise to superconductivity, since according to (4.226) the missing area is equal to the degrees of freedom which condense at zero frequency, thus forming a new coherent macroscopic ground state with off-diagonal long range order. Semi-conductors for instance are systems exhibiting an energy gap in their spectrum as well, but are
σDCTc
ω2 P
105 106
107 108
Splines Linear Fit Fit in [43]
σDCTc
ω2 P
105 106 107
107 108 109
Splines Linear Fit Fit in [43]
Figure 4.7. Plots of the complete data in Table 1 in [43]. The error bars are calculated by
∆pω2Pq “ 2ωP∆ωP and∆pσDCTcq “ Tc∆σDC. The left plot shows data from high Tcsuperconductors and for Ba1-xKxBiO3, while the one on the right includes three data points from elemental superconductors. As shown on the right the elemental conventional superconductors, two data points for Nb and one coming from the Pb superconductor, actually give Homes’ law as stated in [43]. If these three data points are ignored, as shown in the left panel, the linear fit is shifted and the data points are below the Homes’ law line.
not necessarily superconducting since there is no missing area and hence no new ground state, let alone a phase transition. On the other hand, superconductors retain their properties even if the energy gap is removed (e.g. by magnetic impurities) due to the missing area under the Reσpωqcurve. As a caveat, let us note that high temperature superconductors may not satisfy the Ferrell-Glover-Tinkham sum rule, while it is expected to hold for dirty BCS superconduc-tors.20
In order to see clearly the relation between superfluid density and the product of the conductiv-ity at the critical temperature and the critical temperature expressed by Homes’ law in (4.220), we have reproduced Table I from [43] with and without the elemental superconductors niobium Nb and lead Pb in Figure4.7. According to [43] the constant isC “35.2. Despite the original claim by Homes et al., the relation (4.220) seems not to be entirely independent of the doping.
Some interesting deviations from the Homes’ scaling line has been discussed in [234] along with possible explanations of the origin of this relation. It would be interesting to test/check if Homes’
law strictly holds for optimally doped superconductors. Furthermore, in [40,42,43,234] some possible explanations are given concerning the origin of Homes’ law: Conventional dirty-limit superconductors, marginal Fermi-liquid behavior, for cuprates a Josephson coupling along the c-axis or unitary-limit impurity scattering. Moreover, the authors discuss limits where the relation breaks down, which is true in the overdoped region of cuprates. For dirty limit BCS superconduc-tors, Homes’ law can be explained by the very broad Drude-peak21which is condensing into the
20Experimentally, it is not possible to “integrate” up toωÑ 8since a measurement cannot be done at arbitrary high frequencies, so in reality we need to introduce a cut-off frequencyωc. For high temperature superconductors this cut-off frequency may be higher than the experimentally accessible frequencies and thus these superconductors may not satisfy the Ferrell-Glover-Tinkham sum rule,c.f. [43].
21The Drude peak is located at zero frequency where the real part ofσpωqreaches its global maximum, see for example Figure4.6(and for more details on the Drude model see Section4.3.3).
superconductingδ-peak atω“0. The spectral weight of the condensate may then be estimated by an approximate rectangle of areaρs « σDC¨2∆ in an optical conductivity plot, similar to Figure4.6, where the gap in the energy of the superconducting state is denoted by2∆andσDC
is the maximum of the curve atω“0. According to the BCS model, the energy gap in the super-conducting phase is proportional to the critical temperatureTc and thusρs„σDCTc. For highTc temperature superconductors the most striking argument can be found in [40] which links the universal behavior to the “Planckian dissipation” giving rise to a perfect fluid description of the
“strange metal phase” with possible universal behavior, comparable to the viscosity of the quark-gluon plasma. The argument, reproduced here for completeness, relies on the fact that the right structure ofρs,σDC andTc may be worked out by dimensional analysis: First, as already stated above (4.221) the superfluid density must be proportional to the density of the charge carriers in the superconducting state. The natural dimension for this quantity isptimeq´2so the product ofσDC and Tc should have the same physical dimension. Second, the normal state possesses two relevant time scales, the normal state plasma frequencyωPn and the relaxation time scale τ, which describes the dissipation of internal energy into entropy by inelastic scattering. One of the simplest combinations is the product of the two time scales which will yield the dimension ptimeq´1. Therefore, we may take the optical conductivity to be of the Drude-Sommerfeld form given by
σDC“ω2Pnτ
4π “ nne2τ
m˚ . (4.227)
The last and most crucial step is to convert the critical temperature into the dimensionptimeq´1. Energy and time are related by Heisenberg’s uncertainty principle and thus quantum physics and the idea of “Planckian dissipation” will enter,
τ¯hpTcq´1“ kBTc
¯
h . (4.228)
This time scale is the lowest possible dissipation time for a given temperature. For smaller time scales the system will only allow for quantum mechanical dissipationless motion. Interestingly, at finite temperature the lower bound can only be reached if the system is in a quantum critical state [6]. This implies that highTcsuperconductors exhibit a quantum critical region above the superconducting dome, which is supported by experimental evidence [7,235]. The “Planckian dissipation” time scale can be compared to the ratio of shear viscosity over entropy density of a perfect quantum fluid. The shear viscosity is the constant connecting the shear force applied to a fluid layer to the resulting velocity gradient and hence its units are given by the units of
force Area velocity
length
“pressureˆtime“energy densityˆtime. (4.229) Therefore, we see that the viscosity is proportional to the relaxation timeτ, whereεdescribes the energy density
η„ετη, (4.230)
so the “Planckian dissipation” describes a perfect fluid in the sense of the viscosity of the quark-gluon-plasma and black branes22. It seems that the quantum critical region, that might be above the superconducting dome, is described by the relaxation time being the “Planckian dissipation”
time (4.228), so the “strange metal” phase is an almost perfect fluid with possible universal
22Note that in [40] there is a confusing statement about the viscosity of a Planckian dissipative process stating that it is maximally viscous. According to [8] the author is aware of this lapse.
behavior. This can be compared to the universal behavior in the quark-gluon plasma [220]
where one finds
η s “ 1
4π
¯ h kB “ 1
4π. (4.231)
Furthermore, to connect the expression on the right-hand-side of (4.220) with the left-hand-side, we can employ Tanner’s law [236] which states that
ns« 1
4nn, (4.232)
relating the superfluid density to the normal state plasma frequency, as can be seen from (4.221) and (4.227). This “explanation” of Homes’ law will guide us in order to find a holographic realization. We will expand on this idea in the following section.