GGeneral Relativity and the Cuprates
Gary T. Horowitz and Jorge E. SantosDepartment of Physics, UCSB, Santa Barbara, CA 93106, USA [email protected], [email protected]
Abstract
We add a periodic potential to the simplest gravitational model of a superconductor andcompute the optical conductivity. In addition to a superfluid component, we find a normalcomponent that has Drude behavior at low frequency followed by a power law fall-off. Boththe exponent and coefficient of the power law are temperature independent and agree withearlier results computed above T c . These results are in striking agreement with measurementson some cuprates. We also find a gap ∆ = 4 . T c , a rapidly decreasing scattering rate, and“missing spectral weight” at low frequency, all of which also agree with experiments. a r X i v : . [ h e p - t h ] M a r ontents Gauge/gravity duality has provided a powerful new tool to analyze certain condensed matter sys-tems. In particular, it can be used to calculate transport properties of strongly correlated systemsat finite temperature. Remarkably, this is achieved by mapping the problem to a gravitationalproblem in one higher dimension. Since the system of interest lives in one dimension less than thegravitational problem being solved, this approach is often called holographic.The optical conductivity in a simple holographic model of a 2 + 1 dimensional conductor wasrecently studied including the effects of a lattice [1]. Earlier studies assumed translation invariancewhich implied momentum conservation. In that case, the charged particles cannot dissipate theirmomentum so the real part of the optical conductivity always contains a delta function at zerofrequency reflecting infinite DC conductivity. With the lattice included, the delta function isresolved. It was found that at low frequency the conductivity follows the simple Drude form, butat intermediate frequency, it follows a power law | σ ( ω ) | = Bω / + C (1.1)The exponent − / B is temperature independent, although the cuprates do not appear to have the off-set C . The origin of (1.1) from the gravitational side is still not understood. It has been shown that asimilar power-law is seen in the thermoelectric conductivity and for a 3 + 1 dimensional conductor2although the exponents differ) [8]. In a recent paper, Vegh [9] studied the optical conductivityin a holographic model in which momentum conservation was broken by a graviton mass term.He finds a similar power-law scaling of the conductivity (including the constant off-set), but theexponent is not fixed. It depends on the mass of the graviton.The goal of this paper is to extend these results to the superconducting regime. We add alattice in the form of a periodic potential to the simplest (original) holographic superconductor[10, 11] and compute the optical conductivity. We find that below T c , in addition to the superfluid,there is a normal fluid component with the following properties • It has simple Drude behavior at low frequency σ ( ω ) = ρ n τ − iωτ (1.2)where the normal component density ρ n and relaxation time τ are temperature dependent(but frequency independent). • The scattering rate 1 /τ drops rapidly below T c . • At intermediate frequency, the normal component again satisfies the power law (1.1) withthe same coefficient B that was seen above T c . • There is evidence for a superconducting gap of size ∆ = 4 . T c . • Despite this, in the limit of low temperature, ρ n does not vanish, indicating the presence ofuncondensed spectral weight. • The Ferrel-Grover-Tinkham sum rule relating the superfluid density to the decrease inRe[ σ ( ω )] is satisfied, but only if large frequencies of order the chemical potential are included.If one considers only the Drude peak and power law region, there is “missing spectral weight”.As we will discuss, these are all observed properties of the bismuth-based cuprates. In particu-lar, experiments have shown that the power-law is unchanged when the temperature drops belowthe critical temperature and the material becomes superconducting [12]. We will work with the simplest holographic superconductor which requires gravity coupled to aMaxwell field and charged scalar Φ. The action is S = 116 πG N (cid:90) d x √− g (cid:20) R + 6 L − F ab F ab − | ( ∇ − i e A )Φ | + 4 | Φ | L (cid:21) , (2.1)where L is the AdS length scale and F = dA . The scalar mass, m = − /L , is chosen since forthis choice, the asymptotic behavior of Φ is simple. From here on we work in units in which L = 1.If the metric asymptotically takes the usual form ds = − dt + dx + dy + dz z (2.2)3hen Φ = zφ + z φ + O ( z ) . (2.3)In the standard interpretation, Φ is dual to a dimension two charged operator O in the dual theorywith source φ and expectation value φ . Since we want this condensate to turn on without beingsourced, we set φ = 0.As Gubser first suggested [13], the above action has the property that (electrically) chargedblack holes become unstable at low temperatures to developing scalar hair. The reason is essentiallythat the effective mass of the scalar gets a contribution e A t g tt < A t must vanish at the horizon and is asymptotically A t = µ − ρz + O ( z ) (2.4)where µ is the chemical potential and ρ ( x ) is the charge density. For these electrically chargedsolutions, the field equations and boundary conditions require the phase of Φ to be constant. Wewill set it to zero and treat Φ as a real field. We introduce the lattice by requiring that the chemicalpotential be a periodic function of x : µ ( x ) = ¯ µ [1 + A cos( k x )] . (2.5)The lattice is only introduced in one direction for computational convenience, and we will computethe conductivity only in the direction of the lattice. This “ionic” lattice has been discussed earlier(see, e.g., [14, 15, 16, 17, 18]) but almost always treated perturbatively. In [8] it was treated exactlyin the theory (2.1) without the charged scalar field.Our model has several parameters. In principle one could vary the mass and charge of the scalarfield Φ in the action. However, we have already fixed the mass for convenience. The boundarycondition (2.5) depends on three parameters: the mean chemical potential ¯ µ , the lattice wavenum-ber k and the lattice amplitude A . In addition, the solution will depend on a temperature T .Due to a scaling symmetry, physics depends on only three dimensionless quantities which can betaken to be A , k / ¯ µ , and T / ¯ µ .For definiteness, we will choose k / ¯ µ = 2 for the results presented later in this paper. Althoughthe physics is scale invariant, when doing calculations, we will also set ¯ µ = 1. Our choice of k is very close to the one which yields a low temperature DC resistivity which is linear in T in theabsence of superconductivity [14, 1]. Even though k corresponds to an unrealistically large latticespacing, the nonlinearity of Einstein’s equation generates structure on much smaller scales.4 Backgrounds
To numerically construct the gravitational dual of the normal phase we employ the ansatz used in[1, 8]: ds = 1 z (cid:20) − H G ( z ) (1 − z ) dt + H dz (1 − z ) G ( z ) + S ( dx + F dz ) + S dy (cid:21) , A = ψ dt, Φ = 0 , (3.1)and G ( z ) = 1 + z + z − µ z /
2. Here G ( z ) controls the black hole temperature given by T ≡ G (1) / π = (6 − µ ) / π and H , , S , , F and ψ are six functions of x and z , that we determineusing the numerical methods described in [1], which were first introduced in [19] and studied ingreat detail in [20].When A = 0 the problem reduces to the original holographic superconductor [11]. It wasshown there that there is a critical value of T /µ such that above this value, the scalar fieldvanishes and the solution is the simple planar Reissner-Nordstr¨om AdS metric. Below this value,the Reissner-Nordstr¨om solution becomes unstable to developing scalar hair.We now increase A and solve the Einstein-Maxwell equations to find rippled versions of theReissner-Nordstr¨om AdS metric. This is exactly the same as what was done for the ionic latticein [8]. To see when these solutions become unstable to forming scalar hair, we look for a staticnormalizable mode of the scalar field. It is intuitively clear that if one increases the charge e onthe scalar field, it becomes easier for this field to condense and the critical temperature becomeshigher. To find the critical temperature for a given charge it is convenient to turn the problemaround. For each A and any temperature T , we find the value of the charge e such that T isthe critical temperature for a field with charge e . At the onset of the instability, Φ is a staticnormalizable mode of the charged scalar field and can be treated perturbatively:( ∇ a − i e A a )( ∇ a − i e A a )Φ + 2 Φ = 0 , (3.2)with Φ being a function of x and z only. Also, the connections ∇ and A are evaluated on therippled Reissner-Nordstr¨om AdS black holes. Finally, one can recast the former equation to takethe following appealing form − ∇ a ∇ a Φ − e ( − A a A a ) Φ , (3.3)which one recognizes as a positive selfadjoint eigenvalue problem for e . At the boundary wedemand Φ to decay as in Eq. (2.3) and at the horizon we demand regularity.The results are shown in Fig. 1. There are seven curves on this plot denoting seven differentvalues of the lattice amplitude A between 0 and 2 .
4. Each curve shows the expected rise in criticalcharge with temperature (or critical temperature with charge). Comparing the different curves fora given charge, we see that increasing the lattice amplitude also increases the critical temperature.This was first noticed in [21] when the lattice was treated perturbatively. It can be understood asfollows. As we said, the critical temperature of a uniform holographic superconductor is propor-tional to µ . By making µ vary periodically, one raises the maximum value of µ which induces the5 .05 0.10 0.15 0.201.01.52.02.53.03.54.0 T ê m e Figure 1: For each
T /µ we plot the charge of the scalar field for which T would be the criticaltemperature. This is repeated for several values of the amplitude of the lattice. From the topdown the lines represent: A = 0 . , . , . , . , . , . , .
4. Setting e = 2 we read off the criticaltemperatures used in this paper.scalar field to condense at a higher temperature. It then leaks into the regions where µ is smaller.This effect was seen in models of a holographic Josephson junction [22]. We will set e = 2 sincethis is a natural value for a superconducting condensate.Having found the critical temperature, one can next solve the coupled Einstein-Maxwell-scalarequations to find the solutions for T < T c . In order to find the rippled superconducting phase wehave to change the ansatz (3.1). The reason being that Eq. (3.1) is adapted to probe geometries forwhich the entropy is nonzero as T →
0. The homogeneous holographic superconductors are knownnot to have this feature. In particular, their entropy decreases towards zero as the temperatureis lowered, thus obeying to the third law of thermodynamics. One ansatz that is adapted to thisproperty is given by ds = 1 z (cid:26) − H y (1 − z ) dt + H dz (1 − z ) + y (cid:2) S ( dx + F dz ) + S dy (cid:3)(cid:27) , A = ψ dt (3.4)where H , , S , , F , ψ and Φ are seven functions of x and z to be determined using the numericalmethods introduced in [1]. Here, y + parametrizes the black hole temperature as T = y + / π and,in the homogenous case, is the black hole horizon size measured in units of the AdS radius.The result are hairy, rippled, charged black holes. From the asymptotic form of the scalar fieldone reads off the expectation value of the charged condensate in the dual theory. The condensate By adapted here we mean that the T → S , . .02 0.04 0.06 0.08 0.10 0.120.00.20.40.60.81.01.21.4 T ê m H e X O \ L ê ê m Figure 2: The mean value of the condensate as a function of temperature for e = 2 and variousvalues of the lattice amplitude. From the inner to outer curves: A = 0 . , . , . , . , . , . , . x , oscillating about a mean value. In Fig. 2 we plot this mean value as a functionof temperature for various values of the lattice amplitude. In each case, the condensate follows afamiliar form, rising rapidly as T drops below T c and then saturating at low temperature. It isclear from Fig. 2 that increasing the lattice amplitude increases the low temperature mean valueof the condensate. In Fig. 3 we show the variation in the condensate by plotting (cid:104) O ( x ) (cid:105) for A = 2and various temperatures. The variation is roughly 50% of the mean and the condensate remainspositive always.The transition at T = T c which turns on the scalar condensate corresponds to a continuousphase transition. One can show this without explicitly computing the free energy as follows.Consider first the homogeneous case with constant chemical potential µ , and compactify x and y .The free energy in the grand canonical ensemble is F = E − T S − µQ , so its variation is δF = δE − T δS − µδQ − SδT − Qδµ (3.5)But the first law says that δE = T δS + µδQ , so at fixed µ , dFdT = − S (3.6)Since the branch of solutions with scalar hair joins the branch of solutions without hair at T = T c ,their entropies must agree, and hence dF/dT is continuous.This argument generalizes to the case of a position dependent chemical potential. The µQ termin the free energy is replaced by (cid:82) µ ( x ) ρ ( x ) dxdy and the µδQ term in the first law is replaced by7 p p p p k H e X O \ L ê ê m Figure 3: The condensate as a function of x for e = 2, A = 2, and T /T c = . , . , . , .
99 from topdown. Note that by
T /T c = . (cid:82) µ ( x ) δρ ( x ) dxdy (see Appendix). Since we are keeping µ ( x ) fixed as we change the temperature,we recover (3.6). The two branches of solutions with ripples still join at T = T c , so the phasetransition is again continuous.Since we have only introduced the lattice in one direction, one could think of our solutions asrepresenting a striped superconductor. However we will not pursue that interpretation here. In the homogeneous case, the zero temperature limit of these hairy black holes is known totake the form [24] ds = r ( − dt + dx i dx i ) + dr g r ( − log r ) (3.7)and Φ = 2( − log r ) / near r = 0. This metric has a null singularity at r = 0. The scalar field onthe horizon of our solutions is becoming more homogenous as T →
0, and at low temperatures,the entropy scales like S ∝ T . independent of the lattice amplitude. However, the coefficient inthe entropy formula does depend on the lattice amplitude, so it is not clear if (3.7) applies to the T = 0 limit of our rippled solutions.The solutions we have discussed so far are not the only static, rippled, hairy black hole solutions.There are radial excitations where the scalar field has nodes in the radial direction. There are alsoexcitations in the direction of the lattice which can be constructed as follows. Given a solutionΦ with period λ = 2 π/k , − Φ is clearly also a solution. One can construct a solution in whichΦ changes sign from one region of size λ to the next, by simply starting with a seed solution in It is not a gravitational dual of the novel striped superconductor introduced in [23] since that required acondensate whose average value was zero. n + m ) λ in which Φ changes sign on m of the regions. This would clearly result in a smallermean value for the condensate. We expect that these other solutions all have higher free energythan the ones we study, in which Φ remains positive everywhere. To compute the optical conductivity in the direction of our lattice we introduce a perturbationwith harmonic time dependence and fix the usual boundary condition on δA x : δA x → Eiω + J x ( x, ω ) z + O ( z ) . (4.1)This corresponds to adding a homogeneous electric field E x = Ee − iωt on the boundary. Thisperturbation induces perturbations in most metric components as well as other components of thevector potential and scalar. Solving these linear equations with suitable boundary conditions allowsus to read off the current J x ( x, ω ) which determines the conductivity via (cid:98) σ ( ω, x ) = J x ( x, ω ) /E .In contrast with the ionic lattice of [8], there are now a total of twelve linear functions to bedetermined. In addition to the perturbations present in the ionic case, we also have two moreperturbations corresponding to the real and imaginary parts of the charged scalar field. We couldhave worked with U (1) gauge invariant variables, such as in [22], however here we follow a standardgauge fixing procedure. For the metric and gauge field perturbations we use the de Donder andLorentz gauges, respectively. Since we impose a homogeneous electric field, we are interested inthe homogeneous part of the conductivity, σ ( ω ), that can be retrieved from the full conductivityvia Fourier mode decomposition: (cid:98) σ ( ω, x ) = σ ( ω ) + + ∞ (cid:88) n = −∞ n (cid:54) =0 σ n ( ω ) e i n k x . (4.2)In this section, we will fix the lattice amplitude to be A = 2, which corresponds to a criticaltemperature T c = . µ .The real and imaginary parts of the conductivity for T /T c = .
71 are shown in Fig. 4. The firstthing to note is that there is a pole in the imaginary part of the conductivity. This implies (fromthe Kramers-Kronig relation) that there is a delta function at ω = 0 in the real part confirmingthat this is a superconductor. So, unlike the normal phase, the holographic lattice does not removethe delta function in the superconducting regime . The coefficient of the pole is the superfluiddensity. The bump in the real part of σ at ω/T ≈
20 is a resonance which is analogous to theone studied in [8] coming from a quasinormal mode of the black hole. We will return to this later,but for now, the most interesting feature of Re( σ ) is the rise at low frequency. This shows thatthere is a normal component to the conductivity even in the superconducting regime. Thus, ourholographic superconductor resembles a two fluid model with a normal component as well as asuperfluid component. This was noticed earlier in a perturbative treatment of the lattice [18, 25].
10 20 30 40 50 600.00.51.01.52.0 w ê T R e H s L w ê T I m H s L Figure 4: The real and imaginary parts of the conductivity for A = 2 , k = 2, and T /T c = . Let us now focus on this low frequency region. The real and imaginary parts of the conductivityfor various temperatures are shown in Fig. 5. There are four curves ranging from
T /T c = 1 downto T /T c = .
7. It should be emphasized that the curve with
T /T c = 1 was done with a completelydifferent numerical code using a different metric ansatz and solving for one fewer unknown function(no scalar). The continuity of the results is a good check on the numerical accuracy. It is clearfrom Fig. 5 that as we lower the temperature, the normal component is decreasing, and the polein the imaginary part is increasing, showing an increase in the superconducting component.A closer examination of the normal component of the conductivity when T < T c shows that itbehaves very much like the conductivity above the critical temperature. In fact, at low frequencyboth the real and imaginary parts of the conductivity are very well fit by simply adding a pole tothe Drude formula: σ ( ω ) = i ρ s ω + ρ n τ − iωτ (4.3)where ρ s is the superfluid density, ρ n is the normal fluid density, and τ is the relaxation time.The three parameters ρ s , ρ n and τ are temperature dependent but frequency independent. On theright hand side of Fig. 5 one can clearly see the Drude behavior in the bottom curve which is stillin the normal phase. As the temperature is lowered, the pole grows rapidly and soon swamps theDrude behavior of the normal component.The temperature dependence of ρ s and ρ n are shown in Fig. 6. On the left one sees that ρ s rises rapidly as T drops below T c , and on the right, one sees that ρ n drops rapidly. The red dashedline is a fit to ρ n ( T ) = a + b e − ∆ /T with ∆ = 4 . T c . (4.4)This is similar to a BCS superconductor in which the normal component consists of thermally10 ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏ Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï Ï ÏÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚ w ê m R e H s L ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ Ê‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ ‡ÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏ Ï ÏÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚ w ê m I m H s L Figure 5: The low frequency part of the conductivity for
T /T c = 1 . σ ) denotes the zerofrequency delta function. ê T c r s ê m ê T c r n ê m Figure 6: The density of the superfluid and normal components as a function of temperature,extracted from the fit to (4.3). The dashed red line on the right is a fit to (4.4).11xcited quasiparticles with a gap ∆, but in BCS theory the gap is smaller, ∆ ≈ . T c . Observationson the cuprates indeed show a gap of order ∆ = 4 . T c [26], showing that they are not weaklycoupled superconductors. In the early work on holographic superconductors, an attempt wasmade to measure this gap. Since the system was homogeneous, there was no Drude peak so peoplelooked at lim ω → Re( σ ). This was found to behave like e − ∆ σ /T over a range of low temperatures.In the probe limit (where the spacetime metric is fixed), one found a value close to what we findhere, ∆ σ = 4 . T c [10, 27], but when backreaction was included, ∆ σ became smaller and dependedon the charge of the scalar field [11]. With the lattice, we have a much better way to measurethe superconducting gap and its intriguing that we get a realistic value even for a charge twocondensate. Another key difference from BCS is the presence of the constant a in (4.4). If we believe thisextrapolation then ρ n remains nonzero even at zero temperature indicating uncondensed spectralweight. Observations on the cuprates indeed show uncondensed spectral weight at T = 0 [29].Of course the cuprates are d-wave superconductors in which the gap has nodes on the Fermisurface which probably contribute to this effect. It is surprising to see ρ n (cid:54) = 0 at T = 0 in ans-wave superconductor, as we have here. In quasiparticle language, it suggests that the normalcomponent consists of two constituents, one of which is gapped and the other remains gapless. Ofcourse quasiparticles may not be the right language for the strongly correlated dual system. (Aprecursor to this was seen in the homogeneous case, where it was found that lim ω → Re( σ ) wasexponentially small but not zero at T = 0 [24].)To check the extrapolation of ρ n , one would like to compute ρ n at lower temperatures. This isvery difficult because the relaxation time τ rises rapidly as the temperature is lowered (see Fig. 7).To see the Drude peak, one needs to probe frequencies ωτ ∼
1. Since τ is rising so rapidly, theDrude peak is squeezed to very small frequencies which become difficult to resolve. The dashedline in Fig. 7 is a fit to τ = τ e ˜∆ /T with ˜∆ = 4 . T c . (4.5)It follows that the scattering rate, 1 /τ , drops rapidly as T is reduced below T c . This is anotherobserved feature of the cuprates [30]. One often distinguishes clean and dirty limits of supercon-ductors by comparing the scattering rate to the gap. It is easy to check that our holographicsuperconductor is in the clean limit: 1 /τ (cid:28)
2∆ for all
T < T c .It is natural to ask if ρ n is related to the charge inside the black hole. This appears not to bethe case. As we lower the temperature, the charge inside the horizon is reduced, and more of thecharge is carried by the scalar field outside the horizon. However, in the limit T →
0, the chargeinside goes to zero as the horizon area vanishes, in contrast to (4.4). Curiously, in the probe limit at low temperature, Re( σ ) is strongly suppressed for ω < ω g ≈ T c , even if onechanges ∆ σ by changing the mass of the scalar field and spacetime dimension [27]. In a very different holographic realization of a two fluid model, it was found that ρ n remained nonzero at T = 0when the charge on the scalar field was small, but vanished when the scalar charge became larger [28]. .0 0.2 0.4 0.6 0.8 1.0020406080100120140 T ê T c t m Figure 7: The relaxation time as a function of temperature, extracted from the fit to (4.3). Thedashed curve is a fit to (4.5).
At slightly larger frequency, the absolute value of the conductivity of the normal component followsthe same power law (1.1) as was found in the normal phase above the critical temperature. Thisis shown in Fig. 8 where we have subtracted the pole in the imaginary part of σ coming fromthe superfluid component, and plotted the absolute value of the result, | ˜ σ | , minus the off-set C vsfrequency on a log-log plot. The four lines correspond to the same temperatures as in Fig. 5. Thefact that the lines are parallel implies that the exponent is the same. The fact that the lines lie ontop of each other implies that the coefficient B of the power law is again temperature independent.Recall that the lowest curve represents the normal phase. This clearly shows that the power lawfall-off is completely unaffected by the superconducting phase transition.Our results are in complete agreement with observations of the bismuth-based cuprates [12].Fig. 9 shows measurements of | σ ( ω ) | (without the pole in the imaginary part) for Bi Sr CaCu O δ ,commonly called BSCCO. The figure includes eight separate log-log plots showing measurementson eight different samples ranging from underdoped with T c = 67 K to the optimally doped with T c = 96 K to overdoped with T c = 60. Each plot shows a variety of temperatures both above andbelow T c . All show power-law behavior at intermediate frequencies which does not change as thetemperature is reduced below T c . The measured exponent is − / − / We thank van der Marel for bringing this paper to our attention. Ê Ê Ê Ê Ê Ê Ê ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊ Ê Ê Ê Ê ÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊÊʇ‡ ‡ ‡ ‡ ‡ ‡ ‡‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡‡ÏÏ Ï Ï ÏÏ Ï Ï Ï ÏÏ Ï Ï ÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÏÚÚ ÚÚ ÚÚ ÚÚ ÚÚ ÚÚ ÚÚ ÚÚ ÚÚ ÚÚ ÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚ w ê m » s é » - C Figure 8: The normal component of the conductivity on a log-log scale showing power law behaviorwith exponent − /
3. The pole in the imaginary part has been subtracted out. The lines correspondto
T /T c = 1 . C . Another difference is that the power lawin BSCCO extends up to a temperature independent frequency (the lower cut-off is temperaturedependent). In our calculations the power law always extends roughly between 2 < ωτ < T > T c [1]. Since τ is temperaturedependent, both the upper and lower cut-offs are temperature dependent.The phase of the complex conductivity computed from the gravitational dual is roughly constantover the range of frequencies where the magnitude follows the power law. However, this phase istemperature dependent and varies between 60 o and 80 o . The data on BSCCO shows a temperatureindependent phase of 60 o . This difference is likely connected with the constant off-set in ourpower law. Without the offset, scale invariance, causality and time reversal symmetry require σ ( ω ) ∝ ( − iω ) α [2], so the phase is related to the exponent of the power law. The Ferrell-Glover-Tinkham (FGT) sum rule states that the reduction in the spectral weight when
T < T c is taken up by the superfluid density ρ s : (cid:90) ∞ + dω Re[ σ N ( ω ) − σ S ( ω )] = π ρ s (4.6)where σ N is the conductivity in the normal phase and σ S is the conductivity in the superconductingphase. We now ask whether this is satisfied in our gravitational model . The left panel of Fig. 10shows Re[ σ ( ω )] over a wide range of frequency in both the normal phase T = T c and the supercon-ducting phase T = . T c . The right panel of Fig. 10 shows that (2 /π ) (cid:82) ω/ ¯ µ + d ˜ ω Re[ σ N (˜ ω ) − σ S (˜ ω )]indeed approaches the superfluid density at large ω . At the end of the integration, they differ byless than 0 . See [31, 32, 33] for general discussions of sum rules from gravity for translationally invariant backgrounds. T c . The power-law fall-off is clearly unaffected by T c . Plot is taken from [12].15 w ê m R e H s L w ê m p ‡ + w ê m R e H s N - s S L Figure 10:
Left panel : The conductivity in the normal phase T = T c (upper blue curve) andsuperconducting phase T = . T c (lower red curve). Right panel : The red dashed line is thesuperfluid density obtained from the pole in the imaginary part of the conductivity. The blackline is (2 /π ) (cid:82) ω/ ¯ µ + d ˜ ω Re[ σ N (˜ ω ) − σ S (˜ ω )]. This integral approaches the red line at large ω showingthat the FGT sum rule (4.6) is satisfied.FGT sum rule, but also a strong test of the numerics. The two curves on the left panel of Fig. 10were computed using different codes, and the fact that an integral of their difference gives preciselythe expected answer is confirmation of the accuracy of the numerical results.Note that the two curves on the left panel of Fig. 10 differ over a range of frequencies of order ¯ µ .In conventional superconductors, Re[ σ ( ω )] is reduced only over a much smaller range of frequency,so it is common to cut off the integral in (4.6) at a convenient low frequency ω . However, whenthis is done in the cuprates, one finds that the integral underestimates the superfluid density [12].In other words, some of the spectral weight represented by ρ s is missing from the low frequencypart of Re[ σ ( ω )]. One must indeed include frequencies of order the chemical potential to recoverthe sum rule, just as we have seen here [34]. The resonance in the conductivity at ω/ ¯ µ ≈ . σ ( ω ) = G R ( ω ) iω = 1 iω a + b ( ω − ω ) ω − ω . (4.7)For T = T c , one finds the resonance is very well fit by this expression with ω / ¯ µ = 1 . − . i .This quasinormal mode changes very little when the temperature drops to . T c .16 Discussion
We have seen that a simple holographic model of a superconductor reproduces quantitative featuresof BSCCO including an intermediate frequency power law with exponent − / ≈ T c . In addition, itreproduces many qualitative features of the cuprates including a rapidly decreasing scattering ratebelow T c , uncondensed spectral weight at T = 0, and a superconductivity induced transfer ofspectral weight involving energies of order the chemical potential.We find it remarkable that so much of the phenomenology of the cuprates can be reproduced bysuch a simple gravity model. Since gravity coupled to a Maxwell field and charged scalar are partof the low energy limit of string theory with many different compactifications, one can perhapsview it as the universal part of the theory. In this respect, its dual description might be calledthe “standard model” of strongly correlated systems. One could start with this and then addother fields to reproduce observed fine scale structure of particular materials. Perhaps the maindrawback of such an approach is that our current model describes an s -wave superconductor. Westill do not have a satisfactory description of a d -wave holographic superconductor. See [36] for adescription of some of the difficulties.We have presented results for the optical conductivity in the superconducting region for onlyone value of the lattice amplitude A = 2 and one value of the wavenumber k = 2 (with ¯ µ = 1).We believe that the − / T c it has been shown to be veryrobust. We have done preliminary calculations with k = 1 which indicate that the behavior ofthe normal component density ρ n ( T ) is also independent of these choices. In particular, although T c increases, the gap remains ∆ = 4 T c and ρ n approaches the same nonzero value as T →
0. Thebehavior of the relaxation time τ is qualitatively the same, but there is an important difference.Although one again finds τ = τ e ˜∆ /T , the value of ˜∆ is half what it was for k = 2. In other words,it appears that 1 /τ ∝ e − ak /T . This is the behavior predicted in [14] where the scattering rate wasrelated to the density-density correlation function at ω = 0 and k = k . Since Poincar´e invarianceappears to be restored in the zero temperature infrared geometry (3.7), low energy physical statesshould have ω ∼ k so states with ω = 0 and k = k would be exponentially suppressed.The fact that we still see -2/3 power law even in the superconducting phase, makes it clearthat this power law has nothing to do with the AdS × R near horizon geometry of the zerotemperature Reissner-Nordstr¨om solution. As we have discussed, with the charged scalar present,the zero temperature solution has a singular horizon.Homes has discovered a remarkable relation between the low temperature superfluid density ρ s , the critical temperature T c and the DC conductivity just above the critical temperature σ dc [37]. He found that for a wide range of high temperature superconductors ρ s ≈ σ dc T c . (5.1)This relation appears to be universal. It holds for the conductivity perpendicular to the CuO planes in the cuprates as well as the conductivity in the plane, and also holds for the iron pnictides.17nfortunately, it is easy to see that Homes’ law cannot hold for the class of holographic supercon-ductors we have discussed here where the critical temperature depends on the lattice amplitude. The reason is that as A → σ dc diverges since the system becomes translationally invariant.However, both ρ s and T c approach finite limits given by the original homogeneous holographicsuperconductor. So (5.1) cannot hold. To reproduce (5.1) one would have to fix a large A andvary T c by other means, perhaps by adding a double trace perturbation [39]. Acknowledgements
It is a pleasure to thank D. Tong for collaboration at an early stage of this project. We alsothank N. Iqbal and the participants of the Simons Symposium on Quantum Entanglement: FromQuantum Matter to String Theory, especially A. Millis, S. Sachdev, and J. Zaanen for discussions.This work was supported in part by the National Science Foundation under Grant No. PHY12-05500
A The first law for black holes with varying chemical po-tential
In this appendix, we derive the first law for electrically charged black hole solutions to (2.1)with nonconstant chemical potential, µ ( x, y ). For convenience we will imagine that the x and y directions are periodically identified so the black holes have finite horizon area and mass. Wefollow the approach of Sudarsky and Wald [40]. As usual for a diffeomorphism invariant theory,the Hamiltonian is a linear combination of the constraints plus surface terms: H = (cid:90) Σ [ N a C a + N a A a ( D i E i )] + surface terms (A.1)where N a is the lapse-shift vector and C a are the usual Hamiltonian and momentum constraintsof general relativity. The surface terms are determined by the requirement that the variations of H with respect to the canonical variables q ij , p ij , A i , E i are well defined. In addition to the usualgravitational surface terms, there is an additional term coming from the Gauss law constraint: (cid:73) ∂ Σ ( N a A a ) E i dS i (A.2)Given a static, electrically charged black hole, we choose Σ to be a constant t surface whichstarts at the bifurcation surface on the horizon and ends at infinity. We also choose N a = ( ∂/∂t ) a .Now under a perturbation of the black hole δH = (cid:90) δq ij ∂H/∂q ij + · · · = 0 (A.3) An earlier unsuccessful attempt to find a holographic realization of Homes law was made in [38]. δH reduces to a sum of the variation of the surface terms. The gravi-tational surface terms yield the usual δM − ( κ/ π ) δA . Since ∂/∂t = 0 at the bifurcation sur-face, the electromagnetic surface term (A.2) only has a contribution from infinity which is simply (cid:82) µ ( x, y ) ρ ( x, y ) dxdy . Since µ ( x, y ) is fixed by our boundary conditions, the variation of this surfaceterm is (cid:82) µ ( x, y ) δρ ( x, y ) dxdy . We thus obtain the first law: δM = κ π δA + (cid:90) µ ( x, y ) δρ ( x, y ) dxdy (A.4)In the homogeneous case with constant µ , this clearly reduces to the familiar form δM = κ π δA + µδQ . (A.5) References [1] G. T. Horowitz, J. E. Santos, and D. Tong,
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