Homes’ law in holographic superconductor with Q-lattices

Journal of High Energy Physics, Oct 2016

Homes’ law, ρ s = Cσ DC T c , is an empirical law satisfied by various superconductors with a material independent universal constant C, where ρ s is the superfluid density at zero temperature, T c is the critical temperature, and σ DC is the electric DC conductivity in the normal state close to T c . We study Homes’ law in holographic superconductor with Q-lattices and find that Homes’ law is realized for some parameter regime in insulating phase near the metal-insulator transition boundary, where momentum relaxation is strong. In computing the superfluid density, we employ two methods: one is related to the infinite DC conductivity and the other is related to the magnetic penetration depth. With finite momentum relaxation both yield the same results, while without momentum relaxation only the latter gives the superfluid density correctly because the former has a spurious contribution from the infinite DC conductivity due to translation invariance.

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Homes’ law in holographic superconductor with Q-lattices

Received: August Homes' law in holographic superconductor with Q-lattices 0 1 Open Access 0 1 c The Authors. 0 1 0 Gwangju 61005 , Korea 1 School of Physics and Chemistry, Gwangju Institute of Science and Technology Homes' law, s = C DCTc, is an empirical law satis ed by various superconductors with a material independent universal constant C, where s is the super uid density at zero temperature, Tc is the critical temperature, and in the normal state close to Tc. We study Homes' law in holographic superconductor with nd that Homes' law is realized for some parameter regime in insulating phase near the metal-insulator transition boundary, where momentum relaxation is strong. In computing the super uid density, we employ two methods: one is related to the in nite DC conductivity and the other is related to the magnetic penetration depth. With momentum relaxation both yield the same results, while without momentum relaxation only the latter gives the super uid density correctly because the former has a spurious contribution from the in nite DC conductivity due to translation invariance. Holography and condensed matter physics (AdS/CMT); Gauge-gravity cor- Q-lattices 1 Introduction 2 3 4 Holographic superconductor on a Q-lattice Metal-insulator transition without Critical temperature and DC conductivity Super uid density Holographic methods Numerical results Homes' law Conclusion and discussions A Equations of motion for super uid density Introduction Holographic methods or gauge/gravity duality have provided novel and e ective ways to analyse strongly correlated systems. In particular, there have been much e ort and some successes in understanding universal properties of strongly coupled systems. Important examples include the holographic bound of the ratio of shear viscosity to entropy density phase [1{4]. In this paper, we study another universal property observed in high-temperature supers(T = 0) = C DC(Tc) Tc ; where s is the super uid density at zero temperature, Tc is the phase transition tempera DC is the DC conductivity in the normal phase close to Tc. The point is that C is a material independent universal number. C 4:4 for ab-plane high Tc superconductors and clean BCS superconductors or C 8:1 for c-axis high Tc superconductors and BCS superconductors in the dirty limit. Here, s, Tc and DC are de ned to be dimensionless and the numerical values of C are computed in [7] based on the experimental data in [5, 6]. It was argued that Homes' law might be related to `Planckian dissipation', which is the quantum limit of dissipation with the shortest possible dissipation time may give a good chance to nd some universal physics in both condensed matter systems and quark-gluon plasma [10]. Even though the holographic models of superconductor have been extensively developed [3, 4, 11, 12] since the pioneering work by Hartnoll, Herzog, and Horowitz in 2008 [13, 14], Homes' law in this context has not been studied much. It is partly because early holographic superconductor models are translationally invariant with nite charge density.1 As a result they cannot relax momentum and yield in nite DC in (1.1) so C is not well de ned. To have a DC several methods were proposed to incorporate momentum relaxation: spatially modulated boundary conditions for bulk elds [15], massive and models with a Bianchi VII0 symmetry dual to helical lattices [19]. Based on these veloped [7, 20{27]. Among the aforementioned holographic superconductors with momentum relaxation, Homes' law has been studied only in two models [7, 27]. For both cases, there are paramthat C is constant independent of momentum relaxation parameters. In [7] a holographic superconductor model in a helical lattice was analysed and Homes' law was satis ed for scalar elds linear in spatial coordinate2 are studied and Homes' law was not satis ed. Here Therefore, it seems that Homes' law is not realized for all holographic models. Because physics behind Homes' law in [7] has not been clearly understood yet, it is important to nd the common physical mechanism for Homes' law in di erent models. For this purpose, in this paper, we study Homes' law in a holographic superconductor model with Q-lattice3 [22, 23]. We choose this model for two reasons. First, our model can be easily compared with two previous works on Homes' law: i) the model has a similar structure to the helical lattice 1See [10] for an early attempt for Homes' law in holographic superconductors without momen2The property of the normal phase and superconducting phase of this model was studied in [24, 28{31] 3The property of the normal phase of this model was studied in [17]. See [32, 33] for a Mott system tum relaxation. and in [23, 24] respectively. based on this model. model [7] in that it has two parameters (amplitude and wavelength of lattice) ii) the model is also similar to the massless scalar model [27] in certain limit. Second, it was argued in [7] state and it was reported that our model also has the metal-insulator transition [34]. We nd that Homes' law is realized also in our Q-lattice model for certain parameter regime, similarly to the helical lattice model in [7]. However, in computing the super uid density, there is an issue that the super uid density is di erent from the charge density at zero temperature (see the end of section 4 for more details.). The same issue was also raised in other holographic superconductor models [7, 27]. To check if the super uid density is identi ed correctly, we compute super uid density in two methods: one is related to the in nite DC conductivity and the other is related to the magnetic penetration depth. Both yield the same results with nite momentum relaxation, but the only latter captures the super uid density in the case without momentum relaxation. This paper is organised as follows. In section 2, we introduce a holographic superreviewed. In section 3, the superconducting transition temperature and electric DC conductivity are computed. In section 4 the super uid density is computed in two methods. In section 5 we discuss the Home's law and we conclude in section 6. Holographic superconductor on a Q-lattice In this section we brie y review a holographic superconductor model on a Q-lattice, which has been studied in detail in [22, 23]. The action is given by S = two lines are the rst holographic superconductor model [13, 14] with the U(1) gauge eld momentum relaxation by assuming a speci c form of as described below. To be concrete, For classical solutions we consider the following ansatz ds2 = z)U (z)dt2 + + V1(z)dx2 + V2(z)dy2 ; A = (1 = z (z) ; = eikxz (z) ; where U; V1; V2; a; are functions of only the holographic coordinate z. The holowith the boundary a(0) = 1. The complex We choose '1 as a source and '2 with m2 = 2 behaves as = '1z + '2z2 + near boundary. (hOi) as a condensate of the scalar operator. For spon is assumed to be 2 the boundary value (0) = to the lattice amplitude and k is the lattice wavenumber. 2z3=4; V1 = V2 = a = 1; = 0. However, for we have to resort to numerical method. Our numerical theory state at various T = 2 (0; 0:4) for a range of = 2 (0; 90) and k= Metal-insulator transition without We are mainly interested in properties of a holographic superconductor with paper. However, in this subsection, let us rst consider a model with = 0 in (2.1) to 6= 0 in this later to understand properties of a holographic superconductor. For our model with DC , can be computed by horizon data [35] DC = Plugging our numerical solutions of (2.2) into (2.3) we have computed the resistivity function of temperature for = = 50 in gure 1. If k= = 8 (a) the resistivity increases and if k= (a) is an insulator and the latter (b) is a metal.4 The metal insulator transition occurs at we obtained a phase diagram = 0 (red lines) translation symmetry is recovered and the system becomes perfect metal without momentum relaxation. MIT can be understood also by (2.3). For small k (insulating phase), as temperature lowers it turns out V2(1) goes to zero, which yields system is 4 pV1(1)V2(1), the entropy vanishes in insulating phase. For large k (metal phase), (1) goes to zero similarly to gure 5, which yields a large DC. In metal phase, DC ! 0. Because the entropy of the the entropy is nite. Critical temperature and DC conductivity To study Homes' law we need three quantities, critical temperature Tc, DC conductivity at Tc ( DC(Tc)) and super uid density. In this section we compute the rst two and in the next section we investigate super uid density in more detail. below Tc as shown by blue lines in 5This phase diagram was rst studied in [34] and here we extended the analysis for a much bigger range (a) Insulator. = 0. If becomes zero below Tc. It was shown as blue lines. Metal-insulator transition with = 0. Red lines at k = 0 and = 0 represent perfect metals. Using pseudo spectral method [36], we numerically constructed classical solutions (2.2) another solution with In this case, the superconducting state has lower free energy than normal state so a phase transition occurs at Tc= . Second, for a increases for large k= . As k= ! 1, it approaches to the critical temperature of the AdS-RN ( scalar model [24] and the helical lattice model [7]. However, this behaviour was not seen in the previous analysis of Q-lattice models [22, 23], where the scalar eld has a smaller charge q = 2 than our case (q = 6). T/Tc=1 ( = respectively in gure 2. ( = respectively in gure 2. = 0, we have in nite DC because the system is translationally invariant. For a xed = , DC(Tc) decreases ! 1, it again goes to in nity. Notice that both Tc and Indeed, as shown in the following section, the super uid density also approaches the value of the AdS-RN as k= For k= ! 1. This universal feature can be understood in two ways. First, becomes suppressed for k= 1 as shown in gure 5, where, for example, j (z)j at T =Tc = 0:1 with = of kc= in gure 2. Figure 5 also shows that there is a qualitative change of j (z)j at the critical value 10:1. That is j (1)j = 0 for k= > kc= and j (z)j 6= 0 for k= < kc= . Interestingly, this critical kc= = 0 In both gures 3 and 4, the curves have the solid part and the dotted part. The former DC (Tc) has some correlation with the MIT. Super uid density In this section we compute the super uid density s in two ways based on the London equation [14]: Ji(!; p~) = sAi(!; p~) ; which is valid when ! and p~ are small compared to the scale at which the system loses p~ ! 0. The two cases can explain the in nite DC conductivity and the Meissner e ect of superconductors respectively. Ji(!; 0) = Ei(!; 0) (!)Ei(!; 0) ; where (!) denotes complex optical conductivity. Thus the super uid density is identi ed which implies the in nite DC conductivity (the delta function in the real part of the conby the Kramers-Kronig relation Im[ (!)] = Re[ (!)] = Im[ (!)] = phase is understood as the spectral weight transferred from nite ! by the Ferrell-GloverTinkham (FGT) sum rule [7, 24] d! Re[ n(!) s(!)] = where n and s denote the electric optical conductivity in the normal phase and superthe system are conserved. Maxwell's equation r B~ = 4 J~, we have J~ = sB~ . With r2B~ = r = 4 r J~ = implying the Meissner e ect. Here 2 is the magnetic penetration depth squared which is inversely proportional to the super uid density. Holographic methods Based on these two limits, the super uid density can be obtained experimentally by measuring optical conductivity or magnetic penetration depth. Corresponding to both cases ai(z; !; p~) = ai(0)(!; p~) + zai(1)(!; p~) + s = corresponding to the optical conductivity and the magnetic penetration depth respectively.6 However, there is a subtle issue in the order of limit. The two limits ! ! 0 and p~ ! 0 may not commute. In the probe limit, it was shown that the two limits commute [4], but in the case of full back reaction as in our set-up, these two limits may not commute. Because of this potential subtlety we will introduce new notations for super uid density: Ks for the case 1) and K~s for the case 2). small uctuation of the gauge eld of the form [22, 23] Ax = e i!tax(z) ; 6Since the gauge eld in the holographic model is external, currents do not source electromagnetic elds and Maxwell's equation can not be applied in (4.7), but we still have a London equation. which is coupled to the uctuations of the metric and the scalar eld : gtx = e i!thtx(z); = ie i!teikxz (z) : The equations of motion for ax(z); htx(z) and (z) are shown in appendix A. Near boundary the asymptotic behaviour of the uctuations are as follows: ax(z) = a(x0) + za(x1) + (z) = htx(z) = s = Ax = eipyax(z) : gtx = eipyhtx(z): htx(z) = s = of which should be a(x0). Thus we cannot set both (0) and ht(x0) to be zero. However, if we With this condition we get and setting ht(x0) = 0 we have Ax that have momentum dependence of the form [37] Unlike [37], we consider the back-reaction so Ax is coupled to the metric uctuation: The equations of motions for these two uctuations are written in appendix A. Near boun■ ◆ (a) No momentum relaxation. (b) No momentum relaxation. ●●●●● ● ● ● ● ● ● ● ● ●●●● ● ● ● ◆■●◆■●◆■● ◆■● ◆■● ◆■● ◆■● ◆■● ■● ● ● ●●●● ● ● ● (c) Large momentum relaxation. (d) Small momentum relaxation. Numerical results sets of parameters = and k= . For example, in gure 6, we show our results for four added for comparison. First, we display the cases with no momentum relaxation in gure 6 (a) and (b): (a) = 0 means (z) = 0. (b) is not AdS-RN, since there is a nite scalar (z) with a boundary value = 5. However, density should be identi ed with K~s. The non-zero Ks for T > Tc may be interpreted as a spurious e ect by the in nite DC conductivity due to translational invariance. This is an interesting and useful observation, since K~s gives a direct way to compute the super uid density even in the case with translation invariance. T/Tc=0.1 xed lattice amplitude ( = to insulator and metal respectively in gure 2. Next, let us turn to the case with momentum relaxation in gure 6 (c) and (d). Here contribution to Ks by translational invariance vanishes. Notice that the super uid density gure 6 (d) is similar to K~s in gure 6 (a). It is because in the limit k ! 1 the translation invariance is e ectively restored as explained at the end of section 3. In this limit the value of becomes irrelevant and the geometry approaches to the AdS-RN not the one for gure 6 (b). For our goal (Homes' law), we need to know s at zero T . s = Ks = K~s near zero temperature for all cases, so we will use the notation s for super uid density. For example, s at zero T can be read from gure 6 (b),(c),(d), for = = 5 and k= = 0; 2; 20 a range of = and k= and our results are shown in gure 7. For a xed = , s= at values of metal in gure 2. Similarly to Tc ( gure 3) and some correlation with the MIT. = n= 2 ( gure 6(a)(b)) weight is transferred to nite frequencies rather than the delta function at zero frequency. ti cation of super uid density in (4.3) or (4.15) may not be correct and it was proposed to cross check it via the magnetic penetration depth, which is (4.20). We have cross checked (a) Contour plot of C = s=( DC Tc) in = k= plane. ▼▲▼▲▼▲▼▲▼▲▼▲▼▲▼▲▼▲▼▲▼▲▼▲▼▲▼▲▼▲▼▼▲▲▼▲▼▲▼▲▼▲▼▲▼▲▼▼▲▼▲▲■ ◆◆■◆■◆■◆■◆■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆ ●◆ 19.0 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 (c) Zoom-in of the grey window in (b). ● ■●◆▲■ Cross-sections of (a) at 1; 5; 10; 30; 50; 70; 90. The solid part and the dotted part correspond to insulator and metal respectively in gure 2. ▼▼▲ ▼ ◆ ▲● 0.007 0.008 0.009 (49; 51) and k= 2 (7; 10). Homes' law Homes' law is given by s(T = 0) = C DC (Tc)Tc ; where C is a universal material independent constant. In our holographic model, k and correspond to the properties of material so we want to check if C is constant irrespective of k and . Having computed Tc ( gure 3), DC (Tc) ( gure 4), and s ( gure 7) as functions of k and we are ready to check Homes' law in our holographic superconductor model. s=( DC Tc) in = -k= 20 and 0 gure 8(a). The black diagonal line is the MIT line in gure 2. In general, in the region close to the MIT, C is larger and below 1 and k= 1 because DC due to the restoration of translational invariance. compared to the other region. In that region, there is a possibility that Homes' law hold. To see it more clearly we make gure 8(b), which is the cross-sections of gure 8(a) for xed = & 50, which means Homes' law holds in that regime. The regime are in the insulating phase near the MIT line, which was also observed in a holographic superconductor with the helical lattice case, even thought C is a little bit di erent for a di erent = .8 In gure 8(c), we zoom in the grey window in gure 8(b) for 49 51. Figure 8 (d) is and (d) are similar to gure 15 in [7]. 51 and 7 10. Figure 8 (c) derstood from gure 3, 4 and 7, where all three quantities s , DC , and Tc show the same However, if we look at closely, the plateau of every s , DC , ann Tc is not strictly at. They are slightly increasing or decreasing, but the combination of them, C, shows a better plateau behaviour. To check it explicitly we have made a plot for B s=Tc without DC and found that B is not as at as C shown in gure 8(c). Physically, this means that the Uemura's law9 does not hold in our model. 5 in gure 8(a). due to numerical instability for = holds only for underdoped cuprates [5, 6]. Conclusion and discussions We investigated Homes' law by computing the critical temperature (Tc), the DC conductivity at the critical temperature ( DC(Tc)), and the super uid density ( s) in a holographic is independent of the amplitude ( ) and/or wavenumber (k) of Q-lattice. We xed = & 50. As = grows, C tends to approach to some universal value. Homes' law holds in insulating phase near the metal i) for a given = , there is k= around the kc= . To compute the super uid density, we employed two methods. One is related to the in nite DC conductivity and the other is related to the magnetic penetration depth. With nite momentum relaxation both give the same results, which serves as a good cross-check 8For large , it seems that C is approaching to the universal value. However, we could not con rm it of our computation. However, without momentum relaxation only the latter correctly captures the super uid density. The former gets spurious contribution from the in nite DC conductivity due to translational invariance. tum relaxation Ks= FGT sum rule (4.6) still holds it seems that some of the low frequency spectral weight is transferred to nite frequencies rather than the delta function at zero frequency. 2 in detail. We have also If m2 increases, it is possible that the MIT does not occur and consequently Homes' law model and it was shown that there is no MIT and no Homes' law in that model if not have z dependence [27]. Homes' law in our model comes from the MIT and strong momentum relaxation. The may occur [38]. However, it turns out that our model does not have a linear in T resistivity in normal (strange metal) phase as shown in gure 1. Because the linear in T resistivity is a universal property of the normal phase of high Tc superconductors and may be related to the physics of Homes' law by the Planckian dissipation [8], it will be important to study Homes' law in a holographic model having linear in T resistivity such as [39, 40]. Equations of motion for super uid density We present the equations of motion for super uid density used in section 4. The rst one 1. p~ = 0 and ! ! 0 2 2(1 z)U (1 z)z2U 0 = a0x0 + 2ikz2((1 z)a)0 ( 0 0 = h0tx +((1 z)a)0 ax + 0 = 2ik(1 z)U ( 0 (1 z)U V1 2. ! = 0 and p~ ! 0 0 = a0x0 + (1 z)U V2 (1 z)U V1 0 = h0t0x + h0tx +((1 z)a)0 a0x + z2((1 z)a)02 4 2(1 z)U (1 z)U V1 p2z2 +2V2(3+z2( 2 + 2)) (1 z)z2U V2 Acknowledgments We would like to thank Tomas Andrade for collaborations at an early stage of this project. We also would like to thank Johanna Erdmenger, Sean Hartnoll, Elias Kiritsis, Yi Ling, Rene Meyer, Andy O'Bannon, and Koenraad Schalm for valuable discussions and correspondence. 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Chao Niu, Keun-Young Kim. Homes’ law in holographic superconductor with Q-lattices, Journal of High Energy Physics, 2016, 144, DOI: 10.1007/JHEP10(2016)144