Ward identity and Homes’ law in a holographic superconductor with momentum relaxation

Journal of High Energy Physics, Oct 2016

We study three properties of a holographic superconductor related to conductivities, where momentum relaxation plays an important role. First, we find that there are constraints between electric, thermoelectric and thermal conductivities. The constraints are analytically derived by the Ward identities regarding diffeomorphism from field theory perspective. We confirm them by numerically computing all two-point functions from holographic perspective. Second, we investigate Homes’ law and Uemura’s law for various high-temperature and conventional superconductors. They are empirical and (material independent) universal relations between the superfluid density at zero temperature, the transition temperature, and the electric DC conductivity right above the transition tem-perature. In our model, it turns out that the Homes’ law does not hold but the Uemura’s law holds at small momentum relaxation related to coherent metal regime. Third, we explicitly show that the DC electric conductivity is finite for a neutral scalar instability while it is infinite for a complex scalar instability. This shows that the neutral scalar instability has nothing to do with superconductivity as expected.

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Ward identity and Homes’ law in a holographic superconductor with momentum relaxation

Received: May Ward identity and Homes' law in a holographic superconductor with momentum relaxation Kyung Kiu Kim 0 1 2 3 5 6 Miok Park 0 1 2 4 6 Keun-Young Kim 0 1 2 5 6 0 85 Hoegiro , Seoul 130-722 , Korea 1 50 Yonsei-ro , Seoul 120-749 , Korea 2 123 Cheomdan-gwagiro , Gwangju 61005 , Korea 3 Department of Physics, College of Science, Yonsei University 4 School of Physics, Korea Institute for Advanced Study 5 School of Physics and Chemistry, Gwangju Institute of Science and Technology 6 Open Access , c The Authors We study three properties of a holographic superconductor related to conductivities, where momentum relaxation plays an important role. First, we nd that there are constraints between electric, thermoelectric and thermal conductivities. The constraints are analytically derived by the Ward identities regarding di eomorphism from ory perspective. We con rm them by numerically computing all two-point functions from holographic perspective. Second, we investigate Homes' law and Uemura's law for various high-temperature and conventional superconductors. They are empirical and (material independent) universal relations between the super uid density at zero temperature, the transition temperature, and the electric DC conductivity right above the transition temperature. In our model, it turns out that the Homes' law does not hold but the Uemura's law holds at small momentum relaxation related to coherent metal regime. Third, we explicitly show that the DC electric conductivity is nite for a neutral scalar instability while it is in nite for a complex scalar instability. This shows that the neutral scalar instability has nothing to do with superconductivity as expected. ArXiv ePrint: 1604.06205 superconductor; with; momentum; relaxation - Holography and condensed matter physics (AdS/CMT), Gauge-gravity correspondence Ward identities: constraints between conductivities Homes' law and Uemura's law Conclusion and discussions A Two-point functions related to the real scalar operator Contents 1 Introduction 2 3 5 Equilibrium state AC conductivities Charged scalar case (q 6= 0) Neutral scalar case (q = 0) General expression for super uid density Analytic derivation: eld theory Numerical con rmation: holography Conductivities at small ! AC conductivities: holographic model and method Conductivities with a neutral scalar hair instabitliy Introduction reviews and references, we refer to [2, 3, 9, 10]. in section 2. developed [42{51]. We explain each issue in the following. formation of the scalar hair may be understood as the coupling of the charged scalar to the charge of the black hole through the covariant derivative DM iqAM , which gives an e ective mass of su ciently negative near the horizon to destabilize the scalar eld. Based on this it does not break a U(1) symmetry, but at most breaks a Z2 symmetry electric conductivity will be nite contrary to the case with a complex scalar hair conductivity that derived from the neutral hairy black hole is indeed nite. may be two constraints relating three transport coe cients, ; , and [3, 55]. The contwo-point functions: , and three two-point functions related to the operator can be obtained algebraically is computed numerically. This is why only is presented in the literature [2]. alone cannot determine and , which must the Ward identities for two-point functions analytically from eld theory perspective. of our numerical method. right above the transition temperature T~c. ~s(T~ = 0) = C DC(T~c) T~c ; limit C where ~s, T~c and universal constant: C DC are scaled to be dimensionless, and C is a dimensionless 4:4 or 8:1. They are computed in [47] from the experimental 8:1. Notice that momentum relaxation is essential here because without momentum relaxation DC is in nite. There is another similar universal relation, Uemura's law, which holds only for underdoped cuprates [56, 57]: ~s(T~ = 0) = B T~c ; relaxation region, related to coherent metal regime. identities giving constraints between conductivities analytically from eld theory perspecin our model. In section 6 we conclude. AC conductivities: holographic model and method Equilibrium state S = 1 F 2 1 X2 (@ I ) eld A, the complex scalar mass m, two massless scalar elds, The action (2.1) yields equations of motion 1 F 2 1 X2 (@ I ) 1 X2 @M I @N I ; rM F MN = = 0 ; 2 I = 0 ; for which we make the following ansatz: A = At(r)dt + I = ( x; y) ; ds2 = + r2(dx2 + dy2) : renormalization, which are explained in [28, 31, 53, 54, 61] in more detail. order parameter, condensate. Near boundary (r ! two undetermined coe cients J , which are identi ed with the source and condensate respectively. The dimension of the condensate is related to the bulk mass of the complex scalar by m2 = 2 to perform numerical analysis. 3). In this paper, we take m2 = I is introduced to give momentum relaxation e ect is the parameter for the strength of momentum relaxation. For = 0, the and Horowitz (HHH) [7, 8]. related to the boundary stress tensor of the model (2.1) when = 0 [28]: i = where ; are indices in the eld theory x U(r) = r2 At = n (r) = 0; 2rh + n24+rhB2 , and n is interpreted as charge density. It is the dyonic black brane [62] 2 modi ed by due to I [54]. The thermodynamics and transport coe cients(electric, the case without magnetic eld, see [31]. Next, if a superconducting state with nite condensate and its analytic formula is not available.3 = 0 and in [46] for consider this case and refer to [2, 63, 64]. 3A nonzero (r) induces a nonzero (r), which changes the de nition of `time' at the boundary so eld theory quantities should be de ned accordingly. momentum (say = x), P_ xi = ExhJ ti F tx and assume a nite density system without current, i.e. First, if analytic formula is given by 6= 0. (a) (r)= 2 (c) At(r)= (b) (r) (d) U(r)= 2 = 0:5). The black curves are for normal phase ( O = 0). In black curves agree to the analytic formula in (2.9), where enters only into U(r). AC conductivities B 6= 0 see [31] for normal phase. Ai(t; r) = gti(t; r) = i(t; r) = e i!tr2hti(!; r) ; the background are4 Near boundary (r ! 1) the asymptotic solutions are t h0tx = 0 ; 0 = 0 ; htx = ht(x0) + ax = a(x0) + x = x(0) + The on-shell quadratic action in momentum space reads Sr(e2n) = 1 Z d! hJ a !Aab(!)J!b + J a !Bab(!)R!bi ; 0a(x0)1 J a = B@h(t(0x0))CA ; 0a(x1)1 Ra = B@h(t(3x3))CA ; A = B@0 2U (1) 0C ; A B = B@0 density. The index ! in J a and Ra are suppressed. uctuations in momentum space by a collectively. i.e. a = ( ai ; hti ; i) : a(r) = (r 1) 4i!T +na ('a + '~a(r motion are doubled too. Every 'ia yields a solution ia(r), which is expanded near boundary as where a 1 and the leading terms Sia are the sources of i-th solutions and Oia are the constructed from a basis solution set f iag: with arbitrary constants ci's. For a given J a, we always can nd ci,5 so the corresponding response Ra may be expressed in terms of the sources J b, a(r) = Ra = Oiaci = Oia(S 1)i J b : b basis 'ia, (i = 1; 2; = BBB ... ... . With (2.22), the action (2.13) becomes 1 Z 1 Z Sr(e2n) = J ! Aab(!) + BacOic(S 1)ib(!) J b i ! retarded Green's functions are explicitly denoted as B GT J GT T GT S AC : GSJ GST GSS grx = 0 [53]. (a) Electric conductivity (b) Thermoelectric conductivity (c) Thermal conductivity AC electric conductivity( (!)), thermoelectric conductivity( (!)), and thermal conductivity( (!)) for GJ J + GJ T GJ J + GT J GT T (! = 0) GJ T + GT J Conductivities with a neutral scalar hair instabitliy derivative DM Charged scalar case (q 6= 0) thermoelectric conductivity ( (!)), and thermal conductivity ( (!)) for = 1 and part of conductivities. One feature we want to focus on in phase the DC conductivity is nite due to momentum relaxation. for the complex scalar hair may be understood as the coupling of the charged scalar to the charge of the black hole through the covariant derivative DM other words, the e ective mass of de ned by m2 q2jgttjAt2 can be compared with the Breitenlohner-Freedman (BF) bound. The BF bound for AdSd+1 is iqAM . In may be su ciently negative near the horizon to destabilize the The e ective mass m2 instability conditions can be summarized by one inequality [46] me2 = 4m2 = m2BF ; which reproduces the result for = 0 in [2]: me2 = = m2BF : but at most breaks a Z2 symmetry momentum relaxation ( . Therefore, it would be interesting to see if the this issue properly. 7We thank Sang-Jin Sin for suggesting this. (a) Electric conductivity (b) Thermoelectric conductivity (c) Thermal conductivity AC electric conductivity( (!)), thermoelectric conductivity( (!)), and thermal conductivity( (!)) for = 1. For q = 0, m2 of gure 3 from = 1 + 4 Q ; = 4 = 4 (a) Electric conductivity (b) Thermoelectric conductivity (c) Thermal conductivity AC electric conductivity( (!)), thermoelectric conductivity( (!)), and thermal conductivity( (!)) for (rh))=2. merical values very well. For a special case with = 0, in gure 5, we see that (!) = 1, conducting case. General expression for super uid density rM F MN = the x-direction, the x-component of the Maxwell equation reads gF xr = @tp = lim p gF tx + iq Ax = gtx = r2 x = a^x(!; r)e i!t ; i 4!T h^x(!; r)e i!t + i!r2 e i!t ; h^x(!; r) A0 + A1(r be expanded near horizon as term goes to zero as8 With the following source-vanishing-boundary conditions9 x = lim e i!t = 0 ; (3.11) except Ax, the current (3.6) can be interpreted as J x = i! xx(!) Axjr=1 : 8More explicitly, by using (3.9) and (2.6), the rst term of (3.8) is expressed as p gF xr = p g Fxrgrrgxx + i! A0te (r rh) (3.10). Thus, O(!)e i!t. Similarly, by using (3.9) and (2.6), the second term of (3.8) is expressed as dr@tp gF tx = drp ggttgxx@t2 Ax = !2 which is of order !2. in counting order ! near horizon as shown in footnote 8. lim !Im[ ] = lim !!0 Ax(r = 1) = lim = lim !!0 Ax(r = 1) rh !!0 Ax(r = 1) rh This shows how the hairy con guration con rms our numerical analysis. Ward identities: constraints between conductivities eld thedensity and normal uid density and the relation between them. regarding di eomorphism from eld theory perspective. In our eld theoretic derivation the stress-energy tensor T , respectively. scalar operator. Our main results are (4.44){(4.45) for nite magnetic eld (B 6= 0) and (4.56){(4.58) for zero magnetic perspective, by solving bulk equations numerically. where h , A , I , , and are the non-dynamical external sources of the stress-energy tensor T , U(1) current J , real scalar operators OJ , and complex operators O y; O J (x)i = (x)i = 2 OI (x) = ; hO (x)i = (Pt) two-point functions: Analytic derivation: eld theory therein to the case with real and complex scalar elds, which are I and in (4.1). Our nal results are (4.44){(4.45) and (4.56){(4.58). eW [h ;A ; I ; ; ] = DXe S[X;h ;A ; I ; ; ] (y)) = 4 (x)J (y))i = 2 (x)OI (y)) = 2 (x)O (y)) = 2 hPt(J (x)J (y))i = Pt(J (x)OI (y)) = Pt(J (x)O (y)) = Pt(OJ (x)OI (y)) = Pt(OJ (x)O (y)) = Pt(O (x)O y(y)) = h (x) h (y) h (x) A (y) A (x) A (y) We consider the generating functional W [h ; A ; I ; ; ] invariant under di eomorphism, x , and the variation of the elds can be expressed in terms of a Lie 10See [66] for a holographic derivation. derivative with respect to the vector eld = (L h) = r For di eomorphism invariance, the variation of W should vanish: W = A = (L A) = I = (L I ) = = (L ) = ) = 0 ; di eomorphism. = 0 ; U(1) gauge symmetry, which is summarized in footnote 12. (x) . Here we also used the Ward identity for By taking a derivative of (4.18) with respect to either h (y), A (y), J (y) or we obtain the Ward identities for the two-point functions: D hPt(J (y)T (x))i + F hPt(J (y)J (x))i y) + h @ Pt(T (y)T (x)) + (x T (x) + h Pt(T (y)J (x)) @ I Pt(T (y)OI (x)) + 2Re h @ Pt(J (y)O (x)) = 0 ; Pt(T (y)O (x))Eo = 0 ; Pt(OJ (y)J (x)) + h Pt(OJ (y)OI (x)) @ I o = 0 ; Pt(O (y)T (x)) Pt(O (y)J (x)) + h Pt(O (y)OI (x)) @ I Pt(O (y)O (x)) @ = 0 ; where the covariant derivatives act only on the operators of x. GE 0 = 0 = I GE iG~JE;I (k) iG~E;I (k) 0 = 0 = k identities (4.19){(4.22) read From here we consider a at space, h , and assume external elds such as , are constant in space-time. We further assume translation invariance is , should be constant in space-time. In momentum space, the Ward 0 = 0 = k functions. Thus, the Ward identities (4.23){(4.26) become 0 = k G~JR; (k) + iF 0 = G~R; + iG~R;I (k) constant expectation values for the energy-momentum and current J i = (n; 0; 0) ; nite or zero condensate external magnetic eld with a background scalar I : F = Bdx ^ dy ; I = ( x; y) : Under these conditions the Ward identities (4.27){(4.29) becomes I GR iG~JR;I (k) !G~JR;0j + iB ij G~JR;i iB ikG~jR;i + ! jkn + i IkG~jR;I = 0 ; = 0 ; Ij = 0 ; = 0 ; = 0 ; = 0 ; B) hJ Qi + !( B) hSJ i = 0 ; = 0 ; = 0 : Finally, using the Kubo formulas for conductivities11 we obtain the relations between the conductivities: de ned as iG~xR;0y ; iG~0Rx;0y ; G~IR=1;0x iG~IR=1;0y : With this notation, (4.33){(4.35) can be rewritten as Ward 1 : Ward 2 : Ward 3 : where we rede ned to subtract a counter term and 0 hT T i ;!=0 = =2 [3]. hT T i ;!=0. In normal phase, if = 0 and B 6= 0, 11The complexi ed conductivities are denoted by X Xxy iXxx, where X = ; ; ; . = 0 ; = 0 ; = 0 ; B) hSJ i hT T i ;!=0 ; ! hJ T i + ! n + i hJ Si = 0 ; ! hT T i + ! + i hT Si = 0 ; ! hST i + i hSSi = 0 ; Using the Kubo formulas we obtain the relations between the conductivities: hQQi + hJ Si) = 0 ; hJ J i + n + i ! hJ Si = 0 ; hSQi + hSJ i + n + i ! hSSi = 0 : Ward 4 : Ward 6 : Ward 5 : hJ Si = 0 ; hJ Si = 0 ; hST i + i hSSi = 0 ; = 0, hT T i!=0 = Numerical con rmation: holography phase, hT T i!=0 = and =2 for = 0 and + hT T i!=0. In normal and (4.56){(4.58) to check if they add up to zero or not. were reported in [54] and reproduced in gure 2. Here in scalar operator, hJ Si, hQSi, and hSSi. Contrary to gure 2 there is no divergence at other cases: 1) B = 0 and = = 0:1, 2) B = 0 and = 0, 3) B 6= 0. It turned out that all cases in the appendix A. !G1;t + kG1;x = 0. 12If W is invariant under U(1) gauge transformations, A , the Ward identity for onegreen, blue). (a) Ward 4: (4.56) (b) Ward 5: (4.57) (c) Ward 6: (4.58) temperatures shown in gure 6 all together. They are almost zero, less than 10 15. Conductivities at small ! model of superconductor we check Homes' law and Uemura's law. Re[ ] = Re[ ] = Im[ ] + Im[ ] = Im[ ] + Im[ ] = for superconducting phase. By these relations, once is obtained, are completely phase there is another contribution due to condensate. Re[ ] + Re[ ] = Im[ ] + Im[ ] = Re[ ] + Re[ ] = Im[ ] + Im[ ] = hQSi + hSSi = 0 ; values linear to !. Ks is introduced as a strength of the pole of Im[ ], numerical data. Ks = lim !Im[ ] ; J . Contrary to the case of = 0, are not determined by only, because there once we know , , and , we can read o hJ Si, hQSi, and hSSi by the Ward identities. part of , , and = 0). At small !, it is inferred that Re[hJ Si] Also Re[hQSi] !2 from (4.62) and Im[hQSi] !2 from (4.60) and Im[hJ Si] ! from (4.61). ! from (4.63).13 Finally, the small ! (see for example the solid curves in gure 2), unlike normal phase, Im[ ] and Im[ ] have parts. In summary, the small ! behaviours can be written as 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T/Tc 0.2 0.4 0.6 0.8 1.0 1.2 1.4 T/Tc = 3; 5; 7; 10 (green, blue, purple, hQSi, hSSi) diverge when agree to the numerical results in gure 6, 12 and 13. If we de ne a normal uid density (Kn) as goes to zero at small !. We have con rmed that (4.65){(4.70) temperature in our numerics. superconductor phase. Ks + Kn. In gure 8, we plot between the dotted and solid curve at a given zero,14 Kn vanishes for = . 2 ( gure 8(a)). . 2 and incoherent state without a Drude peak for = . 2.15 In coherent state, the normal uid density Kn can be Homes' law and Uemura's law DC is in nite and Homes' law cannot be satisi ed. 56{58]. Uemura's law appearing in underdoped cuprates is ~s(T~ = 0) = B T~c ; and Homes' law satis ed in a broader class of materials is ~s(T~ = 0) = C DC(T~c) T~c ; and q. To check this it is law respectively. To compute B and C, B = s = C = the super uid density ~s(= = 2 in our model [28]. results of ~s; T~c and DC for q = 3 are shown in gure 9. gure 9 we may expect that there is a linear relation between ~s and T~c at least for large small = nd that Uemura's law holds only for = & 2, of which data are red dots. Interestingly, the parameter regime behaviour. They correspond to gure 8(a) and there is a gap between charge density nd that Uemura's law is satis ed for large but with a di erent constant B. For example, for q = 2, B 6:87 and for q = 6, B 4:64 in the regime of = & 2 ( gure 10(c)). Since Uemura's law is observed in underdoped regimes, if can be interpreted as a doping parameter our result will be consistent with phenomena. DC for q = 3. q=2 q=3 q=6 (a) B(= ~s=T~c) , q = 3 (b) B(= ~s=T~c) for q = 2; 3; 6 = 0:3; 0:4; 0:5; 0:7; 1). In (a) the black line is drawn for B 5:47, and in (b) the black lines are drawn for B (a) C(= ~s=( DCT~c)), q = 3 (b) ~s vs DCT~c, q = 3 regime ( = = 2; 3; 5; 7; 10). In Based on our results on Uemura's law ( gure 10(a)) and DC ( gure 9(c)), we may anticipate if Homes' law is satis ed. If DC is quickly decreasing function approaching to constant for & 2 we may have a chance to obtain Homes' law. However, our does not show that behaviour. Therefore, as shown in gure 11, Home's law does not hold in both coherent regime (red dots) and incoherent regime (blue dots). In gure 11(a), for representation, a plot of ~s versus relation between between ~s and DCT~c, where it is also clear that there is no linear satis ed for di erent values of q either. temperature (Planckian dissipation): nN (Tc) ; ( = 2), where momentum relaxation is weak. In gure 8(b), all curves coincide and it Homes' law. The relaxation time for our model can be written as f (T = ; = ; q) f is not universal near Tc. Furthermore, we may induce that f = 2 because Tc gure 9(b) and model is isotropic four dimensional. Uemura's law. as expected. rM F tM = iq we may de ne the charge density of hair outside the horizon, nhair, as gF trjr=1 gF trjr=rh = iq Acknowledgments of POSCO TJ Park Foundation. Two-point functions related to the real scalar operator for other cases too: 1) B = 0; = = 0:1, 2) B = 0; = 0, 3) B 6= 0. For completeness, we show here the numerical data of hJSi, hQSi, hSSi for (1) and (2) in gure 12 and 13 gure 14 we show the numerical results of Ward identites for (3). green, blue). red, orange, green, blue, purple). (a) Ward 1: (4.44) (b) Ward 2: (4.45) (c) Ward 3: (4.46) led case: we plotted all components of the Ward identities (4.44){(4.46) = 0; 0:5; 1; 1:5. Open Access. 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Kyung Kiu Kim, Miok Park, Keun-Young Kim. Ward identity and Homes’ law in a holographic superconductor with momentum relaxation, Journal of High Energy Physics, 2016, 41, DOI: 10.1007/JHEP10(2016)041