Holographic s-wave and p-wave Josephson junction with backreaction

Journal of High Energy Physics, Nov 2016

In this paper, we study the holographic models of s-wave and p-wave Josephoson junction away from probe limit in (3+1)-dimensional spacetime, respectively. With the backreaction of the matter, we obtained the anisotropic black hole solution with the condensation of matter fields. We observe that the critical temperature of Josephoson junction decreases with increasing backreaction. In addition to this, the tunneling current and condenstion of Josephoson junction become smaller as backreaction grows larger, but the relationship between current and phase difference still holds for sine function. Moreover, condenstion of Josephoson junction deceases with increasing width of junction exponentially.

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Holographic s-wave and p-wave Josephson junction with backreaction

Received: August Holographic s-wave and p-wave Josephson junction Yong-Qiang Wang 0 1 2 3 Shuai Liu 0 1 2 3 0 Open Access , c The Authors 1 Lanzhou 730000 , People's Republic of China 2 Institute of Theoretical Physics, Lanzhou University 3 [16] Z.-Y. Nie , R.-G. Cai, X. Gao, L. Li and H. Zeng, Phase transitions in a holographic s In this paper, we study the holographic models of s-wave and p-wave Josephoson junction away from probe limit in (3+1)-dimensional spacetime, respectively. With the backreaction of the matter, we obtained the anisotropic black hole solution with the condensation of matter elds. We observe that the critical temperature of Josephoson junction decreases with increasing backreaction. In addition to this, the tunneling current and condenstion of Josephoson junction become smaller as backreaction grows larger, but the relationship between current and phase di erence still holds for sine function. Moreover, condenstion of Josephoson junction deceases with increasing width of junction exponen- 1 Introduction 2 (2+1)-dimensional s-wave Josephson junction 2.1 2.2 The ansatz and asymptotic forms The numerical results Holographic setup The numerical results 3 (2+1)-dimensional p-wave Josephson junction The AdS/CFT duality [1{3], which originates from string theory, provides a novel and powerful tool to study the strongly coupled eld theories in a weakly coupled gravitational system. It states a d-dimensional conformal eld theory on the boundary is equivalent to a (d + 1)-dimensional dual gravitational description in the bulk. Since the AdS/CFT correspondence can provide a holographically dual description of the strongly coupled system, it has received a broad range of attention. In recent years, the application of AdS/CFT correspondence to condensed matter physics has been quite successful. Especially, one of hot points is the study of holographic superconductor. In [4, 5], the authors studied the s-wave superconductor by coupling anti-de Sitter gravity to the U(1) gauge eld and a complex scalar eld. Ones found that when the Hawking temperature of black hole was below a critical temperature, the U(1) gauge symmetry would be broken spontaneously via the charged scalar eld condensated outside the horizon. In [6{8], the authors investigated the p-wave superconductor by considering SU(2) gauge eld, and studied the d-wave superconductor by a symmetric, traceless second-rank tensor and a U(1) gauge eld in the background of the AdS black hole. Moreover, one also studied the coexistence and competition of order parameters by holographic approach in [9{16]. More detailed introduction of holographic superconductor can be found in [17{20]. In addition to the study of holographic superconductor in the probe limit, the s-wave superconductor with backreaction is also researched in [21]. Furthermore, in [22], the authors constructed an SU(2) Einstein-Yang-Mills theory with (4+1)-dimensional asymptotically anti-de Sitter charged black hole to describe p-wave super uids with backreaction. The authors investigated the s-wave superconductor in (3+1)-dimensional with backreaction in the cases of pure Einstein and Gauss-Bonnet gravity, respectively in [23]. The papers about holographic p-wave phase transition in Gauss-Bonnet Gravity and the swave superconductor in (3+1)-dimensional AdS spacetime with backreaction are presented in [24] and [25], respectively. There are another papers [26{31] about holographic superconductors with backreaction by semi-analytic and numerical computation method. In these papers, ones nd that critical temperature will become lower and condensation will become harder if the strength of backreaction becomes stronger. Furthermore, AdS/CFT correspondence has been application in the study of holographic lattice. For example, in [32], the authors studied the optical conductivity by adding a gravitational background lattice. Other papers about holographic lattice can be viewed in [33{47]. The DeTurck method provides a good tool for solving Einstein equations in these papers. The model of the holographic superconductor can also be extended to study the Josephson junction which are associated with the experiments. As we know, the Josephson junction consists of two superconductor materials and a weak link barrier between them [48]. The weak link can be a thin normal conductor (S-N-S) or a thin insulating barrier (S-I-S). Horowitz et al. in [49] studied the s-wave Josephson junction in probe limit by the Maxwell eld coupled with a complex scalar eld in a (3 + 1)-dimensional Schwarzschild-AdS black hole background, and observed that the current is proportional to the sine of phase di erence with AdS/CFT. A holographic model of 4-dimensional Josephson junction has been investigated in [50, 51]. With the model of a designer multigravity, the holographic mode of a Josephson junction array has been constructed in [52]. The p-wave Josephson junction was discussed by an SU(2) gauge eld coupled with gravity in [53]. In [54], the authors studied (1+1)-dimensional S-I-S Josephson junction in the four-dimensional anti-de Sitter soliton background. A holographic model of superconducting quantum interference device (SQUID) was studied in [55, 56]. In [57], authors investigated the holographic Josephson junction with Lifshitz geometry. A holographic model of hybrid and coexisting s-wave and p-wave Josephson junction was constructed in [58]. The authors constructed a holographic model of s-wave Josephson junction with massive gravity in [59]. The previous studies on holographic Josephson junctions are in the probe limit, however, it would be of great interest to further explore what role the backreaction plays in Josephson e ect beyond the probe limit. For example, turning on the backreaction, we wonder that whether the current is proportional to the sine of phase di erence and condensation decreases with increasing width of junction. Inspired by the previous work, we took advantage of a complex scalar eld coupling to the U(1) gauge eld and SU(2) Yang-Mills eld with (3+1)-dimensional RN-AdS black hole to construct the holographic models of s-wave and p-wave Josephson junction with backreaction, respectively. In this paper we will proceed as below. In section 2, we set up the model of s-wave Josephson junction and analyze our numerical results. In section 3, we write the action of the model of p-wave Josephson junction, and discuss our numerical results. We take conclusion nally in section 4. (2+1)-dimensional s-wave Josephson junction Let us begin with the Maxwell eld and a charged complex scalar eld in the (3+1)dimensional Einstein gravity spacetime with a negative cosmological constant. The Lagrangian density reads L = R and Einstein equations strength of the U(1) gauge eld is F = @ A @ A , m and q represent the mass and the charge of the complex scalar eld , respectively. The charge q appears in the covariant derivative and controls the strength of the backreaction of the matter elds on the metric. Since we are interested in the e ect of the backreaction and to see that how it varies =q. The Lagrangian density (2.1) changes into L = R + Lm = the above, we can see that the backreaction on the gravity will decrease when q increases, and the large q limit (q ! 1) corresponds to the probe limit (non-backreaction) of the matter sources. The equations of motion of the scalar and the electromagnetic elds which can be derived from the Lagrangian density (2.1) are as follows = 0; + iA ) ] = 0; = 0: For the charged scalar eld known Reissner-Nordstrom-AdS (RN-AdS) black hole. The solution with a spherically symmetric can be written as follows ds2 = 1)H(z)dt2 + where H(z) = 1 + z + z2 2z3, and 0 is the chemical potential for U(1) charge. The holographic superconductors, is given by T = scalar condensation begins to occur. For T < Tc, the black hole will have a scalar hair with breaking the U(1) gauge symmetry spontaneously and brings about superconducting phenomena of the (2+1)-dimensional dual theory on the boundary. As for T > Tc, the black hole with a scalar hair degrades into RN-AdS black hole. The ansatz and asymptotic forms In order to build a holographic model of the s-wave Josephson junction, we introduce two bulk gauge elds Az and Ax on the basis of the model of s-wave superconductor. The gauge eld Ax is related to the non-vanished current on the boundary. In addition, the matter elds must depend on spatial coordinates. Considering the above reasons, an ansatz of matter elds should be described as below = j jei ; A = Atdt + Azdz + Axdx; where j j, , At, Az and Ax all depend upon the spatial coordinate x and the radial coordinate z. Furthermore, we take the metric ansatz as ds2 = 1)H(z)E1(dt + E7dx)2 + + E3(dx + E5dz)2 + E4dy2 ; E2(dz + (1 z)H(z)E6dt)2 non-diagonal one, in which the non-diagonal function E5 is required due to the x dependent spatial coordinate, and the functions E6 and E7 are associated with non-vanishing Az and Ax, respectively. Thus the holographic model of s-wave Josephson junction would be along the x direction. The function can be taken to be real by introducing the new U(1) gauge The scalar eld, Maxwell and Einstein equations of motion with the ansatzs (2.9) and (2.10) are a set of non-linear coupled partial di erential equations. Seven equations which come from the Einstein equations (2.6) are second-order PDEs with respect to Ei. Two equations from eq. (2.4), one is a second-order PDE and the other one is a rst-order PDE, are called as constraint equations. The remaining three equations which are from the eqs. (2.5) are also second-order PDEs. It is not convenient to write down all of the twelve equations in our paper, for each equation contains hundreds or thousands of terms. It is obvious that we should solve them numerically instead of seeking the analytical solutions. Before numerical program, we should obtain the asymptotic behaviors of j j, Mt, Mz, Mx conditions we need. On the AdS boundary, the asymptotic behaviors of the functions Ei take the Tc = T Mz(z; x) ! O(z); ( )(x) + z + (+)(x) + O(z1+ + ); of junction, respectively. proportional to (1) where m is the mass of the scalar eld and the values of m2 must satisfy the BreitenlohnerFreedman (BF) bound m2 to AdS/CFT duality, ( )(x) are the corresponding expectation values of the dual scalar operators hO i, respectively. In this paper, we will set ( ) = 0 and take hOi = hO+i = (+) to describe the scalar condensation. The coe cients (x), (x), (x) and J are the chemical potential, charge density, the velocity of super uid and the constant current in the dual eld theory, respectively [61{68]. In order to describe a Josephson junction, we still adopt the chemical potential (x) in [49] as follows (x) = (1) 1 where the chemical potential (x) is proportional to (1) and L is the width of junction. The parameters (+1) = ( 1) at x = control the steepness and depth Next, we introduce the critical temperature Tc of the Josephson junction, which is Obviously, when z goes to zero, the metric (2.10) will approach to the Reissner-NordstromAdS metric with the above asymptotic forms. asymptotic forms m2 = - 2 m2 = - 5 4 When m2 = From eqs. (2.16){(2.23), we can see that for the value of m2 is xed, the critical temperature Tc drops with increasing strength of backreaction , and Tc decreases with increasing m2 5=4(green). When m2 grows, T = c will decrease. c is the critical chemical potential of the superconductor. In gure 1, we plot m2 = 2; 5=4. It indicates that T = c decays as and when m2 grows, T = c will decrease. When m2 = increases with xing the value of m2, 0:0588 (1); 0:0581 (1); 0:0550 (1); 0:0505 (1); 0:0413 (1); 0:0403 (1); 0:0358 (1); 0:0296 (1); = 0; = 1=7: = 0; = 1=7: m2 = 2, 0 = 1, 1 = 6, L = 3, = 0:6, = 0:5 and In condensed matter physics, the current of Josephson junction has to be gauge invariant because of the presence of a gauge eld A . So, we need to consider the gauge invariant phase di erence. Thus, in our holographic model case, the gauge invariant phase across the junction can be de ned as The numerical results For completeness of our study, we carry on numerical computation in this subsection. To make our work easier, we choose m2 = 2. We show the pro les of the functions Mt and E5 in gure 2, respectively. We can see that Mt is even, and E5 is odd. and the phase di erence as shown in gure 3. We take the strength of backreaction J=Tc2 J=Tc2 J=Tc2 J=Tc2 1:39981 sin ; 1:33521 sin ; 0:97168 sin ; 0:51218 sin ; = 0; = 1=7: from top to bottom are for choosing = 0, = 1=16 and = 1=7. The parameters are m2 = 2, 0 = 1, 1 = 6, L = 3, = 0:6 and = 0:5. L with increasing strength of backreaction = 0; 1=100; 1=16; 1=7 in the gure 4(a). The tting results are as follows Jmax=Tc2 Jmax=Tc2 Jmax=Tc2 Jmax=Tc2 17:8455e L=1:17112; 18:4674e L=1:13660; 19:9195e L=0:994678; 17:0124e L=0:860052; hOi=Tc2 hOi=Tc2 hOi=Tc2 hOi=Tc2 33:4411e L=(2 1:25493); 34:8072e L=(2 1:20773); 39:5984e L=(2 1:02245); 41:0873e L=(2 0:863886); = 0; = 1=7: = 0; = 1=7: It is surprising that when results are as below xed. In addition, hOi=Tc2 goes down as 2. For the junction L for m2 = 2. For the xing , hOi=Tc2 varies as a function of L. Curves from top to bottom are for choosing = 0, = 1=16 and = 1=7. The curves t exponential curves = 0:5. with the backreaction = 1=7; 1=16; 0 is in gure 5. From the gure, we can see that m2 = From the above results we nd that the backreaction can hinder the generation of condensation and current, and make the critical temperature lower. In essence, when the intensity of backreaction becomes larger, the Cooper pairs generate harder, the charge , it makes sense that stronger backreaction weakens condensation and lower temperature enhances it, but the latter plays a dominate role in this competition. In the next moment, we take the value of m2 = 5=4 to compare with the case of = 1=100. In gure 6, we represent the relationship between the current J=Tc + and the phase di erence with di erent value of m2. The tting results are shown J=Tc + m2 = m2 = From the above, we see that for the xing , the current goes down as m2 increases. Figdecrease with increasing L, respectively. The tting results are Jmax=Tc + 15:5660e L=1:3160 21:7297e L=0:5570 m2 = m2 = will go up, when increases. We use m2 = 2, 0 = 1, top to bottom, curves correspond to = 1=7; 1=16; 0. 1 = 6, L = 3, = 0:6 and = 0:5. From The value of m2 = 5=4(green). The parameters are 0 = 1, = 0:5 and and goes down as m2 increases. 1 = 7:5, L = 3, = 0:5, hOi=Tc + will decrease, respectively. The value of m2 = 0 = 1, 1 = 6, = 0:5, = 0:5 and 5=4(green). The parameters hOi=Tc + 26:5676e L=(2 1:448) 67:2095e L=(2 0:5360) m2 = m2 = is shown in The above results indicate that for xed backreaction, the larger mass of the scalar eld can hinder the scalar hair to form. (2+1)-dimensional p-wave Josephson junction Holographic setup We consider the non-Abelian SU(2) gauge led in the (3+1)-dimensional Einstein gravity spacetime. The Lagrangian density is as follows L = R F a = @ A The coupling constant of the SU(2) gauge eld is gY M . where , !, Az and Ax are all the real functions of coordinate x and z. The EoMs (3.5) and (3.6) are also a set of non-linear coupled partial equations with the matter ansatz (3.7) of m2 = = 0:5 and L = R F a = @ A @ Aa + "abcAb Ac : see that the e ect of backreaction decreases as gY M grows. The large limit (gY M ! 1) corresponds to the probe limit. The equations of motion from the Lagrangian density (3.1) are 1 F a F a c = 0; 1 F a F = 0: In order to construct a holographic model of the p-wave Josephson junction, we bring in We consider the following ansatz of the SU(2) gauge eld A = A0 3dt + ! 1dy + Az 3dz + Ax 3dx; and the metric ansatz (2.10). These ve equations derive from eq. (3.5), four of them are second-order PDEs and the remaining one is a rst-order PDE which is constrain equation. Seven equations from eq. (3.6) are second-order PDEs with respect to Ei. Because each equation has thousands of terms, we can not write them down in paper. Obviously, we can only solve them numerically. the AdS boundary (z = 0) In the next step, we will study the following asymptotic forms of the matter eld on !(z; x) ! z!(1)(x) + z2!(2)(x) + O(z3); Ax(z; x) ! (x) + Jz + O(z2): shown as below ductor. The asymptotic behaviors of Ei take the same forms as (2.11a){(2.11g) on the AdS boundary. The chemical potential and the phase di erence take (2.14) and (2.24), The critical temperatures of p-wave Josephson junction with di erent values of 0:0654 (1); 0:0613 (1); 0:0578 (1); = 0; = 1=8: From the above, we can see that the critical temperature Tc drops with increasing strength of backreaction . The numerical results In gure 9 (a) and (b), we plot the pro les of the functions E6 and E7 . From these gures, we can see that E6 is odd and E7 is even . gure 10. We take J=Tc2 J=Tc2 J=Tc2 1:94186 sin ; 1:50461 sin ; 1:10894 sin ; = 0; = 1=8: with xing . Furthermore, J=Tc2 will decrease when width of junction L with increasing varies a function of L with increasing = 0; 1=15; 1=8. The p-wave condensation hOi=Tc2 in gure 11 (b). The tting results are shown 1 = 5:5, L = 3, = 0:6, = 0:5 and top to bottom, curves correspond to = 0; 1=15; 1=8. We use the parameters 0 = 1, 1 = 5:5, L = 3, = 0:6 and = 0:5. increases, Jmax=Tc2 and = 0; 1=15; 1=8. The parameters are 0 = 1, 1 = 5:5, = 0:6 and = 0:5. Jmax=Tc2 Jmax=Tc2 Jmax=Tc2 15:8458e L=1:41421; 18:5319e L=1:19729; 18:9418e L=1:06426; hOi=Tc2 hOi=Tc2 hOi=Tc2 39:4108e L=(2 1:61943); 48:4891e L=(2 1:28419); 54:6341e L=(2 1:10103); = 0; = 1=8: = 0; = 1=8: T =Tc by xing , and Jmax=Tc2 goes down as above results is the same as s-wave junction. and L is the same as the case of Jmax=Tc2. increases. The reasons that lead to the Motivated by the e ect of backreaction on the s-wave and p-wave holographic superconductors, we studied backreaction to the s-wave and p-wave Josephson junction, respectively. For s-wave Josephson junction, we investigated the the full, dynamic equations of motion will go up, when increases. From top to bottom, curves correspond to = 1=8; 1=15; 0. The parameters are 0 = 1, L = 3, = 0:6, = 0:5. including Einstain equations in the (3+1)-dimensional spacetime from the action of U(1) gauge eld and complex scalar eld, and for p-wave Josephson junction, we considered the backreaction from the SU(2) gauge eld. We get a series of partial di erential equations of elds that are nonlinear and coupled and solve them numerically. For the s-wave Josephson junction, we take two value of m2 = that when the value of m2 and the strength of backreaction are xed (m2 = that in the probe limit ( and the scalar eld) do not in uence each other, the current is proportional to the sine of phase di erence; when a ects the whole of the matter instead of the gauge eld or the scalar eld respectively, the relationship between the gauge eld and the scalar eld is not changed, so the current is still proportional to the sine of phase di erence; in addition, the parameter = 1=q2 is related to the charge q, which is the coupling coe cient between the gauge eld and the scalar eld and represents the charge of Cooper pair in superconductor, when charge q will change, thus the backreaction only a ects the amplitude of the current, so the relationship between current and phase di erence holds for sine function with backreaction. the mass of scalar eld is, the harder the scalar hair forms. When the value of m2 is xed (m2 = The reasons are that the backreaction makes the Cooper pairs generate harder, and the varies with T =Tc goes up as increases, this is because the e ection of low temperature is stronger than that of backreaction. For p-wave Josephson junction, the property is the same as the case of s-wave Josephson junction. Now, we have studied holographic models of s-N-s and p-N-p Josephson junctions away from probe limit, moreover we would like to investigate the s-I-s and p-I-p junctions with backreaction in the future. In [57, 59], the s-N-s Josephson junction in the probe limit have been studied in the Lifshitz gravity and massive gravity, respectively. Thus our next study would be to generalize the above work to the case with backreaction . Acknowledgments YQW would like to thank Rong-Gen Cai and Yu-Xiao Liu for very helpful discussion. SL and YQW were supported by the National Natural Science Foundation of China. Open Access. 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Yong-Qiang Wang, Shuai Liu. Holographic s-wave and p-wave Josephson junction with backreaction, Journal of High Energy Physics, 2016, DOI: 10.1007/JHEP11(2016)127