Holographic superconductor on Q-lattice

Journal of High Energy Physics, Feb 2015

We construct the simplest gravitational dual model of a superconductor on Q-lattices. We analyze the condition for the existence of a critical temperature at which the charged scalar field will condense. In contrast to the holographic superconductor on ionic lattices, the presence of Q-lattices will suppress the condensate of the scalar field and lower the critical temperature. In particular, when the Q-lattice background is dual to a deep insulating phase, the condensation would never occur for some small charges. Furthermore, we numerically compute the optical conductivity in the superconducting regime. It turns out that the presence of Q-lattice does not remove the pole in the imaginary part of the conductivity, ensuring the appearance of a delta function in the real part. We also evaluate the gap which in general depends on the charge of the scalar field as well as the Q-lattice parameters. Nevertheless, when the charge of the scalar field is relatively large and approaches the probe limit, the gap becomes universal with ω g ≃ 9T c which is consistent with the result for conventional holographic superconductors.

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Holographic superconductor on Q-lattice

Received: November Holographic superconductor on Q-lattice Yi Ling 0 1 2 3 5 6 Peng Liu 0 1 2 5 6 Chao Niu 0 1 2 5 6 Jian-Pin Wu 0 1 2 3 4 6 Zhuo-Yu Xian 0 1 2 5 6 0 Jinzhou , 121013 China 1 Chinese Academy of Sciences , Beijing, 100190 China 2 Beijing , 100049 China 3 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics 4 Department of Physics, School of Mathematics and Physics, Bohai University 5 Institute of High Energy Physics, Chinese Academy of Sciences 6 Open Access , c The Authors lattices. We analyze the condition for the existence of a critical temperature at which the charged scalar field will condense. In contrast to the holographic superconductor on ionic lattices, the presence of Q-lattices will suppress the condensate of the scalar field and lower the critical temperature. In particular, when the Q-lattice background is dual to a deep insulating phase, the condensation would never occur for some small charges. Furthermore, we numerically compute the optical conductivity in the superconducting regime. It turns out that the presence of Q-lattice does not remove the pole in the imaginary part of the conductivity, ensuring the appearance of a delta function in the real part. We also evaluate the gap which in general depends on the charge of the scalar field as well as the Q-lattice parameters. Nevertheless, when the charge of the scalar field is relatively large and approaches the probe limit, the gap becomes universal with g ' 9Tc which is consistent with the result for conventional holographic superconductors. ArXiv ePrint: 1410.6761 Gauge-gravity correspondence; Holography and condensed matter physics 1 Introduction 2 3 4 The holographic setup The optical conductivity Superconductivity over Q-lattices dual to a metallic phase Superconductivity over Q-lattices dual to an insulating phase The Gauge/Gravity duality has provided a powerful tool to investigate many important phenomena of strongly correlated system in condensed matter physics. One remarkable achievement is the building of a gravitational dual model for a superconductor [13]. This construction has been extensively investigated in literature and more and more evidences in favor of this approach have been accumulated. In particular, the holographic lattice technique proposed recently has brought this approach into a new stage to reproduce quantitative features of realistic materials in experiments [46]. One remarkable achievement in this direction is the successful description of the Drude behavior of the optical conductivity at low frequency regime [424] and the exhibition of a band structure with Brillouin zones [25, 26]. Inspired by holographic lattice techniques people have also developed numerical methods to construct spatially modulated phases with a spontaneous breaking of the translational invariance [2735]. The original holographic lattice with full backreactions is simulated by a real scalar field or chemical potential which has a periodic structure on the boundary of space time [46]. We may call these lattices as scalar lattice and ionic lattice, respectively. This framework contains one limitation during the course of application. Namely, the numerical analysis involves a group of partial differential equations to solve, while its accuracy heavily depends on the temperature of the background which usually are black hole ripples. Thus, it is very challenging to explore the lattice effects at very low or even zero temperature(for recent progress, see [36]). Very recently, another much simpler but elegant framework for constructing holographic lattices is proposed in [37], which is dubbed as the Q-lattice, because of some analogies with the construction of Q-balls [38].1 In this framework, one 1Another sort of simpler holographic models with momentum relaxation can be found in [21, 22], where a family of black hole solutions is characterized only by T and k, while the parameter representing the lattice amplitude in Q-lattice is absent, thus no metal-insulator transition at low temperature. only need to solve the ordinary differential equations to compute the transport coefficients of the system, thus numerically one may drop the temperature down to a regime which may exhibit some new physics. Indeed, one novel feature has been observed in this framework. It is disclosed that black hole solutions at a fairy low temperature may be dual to different phases and a metal-insulator transition can be implemented by adjusting the parameters of Q-lattices [37, 3941]. Another advantage of Q-lattice framework is that the charge density as well as the chemical potential on the boundary can still be uniformly distributed even in the presence of the lattice background, which seems to be closer to a practical lattice system in condensed matter physics. However, in the context of ionic lattice the presence of the lattice structure always brings out a periodically distributed charge density and chemical potential on the boundary, which looks peculiar from the side of the condensed In this paper we intend to investigate the Q-lattice effects on holographic superconductor models. We will show that in general the presence of Q-lattices will suppress the condensation of the scalar field and lower the critical temperature, which is in contrast to the holographic superconductor on ionic lattices, where the critical temperature is usually enhanced by the lattice effects [6, 42]. In particular, when the black hole background is dual to a deep insulating phase, the condensation would never occur for some small charges. Furthermore, we will numerically compute the optical conductivity in the superconducting regime. It turns out that the Q-lattice does not remove the pole in the imaginary part of the conductivity, implying the appearance of a delta function in the real part and ensuring that the superconductivity is genuine and not due to the translational invariance. We also the parameters of Q-lattices. We organize the paper as follows. In next section we present the holographic setup for the superconductor model on Q-lattice, and briefly review the black hole backgrounds which are dual to metallic phases and insulating phases, respectively. Then in section three we will analyze the instability of these solutions and numerically compute the critical temperature for the condensate of the charged scalar field. The optical conductivity in the direction of the lattice will be given in section four and the gap will be evaluated as well. We conclude with some comments in section five. The holographic setup Recently various investigations to the inhomogeneous effects or lattice effects on holographic superconductors have been presented in literature [8, 4250], but these effects are almost treated perturbatively and the full backreaction on the metric is ignored. As far as we know, the first lattice model of a holographic superconductor with full backreaction is constructed in [6], in a framework of ionic lattice. Here, inspired by the recent work on Q-lattices in [37], we will construct an alternative lattice model of holographic superconductor closely following the route presented in [6]. As the first step we will construct the simplest model with the essential ingredients in this paper, but leave all the other possible constructions for further investigation in future. We start from a gravity model with two complex scalar fields plus a U(1) gauge field in four dimensions. If we work in unit in which the AdS length scale L = 1, then action is S = 4 g R + 6 2 1 F F + 2 |( ieA)|2 ||2 m2||2 , charged under the Maxwell field and will be responsible for the spontaneous breaking of the U(1) gauge symmetry and the formation of a superconducting phase. For convenience, S = 1 Z 4 g R + 6 2 1 F F + (2 e2AA)2 ()2 ||2 m2||2 , 3 + tric Reissner-Nordstrom-AdS (RN-AdS) black hole solutions on Q-lattice which have been constructed in [37]. This is a three-parameter family of black holes characterized by the chemical potential of the dual field theory and can be treated as the unit for the grand canonical system. The ansatz for the Q-lattice background is ds2 = z2 (1 z)p(z)U dt2 + (1 z)p(z)U solutions can be obtained by setting a non-trivial boundary condition at infinity for the It is shown in [37] that at low temperature the system exhibits both metallic and insulating phases. In metallic phase the conductivity in the low frequency regime is subject to the Drude law and the DC conductivity climbs up with the decrease of the temperature, while in insulating phase one can observe a soft gap in the optical conductivity and the DC conductivity goes down with the temperature. Numerically, one finds a small lattice phase. In next section we will discuss the condensate of the charged scalar field over Q-lattice background. In this section we construct a Q-lattice background with a condensate of the charged scalar unstable by estimating the critical temperature for the formation of charged scalar hair. For to find static normalizable mode of charged scalar field on a fixed Q-lattice background. As argued in [6], it is more convenient to turn this problem into a positive self-adjoint eigenvalue problem for e2, thus we rewrite the equation of motion as the following form Before solving this equation numerically, we briefly discuss the boundary condition for from the beginning, such that its asymptotical behavior at infinity is paper. Now imposing the regularity condition on the horizon and requiring the scalar field to decay as in (3.2), one can find the critical temperature for the condensate of the scalar field by solving the eigenvalue equation (3.1) for different values of the charge e. Our results are shown in figure 1. There are several curves on this plot, and each of them denotes the interesting to compare our results here with those obtained in the ionic lattice model [6]. As expected, each curve shows a rise in critical temperature with charge, which is consistent with our intuition that the increase of the charge make the condensation easier, thus the critical temperature becomes higher. This tendency is the same as that in [6]. For a given charge, we find that increasing the lattice amplitude lowers the critical temperature, which means that the condensate of the scalar field is suppressed by the presence of the Q-lattice. Later we will find this tendency can be further confirmed by plotting the value of the condensate as a function of temperature. Such a tendency is contrary to what have been found in ionic lattice and striped superconductors, where the critical temperature is enhanced by the lattice effects [6, 42]. Preliminarily we think this discrepancy might come from the different behaviors of the chemical periodic while in Q-lattice model all the fields does not manifestly depend on x except At the zero temperature limit, not all the curves have a tendency to converge to the same point as depicted in ionic lattices [6]. On the contrary, we find when the amplitude of the lattice is large enough (which may correspond to an insulating phase before the occurrence of the condensation), these lines do not converge at least at the temperature regime that our numerical accuracy can reach (see the inset of figure 1). It implies that for a given charge if its value is relatively small, the system with large lattice amplitude would not undergo a phase transition no matter how low To see more details on the dependence of the critical temperature on the lattice parameters, phase transition occurs more easily in the region with small lattice amplitudes but large wavenumbers, which corresponds to the metallic phase before the transition (as is shown in the left plot of figure 2). For a given wavenumber, the condensate becomes harder with the increase of the lattice amplitude, as illustrated in the right plot of figure 2. While for a given lattice amplitude, the condensate becomes harder with the decrease of the wavenumber (or with the increase of the lattice constant). Having found the critical temperature in a perturbative way, next we will solve all the coupled equations of motion in eq. (2.3) to find Q-lattice solutions with a scalar hair at c 2T/ 40 O e< 20 ( amplitude. The left plot is shown in the unit of the chemical potential while the right illustrates iteration. We plot the value of the condensate as a function of the temperature in figure 3. From this figure it is obvious to see that the critical temperature for the condensation goes down when the lattice amplitude increases. Such a tendency also implies that the condensation would never occur when the amplitude is large enough and beyond some critical value. Moreover, from the right plot in figure 3 we find the expectation value of the condensate becomes much larger in the unit of the critical temperature, implying a larger In the end of this section we address the issue of the relation between the value of the condensate and the charge of the scalar field. It is known in literature that when the back-reaction is taken into account, the expectation value of the condensate which may be has been testified in various Einstein gravity models with translational invariance. The presence of the ionic lattice drops the critical charge down to a lower value and it is found 2It corresponds to e ec ' 3 in the convention adopted in [2, 3]. that even with e ' 2, a gap with 8Tc can be reached [6]. Now for Q-lattice background, we plot the condensate as a function of the temperature for different values of the charge Firstly, we have tested that this value, as found in literature, is still universal in the probe limit (namely e parameters. Secondly, in comparison with the previous models we find the critical value of Qualitatively, we find the larger the lattice amplitude is, the larger the critical charge ec. This tendency is consistent with the fact that the presence of Q-lattice suppresses the condensate of the scalar field. The optical conductivity Now we turn to compute the optical conductivity in the direction of lattice. It turns out that it is enough to consider the following consistent linear perturbation over the Q-lattice As stressed in [37], htx, ax and are real functions of (t, z) such that the perturbation equations of motion will be real partial differential equations. Moreover, we suppose the to three ordinary differential equations for htx(z), ax(z) and (z). Before solving these of the system, while in previous literature one has a fixed charge density and takes its square root as the unit. Thus Tc and Tc have different units. We have checked that our results are consistent with those in literature indeed once we change the units. condition is to guarantee what we extract on the boundary for the dual field is just the current-current correlator, as investigated in [37].4 The optical conductivity is given by zax(0) Superconductivity over Q-lattices dual to a metallic phase In this subsection we discuss the optical conductivity as a function of the frequency over a Q-lattice background which is dual to a metallic phase before the phase transition. For subsection. We show the real and imaginary parts of the conductivity as a function of frequency for various charges in figure 5. Our remarks on the behavior of the optical conductivity as a function of frequency can be listed as follows. Superconductivity. First of all, in all plots we notice that once the temperature falls below the critical temperature, the imaginary part of the conductivity will not be lattice does not remove the delta function in the real part of the conductivity below the critical temperature, confirming that it is dual to a genuine superconductor. DC conductivity due to normal fluid. The real part of the conductivity will rise at the low frequency regime as well due to the lattice effects, indicating that there is a normal component to the conductivity such that our holographic model resembles a two-fluid model. Moreover, we notice that the DC conductivity will go down at first with the decrease of the temperature and then rise up to a much larger value, which can become more transparent in a log-log plot as we show in figure 6. It means that normal component of the electron fluid is decreasing to form the superfluid component, but this normal component will not disappear quickly. The raise of the DC conductivity comes from the increase of the relaxation time, as we will describe below. In addition, when the charge becomes larger, we find that the DC conductivity starts to rise up at much lower T /Tc. Low frequency behavior. In low frequency region we notice that the conductivity exhibits a metallic behavior with a Drude peak even at much lower temperature. We may fit the data at low frequency with the following formula 4Such a boundary condition is obtained by requiring that the non-zero quantities of htx and on the boundary can be cancelled out by the diffeomorphism transformation generated by the vector field and invariant under this sort of diffeomorphism transformation, this boundary condition remains in the superconducting case. Im 4 ) 0.01 a fit to this equation near the critical temperature in figure 7 and plot the values of these parameters as a function of the temperature in figure 8. In figure 7 the imaginary part of the conductivity exhibits a sudden change in low frequency region lines are fits to eq. (4.3). 2 4 when the temperature drops through the critical point. From figure 8 we find that Ks which is related to the superfluid density increases as the temperature goes down and becomes saturated around T /Tc ' 0.6, while Kn which is related to the normal fluid density decreases rapidly below the critical temperature. However, the relaxation time does not have such a monotonous behavior. As the temperature goes down from the critical one, the relaxation time will decrease at first and then rise up quickly in low temperature region, which looks peculiar in comparison with other lattice models, where the relaxation time monotonously increases with the decreasing of the temperature. In particular, when the charge of the scalar field becomes large, the turning point moves to lower temperature region. Such a phenomenon might explain why we have a smaller DC conductivity at lower temperature as described above, since it is proportional to the relaxation time. But definitely, the issue of why the relaxation time becomes smaller at lower temperature calls for further understanding in the future. Finally, we remark that in log-log plot the Drude behavior can also be conveniently captured by a straight line with a constant slope as shown in figure 6, which has previously been described in p-wave superconductors as well [51, 52]. Next we are concerned with the energy gap of the superconductor in Q-lattice model, which may be evaluated by locating the minimal value of the imaginary part of the conducand 9.247Tc, respectively. Firstly, we find these values are comparable with the values of 0.00 0.0 0.20.4 0.6 T /Tc, where the derivative is with respect to T /Tc. The dashed black line in the last plot is for the holographic superconductor in the absence of the Q-lattice (Note that in this case the Drude law is absent, Kn is identified with the DC conductivity.). the condensate we obtained in the previous section and indeed they are close. Secondly, in zero temperature limit does depend on the lattice parameters as well as the charge of always exists a critical value ec for the charge such that the energy gap approaches the previous section on the condensate of the background. In literature another way to evaluate the energy gap is to fit the temperature depenIn this thermodynamical method one need to know the normal fluid density at first. Usually which has been testified in various holographic models [2, 3, 6]. However, in the context of Q-lattice we find such exponential behavior described by eq. (4.4) is not clearly seen. The data can not be well fit in the entire region. To see if the data would have an exponential behavior in any possible interval, we would better take an alternative plot as follows. If the exponential behavior would present in some region, then from eq. (4.4) one would find the the parameter a, where the derivative is with respect to T /Tc. Therefore, an exponential behavior like eq. (4.4) would be featured by a horizontal line in the plot of TTc versus T /Tc. Our results are shown in the second plot of figure 9. In Q-lattices we do not find such behavior in any temperature region. The value of TTc constant but varies with the temperature and is obviously below the value for a system without Q-lattice in zero temperature limit. In comparison, we notice that a holographic superconductor without Q-lattice does exhibit such behavior in low temperature region, we think this discrepancy may imply that the factor Kn might not be related to the density would be true when the effective mass of quasiparticles is also temperature dependent. This issue deserves further study in the future. In the end of this subsection we briefly address the issue of the scaling law at the mid-frequency regime. This issue has previously been investigated in both normal phase [4, 19, 37, 41, 53] and superconducting phase [6, 24]. It was firstly noticed in the context of scalar lattices and ionic lattices that in an intermediate frequency regime, the magnitude of the conductivity exhibits a power law behavior as This rule has been testified in various models and in particular, its similarities with the Cuprates in superconducting phase are disclosed. However, later it is found that such a power law is not so robust in other lattice models. In particular, it was pointed out that in the context of Q-lattices there is no evidence for such an intermediate scaling [37]. Now for the holographic superconductors in Q-lattices, we may treat it in a parallel way and our result is presented in figure 10. From this figure, we find the intermediate scaling law is not manifest. Superconductivity over Q-lattices dual to an insulating phase In this subsection we briefly discuss the superconductivity over a Q-lattice which is dual to an insulating phase before the phase transition.5 We present a typical example with 5We mean the Q-lattice would exhibit an insulating behavior in zero temperature limit in the absence of the charged scalar field. ) (0.010 e In comparison with the superconductors over the Q-lattice in metallic phase, we present some general remarks as follows. With the decrease of the temperature, DC conductivity goes down at first and rises up again, which shares the temperature-dependence behavior with the case dual to the metallic phase. At low frequency, the lattice effects drive the normal electron fluid to deviate from Drude relation and exhibit an insulating behavior. This can be seen manifestly in the log-log plot in which the straight line with a constant slope becomes shorter at lower temperatures, as illustrated in the last plot of figure 11. The energy gap has the same universal behavior in the probe limit, namely Discussion In this paper we have constructed a holographic superconductor model on Q-lattice background. We have found that the lattice effects will suppress the condensate of the scalar field and thus the critical temperature becomes lower in the presence of the lattice. In particular, when the Q-lattice background is dual to a deep insulating phase, the condensate would never occur when the charge of the scalar field is relatively small. This is in contrast to the results obtained in the context of ionic lattice and striped phases, where the critical temperature is enhanced by the lattice effects. In superconducting phase it is found that the lattice does not remove the pole of the imaginary part of the conductivity, implying the existence of a delta function in the real part. The energy gap, however, depends on the lattice parameters and the charge of the scalar field. Nevertheless, in the probe limit, is consistent with our knowledge on other sorts of holographic superconductor models. For convenience we have only computed the optical conductivity of a superconductor where the Q-lattice background is dual to a typical metallic phase or a typical insulating phase prior to the condensation. In practice, many interesting phenomena have been explored by condensed matter experiments in the critical region where metal-insulator transition occurs in zero temperature limit. From this point of view one probably shows more interests in the superconducting behavior of the Q-lattice model with critical parameters This simplest model of holographic superconductors on Q-lattice can be straightforwardly generalized to other cases. For instance, we may consider to input Q-lattice structure in two spatial directions with anisotropy [54]. We may also construct the superconductor models on Q-lattice in other gravity theories, such as the Gauss-Bonnet gravity and Einstein-Maxwell-Dilaton gravity. These works deserve further investigation. This work is supported by the Natural Science Foundation of China under Grant Nos. 11275208, 11305018 and 11178002. Y.L. also acknowledges the support from Jiangxi young scientists (JingGang Star) program and 555 talent project of Jiangxi Province. J.P. Wu is also supported by Program for Liaoning Excellent Talents in University (No. LJQ2014123). Open Access. 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Yi Ling, Peng Liu, Chao Niu, Jian-Pin Wu, Zhuo-Yu Xian. Holographic superconductor on Q-lattice, Journal of High Energy Physics, 2015, 59, DOI: 10.1007/JHEP02(2015)059