#### 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.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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