# A holographic model of -wave superconductor vortices with Lifshitz scaling

###### Abstract

We study analytically the -wave holographic superconductors with Lifshitz scaling in the presence of external magnetic field. The vortex lattice solutions of the model have also been obtained with different Lifshitz scaling. Our results imply that holographic -wave superconductor is indeed a type II one even for different Lifshitz scaling. This is the same as the conventional -wave superconductors in the Ginzburg-Landau theory. Our results also indicate that the dynamical exponent has no effect to the shape of the vortex lattice even after higher order corrections (away from the phase transition point ) are included. However, it has effects on the upper critical magnetic field through the fact that a larger results in a smaller and therefore influences the size (characterized by ) of the vortex lattices. Furthermore, close comparisons between our results and those of the Ginzburg-Landau theory reveal the fact that the upper critical magnetic field is inversely proportional to the square of the superconducting coherence length , regardless of the anisotropy between space and time.

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## I Introduction

The gauge-gravity duality ads/cft ; gkp ; w offers a very promising way to explore the possible dynamics of strongly interacting matter in field theory. It provides a well-established method for calculating correlation functions in a strongly interacting field theory in terms of a dual classical gravity description. While its relevance to any specific strong coupling system that can realized in the laboratory is not well-understood, it nonetheless provides a window through which we might hope to obtain insight into the properties of some condensed matter systems that defy description by traditional approaches. One of the unsolved mysteries in modern condensed matter physics is the mechanism of the high temperature superconductors cuprates(HTSC). These materials are layered compounds with copper-oxygen planes and are doped Mott insulators with strong electronic correlations which the pairing symmetry is unconventional and there is a strong experimental evidence showing that it is the -wave superconductor. This makes the -wave superconductor particularly attractive for physicists.

It was Gubser who first noticed that by coupling the Abelian Higgs model to gravity with a negative cosmological constant, one can find solutions that spontaneously break the Abelian gauge symmetry via a charged complex scalar condensate near the horizon of the black holegub1 ; gub2 . This model exhibits the key properties of superconductivity: a phase transition at a critical temperature, where a spontaneous symmetry breaking of a gauge symmetry in the bulk gravitational theory corresponds to a broken global symmetry on the boundary, and the formation of a charged condensate. Based on this observation, Hartnoll et al proposed a holographic model for -wave superconductors by considering a neutral black hole with a charged scalar and the Maxwell fieldhorowitz . Since then this correspondence has been widely explored in order to understand several crucial properties of these holographic superconductors (see Ref. HH for reviews). The gravitational model that dual to the -wave superconductors was proposed in wen ; be where the complex scalar field for the -wave model is replaced by a symmetric traceless tensor.

One of the major characteristic properties of superconductors is that they expel magnetic fields as the temperature is lowered through the critical temperature. In the presence of an external magnetic field, ordinary superconductors may be classified into two categories, namely type I and type II. It was found that at , magnetic field expels the wave condensation for holographic -wave superconductormps ; Maeda2 , holographic -wave superconductor murray , and holographic -wave superconductorarXiv:1006.5483 as well, along with the formation of Abrikosov vortices. This indicates that these holographic models of superconductor belong to type II ones. However, all these holographic models were constructed only in the relativistic spacetimes. Thus we wonder whether the holographic -wave superconductor still be the type II one in non-relativistic spacetimes, for example, Lifshitz spacetime, which is our main motivation in this paper.

It is often observed that the behaviors of many condensed matter systems are governed by Lifshitz-like fixed points. These fixed points are characterized by the anisotropic scaling symmetry

where is called the dynamical critical exponent and it describes the degree of anisotropy between space and time. The nonrelativistic nature of these systems makes the dual description different and a gravity dual for such systems can be realized by nonrelativistic CFTs dtson ; kbm ; wdg ; taylor . Recently, Bu used the nonrelativistic AdS/CFT correspondence to study the holographic superconductors in the Lifshitz black hole geometry for in order to explore the effects of the dynamical exponent and distinguish some universal properties of holographic superconductors Bu . It is found that the Lifshitz black hole geometry results in different asymptotic behaviors of temporal and spatial components of gauge fields compared to those in the Schwarzschild-AdS black hole, which brings some new features of holographic superconductor models. More recently, Lu discussed the effects of the Lifshitz dynamical exponent on holographic superconductors and gave some different results from the Schwarzschild-AdS background Lu . To date, there have attracted considerable interest to generalize the holographic superconducting models to nonrelativistic situations CaiLF ; JingLF ; Sin ; Brynjolfsson ; MomeniLifshitz ; Schaposnik ; Abdalla ; Tallarita .

In this paper,we analytically study the spatially dependent equations of motion for the -wave holographic superconductor with Lifshitz scaling when the added magnetic field is slightly below the upper critical magnetic field. We want to distinguish the effects of the dynamical exponent to the vortex lattice and explore the behavior of the upper critical magnetic field. In particular, according to the Ginzburg-Landau (GL) theory, it should be noted that the upper critical magnetic field has the well-known relation Poole . A number of attempts have been made to investigate the effects of applying an external magnetic field to holographic dual models SetareEPL ; GeBackreac ; Montull ; Gao ; Ged ; Roychowdhury ; Momeni ; Roychowdhury2 ; Cui ; RoychowdhuryJHEP2013 ; Cai2013 ; Gangopadhyay ; Lala . All these papers are made in relativistic situations. It is therefore very natural to consider the nonrelativistic situations, such as Lifshitz black hole. Furthermore, we constructed the vortex lattice solution, or the Abrikosov lattice which is characterized by two lattice parameters, and , perturbatively near the second-order phase transition. There is an observationLala that the dynamical exponent has no effect to the shape of the vortex for wave superconductor. In this paper, one of main motivation is to see whether it is still correct for the wave superconductor. In addition, we would pay much attention to see this point a little bit away from the second-order phase transition point. Our result is very interesting, it seems that the exponent does not influence the shape of the lattice for higher order corrections.

The organization of this paper is as follows. In section 2, we will study the -wave holographic superconductors with Lifshitz scaling. In section 3 we investigate the properties of the holographic superconductors with Lifshitz scaling in an external magnetic field. Section 4 is devoted to the construction of vortex solution of the -wave model and to show that the dynamical exponent does not influence the shape of the vortex. And we will conclude in the last section of our main results.

## Ii The -wave holographic superconductor models with Lifshitz scaling

In this section we first give the spatial dependent equations of motion for the -wave model in the presence of a uniform magnetic field, then we will study the condensate solution and discuss the critical temperature.

### ii.1 Holographic -wave superconductor: the model

The action of the -wave superconductor in dimensions is the following^{1}^{1}1In principle, it is possible to generalize our analysis to higher dimensions.wen

(1) | |||

(2) |

where is a symmetric traceless tensor, is the Ricci scalar, is the negative cosmological constant with the AdS radius, and is the gravitational coupling. is the covariant derivative, and are the charge and mass squared of , respectively.

Working in the probe limit in which the matter fields do not backreact on the metric and taking the planar Lifshitz-AdS ansatz, the black hole metric reads:

(3) |

where the metric coefficient

(4) |

and is the horizon radius of the black hole. The Hawking temperature of the black hole is . Setting , the metric can be rewritten in the form

(5) |

where .

The -wave superconductors is translational invariant on the plane and condensate on the boundary, while the rotational symmetry is broken down to due to the condensate change its sign under a rotation on the plane. To fulfill these features we take the following ansatzwen

(6) |

where we keep a nonvanishing so as to have an external magnetic field along direction.

After variation of the action with this ansatz, the equations of motion for the tensor field , the gauge field components and are given, respectively, by

(7a) | |||

(7b) | |||

### ii.2 Condensate in holographic -wave superconductors without external magnetic filed

The equations of motion (7)-(7) of the -wave superconductors are very similar to the -wave model and the matching method should be valid. In order to solve the above equations, let us impose the boundary condition near the horizon and in the asymptotic AdS region, respectively:

1). On the horizon , as usual, we must have so that is well defined and the other fields are regular.

2). At infinity , the solution of fields behaves like

(8a) | |||||

(8b) | |||||

(8c) |

where . The coefficients represents as the source of the dual operator and correspond to the vacuum expectation values of the operator that couples to at the boundary theory. BF bound requires (thus ) such that the term is a constant or vanishes on the boundary.

To solve the critical temperature with the spatial dependent equations of motion, we ignore the influence of the external magnetic field so as to get the equations of motion only with the reaction of radial coordinates:

(9a) | |||||

(9b) |

As we can see, the change of the equations does not affect the boundary conditions. We impose boundary condition in the following discussion. For clarify, we set and in this work.

It should be noted that Frobenius analysis of the equations of motion near the boundary reveals that for the case . For simplicity, we will not consider this case in the following studies.

Following the matching method applied in Pan , which expands the fields and near the horizon , reads off the expanded solutions from the equations of motion with the above boundary conditions, then matchs the asymptotic solutions at some intermediate point , in the end we obtain

(10) |

where

(11) |

and we have defined the critical temperature

(12) |

The parameter in (12) is given by

(13) | |||||

In order to avoid a breakdown of the matching method, we take the value of to ensure that is real and find the range of the matching point

(14) |

It is interesting to observe that the value of is smaller than when , which means that the value of is real all the while when we set the range of

(15) |

so it is convenient to use the range in this work. In addition, Eq. (12) implies that the larger dynamical exponent makes the condensation harder to form.

According to the AdS/CFT dictionary, near the critical temperature we can express the relation for the condensation operator as

(16) |

The analytic result shows that the phase transition of holographic superconductors with Lifshitz scaling belongs to the second order. It also indicates that condensation versus temperature have a square root behavior near , which suggests that the critical exponent is 1/2, as expected from the mean field theory. The Lifshitz scaling and spacetime dimension will not influence the resultLu .

In Fig.1, we visualize the condensate of the operator as a function of temperature with different dynamical exponent for the mass of the traceless tensor field . It is observed that corresponding to the lower critical temperature, the gap becomes increasingly smaller as increases than the results in Pan .

## Iii Effects of external magnetic filed on the holographic -wave superconductor

In this section we would like to study the effect of external magnetic field on the holographic superconductors with Lifshitz scaling. From the gauge/gravity correspondence, the asymptotic value of the magnetic field corresponds to a magnetic field added to the boundary field theory. Near the upper critical magnetic field , the tensor field can be regarded as a perturbation.

### iii.1 Perturbative expansion of the equations of motion

It is very difficult to exactly solve the above nonlinear coupled partial differential. However, as the magnetic field is slightly below the upper critical field , it is possible to solve them perturbatively, as what have done inMaeda2 . For this purpose, we introduce a small parameter , and the fields can be then expanded as follows

(17a) | |||||

(17b) | |||||

(17c) |

where . The zeroth order solution is

(18) |

where the rotational symmetry keeps unbroken and therefore it corresponds to the normal state. The magnetic field on the boundary is given by as expected.

Without loss of generality, we assume (with and some constants), then the equation of motion for reduces to

(19a) | ||||

(19b) |

where . The distribution of the order parameter on the plane is given by the solution of the transverse equation Eq. (19a). On the other hand, the radial equation Eq. (19b) determines superconducting phase transition.

### iii.2 The upper critical magnetic field

There is critical value above which Eq. (19b) only has vanishing solutions. As one lowers the magnetic field below , we lead to a phase transition. The maximum upper critical magnetic field is given by where takes the minimum value. Thus, we can express the equation of as

(20) |

Near boundary (), in (20) behaves like

(21) |

As before we let and set and in the following discussions.

Using the matching method just what we did in the last section, one can get from Eq.(20)

(22) |

in which and .

In order to get the external critical magnetic field which is very close to the critical magnetic field we find the solution

(23) |

with

(24) |

(25) |

As , we can express the critical magnetic field as

(26) | |||||

which has the same form as Pan . It is convenient to observe that there is a superconducting phase transition when at where

(27) |

This is equivalent to

(28) |

which is related to Lifshitz scaling but independent of the tensor field mass. In order to ensure the condition at , we calculate the Eq.(27) and Eq.(28) with the requirement of . As a consequence, we get

(29) |

What can be noted is that all of the is selected in the range with the situation so we can choose the matching point arbitrarily for this case. And the results also show that the allowable range of is restricted at the point when . So we clearly find that the range of the matching point depends on Lifshitz scaling and tensor field mass .

It is subtle for where the matching point leads to vanishing critical temperature as shown in (12) and (13), which implies a breakdown of the matching method. Our strategy is to matching the result by shifting the matching point to a small value from . However, cannot be arbitrarily small. As , the value of approaches to zero, which causes divergent. As an example, we choose the Lifshitz scaling with , and , the left graph of Fig.2 shows that there exists a breakdown of matching method when .

Keep this in mind, we should choose is large enough so as to keep finite. As another example, we set which can relax the breakdown when the matching point approaches the allowable region. The right graph of Fig.2 proves this point and shows that the critical magnetic field decreases as we amplify which is qualitatively in good agreement with the work of Pan . When we can have for different Lifshitz scaling which agrees well with the Ginzburg-Landau theory. And it is also noted that the dynamical exponent cannot modify the relation. Thus, for the case , the Ginzburg-Landau theory still holds in Lifshitz black hole.

## Iv Vortex lattice of the -wave superconductor.

Based on our previous observations in the last sections, in this section we would like to construct the vortex solution of -wave superconductor model, following the work Maeda2 . The main motivation is to see whether and how the dynamical exponent could impose its influences on the formation of the lattice.

### iv.1 Leading order solution

As a first step, we will consider the leading order () solution for the field . The next order corrections will be discussed in the next subsection. Our start point is Eq. (19a), whose solution of that satisfies the boundary conditions is the following

(30) |

where are the corresponding eigenvalues. It was noted in Maeda2 that a vortex lattice solution of Eq. (19a) can be constructed by linearly combining the lowest solution of (30) through

(31) |

where is the solution of radial equation (20) and is the droplet solution which is given by

(32) |

with .

It should be noted that the above solution (32) is very similar to the expression of the order parameter in GL theory for the type II superconductor in the presence of a magnetic field

(33) |

where is the superconducting coherence length, , and is the flux quantum. As a result, Eqs. (31) and (33) gives

(34) |

which is the same as the result of the GL theory. In the previous section we have near , which indicates that and is also the same as that of the GL theory.

Just like what has been made for the -wave model in Ref. Maeda2 , one can construct the vortex lattice from droplet solutions by considering the following superposition:

(35a) | |||

(35b) |

for arbitrary parameters and . The function can be written as the elliptic theta function and has translation invariance (up to a phase) in two directions and , and hence is called a vortex lattice. The area of a unit cell for this vortex lattice is , and the magnetic flux penetrating the unit cell is given by , implying quantization of the magnetic flux penetrating a vortex.

Fig.3 shows the configuration of in the plane for the rectangular lattice. In this plot we have chosen , , and for the Lifshitz scaling and respectively. These plots reveal a very interesting fact that the critical dynamical exponent cannot affect the shape of the lattice, instead it only changes the characteristic length (which is proportional to the coherence length of the superconductor) of the unit cell. Specifically, as shown in Fig. (4), the characteristic length (equivalently the coherence length ) increases with monotonously.

### iv.2 Higher order corrections

One of the most interesting result in the last subsection is that the dynamical exponent , at least in the leading order, dose not influence the droplet solution and hence the form of the vortex lattice. There has the same result in holographic wave superconductor as shown in Lala . However, though it is correct in the first order solution, whether it is still hold or not in higher order expansions is not clear. As a preliminary check, in this subsection we will pay our attention to the question to the next order, i.e., in expansions. To do so, we expand the fields in the way like (17) and write down the equations of motion to the order :

(36) |

where the zeroth order fields and have been obtained in the previous sections and the order solution is given by Eq. (35a). In addition, there are order fields and which can be obtained by solving the following differential equations:

(37a) | |||||

(37b) |

where . and as before

(i) We first note that the right sides of these two equations are independent of , implying both and are -independent.

(ii) Notice that the lattice has periodicity as in direction, we therefore expand and as a Fourier series in coordinate^{2}^{2}2Theoretically one cannot determine the lattice shape due to the free parameters and . For simplify, we would like to set in the following discussions. By doing so, we have chosen a rectangular lattice for the following consideration.
,

(38a) | |||||

(38b) |

where .

(iii) Using the Fourier series relations (47) and (A), the set of equations (37a) (37b) reduce to

(39a) | |||||

(39b) |

where prime denotes differentiate w.r.t. and again is the solution of (20). In general, it is not easy to analytically solve these differential equations. However, the asymptotic behaviour near the boundary is very straightforward

(40) |

(41) |

which are independent of and . Their behaviors far from the boundary can be obtained by solving Eqs. (40) numerically. One thing should be mention is that the coefficients in the Fourier series (38) are exponentially suppressed as a function of and . This point and the fact that lead to a consequence that one can neglect the terms proportional to in Eqs. (40), then they become

(42a) | |||||

(42b) |

The above two equations can be solved analytically. Let us denote them by and (their exact expressions are irrelevant to our present discussion). After insert them into (38), one obtains the full functions of and

(43a) | |||||

(43b) |

The function can be solved in the same way. We first expand as a double Fourier series in and

(44) |

Then we put Eq. (35a) and the series (38) into Eq.(IV.2), making use of the relations of the infinite sum of Gaussians and the infinite sum of exponential in the Appendix, neglecting the terms proportional to and , then we lead to the following differential equation for in the radial coordinate