Higher derivatives driven symmetry breaking in holographic superconductors

Hai-Li Li1,∗ Guoyang Fu2,† Yan Liu1,‡ Jian-Pin Wu2,3,§ and Xin Zhang1,4,5¶ 1 Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China 2 Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, Yangzhou 225009, China 3 School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China 4 Ministry of Education’s Key Laboratory of Data Analytics and Optimization for Smart Industry, Northeastern University, Shenyang 110819, China 5 Center for High Energy Physics, Peking University, Beijing 100080, China Abstract In this paper, we construct a novel holographic superconductor from higher derivative (HD) gravity involving a coupling between the complex scalar field and the Weyl tensor. This HD coupling term provides a near horizon effective mass squared, which can violates IR BreitenlohnerFreedman (BF) bound by tuning the HD coupling and induces the instability of black brane such that the superconducting phase transition happens. We also study the properties of the condensation and the conductivity in the probe limit. We find that a wider extension of the superconducting energy gap ranging from 4.6 to 10.5 may provide a novel platform to model and interpret the phenomena in the real materials of high temperature superconductor.


I. INTRODUCTION
Based on AdS/CFT (Anti-de Sitter/Conformal Field theory) correspondence [1][2][3][4], a holographic superconductor model is suggested in [5]. In this model, a complex charged scalar field is introduced in Schwarzschild-AdS (SS-AdS) black brane to spontaneously break the U(1) symmetry and transform to a charged scalar hair black brane. The charged scalar field in the bulk is dual to the "Cooper pair" operator at the boundary and the vacuum expectation value is the order parameter. The symmetry breaking is introduced by a negative mass squared of the scalar field, which is allowed due to the Breitenlohner-Freedman (BF) bound in AdS spacetime [5,6]. In [7], a positive mass squared situation was studied. It was shown that as m 2 increase the phase space folds due to the non-linearity of the equations of motion, so the two nearby points in the phase space can represent symmetry breaking. And they show that for a small positive mass squared, the results is not much different from the negative case which has been studied in [8][9][10][11][12].
In [13], they discuss the superconductivity instability by studying the normalisable solution of equations of motion (EOMs) for the charged scalar field on top of RN-AdS black brane geometry. In essence, it is that the near horizon effective mass squared is below the AdS 2 BF bound, which induces the instability. At the same time, we also require that the mass squared is above the boundary AdS 4 BF bound, which guarantees the stability of scalar field at the boundary. In this paper, we construct a novel holographic superconductor model by introducing a higher derivative (HD) term, which couples the scalar field and Weyl tensor. This HD term provides a near horizon effective mass squared but doesn't modify the boundary AdS 4 BF bound.
The superconducting energy gap is an important characteristic of superconductor models.
In the weakly coupled BCS theory, this value is 3.5. In the usual holographic superconductor model [5,11,14], this value is approximately 8, which is more than twice the one in the BCS theory, but roughly approximates the value measured in high temperature superconductor materials [15]. The HD term introduced in holographic superconductor model drives the superconducting energy gap running, ranging from 5.5 to 16.2 [16][17][18][19][20] 1 . In this paper, we 1 The holographic superconductor models from HD theory coupling Weyl tensor are also constructed in [21][22][23][24][25][26][27][28]. But the study of conductivity is absent. Also, the extension of superconducting energy gap is also observed in other holographic superconductor models from the Gauss-Bonnet gravity [29,30] and the also study the properties of the conductivity of our present model and in particular the running of the superconducting energy gap.
Our paper is organized as what follows. We will introduce the framework of holographic superconductivity in Section II. In Section III, we make a deep analysis for the instabilities.
Then we move on to investigate the condensation in Section IV. In Section V we give the results of the conductivity. Conclusion and discussion are given in Section VI.

II. HOLOGRAPHIC FRAMEWORK
Our starting point is the following actions derivative and Ψ is the charged complex scalar field with mass m and the charge q of the Maxwell field A. We can write Ψ = ψe iθ with ψ being a real scalar field and θ a Stückelberg field. And then, for convenience we choose the gauge θ = 0 in what follows. In the action S Ψ , a new interaction, which couples the complex scalar field to the Weyl tensor, is added in S Ψ . Since the pure AdS geometry is conformally flat, the Weyl tensor vanishes in the UV boundary. But C 2 provides a nontrivial source for the scalar in the bulk geometry and gives an effective mass of the scalar. This interaction provides a new mechanism of the symmetry breaking 2 , for which we shall do in-depth studies in this paper. In the action S A , the tensor X is . . . C µν α n−1 β n−1 with C µνρσ being the Weyl tensor. When X ρσ µν = I ρσ µν , the action S A reduces to the standard quasi-topological gravity [31,32]. But the value of the superconducting energy gap is always greater than the value of the standard version holographic superconductor. 2 This interaction for a neutral scalar field has been study in [33][34][35][36][37][38]. the gauge field. Starting from the third term, they are an infinite family of HD terms [39].

III. SUPERCONDUCTING INSTABILITY
When Ψ = 0, the system (1) achieves a charged black brane solution, which corresponds to the normal phase. In this section, we shall explore the instability condition for the normal phase towards the development of a hairy black brane under small charged scalar field perturbations. This allows one to determine the superconducting phase structure in the dual boundary field theory. The condition for the formation of the hairy black brane depends on the charge and the mass of the scalar field as well as the background which is specified by the model parameters. For clarity, we first study the role of α 1 term in the formation of the hairy black brane. And then, we further explore the joint effect from the α 1 term and the γ term.
A. Case I: α 1 HD term In this subsection, we first want to see what role the α 1 term plays in the formation of the hairy black brane. So we only turn on α 1 term and turn off other coupling parameters in this section. In this case, the background geometry of the normal phase is RN-AdS black brane, Here µ is the chemical potential of the field theory. The Hawking temperature is We can estimate the critical temperature for the formation of the superconducting phase by finding static normalizable modes of the scalar field ψ in the above background (3). The procedure has been used in [42][43][44][45]. This problem can be casted into a positive self-adjoint eigenvalue problem for q 2 and so we write the equation of motion for the scalar field as the following form: Without loss of generality, we set the mass of the charged scalar field as m 2 = −2 here 3 .
For this case, its asymptotical behavior at infinity is Here, we will treat ψ 1 as the source and ψ 2 as the expectation value in the dual boundary theory. Also we set ψ 1 = 0 such that the condensation is not sourced. Now we can numer- for different q and the right plot is the quantum phase transition (QPT) diagram in (α 1 , q) space. There are several characteristics are summarized as what follows: • For a given α 1 , we find that the critical temperature becomes higher with the increase of the charge (left plot in FIG.1), which means that the increase of the charge make the condensation easier. This tendency is consistent with our intuition and have been observed in [42,43].
• For a given charge, we find a rise in critical temperature with the coupling parameter FIG.1), which means that the higher derivative term α 1 plays the role of driving the superconducting phase transition.
• From the left plot in FIG.1, we see that for a given charge, if the coupling parameter α 1 is relatively small, the system would not undergo a superconducting phase transition no matter how low the temperature. It means that there is a QPT at zero temperature.
We show the QPT diagram in (α 1 , q) space in the right plot in FIG.1. The blue line is the QPT critical line and the blue zone is the superconducting phase at zero 3 We have set L = 1. temperature. The QPT can be also understood by the BF bound. We shall further address this problem in what follows. Now, we shall analyze the superconducting phase transition at zero temperature by BF bound. To have a superconducting phase transition, the near horizon effective mass squared shall be below the corresponding BF bound, but the UV boundary effective mass squared shall be above the boundary AdS 4 BF bound, which guarantees the stability of scalar field at the boundary.
For the RN-AdS black brane, the extremal limit can be arrived at when µ = 2 √ 3. At this limit, the near horizon geometry and the gauge field are [46] where L 2 ≡ L/ √ 6. In deriving the above equations, we have made the transformation ζ and t = −1 τ in the limit → 0 with finite ζ and τ . It is obvious that at the zero-temperature limit, the near horizon geometry is AdS 2 × R 2 with the AdS 2 curvature radius L 2 .
The effective mass of this system is Since q 2 A 2 = q 2 g tt A 2 t , which contributes with minus sign, it tends to destabilize the normal state and induces the superconductivity. While the role plays α 1 term depends on the sign of α 1 , which can stabilize or destabilize the normal state. To have a superconducting phase transition, we require that the near horizon effective mass squared shall be below the corresponding BF bound, but the UV boundary effective mass squared shall be above the corresponding BF bound. Thus we have where m 2 IR and m 2 U V denote the near horizon effective mass squared and the bulk mass squared, respectively. Using the expression of the effective mass squared (8) combining with the near horizon geometry (7) and the AdS 4 geometry, the above equations can be specifically expressed as It has been well studied that for the usual holographic superconductor (α 1 = 0 here), the superconducting instabilities are determined by both the bulk mass squared m 2 and the charge q, equivalently, the chemical potential, which can be clearly seen from the above instability condition (10) and have been explored in [7,13]. Here, we also exhibit the allowed region (blue zone) for the superconducting phase transition with α 1 = 0 at m 2 vs q plane at the zero temperature (left plot in FIG.2). For small q, the superconducting phase transition is forbade if the mass squared is positive. But provided the charge is large, the superconducting phase transition still happen for m 2 = 0 or even positive m 2 , which is called the density driven symmetry breaking in holographic superconductor in [7].
Subsequently, we turn to explore what role α 1 plays. We plot the α 1 as the function of q (i.e., QPT diagram) for m 2 = −2 at zero temperature, which is shown in the right plot in can infer that for small q, the superconducting phase transition still happen for m 2 = 0 or even positive m 2 , provided that α 1 is large. It is just the role α 1 plays and we call the HD driven symmetry breaking in holographic superconductor, which is the focus of our present paper and we shall thoroughly explore this issue in next section.

B. Case II: γ HD term
In this subsection, we simultaneously turn on α 1 and γ. Since γ term involves the coupling between the Weyl tensor and gauge field, the background geometry of the normal phase is no longer a RN-AdS black brane. We need to solve a set of third order differential equations to obtain the background solution. It is a hard task. However, we can obtain the perturbative solution up to the first order of γ as [47] ds 2 = f (u) For this black brane, we can obtain the dimensionless Hawking temperatureT ≡ T /µ with When γ → 0, µ = 2 √ 3, which reduces the case of RN-AdS black brane. At extremal limit, the near horizon geometry of the perturbative solution (11) is also AdS 2 × R 2 as RN-AdS black brane but with a different AdS 2 curvature radius L 2 as which explicitly dependens on the Weyl parameter γ.
Following the same procedure in the above subsection, in this case, the constraint on the the near horizon effective mass squared gives For convenience, in the above equation, we have defined w as In addition, the bulk mass squared satisfies Eqs. (15) and (17) give the conditions that the superconducting phase happens at zero temperature. We summarize the roles α 1 and γ play in the superconducting phase transition as what follows.
• In previous subsection, we have observed that for γ = 0, in the phase diagram  • FIG.7 exhibits 3D plot of the allowed region for the superconducting phase transition at the zero temperature for a fixed α 1 . When α 1 is negative, there is a forbidden region of superconducting phase transition in the phase diagram (m 2 , q, γ). As α 1 increases, this region shrinks and α 1 is large, this region vanishes.
In this section, we have made the superconducting instability analysis, from which we clearly see what roles of the HD terms α 1 and γ play in the superconducting phase transition.
In next section, we shall numerically solve the system (1) to study the superconducting phase transition. However, it is hard to solve the system (1) with backreaction because it involves solving a set of third order differential equations with high nonlinearity. Therefore, we shall work in the probe limit, i.e., we don't consider the backreaction of the gauge field and the scalar field on the geometry.

IV. CONDENSATION
In the probe limit, we consider the metric as which is the SS-AdS black brane. u = 1 denotes the horizon and u = 0 is the asymptotically AdS boundary. The Hawking temperature of this system is T = 3/4π. And then, the EOMs of gauge field and scalar field can be derived as The ansatz for the scalar field and gauge field is taken as Under the above ansatz (Eqs. (18) and (20)), the EOMs for ψ and φ (Eq. (19)) can be explicitly expressed as follows, Here, we only consider the HD terms up to 6 order. Regardless of any details of the above EOMs, the asymptotical behaviors of φ and ψ at the conformal boundary are In the dual boundary field theory, µ and ρ are the chemical potential and the charge density, respectively, as has been mentioned above. This system is depicted by a dimensionless quantityT ≡ T /µ. We take standard quantization and so ψ 1 is treated as the source and ψ 2 as the expectation value of the scalar operator, for which we denote O 2 . We expect that the condensation is not sourced and so we set ψ 1 = 0. Subsequently, we numerically solve EOMs (21) by the shooting method and study the properties of the condensation O 2 with HD derivative terms 4 . We shall firstly study the condensation by considering the scalar field with negative mass, m 2 = −2, which has been well studied in the usual holographic superconductor model.
FIG .8 shows the condensation O 2 /T c as a function of the temperature T /T c for different α 1 . We observe that as α 1 decreases, the condensation value becomes much larger. It indicates that a larger superconducting energy gap ω g /T c emerges, which shall be explicitly addressed in the following study of optical conductivity. Furthermore, we find that with the increase of the α 1 , the critical temperature T c goes up (see TABLE I  In previous works [16,19], we have studied the holographic superconductor from the coupling between Weyl tensor and gauge field. We find that the condensation value at low temperature runs and the critical temperature T c changes as the coupling parameters γ or γ 1 . The tendency is consistent with that from the instability analysis in Section III. Here we further explore the joint effect from α 1 term and γ or γ 1 term. We find that though α 1 term and γ or γ 1 term modify the EOMs (21) by different ways, the joint effect augments the running range of the condensation (see FIG.9). In addition, for fixed γ or γ 1 term, the critical temperatureT c goes up as α 1 increases (see TABLE I and TABLE II). The result further confirms that the conclusion on the superconducting phase diagram (m 2 , q, γ) in Section III B.

V. CONDUCTIVITY
In this section, we study the optical conductivity of our holographic system with HD terms. For this, we turn on the perturbation of the gauge field along x direction as . And then we can derive the following perturbation equation We can numerically solve the above equation with the ingoing boundary condition. Once the solution is at hand, we can read off the conductivity in terms of There is no doubt that as the standard version of holographic superconductor model [11], In previous works [16][17][18][19], the authors have found that after introducing the coupling between the Weyl tensor and the gauge field, the ratio of the superconducting energy gap frequency over critical temperature ω g /T c ranges from about 5.5 to 16.2. Here we would also like to explore the running of the superconducting energy gap after the coupling term α 1 , which is the coupling between the complex scalar field and the Weyl tensor. From   FIG.10, we clearly see that the superconducting energy gap runs with the parameter α 1 .
Quantitatively, it ranges from about 7.9, which is less than the value of the standard version holographic superconductor in [11], to 9.5, which is beyond the value of the standard ver-

VI. CONCLUSION AND DISCUSSION
In this paper, we construct a novel holographic superconductor from HD gravity, for which we introduce a coupling between the complex scalar field and the Weyl tensor. The α 1 coupling term provides a near horizon effective mass squared but doesn't modify the boundary AdS 4 BF bound. The instability analysis indicates that a quantum phase transition can be triggered by tuning the coupling parameter α 1 . In particular, even for the  positive mass squared, the superconducting phase transition also happens by tuning the α 1 .
Therefore, the α 1 HD term plays the role of driving the symmetry breaking and results in the superconducting phase transition. We also explore the instability from the HD terms α 1 and γ, which involves the coupling between the Weyl tensor and gauge field. γ coupling term modifies the near horizon AdS 2 curvature and thus the near horizon effective mass squared, which provides a mechanism to result in the superconducting phase transition.
The properties of the condensation and the conductivity in our holographic model are also studied. Although in the probe limit, we have the same tendency as the instability analysis. We summarize the main properties of our present model as what follows: • For the positive α 1 , the condensation becomes easy. While for the negative α 1 , the result is just opposite.
• A wider extension of the superconducting energy gap, ranging from 4.6 to 10.5, is observed. We expect that our model provides a novel platform to model and interpret the phenomena in the real materials of high temperature superconductor.
As a novel mechanism, there are some interesting topics deserving further pursuit in future.
• We want to explore whether there is the phenomena of the phase space folding as Ref.
[7] due to the introduction of α 1 term.
• The Homes' law can be observed in the holographic superconductor model from HD theory [19], which involves the coupling between Weyl tensor and the gauge field. It is interesting to explore this issue in our present model.
• The transports at full momentum and energy spaces can provide far deeper insights into the holographic system than that at the zero momentum [48,49]. In future, we shall study this issue in the framework of our present model.