Zeroth Order Phase Transition in a Holographic Superconductor with Single Impurity

We investigate the single normal impurity effect in a superconductor by the holographic method. When the size of impurity is much smaller than the host superconductor, we can reproduce the Anderson theorem, which states that a conventional s-wave superconductor is robust to a normal (non-magnetic) impurity with small impurity strength. However, by increasing the size of the impurity in a fixed-size host superconductor, we find a decreasing critical temperature $T_c$ of the host superconductor, which agrees with the results in condensed matter literatures. More importantly, the phase transition at the critical impurity strength (or the critical temperature) is of zeroth order.


Introduction
Duality between a large N d-dimensional strongly coupled quantum field theory and a (d + 1)-dimensional classical gravity theory (the AdS/CFT correspondence) [1] has become a very powerful tool to study the condensed matter phenomena [2,3,4,5]. In particular, a black hole background coupled to a charged scalar theory was constructed in [6]. There, the author found that in the probe limit, there is a critical temperature T c below which the bosonic operator of the boundary field theory has a finite expectation value, which corresponds to a homogeneous s-wave superconductor. Reviews of the holographic superconductor can be found in [7,8,9]. In this paper we will extend this homogeneous construction to a single normal impurity effect 1 in a holographic superconductor, in which the order parameter becomes spatial dependent due to the impurity. Other studies of inhomogeneous holographic superconductors can be found in [10,11,12,13,14,15,16,17,18,19,20].
To study a superconductor with an impurity substitution is important in order to understand superconductivity in condensed matter physics, for reviews see [21]. Early important experimental results show that the conventional superconductivity is robust to small concentrations of normal impurity, especially a single normal impurity. These results can be understood by the Anderson's theorem [22]. In which, Anderson found that at the mean field level with small impurity concentration, if one assumes that the gap is still uniform in the presence of an impurity, the gap equation keeps the same if the density of the states is unchanged compared to the case without an impurity. Thus the critical temperature T c remains to T c0 , which is the critical temperature of the pure host superconductor. Anderson's theorem is however an approximate statement, in fact even there is only a small impurity, the local properties of the impurity point will change a lot [23,24]. In these two papers, the order parameter of a superconductor in the presence of a single impurity was obtained by solving the self-consistent Bogoliubov-de Gennes (BdG) equations, although the host condensation will not be affected by the impurity, the condensation at the impurity point is suppressed a lot. Hence, one can naturally expect that if the size of the impurity is increased, the host superconductor properties will change as well. This phenomenon requires us to study the single impurity effect on different length scales, from lattice spacing to coherence length, even to the host superconductor size. Specifically, when the impurity size is of lattice spacing, or in other words, in the limit of the localization size, the host superconductor will keep the same as the pure case [25]; When the impurity size approaches to the coherence length, which is smaller than the host superconductor, properties of the host sample will change; However, if we keep increasing the impurity size to the host superconductor size, superconductivity are expected to reduce substantially [26,27].
The interesting question is how to understand the single impurity effect from AdS/CFT correspondence. In this paper, we construct a gravity dual of a superconductor with a normal impurity in the center of the superconducting host. We reproduce the Anderson theorem that T c of the host superconductor will not be affected by the impurity when the size of the impurity is smaller compared to the host; However, we find it does reduce the gap at the impurity point as studied in [23,24]. This contradiction to Anderson theorem can be understood since Anderson theorem is an approximate statement about the thermodynamic average of the system in the mean field theory level, while we are solving the whole space dependent gap equation in the gravity dual which corresponds to strongly coupled field theory. For a larger size impurity , we find that T c decreases dramatically and finally the impurity can destroy superconductivity when T is close to T c0 , which is the pure host critical temperature.
The paper is arranged as follows: In Section 2 we basically set up the model holographically; The numerical results of the suppression of the superconductivity can be found in Section 3; We draw our conclusions and discussions in Section 4.

The Holographic Set-up
Even the substitution impurity concentration is small, the potential scattering induced by a local or finite size normal impurity in a homogeneous superconductor will modify the properties (for example the gap and the charge density) of the host superconductor at the impurity point. For instance, a self-consistently determined non-uniform gap function had already been obtained in [23,24] by solving the spatial dependent gap equation with a local impurity, the gap was strongly suppressed and localized in a small region. For simplicity, we can consider this model by coupling a superconducting host to a small normal impurity in its center. From the gravity side, the equations of motions (EoMs) of the scalar field and gauge fields in the bulk correspond to the gap equations in the BCS theory. Moreover, from the AdS/CFT dictionary, chemical potential and charge density of the boundary field theory are dual to the coefficients of the expansions of the gauge field A t near the boundary, i.e., A t (r) = µ(r) − ρ(r)z, in which z is the bulk radial coordinate while r is the polar radial coordinate of the boundary spacetime. In particular, we introduce a finite size impurity in the middle of the host by imposing a boundary condition that at the center of the host (with a small finite size), µ(r) (or ρ(r)) takes a smaller value while outside the impurity point they take a larger value above the critical point. Thus the host is in the superconducting phase.
Our method to include a localized impurity in holographic superconductor is somewhat different from [28,29], in these two papers, the average effect of impurity is studied by introducing another massive gauge field which is supposed to dual to the added impurity.
Concretely, we adopt the action in the bulk which is dual to a holographic superconduc- is the strength of the gauge field. The metric is an AdS-Schwarzschild black hole, , and r, θ are the boundary coordinates( we use polar coordinates on the boundary in order to put an impurity at the center of the host) . Without loss of generality, we set = 1. The temperature of the black hole is T = 3 4πz 0 , where z 0 is the position of the horizon. We use the ansatz of ψ = ψ(z, r), A = (A t (z, r), 0, 0, 0), and choose m 2 = −2. In the probe limit, with the rescaling of ψ → ψz, we have the following EoMs: The expansions of ψ and A t near the infinite boundary are: From the AdS/CFT dictionary, ψ (0) is interpreted as the source of the boundary scalar operator while ψ (1) can be regarded as the condensation value of the operator. In the holographic superconductors, we usually turn off the source of the scalar operator, i.e., ψ (0) = 0. In order to simulate the single normal impurity effect in the middle of the sample, in the polar coordinates we introduce a chemical potential that where µ max is the value of the chemical potential µ outside the impurity, and the parameters L/2, σ and are the radius, steepness and depth of the impurity that couple to the host superconductor, respectively. The maximal value of is one in order to make sure µ(r) is always positive. We can also introduce a charge density with similar form to get similar results. We emphasize that the exact form of µ(r) or ρ(r) is not important.
We discretize the EOMs on a two dimensional Chebyshev grids with 20 points along the z direction while 80 points in the r direction. A sample plot of the order parameter configuration is shown in Fig.1 with µ max = 4.2, σ = 0.5 and = 0.2, the size (radius) of the host is r max = 20 and the impurity size is L/2 = 1. It is clear that at the impurity point (the center of the host) the gap is suppressed a lot compared to the host superconductor. This gap configuration is very similar to the Fig.2 in [24], in which the inhomogeneous gap was obtained by solving the self-consistent Bogoliubov-de Gennes (BdG) equation with an impurity at the center. In order to see that the host superconductor will not be affected by an impurity of any depth, we fixe the radius L/2 = 1 and the host size r max = 20. The gap configuration for any depth can be found on the right panel of Fig.2. We can see that around r > 9 the order parameters have the same values whatever the depths are. This also indicates that the critical temperature T c for the host superconductor dose not change with respect to the depth, as Anderson theorem stated.

Suppression of Superconductivity
Since the order parameter is reduced a lot at the impurity point, it is nature to expect that by increasing the impurity size with fixed host superconductor size, or by reducing the host size while fixing the impurity size, we can expect that the order parameter of the whole host superconductor will be suppressed. This phenomenon is shown in Fig.3, in which we plot the condensation at the r max for different host superconductor size r max with fixed impurity size L/2 = 1 and µ max = 4.2. We can see that for different = 0.3; 0.5; 0.7; 1, there is a critical r c max 8, below which the condensation of the host superconductor will become suppressed. The value of r c max is independent of , which is expected to be decided by the impurity size L.

The critical depth of impurity c
With the realizations above, we take r max = 5 and fix the impurity size L/2 = 1 for example, by increasing the strength of the impurity (associated to the depth ) will finally induce a phase transition from superconducting state to normal state, see Fig.4. From which we can see that for a host superconductor with temperature T ∝ 1/µ max = 4.1, which is close to T c0 ∝ 1/µ c0 , with µ c0 = 4.06, the increasing impurity depth will suppress the host  In order to find the exact value of c where the superconductor/metal phase transition occurs, we scanned 100 points from = 0.14 to = 0.15 with every step as 10 −3 . The results are shown in Fig.5, we can see that the c ∼ 0.143 when µ max = 4.1. Another case is shown in Fig.6, with a larger µ max = 4.11, the critical depth of impurity is c = 0.265.
The phase diagram with L = 2, σ = 0.5 and fixed r max = 5 is plotted in Fig.7. The critical temperature decreases with increasing . Though the reduction is small, we can still see a phase transition when the host superconductor condensation is small.
The discontinuous order parameter shown in Fig.5 and Fig.6 indicate that the phase transition at the critical c is of zeroth order. In order to prove the order of phase transition we need to compute the free energy.

Discontinuous free energy at the phase transition point
The discontinuous order parameter indicates that the phase transition at the critical c is of zeroth order. To prove the order of phase transition we need to compute the free energy.
According the AdS/CFT dictionary, the free energy of the boundary theory is given by the on-shell action of the bulk theory, F = −T S os [31]. In the holographic superconductor model, S os can be evaluated by integrating by parts and using the equations of motion, plus the counter term to cancel the divergence, which results in [32], In which ψ is from rescaling ψ → ψz. The results shown in Fig.5 and Fig.6 tell us that at the critical depth where the phase transition occurs, the free energy is also discontinuous, which confirms that the phase transition is of zeroth order. Other observations of zeroth order phase transition in holographic superconductor can be found in [16,33]

Conclusions and Discussions
In this paper, the single normal impurity effect is investigated in the holographic s-wave superconductor. We uncover that the host superconductor is robust to the small size impurity, as the Anderson theorem stated [22,25]. However, although the host superconductor is robust to the impurity, the gap at the impurity site is strongly suppressed, this phenomenon agrees with the studies by solving the BdG equation in the presence of a normal impurity [23,24]. The suppression of gap at the impurity point indicates a decrease of host order parameter if we have a larger size impurity when fixing the host size, or decreasing the host superconductor size while fixing the impurity size. This is similar to the studies in superconductors with ultrashort coherence length, in which the host order parameter will be reduced substantially when an impurity is presented [26,27]. Moreover, if we have a small host superconductor with small condensation we obtain a zeroth order phase transition from superconducting state to normal metal state by increasing impurity depth. All these phenomena are similar to the results in the condensed matter literatures about the normal impurity effect in a superconductor. [21] In order to understand the zeroth order phase transition in terms of the impurity depth, we should notice that the boundary field theory is compactified on the angular direction. We can expect that this compactification may form a bound state, which may give rise to the zeroth order phase transition with respect to the impurity depth on the boundary field theory.
As another important topic in condensed matter physics, to understand the magnetic impurity effect in a superconductor holographically is also important, a possible way to study the magnetic impurity effect in holographic superconductor is to adopt the holographic paramagnetism-ferromagnetism phase transition [34] in a small region to a large host superconductor. We expect that by coupling a holographic superconductor to a magnetic impurity, the Kondo effect [35] can be realized holographically as in [36,37].
during the completion of this work. This work is supported in part by the National Natural Science Foundation of China (Grant No. 11205020 and No.11205097) and in part by the fund of Utrecht University budget associated to Gerard 't Hooft.