Holographic Lifshitz superconductors: Analytic solution

We construct an analytic solution for a one-parameter family of holographic superconductors in asymptotically Lifshitz spacetimes. We utilize this solution to explore various properties of the systems such as (1) the superfluid phase background and the grand canonical potential, (2) the order parameter response function or the susceptibility, (3) the London equation, (4) the background with a superfluid flow or a magnetic field. From these results, we identify the dual Ginzburg-Landau theory including numerical coefficients. Also, the dynamic critical exponent $z_d$ associated with the critical point is given by $z_d=2$ irrespective of the value of the Lifshitz exponent $z$.

The holographic superconductors arise in a broad range of gravitational theories with matter fields. From field theory point of view, this is natural since a superconductor is a robust phenomenon at low temperature. For example, it arises not only in asymptotically AdS spacetimes but also in asymptotically Lifshitz spacetimes [18,19], which is our main focus in this paper.
A holographic superconductor is typically an Einstein-Maxwell-complex scalar system. Such a system is hard to solve in general. One often needs either a numerical computation or an approximation method, and there are only a few analytic solutions [20][21][22]. In this paper, we present an analytic solution for a one-parameter family of holographic Lifshitz superconductors.
A holographic Lifshitz superconductor has three parameters (p, z, ∆), where p is the number of boundary spatial dimensions, z is the Lifshitz exponent 1 , and ∆ is the scaling dimension of the order parameter. In this paper, we consider the case where 1. p = 3z, and (GL) theory or the φ 4 mean-field theory. We identify the dual GL theory including numerical coefficients (Sec. VII A).
The holographic Lifshitz superconductors have been studied previously, but it is still nice to analyze these properties all at once analytically for an infinite number of theories. First, in previous works, the system was studied mostly using numerical methods. Second, the system was studied only for some specific values of (p, z, ∆) 2 . Third, some of the above properties were studied but not all were studied.
In particular, previous works typically have shown that (i) there exists a Ψ = 0 solution at low temperatures, (ii) the solution is favorable from the free energy or from the grand canonical potential, (iii) the spontaneous condensate has the standard φ 4 mean-field exponent β = 1/2, and (iv) the diverging DC conductivity.
On the other hand, the other properties are newly investigated, e.g., the other critical exponents as well as exact expressions for various numerical coefficients including critical amplitudes. Also, the critical dynamics of a holographic Lifshitz superconductor has never been investigated 3 . At a finite-temperature critical point, the correlation length ξ and the relaxation time τ of the order parameter obey a scaling law: We obtain z D = 2 irrespective of the value of the Lifshitz exponent z. We discuss the relation between z and z D in Sec. VII B.
There are various Lifshitz black hole solutions known in the literature, both analytically and numerically, depending on bulk theories. We use the solution in Refs. [34,35]. The metric is given by where u := r h /r, and r h is the horizon radius. The metric is invariant under the Lifshitz scaling (2.2) with the scaled horizon radius r h → r h /a. The Hawking temperature is given by The metric can be obtained as a solution of an Einstein-Maxwell-dilaton system 4 : where G p+2 is the (p + 2)-dimensional Newton's constant and The matter field solutions are given by (2.7b) But for our purpose, the point using this solution is that (i) it provides an analytic Lifshitz black hole solution, and (ii) a class of holographic superconductors in this background admits an analytic solution.

B. Holographic Lifshitz superconductors
We couple an additional matter system, a Maxwellcomplex scalar system in addition to the above system [18,19]: (2.8) 4 We use capital Latin indices M, N, . . . for the (p + 2)-dimensional bulk spacetime coordinates and use Greek indices µ, ν, . . . for the (p + 1)-dimensional boundary coordinates. The boundary coordinates are written as x µ = (t, x i ) = (t, x, y, · · · ). where The U (1)-field A M is different from A M in Eq. (2.5).
We take the probe limit e ≫ 1, where the backreaction of these matter fields onto the geometry is ignored. Namely, we solve the system (2.8) in the background (2.3). The equations of motion are given by In the A u = 0 gauge, the asymptotic behaviors of the matter fields are given by represents the charge density ρ, and A (0) t = µ is the chemical potential. Similarly, A (1) i represents the current density J i , and A (0) i is the vector potential. For Ψ, Ψ (1) represents the order parameter O, and Ψ (0) is the external source for O. (See Appendix A for the precise dictionary.) Then, the BF bound in the asymptotically Lifshitz geometry is given by When the BF bound is saturated, the asymptotic behavior is replaced by The equations of motion (2.10) admit a solution where boldface letters indicate background values. But, at the critical point, the Ψ = 0 solution becomes unstable and is replaced by a Ψ = 0 solution. We see this in detail below.

III. CRITICAL POINT
Below we consider the case p = 3z. It is convenient to introduce a new coordinate s := u 2z . The metric then becomes 5 We consider the scalar which saturates the Lifshitz BF bound m 2 = −(2z/L) 2 . The scaling dimension ∆ is given by ∆ = 2z. First, consider the static homogeneous solution Ψ = Ψ(s), and approach the critical point from high temperature. Near the critical point, the scalar field Ψ remains small, and one can ignore the backreaction of Ψ onto the Maxwell field. In this region, one can use Eq. (2.14) for the Maxwell field, and it is enough to solve the Ψ-equation 6 Thus, the solution is parametrized by a dimensionless parameter µ/T . The equation admits a solution This is the solution at the critical point.
The z-dependence disappears in Eq. (3.2), and it only appears implicitly in the definition of T . One can understand this as follows. For the static homogeneous solution, the Laplacian becomes The z-dependence appears in the boundary spatial metric g ij , and it appears only through det g. But in our case, so p and z disappear. The metric g ss is also proportional to z 2 , but it is factored out in the Ψ-equation since m 2 ∝ −z 2 and 1/T 2 ∝ 1/z 2 . It then follows that the Ψ-equation formally reduces to the same equation for all z. Note that Eq. (3.4) is the solution directly at the critical point. As one lowers temperature further, the solution is modified, and we construct the background solution Ψ, A t in Sec. V. The z-dependence can also be eliminated from the A t -equation [by redefining Ψ and A t as in Eq. (3.9).] Thus, the static homogeneous solution is essentially the same as the z = 1 case apart from various factors of (2z). Then, from the analysis of Ref. [22], Eq. (3.4) is the solution at the critical point, and the solution has a lower grand canonical potential than the Ψ = 0 solution at low temperature.
However, the full equations of motion do not reduce to the same equations as the z = 1 case. In general, more nontrivial z-dependences appear. For example, they appear when one considers • inhomogeneous perturbations in the boundary spatial directions e.g., δφ ∝ e iqx (Sec. IV), or • perturbations or solutions with a vector potential A i (Sec. V D and Sec. VI).
Below we construct the background solution Ψ, A M . We also consider the linear perturbations from the background: We take the gauge A s = δA s = 0. We consider the perturbations of the form where k µ = (ω, q, 0, · · · ). Then, the Maxwell perturbations are decomposed as • vector modes, e.g., δA y , and • scalar modes δA t , δA x which can couple to δΨ in general.
For simplicity, we set e = L = r h = 1 below. In this unit, µ c = 2z, and we vary the chemical potential µ. Also, we often use quantities with "¯". All quantities with "¯" are defined byφ when r h = 1. For example,μ c = 1. We restore units for some of our main results in Appendix C.

IV. HIGH-TEMPERATURE PHASE
At high temperature, the background solution is given by The interesting quantity in the high-temperature phase is the "order parameter response function," the susceptibility, or the correlation function of the order parameter. We show that the response function takes the form for a small (ω, q, ǫ µ ), where ǫ µ := µ − µ c , and c K and Γ are parameters we use to compare with the GL theory (Sec. VII A). The function gives the following information: • The ω = q = 0 limit is the thermodynamic response function where the coefficient A is known as the critical amplitude. Then, the exponent γ defined by χ T ∝ |ǫ µ | −γ is γ = 1.
• The ω = 0 case is the dynamic response. Then, the relaxation time behaves as τ q=0 ∝ ξ 2 , (4.3c) and the dynamic critical exponent z D defined by τ q=0 ∝ ξ zD is z D = 2.
Thus, the computation determines the exponents (γ, ν, η, z D ) as well as the critical amplitude A. An explicit solution is not really necessary to compute critical exponents, and analytic arguments are possible [13,16]. On the other hand, an explicit solution is useful to obtain various numerical coefficients such as A.
The response function can be obtained from the bulk scalar field Ψ. Consider the linear perturbation from the background Ψ = Ψ + δΨ. From the bulk point of view, the response function pole corresponds to a quasinormal pole of δΨ. When Ψ = 0, Maxwell scalar modes δA t and δA x decouple from the δΨ-equation 7 . Thus, to determine the order parameter response, it is enough to consider the δΨ-equation: Asymptotically, we impose the boundary condition δΨ(s → 0) = δΨ (0) s ln s/(2z). At the horizon, we impose the incoming-wave boundary condition.
The δΨ perturbation cannot be solved for a generic µ, so we setǭ µ =μ − 1 < 0 and employ theǭ µ -expansion as well as the (ω, q)-expansion: This form is taken to implement the incoming-wave boundary condition. Then, the boundary condition reduces to the regularity condition for ψ c and so on. The equation of motion reduces to and the index i collectively represents ǫ, ω, and q. The homogeneous equation L ψ ψ c = 0 can be solved as (4.7) From the regularity at the horizon, c 2 = 0. The source terms of inhomogeneous equations then become The ψ ǫ and ψ ω solutions are For ψ q , we discuss the z = 1 and z = 1 cases separately.
The ψ q solution is given by The asymptotic behavior then becomes The asymptotic boundary condition determines c 1 . Then, the order parameter response function becomes where we use the dictionary in Appendix A. The response function indeed takes the form of Eq. (4.2), and The dispersion relation is given by The relaxation time τ then becomes In this case, we are not able to obtain the generic expression for ψ q . Besides, even when the analytic expression is available, it is too cumbersome to write here. However, the slow falloff has a simple expression: where ψ 0 (x) is the digamma function: A few examples of I(z) are The combination I(z)/z monotonically increases with z and reaches 1 for z → ∞. In order to obtain the falloff, we essentially used the standard method to solve an inhomogeneous differential equation (Appendix B). Then, the order parameter response function becomes which gives The dispersion relation is given by The relaxation time is given by For large z, In the low-temperature phase, our task is 1. to construct the background, 2. to show that the Ψ = 0 solution has a lower grand canonical potential, and 3. to derive the London equation. (This establishes that the Ψ = 0 phase is a superconducting phase.)

A. Background solution
The solution (1.2) is the solution only at the critical point, and we first construct the background solution in the low-temperature phase. As mentioned in Sec. III, the construction is essentially the same as the z = 1 case [22].
Consider the solution of the form The equations of motion are given by One can set Ψ to be real. We construct the background perturbatively: where ǫ is a small parameter whose meaning will be clear in a moment. From Sec. III, we already know To proceed to higher orders in ǫ, we impose the boundary conditions following Ref. [22]: • Ψ n : Asymptotically, no slow falloff and no fast falloff, or Ψ n = 0 (for n ≥ 2). The former means the condition for a spontaneous condensate. The latter means that O comes only from Ψ 1 . At the horizon, we impose the regularity condition.
Namely, we fix the fast falloff O, but the chemical potential is corrected as Under these boundary conditions, so ǫ 1/2 represents the order parameter O. We impose 4 boundary conditions in total, which completely fixes the solution. For example, Φ 1 and Ψ 2 have 4 integration constants, and they are determined by the 4 conditions. At O(ǫ), where we imposed the boundary condition Φ 1 (s = 1) = 0, and δμ 1 is the remaining integration constant. It is fixed at O(ǫ 3/2 ) from the condition that Ψ (0) 2 = 0. At O(ǫ 3/2 ), there are 2 more integration constants and δμ 1 . After imposing the boundary condition at the horizon and the Ψ (1) 2 = 0 condition, one obtains so the remaining no slow falloff condition Ψ (0) 2 = 0 gives δμ 1 = 1/12. Then, at O(ǫ), the chemical potential becomesμ so ǫ is determined as At O(ǫ 2 ), Again, we determine an integration constant δμ 2 at O(ǫ 5/2 ) from the condition Ψ (0) 3 = 0. The expression for Ψ 3 is too cumbersome to write here.

B. Grand canonical potential
We use the Lorentzian formalism to evaluate the grand canonical potential Ω. (Note S E = βΩ = −S L .) The matter on-shell action is given by We are interested in the grand canonical potential of the spontaneous condensate, or the solution with Ψ (0) = 0, so the boundary term from Ψ vanishes. We evaluate the difference of the grand canonical potential between the Ψ = 0 solution and the Ψ = 0 solution. We fix the chemical potential asμ = 1 + ǫδμ 1 + ǫ 2 δμ 2 + · · · , where δμ 1 and δμ 2 are obtained in the previous subsection. It turns out that δS OS = 0 at O(ǫ), so we evaluate the difference at O(ǫ 2 ). This implies that one has to take into account up to O(ǫ 2 ) of A t in order to evaluate the above boundary action.
For the Ψ = 0 solution, In this case, only the boundary action contributes since Ψ = 0. The on-shell action becomes where β is the inverse temperature, and V p is the boundary spatial volume. For the Ψ = 0 solution, δΩ < 0, so the Ψ = 0 solution is favorable. The difference is proportional to ǫ 2 µ = (µ − µ c ) 2 ∝ (T c − T ) 2 , which implies the second-order phase transition. (The difference and its first derivative are continuous, but the second derivative is discontinuous.) The specific heat C µ behaves as C µ = −T ∂ 2 Ω/∂T 2 ∝ T , which determines the critical exponent α = 0, where α is defined by

C. Background with source
We construct the background without the source of the order parameter, but it is straightforward to extend the construction to the background with the source. Going back to Eq. (5.27), we obtained so the asymptotic behavior becomes Previously, we imposed the source-free condition Ψ (0) = 0, which gives δμ 1 = 1/12. We now allow Ψ (0) = 0. The chemical potential is given byμ = 1+ǫδμ 1 . At the critical point,μ = 1, so δμ 1 = 0. From the asymptotic behavior (5.28), O ∝ ǫ 1/2 and Ψ (0) ∝ ǫ 3/2 . Then, the exponent δ defined by O ∝ (Ψ (0) ) 1/δ (at µ = µ c ) is δ = 3. One can evaluate the thermodynamic response function at low temperature. By imposing our boundary conditions,Ψ The chemical potential is then determined as which is rewritten as This is essentially the GL equation of motion (Sec. VII A). For a fixed µ, this gives Thus, We obtained χ < T from the background solution, but it should also be possible to obtain it from the scalar perturbation as in Sec. IV. One would also obtain the full response function χ < ω,q using the (ǫ, ω, q)-expansion. But, in the low-temperature phase, δΨ couple with δA t and δA x , and the computation is more involved, so we leave it to a future work.

D. Vector modes
From the vector mode, one can show the London equation and compute the conductivity. The δA y -equation is given by whereΨ was constructed in Eq. (5.3a). We impose the incoming-wave boundary condition at the horizon and δA y | s=0 = A (0) y asymptotically. We again employ the (ǫ, ω)-expansion: The equation of motion reduces to L a a c = 0 , The homogeneous equation L a a c = 0 can be solved as From the regularity at the horizon, c 2 = 0. The source terms of inhomogeneous equations then become Again, we discuss the z = 1 and z = 1 cases separately.
The solution is The asymptotic behavior then becomes We determine the constant c 1 from the asymptotic boundary condition δA y | s=0 = A y . So, The ω → 0 limit gives the London equation with the London penetration depth λ −2 = 2ǫ. The conductivity is then given by (5.46) Im(σ) has the 1/ω-pole which implies the diverging DC conductivity. A superconductor has singular behaviors in the current, but its essence is not in the diverging DC conductivity but in the London equation. A diverging DC conductivity also appears in a perfect conductor, but the London equation is unique to superconductors.
When one combines the London equation with the Maxwell equation, one obtains the Meissner effect. However, for usual holographic superconductors, the boundary Maxwell field is added just as an external source and is not dynamical in the boundary theory, so the Meissner effect does not arise; the magnetic field can always penetrate into the material. In this sense, a holographic superconductor may be regarded as a superfluid. (In low spatial dimensions p ≤ 2, one can obtain a boundary theory with a dynamical Maxwell field. See, e.g., Ref. [36].) However, the London equation must hold even in this case if the system is really a superconductor or a superfluid. The London equation is the response of the current under the external source, and whether the source is dynamical or not is irrelevant to the issue.

z > 1
For a ω , one can get the generic expression 8 : For a ǫ , the generic expression is either difficult to obtain or too cumbersome, but again the fast falloff has a simple expression: The asymptotic behavior then becomes Thus, Again, the ω → 0 limit gives the London equation with the London penetration depth λ −2 = 4zI(z)ǫ. The conductivity is then given by The GL parameter κ is defined by (5.54) In conventional superconductors, κ 2 < 1/2 for type I and κ 2 > 1/2 for type II superconductors. For z = 1, κ 2 = 1/6, so one may conclude that our system is type I (in the sense of κ), but whether our system is type I or II is more subtle. Physically, 1/λ represents the Maxwell field mass, so we should determine the normalization of λ by comparing with normalization of the boundary Maxwell action. However, as mentioned above, the boundary Maxwell field is added as an external source here and is not dynamical in the boundary theory, so the normalization cannot be determined 9 . (Holographic superconductors are type II superconductors in the sense that there is no Meissner effect.)

VI. BACKGROUND WITH VECTOR POTENTIAL
In this section, we add a vector potential A i as a background. We again consider the perturbative expansion:

1b)
A y = A y,0 + ǫA y,1 + · · · . (6.1c) Note that we take into account (1) A y as a background and (2) the backreaction of A y onto the other fields.
(That is why we use variables with primes.) The former is the difference from the perturbative expansion in Sec. V A, and the latter is difference from the vector mode computation in Sec. V D.
At O(ǫ 0 ), the Maxwell equation becomes The former corresponds to adding a constant superfluid flow a y , and the latter corresponds to adding a constant magnetic field B.

A. Superfluid flow
For the superfluid flow, it is enough to consider homogeneous perturbations. The equations of motion are given by We impose the same boundary conditions as Sec. V A. Our main interest is the phase diagram, i.e., the deviation of the critical point by the vector potential. Then, we evaluate how A y at O(ǫ 0 ) affects Ψ ′ 1 at O(ǫ 1/2 ). This in turn affects the value of µ c . We employ the a yexpansion as well as the ǫ-expansion [22]. Namely, where c 0 is a constant. This expansion is consistent with the above equations of motion.
The equation is hard to solve in general. However, to determine theā y -dependence on the chemical potential, it is enough to obtain the slow falloff of Ψ a . The slow falloff has a simple expression: We impose the boundary condition Ψ where µ c,0 = 2z is the critical point without superfluid flow.
To obtain J µ , one needs to obtain A µ . This is necessary to derive the second sound c 2 [22,39]: To derive c 2 , it is enough to use the results obtained in the previous section. The A y,1 -equation is the same as the vector mode perturbation a ǫ . So, J y is given bȳ J t is given bȳ Thus,

The equation then takes the form
The solution is parametrized byBλ n , so one has the largest magnetic field B c2 when λ n takes the minimal value, namely the n = 0 solution. 10 For simplicity, we set the other momenta as k 3 = k 4 = · · · = 0.
The ρ 0 -equation is given by Then, the problem formally reduces to the same problem as the superfluid flow one with the replacementā 2 y bȳ B c2 . Thus, the critical point is given by Using the result of ξ 2 > in Sec. IV, we get

A. Identifying the dual GL theory
We thus obtained all critical exponents and critical amplitudes (α, β, γ, δ, ν, η, z D ) = 0, The results are consistent with the standard GL theory or the φ 4 mean-field theory. In fact, the following GL theory reproduces all our results 11 : In the dynamic case, consider the time-dependent GL equation (for Model A dynamic universality class): We determine the GL parameters (c 2 , c 4 , c φ , c K , c A , Γ) to reproduce our holographic results.
In the static homogeneous case, the φ-equation becomes Substituting the J = 0 solution |φ| 2 = c 2 ǫ µ /c 4 into H GL , one obtains the grand canonical potential: The current is given by In the high-temperature phase, the response function is given by Add a background vector potential. When a constant A y = a y is added, the critical point is shifted as When a magnetic field is added, the critical magnetic field is given by In the presence of a background vector potential, Eqs. (6.9) and (6.23) agree with Eqs. (7.11) and (7.12), respectively. One would be tempted to ask how various results change as we vary z. But to make such a comparison, one must keep in mind that (1) we consider a special class of Lifshitz theories and (2) we must specify what quantities to fix as we vary z.
We consider a special class of theories where p = 3z and m 2 = −4z 2 . Even the spatial dimensionality p is different for a different z, and it is unclear if the comparison with a different z is physically meaningful. Also, some results may be generic for Lifshitz theories in general but some are not. As a simple example, in our case, (µ/T ) c is independent of z. This is so by construction of our theories as discussed in Sec. III and is certainly not a generic behavior. It simply means that holographic Lifshitz superconductors have enough parameters to fix (µ/T ) c as we vary z.
We also have to specify what quantities to fix. One natural candidate is µ c (or T c ) since (µ/T ) c is z-independent, but it is unclear if this is really appropriate. We do not have the answer to this question. So far we set r h = 1 just for simplicity, so here we simply fix r h (and ǫ µ ) and how various results change as we vary z. Again, we do not mean that fixing r h is natural from the boundary point of view. Rather, the following comparison should be regarded as a handy way to understand the z-dependence of our holographic results or the dual GL theory.
1. In the dual GL theory, the coefficient of the φ 4term becomes smaller as we increase z. So, the condensate increases as |φ| 2 ∝ z.
2. The z-dependence appears only in the kinetic term and the φ 4 -term. Thus, the relaxation time τ q=0 of the order parameter does not depend on z.
3. The correlation length ξ depends on the kinetic term so does depend on z. It monotonically increases as I(z)/z but increases slowly and reaches a constant value for z ≫ 1.
4. On the other hand, the London penetration depth λ decreases. This is because and because φ increases. (λ also depends on the kinetic term so has the factor I(z)/z, but it is not a dominant factor.) Then, the GL parameter κ decreases.

As usual, the presence of a background vector potential A
(0) i increases the critical chemical potential µ c . From the gravity point of view, this is because A i increases the effective mass of Ψ. µ c monotonically increases as I(z)/z since A (0) i comes from the covariant derivative in the kinetic term.

B. Lifshitz exponent and dynamic critical exponent
We already mentioned that some results are not generic to Lifshitz theories in general. Then, what results are expected to be generic? An obvious answer is critical exponents and the ratio of critical amplitudes. The φ 4 mean-field critical exponents are likely to hold for theories of Eq. (2.8). The dynamic critical exponent z D = 2 is also likely to hold.
For a Lifshitz geometry, one would expect a dispersion relation of the form ω ∝ q z . (7.15) This form is expected from the Lifshitz scaling (2.2). But from the analysis of the high-temperature phase, the order parameter obeys the dispersion relation Conservation laws play important roles to determine the dynamic universality class since a conservation law forces the relaxation to proceed more slowly. When only the order parameter matters in critical dynamics, a nonconserved order parameter gives Model A, and a conserved order parameter gives Model B. The Lifshitz geometry is conjectured to describe a quantum critical point. Using holographic Lifshitz superconductors, one prepares a new finite-temperature critical point in addition to the Lifshitz critical point. What we have shown is that the dynamic critical exponent z D associated with the new critical point can differ from z. Instead, the value of z D is determined by the critical dynamics of the new critical point.

A. Lifshitz geometry and holographic superconductors
The Lifshitz geometry appears even in the context of the standard z = 1 holographic superconductor [42,43].
Consider the backreaction of matter fields onto the geometry. In the high-temperature phase, Ψ = 0, so the geometry is the Reissner-Nordström AdS black hole. In the low-temperature phase, Ψ = 0, but one may expect that the geometry is somewhat similar to the Reissner-Nordström AdS black hole. However, the T = 0 geometry is conjectured to be a Lifshitz geometry in the IR and the AdS geometry in the UV. The solution in IR has been constructed, but the full geometry remains an open question.
It is unclear what happens at low temperature, but it is natural to expect that a Lifshitz-like black hole appears at low enough temperature. (Unfortunately, the Lifshitz black hole used in this paper is not a solution of the Einstein-Maxwell-complex scalar system.) Then, one should consider the Einstein-Maxwell-complex scalar system in a Lifshitz (IR)-AdS (UV) black hole. This is not an easy task however. First, the full geometry is not constructed even in the T = 0 limit. Second, the stability of the geometry is an different issue. Finally, one has to solve perturbations in the full geometry to explore various properties.
What we have done in this paper is one small step towards this program; we solved matter fields in a simple Lifshitz black hole background. As we have seen in this paper, qualitative behaviors of holographic Lifshitz superconductors are the same as the ones of the standard holographic superconductors. In particular, static and dynamic critical exponents are the same. One would expect those behaviors are common even in the full problem. Critical dynamics is governed by dynamics of the critical point itself (such as criteria 1 and 2 in the previous subsection) and is not governed by the Lifshitz exponent z in the underlying geometry.

B. Implications to quantum criticality
We briefly discuss the implications of our result on quantum criticality. The Lifshitz geometry is conjectured to describe a quantum critical point. In this sense, our system has two critical points: • One is the T = 0 quantum critical point. Its dynamic scaling is determined by z.
• The other is the T = 0 superconducting critical point explored in this paper. Its dynamic scaling is determined by z D as we have shown in this paper.
It has been proposed that quantum criticality explains strange metallic behaviors of high-T c superconductors. According to the proposal, a quantum critical point is "hidden" inside the superconducting dome, and the quantum criticality explains scaling behaviors of various transport coefficients even in the normal phase.
Our model is far from real materials, but roughly speaking, the quantum critical point could correspond to the T = 0 Lifshitz geometry, and the superconducting dome could correspond to the holographic Lifshitz superconductor. The Lifshitz scaling may determine the scaling behaviors in the normal phase. But our result implies that the Lifshitz scaling does not determine the scaling behavior of the order parameter near T c . Rather, the T = 0 critical point has its own scaling. Whatever the value of z a quantum critical point has, the T = 0 critical point is likely to have z D = 2 at the mean-field level. The asymptotic behaviors of matter fields are given by (In expressions after "⇒", we set p = 3z = 3∆/2 and useds :=ũ 2z .) The field/operator correspondence is derived by evaluating the on-shell action. The bulk on-shell action, in general, diverges, and one needs to add counterterm actions. We take the probe limit, so we discuss counterterm actions for matter fields only. We use the Lorentzian formalism.
In the static homogeneous case, or at the leading order in the (ω, q)-expansion, the scalar action diverges, and the counterterm action is where γ µν is the (p+1)-dimensional boundary metric and u = δ (or s = δ s := δ 2z ) is the UV cutoff. As usual, the second term is necessary for the scalar which saturates the BF bound.
Using the standard holographic technique, one then gets (More precisely, left-hand sides represent expectation values such as O .) The Lifshitz scaling (2.2) is just a coordinate transformation from the bulk point of view. The Maxwell field is a one-form, and Ψ is a scalar, so they transform as under the scaling. Then, the scaling dimensions are On the other hand, the mass dimensions are Continuing higher orders in the derivative expansion, one has additional counterterms: M F and M Ψ are power series in Ψ whose explicit forms are not necessary in the discussion below. In the text, we take into account O(ω, q 2 ) terms in the scalar perturbation and O(ω) term in the vector perturbation, so it is enough to consider L 2 and L 3 , but they make no contribution. For the scalar perturbation in the hightemperature phase, F µν = 0, so L 2 = 0, and so L 3 makes no contribution as s → 0 (for a finite z). For the vector perturbation, L 2 ∼ O(ω 2 , q 2 ), so L 2 makes no contribution 12 , and where we consider the case of the spontaneous condensate forΨ.

Appendix B: Extracting falloffs
We solve the following differential equation: Denote two independent solutions of the homogeneous equation Lϕ = 0 as ϕ 1 and ϕ 2 . We assume that ϕ 1 satisfies the boundary condition at the horizon s = 1.
The solution of the inhomogeneous equation (B1a) which is regular at the horizon is given by where W is the Wronskian W (s) := ϕ 1 ϕ ′ 2 − ϕ ′ 1 ϕ 2 . For example, for δΨ and δA y , Even if the integral (B2) is difficult to evaluate or has a cumbersome expression, one can extract a falloff. Suppose that ϕ 2 has the appropriate falloff. Then, near the AdS boundary s → δ s , ϕ(δ s ) ∼ −ϕ 2 (δ s ) 1 δs ds j(s)ϕ 1 (s) .
(B3) 12 Since L 2 is a relevant operator for z ≥ 1 in our theories, it should be taken into account at O(ω 2 , q 2 ).
This integral essentially gives the falloff coefficient we want.
The δ s -dependence in the integral essentially has no contribution from the following reason. First, the integral may or may not converge: 1. When it converges, one can take the δ s → 0 limit since the δ s -dependence in the integral does not produce an appropriate falloff when it is combined with ϕ 2 (δ s ); it gives a subleading falloff.
2. When it diverges, simply discard the δ s -dependence in the integral since again it does not produce an appropriate falloff 13 . Even if it diverges as δ s → 0, the expression (B2) itself does not.
For example, the slow falloff of ψ q becomes ψ (0) q = − We set e = L = r h = 1, but here we present some of our main results by restoring units.
• The scalar mode (high-temperature phase): the dispersion relation, the relaxation time, and the correlation length are given by ω = 3 − i 5 |ǫ µ | + I(z) z L r h L z−2 q 2 + · · · , (C1a) • The order parameter: • The current (low-temperature phase): In our conventions, it is natural to define λ as Then, λ and κ are given by (C5b) • The dual GL theory: