A Holographic Model For Hall Viscosity

We have modified the holographic model of Saremi and Son by using a charged black brane, instead of a neutral one, such that when the bulk pseudo scalar potential is made of quadratic and quartic terms, parity can still be broken spontaneously in the boundary theory. In our model, the 3+1 dimensional bulk has a pseudo scalar coupled to the gravitational Chern-Simons term in the anti de Sitter charged black brane back ground. Parity could be broken spontaneously in the bulk by the pseudo scalar hairy solution and give rise to non-zero Hall viscosity at the boundary theory.


I. INTRODUCTION
In recent years, the AdS/CFT correspondence [1][2][3] has been applied to study strongly coupled phenomena in condensed matter physics at finite temperature and chemical potential. In particular, inspired by the idea of spontaneous symmetry breaking in the presence of horizon [4,5], holographic superconductors [6,7] and superfluids [8] are two remarkable examples where the Gauge/Gravity duality plays an important role.
On the other hand, the hydrodynamic limit of AdS/CFT correspondence has also attracted much attention recently. Computations of the ratio of shear viscosity to entropy density for a big class of gauge field theories with gravitational duals yields the same number 1/4π which is not far away from that observed in the strong interacting quarkgluon plasma created in RHIC [9,10]. Later it has been shown that by using the boundary derivative expansion, one can consistently solve the Einstein equation order by order and compute various hydrodynamics transport coefficients of the boundary fluid [11].
Recently, a holographic model for the parity violating Hall viscosity was proposed. Like the other transport coefficients, Hall viscosity is also found to be uniquely determined by the near horizon data of the bulk black brane [12]. This is yet another example of the membrane paradigm. In the original construction the (3 + 1) dimensional bulk action has a negative cosmological constant, a real scalar field coupled to the gravitational Chern-Simons term * .
While it has been shown that a non-trivial profile of the bulk scalar field is important to obtain a nonvanishing Hall viscosity of the (2+1) dimensional boundary field theory, from the holography point of view it would be interesting to further investigate what role this bulk scalar plays at the boundary. One possible interpretation is to identify the boundary value of this scalar as an order parameter field which condensates at low temperature in the boundary field theory. From the condensed matter point of view the physical realization of this order parameter, which leads a system to the spontaneously parity breaking phase is not clear. But interestingly in terms of physical quantity such as hall viscosity one might get information about how the system breaks parity spontaneously. So, effectively * There exist early studies of Chern-Simons term in the Holographic models. To mention a few: the effect of Maxwell Chern-Simons term θ * F F was studied in the Holographic superconductor [13,14]. The spectrum of quasinormal modes was studied in the dynamic Chern-Simons gravity and correction to some hydrodynamic quantities was discussed [15].
in the hydrodynamic regime, Hall viscosity can play the role of order parameter which is non-zero only below a critical temperature. To be ready for such a boundary theory interpretation, one shall look for a sourceless boundary condition for the hairy scalar if parity is only broken spontaneously.
However, it has been shown that a neutral scalar hair with quadratic and quartic potential that satisfies the usual sourceless boundary condition in a Schwarzschild-AdS black hole spacetime does not satisfy the positive energy theorem [16]. This essentially means that a Schwarzschild-AdS black hole with a sourceless neutral scalar hair is intrinsically unstable.
While it is still possible to find a sourced solution which minimizes the free energy, we will take a different approach to modification of the model by including a gauge field in the bulk. The scalar in the original theory is identified as a pseudo scalar now, so its coupling to the gravitational Chern-Simons term does not break parity. The pseudo scalar hair, however, breaks parity spontaneously and gives a pseudo scalar condensate in the boundary field theory which, as we will demonstrate in the next section, is important for Hall viscosity. In the probed limit, this pseudo scalar hair solution in the charged black brane background is known to be stable [17].
The paper is organized as follows: in section two, we present a general discussion of the parity violating viscosities and set up the holographic model. We then compute the Hall viscosity and commend on the boundary field theory in section three. We then conclude our results in section four. A detailed derivation of Hall viscosity together with an analytical approximation are given in the appendix.

II. GENERAL PROPERTIES OF VISCOSITIES
It is instructive to classify viscosities by considering the general relation between the energy momentum tensor and the spatial derivative of the fluid velocity where i, j,k,l are spatial indices and just the symmetric and anti-symmetric combinations of the derivatives, respectively. We have T ij = T ji . In two spatial dimensional systems, η ijkl and ξ ijkl can be constructed by Notice that the force flips its direction when the direction of the velocity field, which is depicted by the arrows outside the spheres, is reversed. and ǫ ik ǫ jl + ǫ jk ǫ il give rise to the usual shear (η) and bulk (ζ) viscosity contributions Taking η ijkl ∝ δ ik ǫ jl + δ jk ǫ il gives rise to the Hall viscosity (η A ) and "curl" viscosity (ζ A ) The curl viscosity can also arise from taking ξ ijkl ∝ δ ij ǫ kl . The curl structure naturally reminds us vortices. It is interesting that the bulk and curl viscosities are associated with the divergence and the curl of the velocity. Both of them can only exist in systems without scaling invariance due to its trace like structure in the energy momentum tensor. It is easy to generalize the above discussion to higher dimensions. However, the Hall and curl viscosities can only exist in two dimensions as depicted in Figure 1.
The Hall and curl viscosities have distinct transformation properties from the shear and bulk viscosities under parity. Under the coordinate reflection ( Since δT ij exists in parity conserving systems, δT A ij only exists in parity violating systems. In summary, we need to work in (2 + 1) dimensional parity violating systems to study the Hall and curl viscosities. In the following section we will explicitly construct a holographic model and calculate the Hall viscosity of the boundary fluid.

III. THE HOLOGRAPHIC SET UP
Following the discussion of the previous section, we will consider a four dimensional bulk action as the holographic dual to a three dimensional boundary theory. It is given by a four dimensional Einstein action with a negative cosmological constant; the matter sector includes an abelian Yang-Mills F µν and a pseudo scalar field θ: The the bulk action conserves parity, so θ is a pseudo scalar from the last term on the Lagrangian which is important to introduce parity violation to the boundary theory through the θ condensate. The F 2 term is the only difference between our model and Saremi and Son's. In our model a charged black hole solution is allowed. We will recover their result by taking the black hole charge to zero.
In this paper, we will only focus on the following form of the potential, As discussed in [17], the second term is necessary to have consistent solution at T = 0.
We will study the probed limit of the scalar field by sending θ → εθ and λ → ελ for small ε. Thus, at leading order in ε, we only need to solve for the equation of motion governed by the upper line of (4). Then the background is exactly a charged black brane in AdS 4 spacetime and the Hall viscosity η A can be recovered at the O(ε 2 ) order, since η A → ε 2 η A in this probe limit.
The charged black brane solution is given by the metric: where and the abelian gauge field † : Here black brane mass and electric charge are M and Q. The horizon is at r = r H . The metric is asymptotically AdS 4 with curvature radius L. It is convenient to work in the units of L = 1 and rescale the horizon to The charged black brane in the bulk corresponds to a boundary field theory at finite temperature T and chemical potential µ, that is We remark that κ = 0 corresponds to a neutral black brane with zero chemical potential and κ = 1 corresponds to an extremal black brane at zero temperature.
The equation of motion the probed neutral pseudo scalar reads, Near the boundary, the asymptotic behavior of pseudo scalar is We remark that in our construction, the mode J can be consistently turned off and O is identified as the condensate in the boundary § . However, this was not possible in † The A r component of gauge potential can be gauged away with no contribution to the equation of motion. However, we keep it here to show that under a proper coordinate transformation, v = t + h(r) where h ′ (r) = 1 r 2 f (r) , this black brane solution can be brought to the usual diagonal coordinate given by (t, r, x, y). ‡ The action and equations of motion are invariant under the following scaling: Here we adopt the convention of [17] where Q → Qr 2 H , r → r H r and (v, x, y) → r −1 H (v, x, y). § For −9/4 < m 2 L 2 < −5/4, J and O are both renormalizable and one is free to choose either one as source and the other as condensate [18]. the original construction with neutral black brane [12] where J can be turned off only if c < − 3 4 [16] which violates the positive energy theorem and hence it is not a stable solution. In our model, the θ 4 term is required to make the θ solution regular at the horizon [17].

IV. THE HALL VISCOSITY
The detail derivation of the viscosities is presented in the appendix. We have first included the back reaction of θ as was done in [12], then take the probe limit (θ → εθ and λ → ελ) to the final result of the viscosity expression. The expression for η A , which appears at O(ε 2 ), is identical to that obtained in [12].
From the derivation in the appendix, we obtain the shear viscosity of the universal value as expected: where s is the entropy density. Theses combinations are dimensionless and are invariant under the scaling of Eq.(9).
The Hall viscosity in our charged black brane background takes the same form as the case of the neutral black brane background [12], that is The dimensionless and scale invariant combination yields In Eq. (17), η A /s vanishes when the solution of θ is trivial (θ(r) = 0), which happened in the symmetric phase, or when θ is a constant field. In the former case, parity is not broken in the bulk. Then by the correspondence, it will not be broken at the boundary either. Likewise, in the latter case, when θ is a constant, the * RR term is just a surface term in the action which has no effect to the bulk equations of motion. Hence it does not contribute to η A either. Therefore, it should not be a surprise that the phase diagram for η A /s is very similar to that with the neutral scalar hair of Ref. [17] with just one difference-η A /s vanishes when T = 0. This comes from the factor f ′ (r H ) ∝ T . One peculiar feature of this model is that the entropy of the charged black hole does not vanish at zero temperature. Perhaps in models with zero entropy at zero temperature, η A /s stays finite at zero temperature. We show η A /sλL as a function of T /µ and m 2 L 2 in Fig. 2. These three quantities are all scale invariant and dimensionless.
In Fig. 3, the dependence of the scale invariant, dimensionless quantities η A /sλL and T /µ is shown for m 2 L 2 = −2. η A /s ∝ T as T → 0 due to f ′ (r H ) ∝ T in Eq. (16). When T → T c , η A /s vanishes. The analytic approximation performed in the Appendix suggests that critical exponent is of mean field value: η A /s ∝ (1 − T /T c ) 1/2 as T → T c . One can also see that η A → 0 as we take µ → 0 and the black hole becomes charge neutral.
In Fig. 4, η A /sλL vs. m 2 L 2 is plotted for T /µ = 7.55 × 10 −6 . The critical m 2 L 2 is smaller than the critical value m 2 L 2 = −1.5 at zero T because it is harder to form the condensate at higher T .
In our model, the non-zero Hall viscosity arises because parity is broken spontaneously.
The non-zero classical solution (or equivalently, vacuum expectation value) of θ yields a pseudo scalar condensate at the boundary which is a necessary condition to have non-zero Hall viscosity.

V. CONCLUSION
We have modified the holographic model of Saremi and Son [12] by using a charged black brane, instead of a neutral one, such that when the bulk pseudo scalar (θ) potential is made of θ 2 and θ 4 terms, parity can still be broken spontaneously in the boundary theory. In our model, the 3+1 dimensional bulk has a pseudo scalar coupled to the gravitational Chern-Simons term in the anti de Sitter charged black brane back ground.
Parity could be broken spontaneously in the bulk by the pseudo scalar hairy solution and give rise to non-zero Hall viscosity at the boundary theory.
This study does not exclude a non-vanishing Hall viscosity in Saremi and Son's model be found with a more general potential. It is interesting to investigate the Hall viscosity in other parity-broken holographic condensed matter systems, such as the D-wave superconductors [19,20]. We will report it in a future project.

A. Derivation of Hall viscosity
Here we detail the Hall viscosity derivation with the charged black brane solution. The hydrodynamics of charged fluid has been extensively studied in the holographic set up [21].
The general procedure to calculate the holographic hydrodynamic transport coefficients has been given in [11]. We largely follow the procedures adopted in [11,12] with the neutral black brane solution. The equations of motion by varying the action (4) with respect to the metric, the scalar and the gauge field are as where and C M N is called Cotton tensor coming from the gravitational Chern-Simons term where ǫ AM BC is the usual four dimensional Levi-Civita tensor.
An ansatz satisfying the equations of motion is This ansatz describes a boosted black brane solution alone the boundary coordinates.
Then following the standard procedure of the Fluid/Gravity correspondence, we perturb the system away from equilibrium by promoting the velocity u µ , mass b and charge q to vary slowly with respect to the boundary coordinates. In the co-moving frame where the fluid two-velocity is zero at the origin of the boundary coordinates (x µ = 0), we Taylor expand quantities near the origin to the first derivative order: Substitute these into the ansatz, we get ds 2 = 2H(r)dvdr − r 2 f (r)dv 2 + r 2 dx i dx i +ǫ 2δHdvdr − r 2 δf dv 2 − 2H(r)x µ ∂ µ β i dx i dr − 2r 2 (1 − f (r))x µ ∂ µ β i dx i dv , θ = θ(r) + ǫδθ, where we have added the parameter ǫ to keep track of how many derivatives on the boundary coordinates each term has.
Note that after we promote the parameter to be dependent on the boundary coordinates, the ansatz no longer satisfies the equations of motion. Hence we add corrections order by order to the metric, scalar and gauge fields such that, order by order, the whole metric, scalar and gauge fields still satisfy the equations of motion. To calculate the Hall viscosity, it suffices to consider the symmetric traceless part of the correction to the metric: ds 2 = ǫ k(r) r 2 dv 2 + 2h(r)dvdr − r 2 h(r)dx i dx i + 2 r a i (r)dvdx i + r 2 α ij (r)dx i dx j , θ = ǫθ cor , A = ǫ(A v cor (r)dv + A x cor (r)dx + A y cor (r)dy).