# Holographic superconductivity in M-Theory

###### Abstract

Using seven-dimensional Sasaki-Einstein spaces we construct solutions of supergravity that are holographically dual to superconductors in three spacetime dimensions. Our numerical results indicate a new zero temperature solution dual to a quantum critical point.

## I Introduction

The AdS/CFT correspondence provides a powerful framework for studying strongly coupled quantum field theories using gravitational techniques. It is an exciting possibility that these techniques can be used to study classes of superconductors which are not well described by more standard approaches Gubser:2008px H31 H32 .

The basic setup requires that the CFT has a global abelian symmetry corresponding to a massless gauge field propagating in the space. We also require an operator in the CFT that corresponds to a scalar field that is charged with respect to this gauge field. Adding a black hole to the space describes the CFT at finite temperature. One then looks for cases where there are high temperature black hole solutions with no charged scalar hair but below some critical temperature black hole solutions with charged scalar hair appear and moreover dominate the free energy. Since we are interested in describing superconductors in flat spacetime we consider black holes with planar symmetry. In order to obtain a critical temperature, conformal invariance then implies that another scale needs to be introduced. This is achieved by considering electrically charged black holes which corresponds to studying the dual CFT at finite chemical potential.

Precisely this set up has been studied using a phenomenological theory of gravity in coupled to a single charged scalar field and it has been shown that, for certain parameters, the system manifests superconductivity in three spacetime dimensions, in the above sense H32 . It is important to go beyond such models and construct solutions in the context of string/M-theory so that there is a consistent underlying quantum theory and CFT dual. Also, as we shall see, the behaviour of the string/M-theory solutions will differ substantially from that of the phenomenological model H32 at low temperature. It was shown in dh that the phenomenological models of H32 arise, at the linearised level, after Kaluza-Klein (KK) reduction of supergravity on a seven-dimensional Sasaki-Einstein space . Here we go beyond this linearised approximation by working with a consistent truncation of the KK reduced theory presented in Gauntlett:2009zw . The truncation is consistent in the sense that any solution of this theory, combined with a given metric, gives rise to an exact solution of supergravity. Here we shall use this theory to construct exact solutions of supergravity that correspond to holographic superconductivity.

## Ii The KK truncation

We begin by recalling that any metric can, locally, be written as a fibration over a six-dimensional Kähler-Einstein space, :

(1) |

Here is the one-form dual to the Reeb Killing vector satisfying where is the Kähler form of . We denote the form defined on by . For a regular or quasi-regular manifold, the orbits of the Reeb vector all close, corresponding to compact isometry, and the is a globally defined manifold or orbifold, respectively. For an irregular manifold, the Reeb-vector generates a non-compact isometry and the is only locally defined.

In the KK ansatz of Gauntlett:2009zw the metric is written

(2) |

while the four-form is written

(3) | |||||

where is a four-dimensional metric (in Einstein frame), are real scalars, and is a complex scalar defined on the four-dimensional space. Furthermore, also defined on this four-dimensional space are a one-form potential, with field strength , two-form and three-form field strengths and , related to one-form and two-form potentials via and . Finally .

This is a consistent KK truncation of supergravity in the sense that if the equations of motion for the 4d-fields as given in Gauntlett:2009zw are satisfied then so are the equations. The equations of motion admit a vacuum solution with vanishing matter fields which uplifts to the solution:

(4) |

where is the standard unit radius metric. When , this solution is supersymmetric and describes -branes sitting at the apex of the Calabi-Yau four-fold () cone whose base space is given by the . When the solution is a “skew-whiffed” solution, which describes anti--branes sitting at the apex of the cone. These solutions break all of the supersymmetry except for the special case when the is the round seven-sphere, , in which case it is maximally supersymmetric. Note that the skew-whiffed solutions with are perturbatively stable Duff:1984sv , despite the absence of supersymmetry. Thus such backgrounds should be dual to three-dimensional CFTs at least in the strict limit. We are most interested in the skew-whiffed case because it is for that case that the operator dual to has scaling dimensions or Gauntlett:2009zw and, based on the work of dh , is when we expect holographic superconductivity.

The equations of motion can be derived from a four-dimensional action given in Gauntlett:2009zw . It is convenient to work with an action that is obtained after dualising the one-form to another one form and the two-form to a scalar as explained in section 2.3 of Gauntlett:2009zw . The dual fields are related to the original fields via

(5) |

where . We now restrict to the (skew-whiffed) case . For this case we can make the following additional truncation of the theory:

(6) |

One can show that provided that we restrict to configurations satisfying we obtain equations of motion that can be derived from the action

(7) |

where ,
and we have defined ,
.
Linearizing in the complex scalar , this gives the action considered in
H32 (with their and their ).
This non-linear action is in the class considered in Franco:2009yz and
in addition to the vacuum with and , which uplifts
to (4), it also admits vacuua
with and constant ,
which uplift to the
solutions^{1}^{1}1There are analogous solutions
of the theory considered in
Gubser:2009qm which uplift to IIB solutions found in romans .
found in pw .

## Iii Black Hole Solutions

The key result of the last section is that any solution to the equations of motion of the action (II) with , gives an exact solution of supergravity for any metric. To find solutions relevant for studying superconductivity via holography we consider the following ansatz

(8) |

where and are all functions of only. Being purely electrically charged this satisfies the condition. After substituting into the equations of motion arising from (II), we are led to ordinary differential equations which can also be obtained from the action obtained by substituting the ansatz directly into (II):

where .

We next observe that the system admits the following exact AdS Reissner-Nordström type solution

(10) |

for some constants . The horizon is located at and for large it asymptotically approaches 1/4 of a unit radius (see (4)). This solution should describe the high temperature phase of the superconductor.

We are interested in finding more general black hole solutions with charged scalar hair, . Let us examine the equations at the horizon and at infinity. At the horizon we demand that . One then finds that the solution is specified by 4 parameters at the horizon , , , . At we have the asymptotic expansion,

(11) |

determined by the data . The scaling

(12) |

leaves the metric, , and all equations of motion invariant.

### iii.1 Action and thermodynamics

We analytically continue by defining . The temperature of the black hole is where is fixed by demanding regularity of the Euclidean metric at . We find:

(13) |

Defining , we can calculate the on-shell Euclidean action

(14) | |||||

where . The latter expression only gets contributions from the on-shell functions at since , while the former expression gets contributions from and . The on-shell action diverges and we need to regulate by adding appropriate counter terms. We define and, for simplicity, we will focus on the following counter-term action :

(15) |

where is the trace of the extrinsic curvature. For our class of solutions we find

(16) | |||||

Notice that under a variation of the action with respect to yields the equations of motion together with surface terms. For an on-shell variation the only terms remaining are these surface terms, and after substituting the asympototic boundary expansion (11) (higher order terms are also required) we find

(17) | |||||

Note that we are keeping fixed in this variation. Hence we see that is stationary for fixed temperature and chemical potential (ie. ) and for either or fixed .

We also find that the on-shell total action is given by

(18) | |||||

where is the entropy density of the solution and is the energy density. The two forms of the on-shell action come from the two ways of writing the action as a total derivative given above. We note that the equality of these expressions imply a Smarr-like relation. Also note that after using (since is held fixed) the equality of (17) and the variation of the first line of (18) imply a first law,

(19) |

Both this Smarr relation and the first law were used to confirm the accuracy of our numerical solutions below.

For simplicity we restrict discussion to solutions with boundary condition ^{2}^{2}2One may also consider fixing H32
with similar results. Non-zero is less interesting as we want the scalar to condense without being sourced.
and we interpret
as a thermodynamic potential, .
Note also that then determines the vev of the operator dual to .
Recall from Gauntlett:2009zw that writing , , the
fields are dual to operators with dimensions , .
The truncation (II) implies that the vevs of these
dual operators are fixed by . The asympotic expansion of
to and to
gives
and .

### iii.2 Numerical Results

Following H32 we solved the differential equations numerically using a shooting method. We used (12) to fix the scale . At high temperatures the black hole solutions have no scalar hair () and are just the solutions given in (10). At a critical temperature a new branch of solutions with appears and moreover dominates the free energy. We refer to these as the unbroken and broken phase solutions, corresponding to normal and superconducting phases, respectively. In the figures we have plotted some features of our solutions and compared them with the solutions of the phenomenological model considered in H32 .

While the results are in agreement near the critical temperature,
as expected, we see marked differences as the temperature goes to zero.
We have calculated
the Ricci scalar and curvature invariant at
which indicate that the solutions of H32 are becoming singular
but our solutions are approaching a
regular zero temperature solution, without horizon,
holographically dual to a quantum critical point.
Indeed as we find
, const, and and fixing gives in the extremal limit.
In particular, the geometry near is consistent with being the exact
solution with , mentioned earlier, which uplifts to the solution found in pw .
For such a solution and , agreeing with the low temperature limit seen in the figures.
The full zero temperature solution thus appears to be a charged
domain wall, of the type considered in Gubser:2008wz ,
connecting two vacua of (II), one
with and the other with .
Interestingly this implies the entropy of the solutions vanish in the low temperature limit, unlike for the
Reissner-Nordström solution (10).^{3}^{3}3We thank Gary Horowitz for a discussion on this point.
The asymptotic charge appears to be derived from the scalar hair, with
the region near carrying no flux.

## Iv Concluding remarks

For any seven-dimensional Sasaki-Einstein space we have constructed solutions of supergravity corresponding to holographic superconductors in three spacetime dimensions. We have studied electric black holes using the action (II) whose solutions lift to when . One may consider adding magnetic charge using the full consistent truncation of Gauntlett:2009zw . Our results indicate the existence of a regular zero temperature solution which is a charged domain wall connecting two vacua of (II) and dual to a new quantum critical point. An important open issue is whether or not there are additional unstable charged modes for skew-whiffed solutions, which condense at higher temperatures. If they exist, and dominate the free energy, then the corresponding supergravity solutions would be the appropriate ones to describe the superconductivity and not the ones that we have constructed. However, it is plausible that we have found the dominant modes for large classes of , if not all. For the specific class of deformations of the four-form that were considered in dh , it was proven that the modes that we consider are in fact the only condensing modes. It would be worthwhile extending this result to cover other bosonic and/or fermionic modes.

Note added: after this work was completed we received Gubser:2009qm which constructs solutions of string theory that are dual to superconductors in four spacetime dimensions.

We are supported by EPSRC (JG,JS), the Royal Society (JG) and STFC (TW). We thank R. Emparan, S. Hartnoll, G. Horowitz, V. Hubeny, M. Rangamani, O. Varela and D. Waldram for helpful discussions.

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