We show that a quartic contact interaction between charged fermions can lead to Cooper pairing and a superconducting instability in the background of a charged asymptotically Anti-de Sitter black hole. For a massless fermion we obtain the zero mode analytically and compute the dependence of the critical temperature on the charge of the fermion. The instability we find occurs at charges above a critical value, where the fermion dispersion relation near the Fermi surface is linear. The critical temperature goes to zero as the marginal Fermi liquid is approached, together with the density of states at the Fermi surface. Besides the charge, the critical temperature is controlled by a four point function of a fermionic operator in the dual strongly coupled field theory.
Cooper pairing near charged black holes
Thomas Hartman and Sean A. Hartnoll
Department of Physics, Harvard University,
Cambridge, MA 02138, USA
The first successful microscopic description of superconductivity, BCS theory , describes spontaneous symmetry breaking due to a charged fermion bilinear condensate. In the original theory the fermion pairing is driven by an attractive exchange of low energy phonons. More generally, the essential feature is a marginally relevant four point interaction between excitations about a Fermi surface . Whether this interaction is generated by phonons or otherwise is not crucial. The key fact is rather that in BCS-like theories, superconductivity emerges from a conventional free Fermi liquid fixed point. An important challenge facing condensed matter theory is to characterise the onset of superconductivity from non-Fermi liquid states of matter, such as the ‘strange metal’ phases of high temperature superconductors, e.g. [3, 4].
Recent developments have shown that charged or rotating black holes can carry Fermi surfaces [5, 6, 7, 8, 9, 10, 11]. We focus on the charged case in 3+1 dimensional asymptotically Anti-de Sitter (AdS) spacetime in what follows; it seems clear that many features of the computation will go through in the rotating case also and, with nonzero fermion mass, in asymptotically flat space. Near to the black hole, charged fermions experience a background chemical potential. If the charge of the fermions is sufficiently large compared to their mass, then at low Hawking temperatures they will build up a Fermi surface. The essential physics is the same as that for free fermions in flat space with a chemical potential. Computations are complicated by the fact that the background spacetime and electrostatic potential are nontrivial and vary on scales of order the Compton wavelength of the fermions. In general the Dirac equation cannot be solved in closed form on the whole spacetime.
It is natural to ask whether black hole Fermi surfaces can have BCS instabilities towards superconductivity. In this paper we will add a four fermion contact interaction between the charged fermions and compute the quadratic term of the one loop effective action for Cooper pairs. We will show that under certain circumstances the quadratic action has negative modes, indicating a superconducting instability. The effective action is not local in general on the curved spacetime background; this complicates finding e.g. the zero temperature gap. However, we have been able to obtain an analytic formula for the critical temperature . Furthermore, for massless fermions we have found the Fermi surface zero mode analytically, allowing explicit results without heavy duty numerical work. The critical temperature is
where is the chemical potential provided by the charged black hole at the AdS boundary, is the energy scale of the four fermion interaction, is the AdS radius, and is the effective density of states at the Fermi surface. From the perspective of the dual field theory, the dimensionless quantity determines the magnitude of a four point fermion correlator. Figure 4 shows the fermion charge dependence of , which is given by
where is the Fermi momentum, is the Fermi velocity, and is the fermion zero mode in the black hole spacetime. The precise formula is given in (6.38) below.
For asymptotically AdS charged back holes the bulk (free) Fermi surface admits a dual interpretation, via the applied holographic correspondence [12, 13, 14, 15, 16], as a strongly interacting (non-)Fermi liquid in 2+1 dimensions [5, 6, 7, 8]. It was understood in  that for fermions with a relatively low charge compared to their mass, the dispersion relation of fermion zero modes near the Fermi surface had a non-Fermi liquid form. Furthermore, these modes were broad resonances in the spectral density, rather than sharp quasiparticle peaks, as is indeed observed in strange metals . The non-Fermi dispersion was shown to lead to, for instance, deviations from the venerable Lifshitz-Kosevich formula for quantum oscillations [18, 19]. The black hole BCS instability we present below therefore has the potential to dually describe the non-BCS emergence of superconductivity from a strongly interacting non-Fermi liquid. Unfortunately, perhaps, we will find that the superconducting instability only occurs at larger values of the fermion charge, where the dispersion relation is linear (i.e. Fermi liquid like). This can be traced directly to the vanishing of the density of states at the Fermi surface in the non-Fermi liquid cases. It may be possible to evade this conclusion via alternate bulk pairing mechanisms with long range interactions.
Recent works have considered instabilities of charged scalar fields in charged black hole backgrounds and the corresponding spontaneous symmetry breaking at low temperatures [20, 21, 22]. If the charge of the boson is sufficiently large compared to its mass  it will condense, again in strong analogy to the behaviour of charged bosons in flat space with a chemical potential. As with the Cooper pairing instability we have just outlined, the dual interpretation is of superconductivity emerging from a strongly interacting non-Fermi liquid. One difference is that the boson condensation is classical in the black hole background whereas for fermions the effect requires a one loop computation, with an ensuing nonlocal (bulk) Landau-Ginzburg action. In the Cooper pairing case of interest here, the superconducting order parameter is directly related to a fermionic operator in the dual field theory. This may be phenomenologically useful and motivates fermion spectroscopy (‘ARPES’) computations in the superconducting state along the lines of , in which the fermion is chosen to couple in a natural way to the bosonic condensate.
2 Selfinteracting Dirac fermion
We consider a charged Dirac fermion with quadratic action
, , is the spin connection, and , with a mostly plus metric. We denote bulk spacetime indices by , abstract tangent space indices by , and specific tangent space indices by underlines as in . Eventually we will take the background to be a charged black hole in AdS, but for now the metric and gauge field are general.
The BCS mechanism requires an attractive force between like-charge particles. We therefore add the simple contact interaction
where is the mass scale of the interaction, , and is the charge conjugate fermion
The interaction (2.3), which also appears in color superconductivity , is the relativistic generalization of -wave BCS theory: it couples time-reversed, opposite spin states [26, 27]. A similar interaction was considered in a closely related context in . However, this choice is not unique. Besides choosing a more general contact term, the pairing mechanism could arise from exchange of scalar particles, the attractive channel in a nonabelian gauge theory, or perhaps graviton exchange. An attractive interaction per se is not sufficient to generate superconductivity, but should be in a ‘Cooper channel’. The contact interaction (2.3) is simpler than an exchange interaction, and can be considered a toy model for these other possibilities which may be more natural from the standpoint of string theory on AdS.
3 Effective action for the condensate
Mimicking the standard procedure in BCS theory, we can now perform a Hubbard-Stratanovich decoupling to make the action quadratic in spinors. As usual, there is a choice of channels to decouple. Given that we are anticipating a superconducting instability of the Fermi surface, we choose to decouple in the Cooper channel. Thus we introduce a charged scalar and write the Lagrangian as
Recall that the fermions anticommute. The equation of motion for sets
and we recover the original action.
Now consider the Coleman-Weinberg effective action for , to look for possible instabilties. Specifically, we will compute the one loop mass term generated for upon integrating out the fermions. The effective action at quadratic order is
Here is the Euclidean Green’s function for the Dirac operator in the gauge field and spacetime background, . is the transpose of the Green’s function in spin indices, i.e. . To derive this expression we used , and . See the representation of the gamma matrices in equation (4.8) below. Note also that the interaction term in the Euclidean action is minus that in the Lorentzian action. We use Lorentzian gamma matrices throughout.
We now choose coordinates , with Euclidean time, and assume the spacetime is translationally invariant along . In AdS, the radial coordinate is and the boundary directions are . Thus we can Fourier transform
where the fermionic Matsubara frequencies at temperature are
We furthermore restrict to configurations in which the condensate only depends on the radial direction. The effective action (3.3) becomes
where is the boundary spatial volume and
The next step is to relate the Euclidean Green’s functions appearing in (3.7) to real time Green’s functions. This is a little subtle, although the bottom line is that the boundary conditions at the black hole horizon mimic the usual effects of finite temperature field theory. In particular, as emphasized in [18, 28], eigenfunctions and eigenvalues, and hence Green’s functions, are not analytic functions of . This is because regularity at the Euclidean ‘horizon’ typically requires behaviour of the form
The positive and negative thermal frequencies must therefore be analytically continued separately. Analytically continuing, by setting , the Euclidean Green’s function from the upper imaginary frequency axis yields the retarded Green’s function , with poles in the lower half frequency plane. Analytic continuation from the lower imaginary frequency axis gives the advanced Green’s function, . This relation between Euclidean, retarded and advanced Green’s functions is a general statement that is particularly transparent in the black hole context, as we recall in an appendix.
The sum over Matsubara frequencies can therefore be rewritten as a contour integral
where the contour has a segment in the upper half plane and a segment in the lower half plane, each going clockwise around the poles of . The analytically continued function has a branch cut on the real axis. In the upper half plane, schematically,
where are the retarded and advanced Green’s functions. This product is analytic in the upper half plane. In the lower half plane and are exchanged. On the real axis the correlators are related by
where the transpose in acts on spin indices. This result follows easily from the definition of the various Green’s functions, see the appendix. Deforming the contours in (3.9) onto the real axis then gives
Our objective now is to evaluate these integrals.
4 The charged AdS black hole
At this point we will specialize to a planar, charged, asymptotically AdS black hole background. This is a solution to Einstein-Maxwell theory
The black hole background is given by
The horizon is at and the conformal boundary is . The chemical potential of the dual CFT is the boundary value of the Maxwell potential
We also introduced the ratio of electric and gravitational couplings
In terms of the above quantities, the Hawking temperature of the black hole (and temperature of the dual field theory) is
The nonzero components of the spin connection are
Finally, we adopt the following gamma matrix conventions of [8, 24]. These are useful for simplifying the Dirac equation once rotational invariance has been used to consider momentum in the direction without loss of generality. Thus
5 The retarded bulk Green’s function and
In order to compute the effective action of the condensate (3.6) in the black hole background, we need the retarded Green’s function. It is the unique solution of
subject to certain boundary conditions discussed below. Transforming into momentum space except in the radial direction,
where is the radial Dirac operator (including the mass term). The Green’s function equation is solved in the usual manner by multiplying together solutions of the homogeneous equation with a discontinuity across the delta function. For a mode with momentum in the direction,
the Dirac equation is
This equation is written out explicitly in (6.2) but here we need only some general properties. Writing the wavefunction as
The retarded Green’s function satisfies ingoing boundary conditions at the horizon. Near the AdS boundary (), the leading solution of the wave equation is
The two spinors and are eigenspinors of with opposite eigenvalues, implying that and are canonically conjugate (in a radial Hamiltonian slicing). See e.g. . Therefore a boundary condition must be imposed on one, with the other allowed to fluctuate. When , we must pick the fluctuating piece to be the normalizable mode proportional to . More generally, for any we will choose to impose the boundary condition on the fluctuating mode.111For , there is another possible choice discussed in e.g. , and for there is a continuous choice of boundary conditions discussed in e.g. [30, 31]. These more general quantization choices will not be considered here.
With these boundary conditions, the solution of (5.2) is constructed as follows. For each , take to be the solution ingoing at the horizon, and to be the solution with near the boundary. Then
with . The Wronskian is a constant related to the conserved charge current:
At general frequency and momentum , the solution satisfying ingoing boundary conditions at the horizon will not satisfy the asymptotic boundary condition, so generically . We will normalize the solutions so that near the asymptotic boundary we have
We have introduced the quantity , which is the retarded Green’s function of the boundary field theory, similar to the original discussion (for bosons) in . The Dirac equation (5.4) is a real equation for , so the boundary condition implies is real. By evaluating the -independent Wronskian near the asymptotic boundary we obtain
We can see immediately that poles of the boundary and bulk Green’s functions occur at the (in general complex) quasinormal frequencies of the black hole background, at which the mode satisfies both the horizon and asymptotic boundary conditions (see e.g. ).
It follows from the gamma matrices (4.8) that the upper and lower spinor projections are related by . Henceforth we drop the subscript and rewrite all quantities in terms of the first projection, denoting
All wavefunctions in this expression have momentum . We have used the identity
or equivalently, using (3.11), . This last statement can be seen directly from (5.8) and the fact that is real and , see the appendix. Note the modulus sign on and that the integral is over both positive and negative . Positive and negative momenta should be thought of as corresponding to the and components of the full Green’s function, respectively.
Our first objective is to perform the frequency and momentum integrals in (5) (with some regulator). In principle all the quantities appearing in this expression could be obtained numerically. However, we can do better at low temperatures . It was observed numerically in [5, 6, 7], and analytically in , that at these low temperatures poles in the retarded Green’s function can move close to the origin of the complex frequency plane. Specifically, the pole location as . This is of course the signature of a Fermi surface in the bulk geometry, as we might expect for charged fermions in a background electrostatic potential. Following the experience of BCS theory, which we are essentially replicating in a curved spacetime background, one might expect that the crucial pairing physics occurs close to the Fermi surface.
Close to the Fermi momentum, we will recall in the next section that 
Here is the perpendicular distance of the momentum from the Fermi surface, and are real constants, and is a zero temperature critical exponent to be described below. The function is the near horizon Green’s function, that will also be characterised in the following section and in an appendix. At low temperatures, ,
where is positive and the phase is such that poles of (5.15) are in the lower half complex frequency plane.
We expect the dominant contribution to be due to the singular locus of the boundary Green’s function (5.15). Thus we restrict consideration to near the Fermi surface222As well as near the Fermi surface, another singular region of the Green’s function occurs near the boundary of the ‘log-periodic’ region of . Here the denominator of the zero temperature Green’s function takes the form: , where and are real constants and as (i.e. when pure imaginary goes to in (6.7) below). By explicitly performing the integral and then bounding the integral of (5) in the dangerous small and small region, one finds that this region does not lead to singular low temperature behaviour. The effect of temperature can be estimated by replacing by an IR cutoff in the frequency integral at .
and set in the remainder of (5). We will check the self-consistency of this approximation a posteriori. It is simple to perform the integral, leading to
All wavefunctions in this expression are evaluated at . In general, as we recall below, there can be multiple Fermi surfaces for a given charge . Their contributions will sum in the above formula for .
The form of the Green’s function (5.15) used in the integral (5) is only correct for . For our computation to be valid we should therefore make sure that the key contribution to the integral comes from sufficiently low frequencies. More precisely, we will be interested in frequencies in the range . This range is the most IR singular, because the in the integral can be replaced by , and will be seen to capture the universal pairing physics. We now evaluate the integral for these frequencies.
For one can take the low temperature limit (5.16) of . The denominator in (5) becomes . The constant is not dimensionless, but rather . Therefore implies, as emphasised in , that the first term () dominates if while the second () dominates when . In both cases, the range implies that the fermion wavefunctions should be simply evaluated at in the extremal black hole background.333Setting and keeping the coordinates of the wavefunctions finite means that we miss the contribution to the frequency integral from e.g. . One should worry about the possibility of IR singular temperature dependence arising from this very near horizon region. In an appendix we check that this region does not give additional low temperature divergences to those discussed in the main text. Furthermore, given that we are then at and (at ), the wavefunction is precisely the Fermi surface zero mode,
Let us consider the cases and in turn.
: The leading behaviour of (5) in an expansion in is
where is Euler’s constant.
: The leading behaviour of (5) in an expansion in is now
where the constant appearing in the second term is
In these expressions is a renormalisation scale which we implement as a hard cutoff on the frequency integral. In any case what will be most important is the temperature dependence and, in the former case (5.20), the coefficient of the logarithm.
The first of the two cases above is particularly interesting. There is a logarithmic divergence associated with low temperatures. The expression in (5.20) is essentially the same as that appearing in BCS theory. It is this first case, , in which we will be able to consistently describe the onset of superconductivity. The logarithmic divergence in this case indicates that the range of frequencies and momenta we have integrated over do indeed pick out the dominant contribution to the integrals at low temperatures.
In the second case there is no low temperature divergence. Recall that the origin of superconductivity is the marginal relevancy of the BCS coupling about the Fermi surface . What has happenned in the second case is that the modified (non-Fermi liquid) low energy dispersion relation, as opposed to , has resulted in the quartic coupling becoming irrelevant and hence not leading to strong IR effects. This is familiar from the bosonic case . The absence of a low divergence means that (5.21) is not a controlled approximation to the integral and one should consider the full problem before concluding beyond doubt that there is no superconductivity in this case. We have shown however that there is no instability near the Fermi surface. Consistent with this observation we will find below that as is approached from above. In some contexts, see e.g. [35, 36, 37] for a sampling including a color superconductivity case, long range pairing interactions lead to additional inverse frequency dependence in the analogue of (5). Such interactions can compensate for the weaker frequency dependence of the fermion propagator, allowing for non-Fermi liquid pairing to occur. In the holographic models under consideration, a nontrivial fermion propagator can result from classical propagation on a curved spacetime, but the leading fermion interactions are mediated by a contact interaction that seems not to allow the non-Fermi liquids to pair.
In both of the above expressions, (5.20) and (5.21), the and dependence of has factorised. This allows us to solve explicitly for the unstable mode and the critical temperature . The critical temperature is defined by the appearance of a zero mode for the condensate. From the quadratic effective action (3.6) this requires
The factorisation of immediately allows us to conclude that the critical zero mode
which seems rather natural. Plugging back into (5.23) leads to a simple formula for the critical temperature (for )
Solving for the critical temperature gives
where the effective density of states at the Fermi surface is a dimensionless function of the fermion charge in units of , as defined in (4.5). The exponent is the most important part of this expression. In the remainder of the paper we will determine this dependence of on , which is a free parameter in our theory. For our various approximations to be reliable we need . From (5.26) this is seen to hold when . This last inequality can also be thought of as the condition for validity of perturbation theory in the quartic fermion interaction of our theory (2.3). That said, the computation of is balancing a classical and one loop mass term at a nonperturbatively (in the coupling) low temperature. Around and below this temperature a marginally relevant coupling is becoming strong and one might worry about the need to resum large logarithms at higher orders in perturbation theory. The one loop computation we have performed is in fact the only fermion loop contribution to the quadratic effective action for within our theory and so our computation is exact close to the critical temperature. More generally, moving away from the quadratic level, in BCS-Eliashberg theory the kinematics of the Fermi surface leads to the absence of further relevant operators and the one loop computation is exact, see e.g. . It is likely that a similar statement holds for our setup.
As well as the dependence in the exponent, there is the dependence on . The Fermi mass does not correspond to a dimensionful scale in the dual field theory (the only scales in the otherwise conformal field theory are and ). We will discuss in the final section how is instead related to a dimensionless four point correlator in the field theory. This correlator therefore controls in our (dual) strongly interacting field theory in the same way the electron-glue coupling controls the critical temperature in a perturbative BCS treatment.
In the prefactor in (5.26) the overall scale is set by . This is a renormalisation rather than physical scale, highlighting the fact that the prefactor is not well defined but is scheme dependent. Presumably , as is the only scale in the theory at low temperatures. Renormalisation ambiguities will cancel in dimensionless quantities such as the ratio of to the zero temperature mass gap. These ambiguities do not affect the coefficient of the logarithmic divergence at low temperatures, due to frequencies , but will shift the order one term in (5.20). Our main interest will be the function in the exponent, which is robust. We have primarily kept the prefactor in (5.26) in order to emphasize the strong similarity with the standard BCS result. The only difference between (5.25) and the analogous expression appearing in flat space BCS theory is the integral over a spatially dependent zero mode determining the effective density of states at the dual field theory Fermi surface.
As noted above, for large enough there will be multiple Fermi surfaces. The condensate (5.24) and critical temperature (5.26) were given for the case of a single Fermi surface, but can be readily generalized to include the contribution from all the surfaces at once. Each Fermi surface contributes a term of the form (5.20) to . Labeling the Fermi surfaces with by , with corresponding zero modes , the solution of the integral equation (5.23) has the form
Plugging this ansatz into the integral equation gives an eigenvalue equation for . This can be written as
The critical temperature is then given by
where is the largest eigenvalue of (5.28). The corresponding eigenvector is , the relative contribution of each zero mode to .
We now turn to the computation of the various quantities appearing in (5.25). We have been able to find the zero mode analytically when , allowing many explicit results.
6 Solution of the massless Dirac equation
Consider a massive, charged fermion in the black hole background with the wavefunction
The combinations are chosen for convenience. As discussed above, the lower two components of the spinor decouple and so can be set to zero. The Dirac equation (5.4) is
The Dirac equation is actually real, and the appears in (6.2) only because it was inserted by hand in (6.1). Generally, this equation must be solved numerically. We will treat the massless case, where we have been able to obtain the wavefunctions analytically for small . Setting , the components of (6.2) can be decoupled to give the second order equations
6.1 Zero modes at zero temperature
Our first objective is to find the fermion wavefunctions at in the background. From this point on we restrict to the massless case as we are able to find an analytic solution here. Define dimensionless quantities and radial coordinate by
Recall from (4.6) that at zero temperature .
Near the horizon, , the wavefunctions behave as with
We will later be interested in a certain and define
This is the we referred to in the previous section. We will mainly be interested in cases with . We require the behavior for regularity.
At zero temperature, the function that appears in the metric can be written as
Plugging this into the decoupled equations (6.4) with gives the equation for the fermion zero modes. It is found to have the exact solution
with a normalization. The second independent solution is obtained by replacing in (6.1),
with a new normalization . The first solution (6.1) has the required regular behavior at the horizon for the solution. The second will also be required when we consider a small nonzero frequency. Inserting the solution into the first order Dirac equation (6.2) gives the relative normalizations
Thus we have obtained the zero modes appearing in the expression (5.25) for the critical temperature. While computing and will require moving to small frequencies, we already have enough information to obtain .
6.2 Fermi momentum
The Fermi surface is characterized by a zero mode that is regular at the horizon and obeys certain falloff conditions near the boundary of AdS. As described in Section 5, the asymptotic boundary condition on the fluctuating mode is
The equation for the Fermi momentum is therefore
The hypergeometric functions can be evaluated numerically to solve for the Fermi surface. For example, when , we find in agreement with the numerical solution of the Dirac equation in . The solutions of (6.15) are plotted in figure 1. For a given there can be multiple Fermi surfaces . When a distinction is necessary, the largest will be called the ‘first’ Fermi surface, the next the ‘second’ Fermi surface, and so on.
We can also use the explicit solution above to make an observation about the Green’s function at . The boundary Green’s function defined in (5.10), evaluated at zero frequency, is
We will now show that the imaginary part of this Green’s function vanishes when is real. For imaginary the imaginary part of the Green’s function will be positive provided that is taken to be positive (this amounts to a choice of sign of a square root in the above). For real , as we are interested in, the wavefunctions can be shown to satisfy
with the real phase
This expression assumes for concreteness that is real and . It follows that for real the spectral density vanishes away from , i.e. , while the real part has a pole at the Fermi momentum.
By substituting the Fermi momentum into the zero mode wavefunction (6.1) we can plot the radial profile of the unstable Cooper pair mode of equation (5.24). The result for various charges is shown in figure 2 below. The most notable feature of these plots is that for large charge the zero modes are supported away from the horizon while as , at smaller charge, the wavefunctions are supported in the near horizon region. This is consistent with the observation, below and in [8, 11], that for the physics of the (non-Fermi liquid) Fermi surface is captured by the near horizon region.