Electron stars are fluids of charged fermions in Anti-de Sitter spacetime. They are candidate holographic duals for gauge theories at finite charge density and exhibit emergent Lifshitz scaling at low energies. This paper computes in detail the field theory Green’s function of the gauge-invariant fermionic operators making up the star. The Green’s function contains a large number of closely spaced Fermi surfaces, the volumes of which add up to the total charge density in accordance with the Luttinger count. Excitations of the Fermi surfaces are long lived for . Beyond the fermionic quasiparticles dissipate strongly into the critical Lifshitz sector. Fermions near this critical dispersion relation give interesting contributions to the optical conductivity.
Fermions and holographic Lifshitz criticality
Sean A. Hartnoll, Diego M. Hofman and David Vegh
Center for the Fundamental Laws of Nature, Harvard University,
Cambridge, MA 02138, USA
Simons Center for Geometry and Physics, Stony Brook University,
Stony Brook, NY 11794-3636, USA
- 1 Context
- 2 The electron star background
- 3 The WKB Dirac equation
- 4 The boundary fermion Green’s function: generic
- 5 The Luttinger count
- 6 The boundary fermion Green’s function:
- 7 The boundary fermion Green’s function:
- 8 The ‘small star’ limit
- 9 Discussion
- A Intermediate matching formulae for cases I, II and III
- B Matching formulae for
- C Matching formulae for
- D The WKB limit for small stars
Known large gauge theories with holographic gravity duals  typically have a ‘microscopic’ field content of gauge bosons, scalars and fermions. These theories often have global symmetries under which a subset of the scalars and fermions can be charged. The theory can then be placed at nonzero chemical potential (equivalently, charge density ) for this symmetry. The dual gravitational description of the nonzero charge density is that there must be an electric flux reaching out to the boundary of the bulk spacetime. The classical dynamics of the gravity dual will then determine the IR nature of the spacetime subject to this UV boundary condition. This general setup is illustrated in figure 1 below.
The framework of the previous paragraph has similarities with difficult open questions in the theory of metallic quantum criticality in 2+1 dimensions. There, one typically considers a finite density of itinerant fermions interacting with gapless bosonic degrees of freedom [2, 3]. The strong effects of forward scattering of the infinitely many excitations at the (putative) Fermi surface invalidate perturbative frameworks for studying the low energy physics of these theories [4, 5, 6], although see also [7, 8, 9]. Taking the symmetry of the large gauge theories as a proxy for the density of fermions (this identification will be literal if there are no massless microscopic bosons charged under the symmetry) we can use the holographic correspondence as a probe of the IR physics of metallically quantum critical states of matter.
The presence of a density of fermions is a defining characteristic of metallic criticality. Fermionic spectral densities are therefore natural observables from which to understand the low energy physics. In particular one is interested in the possible smearing of fermionic quasiparticle poles at a Fermi surface due to interactions with the critical bosonic degrees of freedom. One should bear in mind that the fermions that can be accessed holographically are not the microscopic fermions but rather gauge-invariant composite fermionic operators.
The simplest answer to the question posed by figure 1 above, that of determining the IR geometry sourcing an electric flux towards infinity, is the extremal Reissner-Nordström-AdS black hole . Here there are no charge-carrying fields in the spacetime and the flux emanates from a black hole horizon. The analysis of fermion spectral functions in extremal black hole backgrounds, initiated by , led to interesting results [12, 13, 14]. The fermion spectral functions exhibit Fermi surface singularities. Furthermore, if the gauge-invariant fermion operator has a sufficiently low scaling dimension relative to its charge, then the pole is broadened into a ‘non-Fermi liquid’ branch cut singularity. The origin of this phenomenon was traced to the behavior of the Dirac equation in the far IR geometry of the extremal black hole  and can therefore be thought of as indeed being due to strong interactions between fermionic excitations and the quantum critical bosonic modes in the IR.
The chemical potential breaks Lorentz invariance and therefore an emergent scaling symmetry in the IR will generically be characterised by a dynamical critical exponent such that under scaling: and . It is natural to think of such emergent scaling in nonzero density holography as a strong coupling version of the Landau damping of bosons weakly coupled to a Fermi surface . The Reissner-Nordström-AdS black hole is then seen to be an extreme case of Landau damping in which . It is of interest to consider holographic duals that admit, presumably more generic, finite IR scaling behavior and to characterize the fermionic spectral functions in such theories. Some general observations about holographic fermions in finite IR geometries were made in . In particular, it was noted that in a certain regime a Fermi surface pole would not only not be broadened, but that the width should be exponentially small at low energies, making the fermion excitations even more stable than those of a Fermi liquid! We shall reproduce this result, embedded within a slightly broader structure.
Various interesting realisations of finite IR geometries due to a nonzero charge density in the UV have recently been found. The first were in superconducting phases [16, 17]. Here the electric field is fully sourced by a condensed scalar field in the bulk, allowing the black hole throat to collapse into a ‘Lifshitz’ IR geometry [18, 19]. We are however interested in non-symmetry broken phases in this paper. A fluid of charged fermions outside of a black hole also absolves the horizon of needing to source the electric field and thereby allows a near horizon Lifshitz geometry . The resulting geometries were dubbed ‘electron stars’ in  and have many features in common with the superconducting phases just mentioned. However, the symmetry is not broken by the fermions as these build up a Fermi surface in the bulk rather than forming a coherent state with a definite phase. Finally, without introducing charged matter explicitly, the extremal near horizon throat can be collapsed by allowing a dilaton coupling to the Maxwell field in such a way that the effective Maxwell coupling goes to zero at the horizon [22, 23]. Our computations will focus on the electron star geometries for concreteness, and also because there is a certain minimalism in considering the spectral function of the same fermion that backreacts on the geometry. Nonetheless it will be clear that our results only depend on qualitative features of the geometry and will apply also to the other realisations of finite we have just mentioned.
Sections 2 and 3 recall the essential features of electron star backgrounds and reduce the computation of the boundary field theory fermion Green’s function to a quantum mechanics scattering problem. The fluid description of electron stars is the WKB limit of the bulk Dirac equation, and therefore the rest of the paper unfolds as a WKB analysis of scattering in a specific potential. Many of our results follow qualitatively from an inspection of the effective Schrödinger potential, as we discuss in section 3.
Sections 4 through 7 characterise different low energy () regimes of the fermion Green’s function. The Green’s function displays a large number of closely spaced Fermi surfaces with very long lived (as anticipated by ) quasiparticle excitations. There is a maximal Fermi momentum . While this extremal momentum dominates the quantum oscillations of the star , we show here it has the smallest spectral weight of all the Fermi surfaces. We show that the sum of volumes of the 2+1 dimensional field theory Fermi surfaces adds up to the total charge density, implying that the Luttinger count holds for this (non-Fermi liquid) system. Excitations of the Fermi surfaces become unstable at small momenta , where they dissipate efficiently into the critical Lifshitz sector. We characterise carefully the poles in the Green’s function close to the critical dispersion and estimate that they give a dominant contribution to the optical conductivity of the system.
Finally in section 8 we take a limit in which the entire star is contained within the Lifshitz region of the spacetime. In this limit and with we can solve for the Green’s function exactly, thereby giving explicit examples of various statements made throughout the paper.
2 The electron star background
This paper will consider the fermionic spectral functions of field theories with electron star gravity duals. As mentioned in the introduction, our results mainly depend on qualitative features of the geometry that are shared by other spacetimes. We will emphasize these aspects throughout.
The background spacetime and Maxwell field take the form
with functions of the radial coordinate . In these coordinates at the spacetime boundary. Throughout, our spacetime will be 3+1 dimensional and the dual gauge theory, 2+1 dimensional. The equations of motion for the metric functions follow from the Einstein-Maxwell-charged fluid Lagrangian density
as described in . Here is the local pressure of a bulk fermion fluid in terms of a local chemical potential and a local entropy density. The fluid is a corse-grained description of free Dirac fermions with unit charge (without loss of generality) and mass . The fluid description is valid when  (analogous in depth discussions are also found in [25, 26])
and the following dimensionless mass is order one
This second condition implies that . The Compton wavelength of the fermion is much smaller than the radius of curvature of the spacetime and therefore we should expect to be in a WKB limit. This WKB limit will greatly simplify our discussion of the fermion Green’s function below. While convenient, some of the essential physics should not depend strongly on this limit. We hope to explore a ‘quantum electron star’ explicitly in the future.
In the deep IR, , the functions are found to tend towards a Lifshitz geometry
with in particular 
The value of the dynamical critical exponent depends on the order one dimensionless parameters and . It is bounded from below by 
This is the finite emergent IR criticality advertised in the introduction.
Integrating outwards from the IR, the local pressure of the fermion fluid decreases monotonically until it vanishes at a radius . This is the boundary of the electron star. Outside the electron star, , the spacetime is that of Reissner-Nordström-AdS
The four constants here are determined by matching at and in turn determine the thermodynamics of the dual field theory.
3 The WKB Dirac equation
Our objective is to compute the boundary Green’s function of the gauge-invariant fermion operators that are making up the electron star. The bulk fermion can be taken to have four components and to satisfy the Dirac equation. With a judicious choice of basis for the Gamma functions, the Dirac equation separates into two decoupled equations for two two-component spinors. These equations differ only by the momentum . Without loss of generality we can focus on one of the two-component spinors. The Dirac equation can then be written 
Here is a two component spinor, the are Pauli matrices, and we have taken the field theory spacetime dependence . The field is rescaled relative to the actual Dirac field: .
In particular, we can write the spinor as
Furthermore, in order to obtain an interesting and dependence, we should rescale
Then to leading order in the Dirac equations become
These equations are simple to solve. The solutions are
The overall function is easily determined by working to next order in the WKB expansion. The zero temperature ‘horizon’ is at . We must impose ingoing boundary conditions at the horizon. The ingoing solution is, for , the upper of the above signs. However, we must be careful to discuss the matching of the solution across the turning points of the exponent .
In order to match across the turning points it is convenient to map the Dirac equation into a Schrödinger form. This will allow us to use the standard machinery from WKB approximations in quantum mechanics. We will thus rederive the expressions (15) and (16) from this perspective. Writing
it is simple to obtain a second order equation for each component of the spinor. The expression is a little clumsy, but it simplifies in our limit to
Thus we recover the leading order exponents (15) in the usual WKB fashion. We will proceed shortly to perform a WKB study of the ‘zero energy’ Schrödinger equation (18) in the following section, but some comments first.
The first of the two equations above has corrections of order .
A second comment concerning (19) is that it appears to become singular when the denominator vanishes. The higher order corrections to the potential in (18) in fact also diverge at such points. These divergences are an artifact of eliminating variables and are not inherent to the Dirac equation (9). The easiest way in practice to avoid this complication is to note that if we instead obtain a Schrödinger equation for and then obtain from , the divergence now occurs at while the effective Schrödinger equation remains the same (18). Thus to avoid possible singularities, we can swap between solving for or first if we approach a singular point. We are interested in real and only, so nontrivial monodromies should not arise. The upshot is that we can just focus on the Schrödinger equation (18). We see that only appears in the effective Schrödinger equation and so in the following we will restrict to without loss of generality.
The nature of the WKB solutions is controlled by the turning points in the exponent. From the turning points alone we will obtain a good picture of the Green’s function we are after. Thus, we need to know the sign of the potential appearing in the effective zero energy Schrödinger equation (18)
In discussing this quantity, it will be useful to introduce the following notions. The local Fermi momentum is
The boundary of the star is determined by 
Note that is negative for , i.e. outside the star. All the interesting features we are about to explore occur inside the electron star. The ‘extremal’ local Fermi surface is determined by 
The extremal radius always exists and is unique with , inside the star. It can be determined numerically . The local Fermi momentum goes to zero at the center of the star as well as at the star boundary.
We are interested in the low frequency Green’s function . Start by setting . At zero frequency
Thus we have, for ,
Here clearly are defined as the two solutions to . Thus we see that the zero frequency WKB Dirac solution is exponential wherever the momentum is greater than the local Fermi momentum and oscillating whenever the momentum is lower than the local Fermi momentum. The oscillations therefore occur when the local states with the given momentum are filled. We may expect some resonant behaviour at separating the two regimes. The two cases are illustrated in figure 2.
Now consider low but nonzero frequencies . This should only affect the zero frequency results (25) and (26) in the near horizon region. This is because the near horizon () and zero frequency limits do not commute. Away from the infinite redshift of the horizon, the effects of small nonzero are uniformly small.
In the near horizon Lifshitz region (5)
In the outer limits of the near horizon region, where , we recover the zero frequency conditions of (25) and (26). This is the regime of overlap of the near and far regions and is consistent with our previous remark that small should not qualitatively change the turning points in the far region. Note that the relations (6) and (7) imply that in the near horizon region . This is the consistency condition that the interior of the star is in fact occupied by fermions.
At any nonzero frequency we see that, assuming , for any mass and momentum, in the far interior . This fact may require another change in sign of at a radius in the near horizon region and leaves us with three possible patterns of signs shown in the following table. The possibilities are also illustrated in figure 3 below.
For the the third case shown in table 1 to occur, it must be that the second intersection point at occurred within the near horizon region. This is because the effects of finite are negligible outside this region.
Furthermore, it must be that the momentum is small, so that its effects in making positive can be offset with a small frequency . In fact, checking for the existence of zeros of (27) we can determine precisely when the third case can occur
It is straightforward to find the maximum analytically, and we shall do so in a later section.
In the following sections we will obtain the boundary fermion Green’s function in some detail. However, a qualitative understanding of the frequency and momentum dependence of the spectral density – the imaginary part of the Green’s function – can be extracted simply from the Schrödinger potentials of figure 3. This gives us a global overview of what to expect in the following.
We can see that in case II there will be many (because we are in the WKB limit) almost bound states that are stable up to an exponentially small tunneling amplitude into the horizon. These correspond to many Fermi surface poles in the retarded Green’s function. At small frequencies the poles are bounded above in momentum by and below by the Lifshitz dispersion . These are the boundaries with cases I and III respectively. The boundary regions themselves are particularly interesting and we will consider them carefully below. In cases I and III we should expect no sharp features in the spectral density as there are no almost stable states. Putting these statements together leads to the schematic figure 4. This plot is very similar to plots describing the fermion spectral density in a superconducting background [28, 29].
This is unsurprising because the background fluid of charged fermions has many features in common with a background charged bosonic condensate. Furthermore, the backgrounds [17, 27] considered in [28, 29] also had an emergent () scaling symmetry in the IR and therefore a ‘light-cone’ within which fermion excitations were efficiently dissipated. One special feature of the case is that the lightcone exists for both positive and negative frequencies. In our figure we have a ‘non-relativistic lightcone’ that only exists at positive frequencies. We will discuss the physics underlying figure 4 in the following section. In a later section we will present a non-schematic version of this figure. We will also be concerned with the extent to which the sharp Fermi surfaces can be ‘smeared’ into a Fermi ball  and with the rôle of the relative residues of Fermi surface poles. Some of the Fermi surfaces will have a spectral weight that is exponentially suppressed relative to others.
4 The boundary fermion Green’s function: generic
The Green’s function of the fermion operator in the quantum field theory dual to our bulk fermion is read off from the asymptotic behaviour of solutions to the bulk Dirac equation. Dropping the subscript for the spinor component, recall that we are interested in solving the Schrödinger equation (18)
with the potential given by (20). Given that , the WKB approximation will be valid as long as the potential and its first two derivatives remain of order one. This follows from the fact that the WKB approximation requires that the derivatives of the potential are small compared to the potential. We can use dimensional analysis to see how the comparison should be made. Because has dimension two, we require:
These inequalities are obviously satisfied if is order one with . Whenever is smaller than order one we should consider a matching strategy over the problematic region. We will come to these cases later.
In the outer region, defined to be the region from the boundary to the first turning point, we have . The two independent solutions of (29) are therefore schematically
Here is a UV cutoff. Outside the star we must use the Reissner-Nordstrom-AdS expressions (8). It is then possible to show that to leading order near the boundary (recall )
To obtain this precise result, including the leading correction to the powers , one needs to include the leading correction, in footnote 1 above, to the Schrödinger potential as well as the prefactor in the WKB expression (31). This is the only place in this paper where we will refer to this footnote and indeed we do not even strictly need this subleading correction here. Imposing ingoing boundary conditions at the horizon and matching across the turning points, we will obtain some combination of these two modes in the outer region. Being careful to solve both (18) and (19) consistently and keeping the leading terms near the boundary
The retarded Green’s function of the boundary operator dual to is then (e.g. )
Because we will be focussing on the single Schödinger equation (29) for it is convenient to re-express the Green’s function in terms of the asymptotic behaviour (32) of this function. Thus, using (19), we obtain
Here and are the boundary chemical potential and speed of light as appearing in (8). has slightly unconventional units due to compatibility with the notation in previous works.
We now proceed to obtain the Green’s function (34). Let us first consider the case in which vanishes linearly in radius at all turning points. This is the generic situation in which we are not at a boundary between two of the three regions above. The matching computation is standard and we have relegated the details to appendix A. The result for the retarded Green’s function at leading order in the WKB expansion is, in cases I and III,
This expression is of course precisely what we should expect. The Green’s function is pure imaginary and given by the probability of the particle tunneling through to the interior oscillatory region and into the horizon. This probability is exponentially small.
In case II the fermion propagates through an intermediate oscillatory region. The Green’s function is found to be
The Green’s function in case II has poles when . Note that both and are large positive numbers when is real. Including the exponentially small imaginary part of the pole to leading order, the poles are at
Here is an integer. This formula has an immediate interpretation. For a given momentum it is to be solved for the ‘Bohr-Sommerfeld’ frequencies . These frequencies are closely spaced and have a real part much greater than their imaginary part. These resonant frequencies correspond to exciting a fermion that is making up the electron star background. The small imaginary part gives the probability for the excitation to tunnel through the barrier and fall into the event horizon. From a field theory perspective this is the probability of dissipating into the IR quantum critical modes. In the background electron star itself, this process is prevented through the electrostatic repulsion between electrons as well as the degeneracy pressure.
Let us emphasize that there are two distinct exponential suppressions here. The first is common to all cases and is due to the fact that the Green’s function is a boundary Green’s function, that the potential is positive at the boundary, and that therefore the fermion needs to tunnel into the spacetime. This suppression is an overall factor, yet it will play an important rôle shortly as it means that some quasiparticle poles have a residue that is exponentially smaller than others, depending on how far the fermion has to tunnel. The second exponential suppression is the imaginary part of the poles appearing in (40). This is the statement that the bound states in the star are very stable as they are separated from the horizon by a potential barrier.
The next step is to translate the above results into explicit statements about the frequency and momentum dependence of the retarded Green’s function. We are interested in low frequencies but arbitrary momentum. In this regime we have noted that there are three distinct cases
See also figure 4 above.
Case I is familiar in essence from Fermi liquids. At zero temperature and at momenta much greater than the Fermi momentum, there are no low energy particle or hole excitations. Therefore the Green’s function should not have an interesting pole structure in the frequency dependence. From the expression for the Green’s function (36) we can obtain the leading low frequency dependence. Outside of the near horizon Lifshitz region, all quantities are analytic as . However, the turning point is inside the Lifshitz region. As , the turning point necessarily has . In this regime we may approximate (27) by
The turning point is therefore
Using the above two expressions for the integral in (36) we can easily extract the singular frequency dependence of the Green’s function.
where we are working up to an overall momentum-dependent constant and the constant in the exponent is
The expressions (44) and (45) are our final answer for the retarded Green’s function in case I. The same formulae were derived, for a slightly different question, in . The spectral density has no features and is exponentially small at low frequencies but has a branch cut at due to the possibility of dissipating (inefficiently) into the quantum critical Lifshitz modes.
Case II is both similar and distinct from Fermi liquids. In a Fermi liquid there is a unique momentum close to which there are low energy particle and hole excitations. This leads to a pole in the Green’s function at and . In the non-Fermi liquids dual to electron stars, the Fermi surface is ‘smeared’  into a near-continuum of Fermi surfaces. These correspond to bulk Fermi surfaces at differing radii in the electron star. This is a large holographic manifestation of the smearing of spectral density due to strong interactions with critical bosonic modes. As we might have anticipated, we see that the Fermi liquid pole at a single momentum and energy is replaced by a sequence of poles closely spaced in energy for any given momentum (40). This is because any given momentum (not too far) below is the Fermi momentum at two radii in the bulk. There are therefore gapless excitations at these radii as well as arbitrarily light excitations at nearby radii.
Here we should emphasize that the above results show (unsurprisingly) that a WKB treatment is powerful enough to identify a large number of closely spaced yet discrete poles, rather than a continuum. We will have to do some work below in order to re-establish contact with the picture of a smeared Fermi surface.
Recall that includes a term plus in (39). The real part of the denominator vanishes at momenta where
These are the many Fermi momenta. Recall that is large in the WKB limit and therefore these poles are reliable for large values of . The poles are therefore closely spaced in momentum. We can now expand and and the overall integral in (37) about and to obtain an expression for the Green’s function that captures all of the low frequency singular behaviour:
Here are order one (in ) positive numbers that can be obtained from the integrals appearing in (37) at and . In particular
Here we added and subtracted a term in the integrand so that we could take the integral all the way to the boundary at . The constant in (48) is again given by (45). The answer agrees precisely with the ‘semi-holographic’ result  with a Lifshitz IR geometry, as it must. The singular frequency dependence of the imaginary part of the denominator follows from the same argument that was outlined for case I above. The numerator has no singular frequency dependence because the turning point for the overall integral in (37), namely , is in the far region where the zero frequency limit may be taken smoothly.
Equation (48) is our final expression for the retarded Green’s function in case II. It describes a large number of Fermi surfaces making up the star, with Fermi momenta bounded between zero and the extremal Fermi momentum of (23). These fermions are exponentially long lived at low frequencies. The spectral density will therefore be well approximated by a sum of delta functions. Note however that the residue of the poles in (48) depends exponentially on times a function of the Fermi momenta.
The plots in figure 5 show the exponent of the Fermi surface spectral weight, in (48), as a function of the Fermi momenta for several values of the electron star parameters. We see that the Fermi surfaces at small momenta, near the top of the potential, have exponentially larger spectral weight than those near , at the bottom of the potential.
We can understand this intuitively from the plot of the effective Schrödinger potential in figure 3; the fermion has further to tunnel from the boundary through to the bottom of the potential compared to high up in the potential. On the other hand, states near the bottom of the potential are exponentially more stable than those at smaller according to (48). This is because they have further to tunnel to reach the interior Lifshitz geometry. Low energy observables can depend on both the lifetime and the spectral weight of the Fermi surface excitations. Depending on which exponential dominates for a given observable (i.e. lifetime or weight), the dominant fermion contribution will come from very different excitations of the electron star.
Case III is distinct from Fermi liquids. At sufficiently small momenta even the ‘smeared’ Fermi surface is ceasing to exist. The appearance of the IR quantum critical dispersion relation suggests an interpretation for the physics at these momenta. Namely, when the fermionic excitations are able to dissipate efficiently into on shell quantum critical modes and therefore lose their quasiparticle residue. In this case, from equation (36) the Green’s function has only analytic dependence. This follows from the fact that is not in the near horizon region, i.e. that, unlike case I, the fermions do not tunnel directly into the Lifshitz sector, and therefore the integrand may be expanded in analytic powers of small over the whole integration range. Our final expression for the low frequency Green’s function in this regime is simply
Here we suppress analytic momentum and frequency dependence, to leading order the exponent is simply a constant . We should recall however that the boundary of case III is itself defined via a non-analytic relation (28) between frequency and momentum: .
The above analysis breaks down at frequencies and momenta for which turning points appear or disappear. These are the boundaries between cases I and II and between cases II and III. We will shortly see how interesting frequency dependences emerge at these boundaries. Before turning to these points however, we can note an interesting way in which the Luttinger count works out from the singularity structure of the fermion Green’s function.
5 The Luttinger count
The Luttinger theorem relates the total charge density of a Fermi liquid to the sum of the volumes of Fermi surfaces . For a spin half fermion in 2+1 dimensions with Fermi surfaces with momentum space volumes
This ‘Luttinger count’, in which the UV charge density is recovered in terms of the IR Fermi surfaces can be violated if the microscopic fermions in the system are coupled to gauge fields. A recent discussion of this fact can be found in . As well as Fermi liquid-like states, in which all the charge is visible in gauge-invariant Fermi surfaces, it is also possible to have ‘Fractionalized Fermi liquids’  in which the sum of gauge-invariant Fermi volumes is less than the total charge. It is also emphasized in  that the best established examples of gauge-gravity duality have the structure necessary to admit Fermi-liquid like (i.e. satisfying the Luttinger theorem) as well as fractionalised Fermi liquid phases. Now that we have characterised the singularity structure of the fermion correlators of the fermions making up the star, we can ask whether the Luttinger count holds true. We can thereby determine what type of phase we are in.
In , an argument was given indicating that the Luttinger count did work out for electron stars.
As shown in , the charge density of the field theory is given by integrating over all the fermions in the star
The local Fermi momentum was defined in (21). Note that there are no hats over any of these quantities. We have also used the fact that defined in  is equal to for spin half fermions. This expression is compatible with a bulk (three spatial dimensional) Luttinger count at each point because we could write it as
where here is the Fermi surface volume at the radius .
From a 2+1 dimensional field theory perspective we need to evaluate
where are the Fermi momenta appearing in our Green’s function. They are defined by poles of the Green’s function at . From (39) and (47), and more generally the discussion of the previous section, we have
There is no overall on the right hand side because there are no hats on the momenta in the square root, cf. (12). The order one contribution to the Luttinger sum comes from highly excited states in our WKB limit and so we can approximate the sum over in (55) by an integral. Letting we have
In the first line, is the maximum of as defined in (23). To get from the first to the second line we differentiated the expression (56) with respect to . From the second to the third line we exchanged the orders of integration, being careful with limits. We then perform the integral over in the third line. The final result then follows from (53).
The computation we have just performed confirms that the Luttinger count holds for electron stars: the sum of the areas of the gauge invariant field theory Fermi surfaces adds up to the total charge density. There is a something slightly magical almost about the way that the square root appearing in the standard WKB formula (56) acts as a projection from the spherical bulk Fermi surfaces to the circular boundary theory Fermi surfaces in such a way that the factors of work out.
We end this section by emphasizing that while the Luttinger count works out, the dual state of matter described by the electron star is not a Fermi liquid. In particular, both the spacetime geometry and the fermion Green’s function indicate that there is a critical Lifshitz sector with gapless degrees of freedom interacting with the fermionic excitations at arbitrarily low energies and momenta.
6 The boundary fermion Green’s function:
This section computes the fermion Green’s function at low frequencies and at momenta on the boundary between cases I and II above. Recall that this corresponds to momenta close to the extremal Fermi momentum . This is the largest of many Fermi momenta of the electron star. It was shown in  that this momentum uniquely determined the period of de Hass-van Alphen quantum oscillations of the system. We expect that the Fermi momentum poles will coalesce into a branch cut in the smearing limit. We may expect that one endpoint of the cut will be precisely at and therefore we need to characterise these poles carefully.
The WKB approximation used in the previous section fails for momenta sufficiently close to . The reason for this is that once the turning point becomes sufficiently close to the minimum of the potential one cannot treat the potential as locally linear. Specifically, near the minimum we can write the potential as
The radius is slightly shifted from as we show in appendix B. Our discussion will be more transparent if we rescale variables and introduce
The Schrödinger problem then becomes
Here dots denote derivatives with respect to . Let us linearise near the turning point and determine whether WKB is valid. Set . To expand and obtain a linear potential we require . Expanding, the equation becomes . Now plugging into the conditions for WKB to hold (30), and recalling to convert the derivatives with respect to into derivatives with respect to , we see that WKB requires . From the two inequalities we have just obtained, it follows that we can only consistently perform a matching to WKB across a linear matching point close to a minimum of the potential when
This condition will not hold for sufficiently close to where tends to zero.
When (65) does not hold we will instead solve the Schrödinger equation in the full quadratic region where (64) holds and match this solution directly onto WKB on each side. The matching solution is now given by Parabolic Cylinder functions (this is just a fancy name for a combination of Confluent Hypergeometric functions), and not the Airy functions of the simpler linear matching. In order for this matching to be valid, we need to be able to neglect third and higher order corrects to the potential (62). Generically, recall the rescaling (63), this will require . Furthermore, in order to be able to match, we need the quadratic form (64) of the potential to hold simultaneously with the WKB conditions (30). A sufficient condition for satisfying the WKB conditions together with is that
To implement the quadratic matching we need to expand the potential (20) about its minimum. At and , see equations (23) and (24) above, the minimum is at and . We are interested in the potential at nearby values of frequency and momentum, and . In appendix B we show that the potential takes the form (62) with
Recall that is defined via (23) and is given by (21). From here we can obtain via (63). Before presenting the full Green’s function, note that the insofar as we are interested in the low lying normal modes and neglect exponentially small tunneling probabilities, we can simply assume that the wavefunction is normalisable in the quadratic region. This corresponds to the familiar Hermite functions and hence we obtain a spectrum of fermionic excitations
For these are the poles we obtained in our previous analysis (46). We have just shown that the approach of this current section allows us to go to lower poles . At larger values of the states probe the deviation of the potential away from the purely quadratic form near the minimum.
If we set we can think of equation (68) as determining the Fermi momenta of each species of particle in our star. Using the above few equations:
The term here indicates that the extremal Fermi momentum receives a quantum correction. As explained below equation (18) above, there are additional corrections at this order that we are not considering, so the quantum shift cannot be considered too literally. In the large limit the spacing between states is suppressed, leading to a pseudo continuum distribution of poles. These poles merge into the poles of (46), with the two descriptions overlapping for . We shall verify the agreement shortly.
Similarly if we set we obtain a spectrum of frequencies . These poles at positive frequencies corresponding to the cost of creating particles of momentum for the different species of particles at different radii in the star. At all Fermi seas of all species are empty, so it is not possible to destroy a particle of this momentum. That is why there are no negative frequency poles.
Our discussion so far has neglected the exponentially small imaginary part of the poles. We will now compute the full Green’s function for , including the real and imaginary part of the poles and their residues. We have relegated the details to appendix B. Here we simply quote the result for the retarded Green’s function
Recall was introduced in (63) in terms of and which are given in (67) and which characterise the minimum of the potential according to (62). The overall exponential terms characterising the tunneling into the spacetime from the boundary are
Here we set . The first line includes terms that regulate the integral near the boundary . The exponential terms that measure the tunneling through the star to the horizon, giving an imaginary part to our poles, have exponents
Here recall that is the (linear in this case) turning point closest to the horizon.
If we ignore the exponentially small imaginary parts in the Green’s function (70) it is clear that we recover the poles at described above. Furthermore, we can ‘smear’ over the poles by looking at the Green’s function in the regime . This is the regime of overlap of the quadratic and linear WKB matching, so we did not have to do a quadratic matching to access this regime. Here the poles should merge and form a branch cut. Using Stirling’s expansion for the gamma functions in (70) and noting that is order one in , one obtains the leading order result
Here and do not depend on the WKB parameter, the frequency or momentum. Thus we see that smearing over the poles does lead to a branch cut emanating from the extremal Fermi surface at
The singularity at this extremal momentum is weak. The spectral weight in the branch cut increases exponentially as we move to lower momentum , away from the branch point. This is consistent, as it had to be, with our discussion around figure (5) above; the residue of the Fermi surface poles close to are exponentially small. Nonetheless, the extremal Fermi momentum does produce a nonanalyticity in the smeared Green’s function (76). This can then lead to important signatures such as quantum oscillations  that pick out momentum space nonalyticities and are independent of the spectral weight. The disjunction between quantum oscillations and spectral weight is an interesting feature of electron stars.
7 The boundary fermion Green’s function:
This section computes the fermion Green’s function at the boundary of cases II and III above. Recall from equation (28) that this corresponds to small frequencies and momenta close to a dispersion relation . For momenta below the dispersion relation, the fermion states become highly unstable to dissipation into the quantum critical sector. In the Schrödinger equation this translates into the fact that we will be considering states close to the top of the effective potential. In contrast, the previous section considered the most stable states, at the bottom of the potential. Our computation is similar to that of the previous section, except that now we must match the WKB region onto an inverted harmonic oscillator potential. A further potential complication is that while we are taking large, as usual, we also need to take and very small as they scale towards the critical dispersion relation.
We will be expanding around the top of the barrier, of for instance figure 3, that prevents fermion states from falling into the black hole. Near the maximum the potential takes the form
The difference from (62) of the previous section is the opposite sign of the quadratic term. Of course take different values as we shall see shortly. Again rescaling as in (63) we obtain the Schrödinger problem near the maximum
Also as previously, linear matching close to the maximum of the potential is legitimate when while quadratic matching, the topic of this section, requires .
We noted above that this maximum necessarily occurs in the near horizon Lifshitz region of the spacetime. The potential we are expanding about the maximum is therefore (27). The maximum is seen to occur at
where we should pick the plus sign to find a solution for positive . We saw above that case III never arises at negative . We have introduced here; it is simply a number with no dependence. Expanding the potential about the maximum we can read off and in (78):
Setting gives us the precise dispersion relation separating cases II and III. The condition for the validity of matching WKB across the quadratic potential is , which is now seen to require that be close to the critical dispersion relation.
In appendix C we perform the matching across the quadratic region to obtain the Green’s function. The result is
Here again gives the amplitude to tunnel in from the boundary
while the integrals characterising the periods of oscillation within the well are
Recall from (63) that
The previous section, in e.g. (69) and below, described how the exponentially stable fermionic excitations ended at the bottom of the potential in a harmonic oscillator fashion. We can now use the Green’s function (83) to see what happens to the poles near the maximum of the potential. In appendix C we find the poles of the Green’s function in the regime . We also derive simple expressions for and in (167) that we use in the remainder. In this regime the Green’s function (83) becomes