Polarized solutions and Fermi surfaces in holographic Bose-Fermi systems
We use holography to study the ground state of a system with interacting bosonic and fermionic degrees of freedom at finite density. The gravitational model consists of Einstein-Maxwell gravity coupled to a perfect fluid of charged fermions and to a charged scalar field which interact through a current-current interaction. When the scalar field is non-trivial, in addition to compact electron stars, the screening of the fermion electric charge by the scalar condensate allows the formation of solutions where the fermion fluid is made of antiparticles, as well as solutions with coexisting, separated regions of particle-like and antiparticle-like fermion fluids. We show that, when the latter solutions exist, they are thermodynamically favored. By computing the two-point Green function of the boundary fermionic operator we show that, in addition to the charged scalar condensate, the dual field theory state exhibits electron-like and/or hole-like Fermi surfaces. Compared to fluid-only solutions, the presence of the scalar condensate destroys the Fermi surfaces with lowest Fermi momenta. We interpret this as a signal of the onset of superconductivity.
The holographic correspondence between field theories in dimensions and gravitational theories in dimension has been extensively used to study properties of strongly coupled systems, obtaining information that is not easily accessible by ordinary methods. In particular, fermionic systems at finite density pose a particularly difficult problem, as there are no theoretical models that can be studied reliably in a controlled approximation and lattice simulations are marred by the “sign problem”. In this context, the holographic method has proved useful by offering a number of insights into possible exotic phases of matter that are not described by Landau’s theory of Fermi liquids or other weakly-coupled descriptions [1, 2, 3, 4].
It was suggested in  that the presence of a charged horizon in the simplest gravity
solutions dual to finite-density states admits an interpretation in terms of fractionalization of the fundamental charged degrees of freedom. When the charge is sourced by
matter fields in the bulk, it corresponds to fermionic operators in the boundary. But when the
charge emanates from the horizon, it cannot be associated to any gauge-invariant observable.
This leads to a violation of the Luttinger’s theorem: the charge contained in the Fermi surface
(or surfaces) does not account for the total charge of the system. The phase transition
corresponding to the onset of fractionalization was studied in [6, 7, 8].
When there are charged scalars in the theory, they can also become excited and condense,
carrying part of the charge. In this case however the symmetry is broken, and
the interpretation is different: the system undergoes a transition to a superconducting
If both components are allowed to be present, there can be a competition between the bosons and the fermions
In the present paper, we continue the study of the system analyzed in , adding a new ingredient, namely a direct coupling between the scalar and the fermions. The reason for introducing this coupling is the following: in the holographic superconductor, the boundary bosonic operator that develops a vacuum expectation value is interpreted as a strongly coupled analogue of the Cooper pair that condenses in the superconducting state in the usual BCS theory. In our model fermions are also present at the same time, so if the boson has to be interpreted as a bound state of fermions, it seems unnatural that they should interact only from the exchange of gauge fields. A natural interaction term would be a Yukawa coupling, as it was considered in . This requires dealing with microscopic fermions and going away from the fluid approximation; moreover the coupling can only be there in the case where the boson has twice the charge of the fermions.
In , an interacting description in the fluid approximation was given that treats the scalar as a BCS-like fermion bilinear. In the fluid approximation, the presence of the condensate modifies the fermionic fluid equation of state. At the macroscopic level, this system too may arise from a Yukawa interaction of a scalar whose charge is twice that of the elementary fermions.
In a strongly coupled system we cannot discard the possibility that the fermions that condense are not the fundamental electrons but some other excitation, perhaps fractionalized, therefore we want to leave the ratio of the charges arbitrary.
If we want to work directly with the fluid approximation, and at the same time leave the scalar and fermion charges generic, the simplest interaction we can write is a current-current coupling. We found that in the presence of this interaction, a surprising phenomenon takes place: in the bulk we can have a polarized charged system, which is constituted of radially separated shells of positively and negatively charged components of the fluid, immersed in a non-zero scalar condensate. We call these solutions electron-positron stars, and they are illustrated schematically in Figure 1.
The reason these solutions may arise is that, due to the current-current interaction term, the local chemical potential can have opposite signs in different regions of the bulk geometry. This leads to both fermionic particles and antiparticles being populated in separate regions. The solution is stable because the scalar condensate effectively screens the negative charge of the positrons so that these do not feel the electric attraction of the electrons but rather they are repelled. The system is kept together by the gravitational attraction, which balances the electromagnetic repulsion.
The simplest boundary interpretation of these solutions appears to be in terms of different flavors of fermions, each of them having a certain band structure but with the zero energy level having a different offset for different flavors, so that a given chemical potential intersects the conductance band for some fermions and the valence band for others (see Figure 2 for a schematic representation of this phenomenon). We do not know of any realistic system that displays such features, it would be interesting to find some real-world realization of this situation.
To shed more light on the physics of the system, we study the spectrum of low-energy fermionic excitations. We consider a probe fermion in the geometry, that is also interacting with the Maxwell field and with the scalar current, since it is supposed to be one of the fermions making up the fluid. We solve the Dirac equation in the WKB approximation and find the normal and quasi-normal modes, that correspond to poles of the boundary fermionic Green’s function. The analysis of the poles at zero frequency and finite momentum reveals the presence of a finite number of Fermi surfaces.
One can compare this situation with that of the unbounded electron star in the absence of the scalar field . In the latter case, there are infinitely many Fermi surfaces, with Fermi momenta accumulating exponentially close to zero. As discussed in , the dual field theory interpretation is that of a Fermi system in a limit where the number of constituent fermions is infinite. In compact stars (both in the electron and the electron-positron version) there are only finitely many Fermi surfaces, meaning that all except for a finite number of Fermi surfaces become gapped due to the scalar condensate. For small frequency, in the electron star the modes have a small dissipation due to the possibility of tunneling into the IR Lifshitz part of the geometry. This was interpreted as the effect of the interaction of the fermions with bosonic critical modes. In compact star solutions, on the other hand, we find that for a certain range of energies above zero, the excitations are stable, so there is no residual interaction at low energies. The comparison with the electron star also reveals that the most “shallow” modes, corresponding to Fermi surfaces with smallest momenta, are the ones that disappear from the spectrum; we interpret this as a signal that the system has become gapped due to the superconductivity; however the gap concerns only part of the system, as other Fermi surfaces remain gapless. Naively one would expect that the smallest Fermi surfaces should be more robust, since they have a lower temperature for the superconducting transition, as predicted by BCS . A possible explanation would be if the mechanism for superconductivity is not described by BCS. In fact, our findings are consistent with the UV/IR duality displayed by holographic Fermionic fluids discussed in , where it was argued that the states corresponding to the smallest Fermi momenta are the last to be filled.
The plan of the paper is as follows: in Section 2 we review the results of our previous work, in particular the compact star solutions. In Section 3 we study the system in presence of the current-current coupling and describe the new solutions, find the free energy and determine the phase diagram. In Section 4 we analyze the probe fermions and determine the low-energy spectrum. We conclude in Section 5 summarizing the results and indicating the open questions and future directions of investigation.
2 Bosonic and fermionic matter in asymptotically AdS spacetimes
We will consider four-dimensional gravitational solutions which are asymptotically and have zero temperature and finite charge density. These solutions arise from models which have an action of the form
where is the field strength of the gauge field , is Newton’s constant, is the asymptotic length and is the coupling. The term represents the Gibbons-Hawking term and the counterterms necessary for the holographic renormalization and is the action for the matter fields.
A charged scalar field with action:
where is the scalar field charge and is negative and satisfies the Breitenlohner-Freedman (BF) bound, .
A fermionic component described effectively by a fluid stress tensor and electromagnetic current:
The form of the fluid energy density , pressure and charge density will be given shortly.
We will restrict to static, homogeneous and isotropic solutions, which by performing diffeomorphisms and gauge transformations can always be written as,
in which is the boundary. As discussed in detail in Appendix A, we rescale all quantities by suitable powers of , and and denote the rescaled quantities with hats. In the UV region we have:
where is the chemical potential of the boundary quantum field theory and the total charge of the system. We will assume that the UV asymptotics of the scalar field corresponds to zero source term for the dual operator, i.e.:
where is the conformal dimension of the field theory scalar operator .
The energy density , pressure , charge density and fluid velocity are assumed to locally satisfy the chemical equilibrium equation of state of a free Fermi gas, as in , with a density of states given by:
where is the constituent fermion mass and is a phenomenological parameter related to the spin of the fermions.
It is convenient to describe the fluid using rescaled energy density, pressure and charge density (defined in Appendix A.1 and denoted by a hat), which under the local chemical equilibrium condition are given by:
where is the local chemical potential in the bulk, defined in Appendix A, and
where is the elementary charge of the constituent fermions.
With respect to the original construction in , we allow for both signs of , i.e. we allow for the possibility of the fluid to be made up of particles
For , due to the Heaviside step functions, and there is no fluid.
The explicit expression of the local chemical potential depends on the way the fermions couple to the Maxwell and scalar field in the bulk. If there is no direct interaction between the scalar and fermionic components, it is given by
and it has the same sign of the electric potential and of the boundary chemical potential (Eq. (2.5)). For this reason, without direct interactions between the fermionic fluid and the scalar field, only positively charged solutions are possible for positive boundary chemical potential. This is the case covered in  and , and the possible bulk solutions are reviewed in the next subsection.
As we will see in Section 3, a direct current-current interaction between the fermion fluid and the scalar field will allow to have both signs, or even to change sign in the bulk, and solutions with both signs of the constituent fermion charge will be possible.
Finally, the local chemical equilibrium and Thomas-Fermi approximation, necessary for this description to be valid, require that
2.1 Taxonomy of non-interacting solutions
The situation where bosonic and fermionic components do not have a direct interaction was analyzed in . In this case, the expression of the local chemical potential is (2.12), and the possible homogeneous, charged, zero-temperature solutions of this system are listed below (the reader is referred to  for more details and to Appendix A for conventions and definitions of the rescaled variables).
Holographic Superconductor (HSC). These are solution with non-vanishing scalar field (with vev-like UV asymptotics) but zero fluid density in the bulk [24, 25]. At zero temperature, the asymptotic solution in the IR () is 
where “hatted” quantities are defined in Appendix A, and . These solutions require the condition
which will be always assumed in this paper.
Electron star (ES). These solutions, first constructed in , have a trivial scalar field, but a non-vanishing fluid density in the bulk region where the local chemical potential exceeds the fermion mass . This happens in an unbounded region , representing the star boundary where . Outside this region (i.e. for ) the solution coincides with the RN-AdS black hole described above. The region occupied by the fluid is unbounded, and in the far IR (), the fluid energy and charge density are constant and the geometry is asymptotically Lifshitz:
where and are constants depending on and , and the dynamical exponent is determined by
Compact electron stars (eCS). Solutions with both a non-trivial scalar profile and a non-zero fluid density were found in . The fluid density is confined in a shell , whose boundaries are determined by the equation
The non-trivial scalar field profile is similar to the one of the holographic superconductor, and causes the local chemical potential to be non-monotonic: this allows equation (2.18) to admit two solutions , which represent the outer and inner star boundaries. Since the star is confined to a shell in the holographic direction, these solutions were called compact electron stars (eCS). In the UV () and in the IR () the fluid density is identically vanishes, and the solution is given by the holographic superconductor described above, with IR asymptotics (2.14).
Different types of solutions with both non-trivial scalar and fluid density were found in , with a different choice of the scalar field potential (which included a quartic term). In this paper we limit ourselves to a quadratic scalar potential, and these solutions will not be considered.
3 Interacting Fermion-Boson Mixtures in AdS
We now generalize the model of  by coupling directly the fermion fluid to the scalar field.
In order to be able to continue working in the fluid approximation for the fermions, we consider a direct coupling between the scalar field and the fluid through their respective electromagnetic currents:
where parametrizes the intensity of the coupling and can have either sign.
In Appendix A, we derive the field equations by considering an action principle for the fluid , including the interaction Lagrangian (3.19) in the fluid action. The local chemical potential now depends on the scalar field through its electromagnetic currrent,
The interaction also contributes to the field equation of the scalar field and Einstein equations. Using the ansatz (2.4), the field equations for the rescaled fields and parameters, derived in Appendix A, are
where primes denote derivatives with respect to the radial coordinate.
The fluid quantities are given by (2.8) where the local chemical potential is now
The fluid parameters , and appear in the field equations (3.22) only through the rescaled quantities (2.11). When working with these rescaled quantities, the fermionic charge drops out of the equations, and its sign is encoded in the sign of as discussed in Section 2.
The bulk fluid is made up of 4d Dirac fermions, thus we can have physical states with either sign of the charge. In our conventions, Dirac particles (which we call electrons) have , antiparticles (or holes, or positrons) have . Thus, a positive chemical potential will fill particle-like states, and a negative one hole-like states.
As noted in Section 2, in the absence of current-current interactions, i.e. for , the sign of the local chemical potential in (3.23) is dictated by the sign of the electric potential. In both the electron star and compact star solutions this is the same throughout the bulk (for example, it is non-negative if the boundary value of the electric potential is positive). The same happens when , as one can see from Eq. (3.23).
On the other hand, if we turn on a boson-fermion coupling , the sign of the chemical potential is not determined, and there can be cases in which has different signs in different bulk regions. This is indeed what happens in the solutions we describe in subsection 3.1.
From equations (2.8), we see that the fluid density is non-zero for . The case corresponds to a fluid made out of positively charged particles (electrons), whereas a negative leads to a fluid of negatively charged particles (positrons). Notice that, in equations (2.8), the energy density and the pressure are positive in both cases, whereas the charge density is positive or negative for the electrons and positrons fluids, respectively.
We will see in the next sections that, depending on the parameters, bulk solutions with various arrangements of differently charged fluids are possible (electrons, positrons, or both).
The relevant parameters of the model are thus:
where the scalar field mass satisfies the BF bound (i.e. the operator dual to is relevant). We restrict the analysis to the case where the scalar parameters satisfy (2.15) and the fermion mass satisfies . Then, the holographic superconductor, with IR asymptotics (2.14), and the electron star are solutions of the system when there is no fluid and the scalar field is trivial, respectively.
We assume that in the UV () the solutions are asymptotically ; the metric, gauge field and scalar field in this region are then given by (2.5) and (2.6) where we have imposed that the non-normalizable mode of the scalar field vanishes. If the normalizable mode is non-trivial, this corresponds to a spontaneous breaking of the boundary global symmetry.
Of the solutions described in Section 2, the ERN, HSC and ES are still solutions for any value of , since in the absence of either the scalar condensate, or the fluid density, or both, the current-current interaction terms drop out of the field equations. The CS configurations are solutions to the system (3.22) when the direct interaction (3.19) between the scalar field and the fluid is turned off. Below, we present new solutions which exhibit a non-trivial profile of the scalar field and a compact electron star, a compact positron star or both at the same time.
3.1 The electron-positron-scalar solution
To see how the new solutions arise, we notice that in the HSC solution, the local chemical potential (3.23) vanishes asymptotically both in the UV and in the IR and admits at least one extremum value in the bulk. In the IR,
For the local chemical potential is negative in the
Similarly to the non-interacting case , solutions with both a non-trivial scalar and a non-trivial fluid exist when at least one local extremum of the local chemical potential in the HSC solution is larger than the mass of the fermions. Three kinds of compact star(s) solutions arise :
Compact positron/electron stars (peCS): In this case the solution exhibits charge polarization in the bulk: two fluid shells of opposite charges are confined in distinct regions of spacetime, bounded respectively by and determined by the equations:
The fluid in one region is made of electrons, the one in the other region of positrons. Clearly, for this solutions to exist, the chemical potential must change sign in the bulk. Due to fixed UV asymptotics of the local chemical potential, the fluid of electrons is situated closer to the UV boundary than the fluid of positrons.
The kind of compact star(s) solutions that may exist depends on the maximum and minimum value of the local chemical potential,
The possible outcomes are summarized in Table 1. We denote the case where no compact star(s) exists by noCS. Notice that for all ; is negative when and vanishes for .
|peCS, pCS, eCS||eCS|
3.2 Phase diagrams of charged solutions
In the previous section we have seen that different arrangements of fermionic fluids are possible at zero temperature. Depending on the parameter values, we can go from the pure condensate with no fermions (holographic superconductor, or HSC), purely positive or purely negative confined fluid shells (electron or positron compact stars, or eCS and pCS respectively), and polarized shells of positive/negative charged fluid regions (compact positron-electron compact stars, or peCS). In all these configurations, the fermionic charges are surrounded by the scalar condensate, which dominates the UV and IR geometry and confines the fluid in finite regions of the bulk.
Here we address the question about which, for a given choice of parameter, is the solution that has the lowest free energy and dominates the grand-canonical ensemble. We work at zero temperature and fixed (boundary) chemical potential . Thus, different solutions will in general have different charge.
As was noted in , comparing the free energy of different solutions is relatively simple, due to the existence of a scaling symmetry of the field equations that allows to change the value of within a given class of solutions. Thanks to this solution-generating symmetry, one can show that, within each class of solution, the free energy has the simple expression
where are -independent constants which depend only on the class of solutions. The index runs over all solutions which exist at a given point in parameter space (HCS, eCS, pCS and/or peCS).
Equation (3.28) shows that there can be no non-trivial phase transitions between the solutions as is varied. On the other hand, the constants depend non-trivially on the parameters of the model, and there can be phase transitions between different solutions as these parameters are varied.
To observe the transition, it is sufficient to compute in units of for one representative of each class of solution, and for a given point in parameter space the solution with larger will be the preferred one, as .
We performed this analysis numerically on the solutions described in the previous section
We will first analyze what happens if we vary while keeping other parameters fixed (Figures (a)a and (b)b). As we know from , for , whenever compact electron star solutions exist, they dominate the ensemble. Otherwise, the preferred solution is the HSC.
Let us first choose the parameters so that, for , the compact electron star is the preferred solution (Figure (a)a): if we dial up a positive interaction term , eventually the effect of the condensate polarizes the star and, at a critical value , we find a continuous transition to the peCS solution, which now is the one dominating the ensemble. The eCS keeps existing beyond the critical point , where we also see the emergence of a new pCS solution which starts dominating over the HSC solution but not over the peCS solution.
On the other hand, if we start from a point where, for , there is no eCS solution (Figure (b)b), we see that dialing up either way one will get to a critical point where either a positive or a negative charged star will be formed, and dominate the ensemble henceforth.
For a given (positive) value of , the type of solution depends on the fermion mass , as shown in Figure (c)c. At small mass, the polarized peCS solution dominates over the eCS and pCS solutions. As the mass is increased, one first encounters a continuous transition from the peCS to the pCS solution at the critical value . At this point, the (subdominant) eCS solution merges into the (subdominant) HSC solution. Then, at the second critical point the charged fluid disappears and the solution merges into the HCS solution.
We have also analyzed the phase diagram as a function of the scalar charge . Phase diagrams of the system are displayed in Figure 7, in the plane at fixed scalar charge (Figure (a)a) and in the plane at fixed (Figures (b)b-(d)d). The critical lines separating the various phases correspond to the points where the maxima and minima of the local chemical potential are equal in absolute value to . The different colors correspond to the dominant phase in each region. Thus, all these transitions are continuous and take place at the points where it is possible to fill the charged fermion states: whenever a fluid solution is possible by the condition , that solution will form. Furthermore, solutions in which the charge is distributed between more fluid components are preferred.
3.3 Charge distribution and screening
Let us briefly discuss how the total charge of the system is divided among the various bulk components. From the boundary field theory point of view, the only invariant definition of the charge is the total charge of the solution, that one can read off from the asymptotic behavior of the gauge field. However, it is still instructive to define, in the bulk, the separate charge of each component by the expression it would have if that were the only component present.
With the ansatz (2.4), the electric charge carried by the scalar field in the bulk is
and the electric charges of the electron and positron fluids are respectively
where and are the boundaries of the electron star, and similarly for the positron star. The charge densities of the electron fluid and the positron fluid are respectively positive and negative, and they are given in Eq. (2.8) with respectively positive or negative.
which reflect the interaction between the scalar and the fluid made of electrons and positrons, respectively, and it gets contributions from the regions where the fluid density is non-vanishing.
The total electric charge of the system
matches the UV asymptotic behavior of the gauge field (2.5),
This has been verified numerically in all solutions we have constructed.
In Eq. (3.32), the second and third terms represent the total contributions from each charged fluid. Despite the possible presence of local negative charge components, we will show below that all three terms in Eq. (3.32) are positive for all solutions under considerations. This is consistent with our choice for the boundary chemical potential in the UV asymptotics (2.5), which implies that the boundary charge, i.e. total charge of the system must be positive in all solutions.
Let us first consider the scalar field condensate contribution to the total charge. As a consequence of the choice in the UV, the electric potential is positive throughout the bulk
Due to the signs of the local charge densities in (3.30), is positive, and is negative. However, the latter is over-screened by the scalar field through the charge of interaction , so that , as can be seen from Eq. (3.30-3.31) and the fact that, inside the positron fluid, , since by Eq. (3.23) this quantity determines the sign of the chemical potential.
By a similar reasoning, is positive (negative) for (), but is always positive. Thus, for electrons, there may be charge screening or anti-screening, but never over-screening.
Given the previous discussion, we can have a qualitative understanding of why the polarized electron-positron compact stars are stable configurations: although the positron and electron parts of the fluid are made up of positive and negative charge fermionic constituents respectively, the screening of the negative electric charge by the scalar condensate renders the total charge positive in both fluids. In particular, for peCS solutions, the two charged shells experience electromagnetic repulsion, rather than attraction. Gravitational and electromagnetic forces are competing, and this makes the solution stable.
In Figures 8-10 (left) we present the distribution of the total electric charge of the system between the scalar field, the fluids of electrons and positrons and the charges of interaction for different values of the parameters (the same that were used in Figures (a)a-(c)c). The boundary condensate is also shown in those figures (right). It is interesting to note that it is lower in the CS solutions than in the HSC solution. In Section 4 we will show that the presence of fermions in the bulk maps to the formation of Fermi surfaces in the dual field theory. If one interprets the scalar operator as being a composite operator of the fermionic operator, decreasing of the condensate in the CS solutions can be thought of as coming from the breaking of part of the scalar operator excitations.
4 Fermionic low energy spectrum
In this section, we compute the Fermi surfaces and the fermionic low energy excitations of our model. This can be done by solving the equation of motion of a probe fermion in the WKB approximation. We assume the probe fermion is a constituent of positive charge .
4.1 Probe fermion and the Dirac equation
The electromagnetic current of bulk elementary fermions with charge is given by
To take into account the current-current interaction between the fermions and the bosons, it is natural to add to the action for free probe fermions the interaction
where and are given by (B.133).
In Appendix B, we obtain in details the Schrödinger-like equation, but we give here the key steps. By choosing correctly the basis of Gamma matrices, the Dirac equation for a probe spinor field on top of the background solution can be written as an equation for the two-component spinor ,
where we have rescaled the momentum and frequency,
by the parameter
which is large in the Thomas-Fermi approximation applied to the bulk fermions. In this limit, the Dirac equation (4.36) is equivalent to the Schrödinger-like equation
together with the expression
for the components and of the spinor
The potential can be expressed as
where the local Fermi momentum is defined as 
Notice that inside the stars only; these are the regions where is relevant for our considerations. The local Fermi momentum is displayed in Figure 11 for an eCS solution and a peCS solution.
The momentum appears only through in the potential (4.42), so we can restrict the analysis to without loss of generality.
The Schrödinger equation in a standard form
The potential (4.42) depends on the momentum . In order to see the physical interpretation of Eq. (4.39), we put it in a Schrödinger form where plays the role of the energy by introducing the new coordinate , defined by
We then obtain the equation
for the rescaled field , where
in the large- limit. Herein and in the following, is the inverse map of
At zero frequency, the potential is, up to a minus sign, given by the local Fermi momentum squared (4.43). It is negative inside the star and positive outside, and the zero-energy turning points are the star boundaries where . Since the local chemical potential vanishes in the IR as in (3.25), the local Fermi momentum behaves in this region as where in the IR, the inverse map of is
and the potential at infinity. The potential for is displayed schematically in Figure (a)a for a compact star solution involving one star (eCS or pCS).
A non-zero frequency affects the behavior of the potential (4.46) in the near-horizon region. Indeed,
so the zero-frequency limit and the near-horizon limit do not commute. This is also observed in the electron star phase  but with a different asymptotic behavior with respect to the potential (4.46), leading to a different conclusion as we shall see below. Outside the near-horizon region, the effects of small frequency are small and do not affect much the shape of the potential. In particular, the zero-energy turning points of the potential are close to the star boundaries. In the UV, the potential (4.46) behaves as
and effects of are subleading. The potential for is displayed in Figure (b)b for a compact star solution involving one star (eCS or pCS).
Our aim is to solve the Schrödinger equation (4.45) for the field to compute the poles of the retarded two-point Green function of the gauge-invariant field operator dual to the bulk fermionic particles. To do so, we shall impose the Dirichlet condition on at the UV boundary and the in-falling condition in the near-horizon region. This second condition can be applied when the solution to Eq. (4.45) is oscillating in the IR, that is when , i.e. for . The dispersion relation of the Green function corresponds in this case to quasinormal modes of the wave equation (4.45). For , the solution is exponential in the near-horizon region and one shall impose the regularity condition, leading to normal modes of the equation (4.45).
The discussion of the previous paragraph allows us to distinguish three different regimes for the spectrum of the Schrödinger equation (4.45). Let us define the extremal local Fermi momentum by
For , the “energy” lies everywhere below the Schrödinger potential, and there are no eigenstates. In the intermediate region , we expect to have a discrete spectrum of bound states, which are normal modes of the Schrödinger equation (4.45). Since the region of spacetime where is compact for any frequency, the number of bound states is finite at fixed frequency and is almost independent of the frequency since only affects the near-horizon region. For , the spectrum of the Schrödinger equation (4.45) is continuous but there is a discrete set of quasinormal modes, which dissipate in the IR region by quantum tunnelling. The number of quasinormal modes is finite at fixed frequency for the same reasons as for the intermediate region. By setting , one can count the number of Fermi surfaces. From the qualitative behavior of the Schrödinger potential in Figure (a)a, the number of boundary Fermi momenta is finite.
In certain parameter regions, although the fluid density is non-zero, there may be no negative energy bound states (thus no Fermi surfaces): this happens for ‘small stars’, for which the potential is not deep enough inside the star to allow any bound state.
We then expect to obtain a finite number of boundary Fermi momenta () satisfying in the large- limit. Each boundary Fermi surface admits particle excitations which are stable at small energy . At larger frequency with , the excitations are resonances, they can dissipate. This dissipation maps in the bulk to the possible quantum tunnelling of the modes into the near-horizon region. The modes dissipate only at sufficiently large frequency due to the fact that the compact star does not occupy the inner region of the bulk spacetime. From the field theory point of view, the bosonic modes, represented in the bulk by the scalar field, the metric and the gauge field, do not interact with the fermions at sufficiently low energy and the excitations around the -th Fermi surface are stable up to .
Let us compare the situation to the electron star phase, studied in detail
At zero frequency the potential is negative for , where is the star boundary, and
For , the potential diverges to in the IR so dissipation occurs
for any non-zero frequency.
This is because the fluid does occupy the inner region of spacetime and the boundary fermions
interact with the bosonic modes at low energy.
For , the potential admits a local maximum at a point
We should notice that for the compact star solutions, also admits a local maximum, as shown in Figure 12. However, this local maximum is positive at zero frequency and not modified at small frequency because it does not belong to the asymptotic IR region. At larger frequencies, the effects of are also relevant outside the near-horizon region. It may happen that for sufficiently large frequencies, this local maximum becomes negative. In this case, the potential would be negative for all larger than the first turning point, leading to unstable states at small momentum.
We have argued above that the number of boundary Fermi momenta dual to the compact star solutions is finite. In the electron star phase on the other hand, at zero frequency we see from (2.16) that the potential asymptotes to zero in the IR as
Here, we focus on the regime with . In this case, all the modes are bound states. We leave the detailed study of the case for future considerations .
Bound states for
In the electron star phase, the dispersion relation of the fermionic excitations around the Fermi surfaces was found to be linear, up to a (imaginary) dissipative term . We will see later that this is also what we obtain for the compact star phases. In the rest of this paper, we will be interested in the behavior of the particle excitations close to the Fermi surfaces, that is for . In this case, and the dependence on of the potential (4.46) can be neglected everywhere, in particular in the IR region, and one can consistently set . The potential is then simply given by
It is easy to generalize the above discussion to peCS solutions. In Figure 11, we display the local Fermi momentum for an eCS solution and a peCS solution. For convenience, we plot it in the original variable . Doing so does not affect the analysis as it only changes the shape of the local Fermi momentum in the radial direction and does not modify the extremal values of it. It is clear from (4.45) that for , the solution is exponential while it is oscillating for . There can exist zero, one or two regions where depending on the background solution and the value of compared to the local maxima of . For eCS and pCS solutions, the extremal local Fermi momentum is
and for peCS solutions, there exist two local extrema
The bounds in the maxima are the star boundaries of the star(s) of the different solutions. The conditions for oscillations are detailed in Table 2.
|and||oscillations in electron and positron stars|
|oscillations in positron star|