Mean field theory for strongly coupled systems: Holographic approach

In this paper, we develop the holographic mean field theory for strongly interacting fermion systems. We investigate various types of the symmetry-breakings and their effect on the spectral function. We found analytic expressions of fermion Green's functions in the probe-limit for all types of tensor order parameter fields. We classified the spectral shapes and singularity types from the analytic Green's function. We calculated the fermions spectral function in the full backreacted background and then compared it with the analytic results to show the reliability of analytic results in the probe limit.


Introduction
Study on strongly interacting system has been the frontier of physics for more than half century.It has been the continuous source of motivation for a new theoretical framework.Along this line, the gravity dual picture [1][2][3][4][5][6] has been suggested as a new paradigm to understand strange metals [7,8] and quantum critical points (QCP) [9][10][11] of strongly coupled systems.A few experimental data were also compared with the theory showing agreements, especially in quark gluon plasma [12], in clean graphene [13][14][15][16][17] and surface of topological insulators [18,19].
On the other hand, QCP itself is a phenomena at zero temperature taking a measure zero sector of the phase diagram.Therefore for the theory to be compared with experiment, it is essential to consider the symmetry broken phases as well as the unbroken phase near the QCP.Such symmetry breaking is an essential step for new collective phenomena, because a phenomenon associated with manybody collective phenonena is possible only when we have an order that leads to a gap which protect the new ground state from the excitations causing disorder.Otherwise, what we will have is just fermi liquid.While gap calculation is usually a consequence of a mean field theory which is seeking for value of the condensation corresponding to the certain fermion bilinear, its validity is limited to the weakly interacting cases.Therefore a dream for theorists is to develop a mean field theory which is valid for the strongly interacting system maintaining the spirit of Landau-Ginzburg and Wilson.
The strongly coupled systems share many similarities with the gravity system in the sense that both have unreasonable speed of equilibration: the Plankian dissipation near the QCP is very similar to the exponentially fast scrambling power [23,24] of black hole horizon.Therefore it is not very surprising to expect a mean field theory for the strongly interacting system out of the gravity dual description.While it is hard to find the exact gravity dual of a given system, it is reasonable to assume the presence of approximate dual theory for a strongly interacting system.It can serve as a representative theory for the purpose of studying various types of the gaps and condensations as well as the singularity types of the Green functions for the strongly interacting systems.
Notice that in the holographic superconductor theory [20,21], the gap calculation was exemplified by considering scalar-vector-gravity theory, which is an important ingredient of the holographic mean field theory (MFT).However, the heart of the MFT is to study the gap creation and the fermion spectrum together, which has not been done systematically yet for different types of condensations.Furthermore, because the holographic theory as a continuum field theory does not encode condensed matter system's detail it is useful to study all possible type of interactions together and classify their spectral behavior to match with physical system's spectral pattern.In ref. [22] we put forwarded a step to such direction by considering all possible couplings of the bulk fermion bilinears with the various tensorial fields representing the different condensations in holographic set-up, which is analogue of the femion bilinear coupled to the Hubbard-Stratonovich field in the usual mean field theory.
In our previous work [22], the fermion spectral function (SF) [25][26][27][28] of each broken symmetry phase was calculated and classified.Subsequently, we realized that [29] the gapless mode in the scalar coupling is the AdS analogue of the surface mode of topological insulator whose topological stability might be related to amazing stability of the Fermi Liquid.The holographic mean field theory has universal structure so that it has the power to accommodate various different phenomena in the same fashion.Just as the superconductivity is described by the condensation of itinerant electrons ∆ S ∼ cc, one may also expect a new quantum ground state of manybody coming from the condensation of the itinerant-localized electrons with spin.Such many Kondo phenomena would be also described by considering the scalar condensation ∆ K ∼ f † c if the mean field theory works for strongly interacting case like Kondo system.Recently, we realized such idea in the holographic set-up [30] by finding that the Kondo condensation ∆ K produces a gap, which was also observed experimentally [30].
However, the ref.[22] has several limitations: i) it is completely numerical so that it is hard to characterize the singularity types of the spectral functions.ii) the back reaction was not taken care of.iii) it was confined to the AdS 4 so that it is not possible to discuss the Weyl semi-metal spectrum from the beginning.In this paper, we will find the analytic expressions of the Green's function for all tensor types of symmetry breaking in the probe limit.Three aspects of the spectral function will be emphasized: i) gap/gapless, ii) presence/absence of flat bands of various types, iii) presence/absence of split Dirac cones.Especially interesting coupling turned out to be the scalar type coupling which can give both gapped and gapless spectrum depending on the sign of the coupling, which can never be possible in flat space theory.
From the analytic expressions, we noticed that most of the Green functions have branch cut singularities but some of them have poles and we can now understand why various different types of flat band exist.The definition of the non-fermi liquid is the vanishing of the quasi-particle weight assuming that the Green function has a pole type singularity.In our holographic mean field theory, it turns out that the generic form of the fermion propagator does not have pole singularity but has the branch cut singularity.That is, the fluid appearing in our theory is generically non-fermi liquid.Interestingly, in some cases, such behavior can happen even after order parameter is turned on.However, for a few types of the order parameter, the pole type singularity appears.Typically, it happens when the flat band appears.
At this moment, our analytic expressions are only for the case where order parameter fields are alternatively quantized where only the leading term is nonvanishing in the pure AdS background.In standard quantization where only the subleading term of the matter field is nonvanishing, we still have not find exact Green functions even in the probe limit.To back up this limitation, we performed and presented the numerical analysis to find fully back reacted solutions of antisymmetric 2-tensor field, which is the most important and complicated case.We calculated the spectral function of the fermions based on such backreacted solution, and compared with the analytic result of the probe limit.In case the singularity is of pole type, we observed a detailed agreement between the approximate analytic result and the fully back reacted numerical result, which suggests the stability of Green function with the pole type singularity.In contrast, for the propagators with branchcut type singularity, the detail of the spectrum is relatively vulnerable to the deformation by back-reaction, although qualitative similarity remains.
The rest of this paper goes as following.In section 2, we give a short description of formalism of the holographic mean field theory and fermion Green function calculation.In section 3, we calculate Green's function analytically and draw spectral function for all types of the Lorentz symmetry breaking.In section 4, we draw spectral functions and discuss features of them.In section 5, we do extensive numerical calculation to find the fully back-reacted order parameter and metric and use them to calculate the fermion spectral function again to support the probe limit analytic results.In section 6, we discuss and conclude.

Holographic mean field theory with symmetry breaking
Our holographic mean field theory has four components: 1st, bulk fermions ψ which are dual to the boundary fermions with strong interactions.2nd, order parameter fields Φ I describing the condensations of fermion bilinear which is the analogue of the Hubbard-Statonovich field of the usual mean field theory.3rd, the gravity describing the interactions between electrons, and finally the gauge field which should be included to describe the density or chemical potential of the fermions.In this work, we do not include the vector for the analytic work but we can do it when we work numerically.For simplicity, we assume that they can be described by local fields in the bulk not in the boundary, because we should allow the strongly interacting system in the boundary be described by nonlocal field theory.The total action is given by [28] where is the order parameter field which couples with bilinear spinor in the bulk, leading to the symmetry breaking under the presence of the source or its condensation.Additionally, we will turn on just one component of field Φ to calculate the spectral function.The gamma matrix convention and the geometry are chosen and given as follows, where the underlined indices represent tangent space ones.Under this convention, the boundary locates at u = 0. Notice that in AdS 5 , including two-flavors of fermions is mandatory because holography projects out half of the fermion degrees of freedom while we need a full 4 component spinor in the 4 dimensional boundary.On the other hand, in AdS 4 , considering one flavor is still allowed since the boundary is of 2+1 dimension where spinors are of two components.
We will analytically determine and analyze the fermions' Green's function in the presence of an order parameter field.So, we consider the absence of gauge field for simplicity.Furthermore, we will begin our analysis in the probe limit and later we will eventually calculate the spectral function in the full back-reacted background.We will compare it with the probe limit analytic results to check the reliability of the latter.

Variational analysis and boundary actions
In this section, we will perform the variational analysis in detail to show the boundary fermions in different quantization choices.The standard-standard (SS) and standardalternative (SA) quantization can be distinguished by the sign of the boundary action (2.4).We first simplify the action by introducing ζ (j) , (2.8) Then, the variation of bulk fermions action (2.2), after the equation of motion is imposed can be written as a boundary term given below. (2.9) If we add the variation of boundary action with sign ± (2.4) depending on SS and SA quantization respectively, the variation of total action is given by δS From this expression, we see what are chosen as independent degree of freedom to makes the variation of total action zero.We call such independent fermions as the source fermions.We define a 4 component spinor ξ's by + ), and ξ − ), (2.12) as the boundary spinors for SS-quantization.The indices S, C in ξ S and ξ C are adopted since they correspond to the source and condensation terms.The ξ C 's supposed to be determined by ξ S 's.Similarly, for the SA-quantization we define − ), and ξ + ). (2.13) One should remember that all ζ ± are two component spinors while ξ's are 4 component ones.The extension ξ(u) of these fermions to the bulk of the AdS can also be considered by the original ζ(u)'s whose boundary values were used above as ζ, so that We can now rewrite the on-shell effective action as

Green's function
We can determine the boundary Green's functions from the the effective actions (2.15)-(2.16)and the definition of source and condensation fermions.Since we have 4 components of ξ(u), there must be 4 independent solutions Ψ i (u) which can span the space of spinor solutions.Our ξ(u) with prescribed boundary value should be a linear combination of these, so that (2.17) By taking the a-th component of this equation, we have ξ a (u) = Ψ a i (u)c i , which can be written as a matrix equation ξ(u) = Ψ(u)c.Here Ψ a i (u) is the a-th component of i-th solution and we considered ξ(u) and c as column matrices.By expanding the matrix Ψ(u) near the boundary, We now can write the source and condensation fermions depending on the quantization choice as follows, the boundary Green's function can be defined as follow Notice that the definition of Green's function remains valid even for the zero bulk fermion mass.Consequently, as far as we can extract the leading-order terms of the bulk fermions near the boundary, the Green's function calculation remains solvable.This is helpful because, in the zero fermion mass, we will be able to obtain Green's function analytically for all Lorentz symmetry breaking interaction with proper choice of scaling dimension of the order parameter field.The term suitable here refers to choosing scaling dimensions for the source term, in which any u-dependence in the interacting term will be eliminated after fully expressing the vierbein and spin connection.This ensures the results remain u-independent at the interaction terms and allows solvable Dirac equations.

Analytic Green's function of fermions in symmetry broken phases
We now consider zero bulk mass fermions with a holographic order parameters having only the leading term by setting ⟨O Φ ⟩ = 0.This is so called alternative quantization of the Φ.This setup allows us to derive Green's function for all types of Lorentz symmetry-breaking analytically.We already performed numerical calculations to study the case of non-zero fermion bulk mass and for the case with condensation in previous works [31][32][33].From the expressions of Γ in (2.7) and our gamma matrices can be expressed in the following decomposed form: where γ µ = (iσ 2 , σ 1 , σ 3 , −i1) with µ = t, x, y, z and γ µν = γ µ γ ν .The complex conjugation appears due to Γ z 's being pure imaginary.In AdS 4 , this decomposition is still valid with µ = t, x, y, because apart from Γ 5 , all gamma matrices are real.So that the complex conjugate disappears in AdS 4 .Such decomposition will be utilized for analytic solutions.

Classification of interaction types
Since the boundary represents the physical world, we will classify the interaction type from the boundary point of view; • 2 types of scalar: 1, Γ u (radial scalar).
Although B u and B µu are component of vector and tensor, respectively, they are scalar and vector from the boundary point of view, For scalar interaction, the solvable solution can be obtained by choosing

SS case:
The bulk equations of motion are given by due to the simple commutation relation between (σ 0 ⊗ γ µ * k µ ), and its conjugate, one can get the simple fully diagonalized decoupled equations [29], which reads The solutions are well-known and decay exponentially since the growing terms are removed by imposing in-falling boundary condition (BC).As a result, the asymptotic solutions near the AdS boundary located at u = 0 are given by ξ where S n , C n are u-independent but momentum dependent 4 × 4 matrices.But apart from the leading term S 0 (k), C 0 (k), they will not contribute to the boundary Green's function.Therefore we will write them simply as S(k), C(k) by deleting the index 0. Since ξ S and ξ C are solved independently, one can plug-in one of the solution on the Dirac equations to find the relation between them [27,28].The condensation term is determined by the source term by solving the Dirac equation and it is given as follows: where we define the matrix T = S 1 S −1 0 .For the scalar interaction, it is given by From the definition of boundary Green's function (2.24), one gets 4 by 4 retarded Green's function as follows, where . It is important to note that Tr K = 2ω.This matrix, K, will play a consistent role in our subsequent calculations.We will discuss the trace result of the Green's functions in the next section.

SA case:
The bulk equations of motion are given by similar to the SS case, one can decouple above equations which again yields (3.7)-(3.8).
Plugging the asymptotic solution into (3.13),we get following the definition of Green's function, and the T (k) given in (3.10), one can get the general form of it as follows.

G(k
One can see that the Green's function contains off-diagonal terms, which are absent in intraflavor interaction case.Moreover, calculating AA (alternative-alternative) or AS(alternativestandard) quantization cases yields the results with the propagator replaced by the complex conjugation.

Radial scalar:
The solvable solution can be obtained by choosing where b is the constant measuring the symmetry breaking strength.

SS case:
The bulk equations of motion are given by Decoupled differential equations (DEs) are given by Then, we can diagonalize the system by using the similarity transformation P defined by the eigenvectors matrix of σ 1 ⊗ σ 0 which yields This yields the solution of the exponential form even after mapping the solution back by the inverse similarity transformation.So, overall, nothing new for B u case.We can still write the Green's function as follows: The surprise of this interaction types is structure of T (k), which is given by So the Green's function reduces into the non-interacting case, SA case: the bulk dirac equation is given by and its complex conjugation is a nondiagonal matrix, a similarity transformation is needed for the diagonalization, which yields The fermion Green's functions for SS and SA quantization with B u interactions are found to be the same.This discovery raises the question of why SS and SA lead to identical Green's functions despite the differences in their equations of motion.The answer lies in two critical factors that influence the structure of the Green's function.Firstly, it depends on the combination of gamma matrices present in the Dirac equation, which varies with the choice of quantization.Secondly, the solutions are affected by the proper in-falling boundary conditions (BC).As a result, despite the apparent difference in the initial appearance of the Dirac equations, the solutions surviving the BC end up with the same Green's function.

Polar vectors:
The solvable solution can be obtained by choosing where µ is a vectors with a single nonzero component b (t,i) which is nothing other than a constant order parameter.

SS case:
The bulk equations of motion are given by Unlike the scalar case, polar vector type interactions cannot be decoupled simply in this basis.However, we can transform the equations to a suitable basis by a u-independent similarity transformation, P, where where Under this transformation, we can get the decoupled equations, where ξS,C = P −1 ξ S,C .We obtain simple solutions even after transforming the solutions back to the original basis, providing exponential decay.Consequently, we can express the asymptotic solution similar to scalar case (3.8).By substituting the solution into the (3.27), we obtain Then, the Green's function is given by The Green's functions can be determined straightforwardly by plugging in T (k) into (3.31).Now let us calculate the explicit expression of the Green's function.
B t /SS : By solving the Dirac equations, one gets where . By plugging the above result into (3.31),we can get the Green's function: with B x /SS : By follwing the same calculation of time-like case, one gets where x .By plugging the T (k) in the above result into (3.31),one gets with One can see that the Green's function of polar vectors together with SS quantization contains branch-cut singularity by the presence of K −1 µ± as the denominator terms.The singularity type does not change after tracing the Green's function matrix, which we will discuss in the next section.
After this case, we will no longer show the full expression of T (k), because it will become more complicated while lacking meaningful content.However, one can obtain the Green's function by following the same logic and calculations.

SA case:
The bulk equations of motion are given by In this case, the differential equations cannot be fully decoupled by similarity transformation.However, the DE is nothing but a linear Ordinary DE system.Similar to other cases, the solutions satisfying the infalling BCs exponential decay (3.8).We, therefore, substitute the solution back to the (3.38) and get the condensation in terms of the source: Therefore, the algebraic Green's function for this case is given by with where t .One can easily check that the pole type singularity k −2 will be canceled out after we take the trace of this Green's function.Since the cancellation of the pole makes the trace of the Green's function becomes non-singularity type Green's function.Therefore, the presence of singularities in the 4 by 4 expression of the Green's function does not guarantee the presence of singularity in spectral function.
B x /SA: By the same calculation in the previous cases, the Green's function reads with where x .In this case, the singularity (k 2 ⊥ − ω 2 ) −1 is not changed or canceled by the trace, so that the trace of the Green's function has pole type singularity.We will back to discuss the trace of these Green's functions in the next sections.

Radial vectors:
The solvable solution can be obtained by choosing where B (−1) µu is a tensor in AdS bulk with a single nonzero component.However, from the boundary point of view, its physical role can be classified as a vector.

SS case:
The bulk Dirac equations are given by µu ]ξ The main procedure of this type is the same with other interactions.We get the condensation in terms of the source as follows, This yields following Green's function,

.49)
B tu /SS: By the same calculation in the previous cases, the Green's function is given by with (3.51) where tu .The trace of the Green's function turn into a simple form which will be discussed in the following section.

SA case:
The bulk Dirac equations are given by The condensation are given as follows, which yields following retarded Green's function matrix, B tu /SA: By the same calculation we have done, the Green's function is given by with where tu .We will discuss the trace of the Green's function in the coming section.
B xu /SA/SS: The decomposition and simplification of the Green's functions for this case pose significant challenges.Consequently, we will focus only on the trace of the Green's function, which included in the following section.

Antisymmetric 2-tensors:
The solvable solution can be obtained by choosing where B (−1) µν is symmetry breaking strength constant.

SS case:
The bulk Dirac equations are given by Following a similar approach to B µ 's SA-case, we can decouple above equation and get a system of decoupled linear equations.So, we then plugin the solution into the above equation to get leads to the Green's function which is given by where xy , and E = k 2 − ω 2 .In this case, the singularity (k 2 z − ω 2 ) −1 is not changed or canceled by the trace, so that the trace of the Green's function has pole type singularity.We will back to discuss the trace of these Green's functions in the next sections.

SA case:
The bulk Dirac equations are given by The above coupled equations can be decoupled to get exact solution of exponential form.

.66)
The Green's function for this case is given by

.67)
B xy /SA: By the same calculation, the Green's function is given by ) where . B tz /SS/SA: The decomposition and simplification of the Green's functions for this case pose significant challenges.Consequently, we will focus only on the trace of the Green's function, which included in the following section.

Features of spectral functions
The spectral functions (SF) can be determined by the imaginary part of the traced Green's functions: Since the analytic results can be obtained when the order parameter fields have only leading terms we will analyze only such cases.

SS
The essential part of the Green's functions is given by the trace (3.12), Tr G (SS) where k 2 = d−1 i=1 k 2 i , and M 0 is the scalar source.The simple pole is located at the surface of the d dimensional cone where d is dimension of the AdS boundary.Notice that the symmetry breaking strength M 0 does not affect the pole structure but only contributes to the gap size.In AdS 4 , the pair of the gapped spectrum with M 0 , M 50 was reported [22].In AdS 5 , we do not have the chiral dynamics of the boundary although we should have corresponding spectrum from the boundary point of view.The difficulty lies in the fact that the chirality cannot be defined in odd dimensions.We postpone this problem to the future work.

SA
The analytic expression is given by Tr G (SA) The main feature of this interaction is the gap generation, as it was noticed in [22,31,32].Therefore, the scalar source in this case can be interpreted as the mass of boundary fermions.In AdS 5 , only scalar SA quantization can generate the gap, while in AdS 4 case, both M 0 , M 2 05 can do that.See figure 1(b,d).

Diagonal interaction in fermion flavors
For the scalar, we consider the case where the fermion-scalar interaction is diagonal type, namely, In this case, we have independent sum of two flavors and the result is following.
Notice that the sign of M 0 is important: for sign(M 0 ) > 0 we have gapless spectrum while for negative case we have gapped one.Therefore, in the intra-flavor case with L int = iΦ( ψ(1) ψ (1) ), the massless-gapped phase transition depends on the changing sign of M 0 [29].However, in our inter-flavor with L int = iΦ( ψ(2) ψ (1) + h.c), there is no phase transition under the sign change of M 0 .See figure 1(a,c).It turns out that for all interaction types other than the scalar-fermion, there is no such phase transition between the gap-gapless phases in the spectral function.

Radial scalar B u
SS and SA For this interaction, there is no effect from the order parameter b due to the cancelation that happened during calculation of the Green's function, see (3.23).In fact, this has been a puzzle from the view of the numerical calculation.As a result, the trace of the Green's function, regardless the quantization choice, is given by Tr G (SS,SA) which is the same as that of critical point where B u = 0.

SS
The trace of the Green's matrix (3.31), by choosing µ = t is given by Tr G (SS) where In this case, above result shows two Dirac cones, shifted along ±ω directions, which are not interacting with each other.The singularities are located at each cone.Notice that there are spherical symmetry in k x -k y -k z .See figure 2(a,b,c)

SA
The trace of the Green's matrix (3.41) is given by Tr G (SA) In this case, the symmetry are the same with SS case.However, there is no singularity in the Green's function.Therefore the entire SF is described by a branch-cut without singularity.See figure 2(d,e,f)

SS
The analytic expression is given by Tr G (SS) In this case, the spectrum is isotropic for each Dirac cone shifted along the entire kspace, and that is why we cannot distinguish the spectrum of B xy(SA) and B tu(SA) in AdS 4 .However, in AdS 5 , the SF has spherical symmetry, while B xy spectrum has planar rotational symmetry in k x -k y plane.See figure 3(a,b,c).

SA
The analytic expression is given by Tr G (SA) where The SF shows the superposition of two Dirac cones shifted along the k x direction, which are non-interacting with each other.(4.10).The distance between the Dirac points is 2b and the surface of the cones are branch-cut type singularity.Notably, the SF in the ω-k x plane exhibits a shifting of 2dimensional Dirac cones, see fig 4(a).In the section of ω-k ⊥ plane; it shows a gap, see figure 4(b).

SA
The trace of the Green's function matrix (3.41) for µ = x is given by Tr G (SA) The main feature of the spectrum is shifted Dirac cones in ±k x direction: two Dirac points is connected by flat band of 1-dimensional pole singularity ( 4(e,f).It is important to note that the residue is zero for k x / ∈ [−b, b], so there is no singularity outside the interval.See figure 4(g,h).

Space-like radial vector, B ux
The analytic expression is given by Tr G (SS) Tr G (SA) where . The structure of the f ± is nothing but shifting of |ω|-radius semispheres in k x direction.It is useful to realize that f − f + is shifting of two |ω|-radius spheres in k x direction.See figure 5.   where , h ± has a semi-torus structure.Here, , which is nothing other than a torus.This structure analogous to B xy with extra branch-cut singularity pieces.For SA case at nonzero ω, we observe a torus with connecting planes.See figure 7(d).For SS case at nonzero ω, it is branch-cut version of B (SS) xy .See figure 7(h).The crucial difference is that there is no singularity at ω = 0 in SS case.
We have successfully obtained spectral functions for all interaction and quantization types where ⟨O Φ ⟩ = 0. From their spectral features, we classify them according to their spectral features: various dimensional of flat bands, semi-metals, singularity types, and ω-shiftings.See figure 8.

Spectrum in the presence of the order parameter's condensation
We have exhibited the SFs corresponding to the condensation in the alternative quantization by employing the analytic Green's functions.Here we mention that the flow equation can numerically compute the case where order parameter fields condensation in the standard quantization [31][32][33] although the analytic expressions are not possible.The spectral features in the probe limit can be generated by employing seven spectral features and modifying the symmetry in k-space.
Our calculation shows that the interactions leading to the simple pole type in the alternative quantization, which are given by Φ, B x , B xy , B tu , also yield the simple pole types singularity in the standard quantizated case as well.But not vice versa.For example, the  In this section, we found three types of Green's function: pole type, branch-cut type singularity and branch-cut non-singularity.In the upcoming section, we will compute the full back-reaction for each order parameter field.It is noteworthy that we have observed a remarkably stable pole-type singularity.This observation agrees with our previous research, wherein we interpreted the pole type, referred to as the zero mode, as a topological mode.The stability of this mode is assured by the boundary conditions of the fermions, which makes it topological [29].

Backreacted spectral functions
So far, our calculations were done in the probe limit, where back-reactions to the metric by the order parameter fields were neglected.Consequently, the reliability of these approximate analytic expressions can be asked and the only way to answer is to actually carry out the full back-reacted solution.In this section, we carry out this program and compare it with the essential probe limit analytic results.Since any of the back reaction calculation in volve full scale numerical work, and preliminary calculation with low numerical grid gave us the tentative result that the Green function with pole type singularity is stable under perturbation while the branch cut type singularity is not.Therefore we also expect that the full scale back reacktion should be similar.In this section, we perform only for the space-like antisymmetric tensor type order parameter, postponing other cases to the future work.We use special lagrangian which permit the non-zero source and zero condensation.In other words, we use the theory that allowed the alternative quantization of the order parameter field.Our result will show that indeed our expectation is correct.
We follow the fundamental action model which is given in [34,35].Additionally, to quantize real B xy alternatively, where only the leading term is nonvanishing, we introduce A t and set the highest order coupling term between A t and B xy as the source of spontaneous symmetry breaking.The model is then given by  here A µ = A t (u)dt and the antisymmetric field-strength tensor H λµν , given by where B µν = B xy (u) dx∧dy.Notice that the choice of our A µ , B xy , leading to F µν B µν = 0, and in this scenario, F 2 B 2 becomes the highest order term.
It is important to note that by setting m 2 = 1, the space-like antisymmetric 2-tensor field B xy (u) diverges near the boundary, 3) The presence of the singularity in the expression causes the numerical challenges.Consequently, we define a new variable, B xy (u) := uB xy (u), significantly enhancing numerical convenience by eliminating the singularity from the expression.With a small value ϵ, we can determine the sub-leading term B xy by B ′ xy (ϵ)/2ϵ .In the back-reaction calculation, we take the following ansatz, Figure 10 shows a calculation result with the back-reaction in alternative quantization of B xy order parameter field.By the given parameters, we numerically get B xy = 8.6 × 10 −6 with T = 0.10 .We also emphasize that in the alternative quantization, the source term of the order parameter is small.So that the effect of the symmetry breaking is located near (ω, k) = 0. To observe the symmetry breaking effect, we amplify the coupling of the fermions by introducing a magnification constant, denoted as λ.This results in the transformation: B xy ψ (1) Γ xy ψ (2) → λB xy ψ (1) Γ xy ψ (2) .
The obtained results reveal the following: i) The degrees of freedom near k ⊥ = |λb|, spread out but the pole singularity structure remains stable.On the other hand, the singularity structure along the branch-cut shows deformation.ii) Regardless of fermion quantization, the main spectral functions maintain the same feature compared to the probe limit results.These results emphasize the reliability of probe-limit results, providing the advantage of avoiding complexities and excessive time calculations associated with full back-reaction calculations.

Discussion
In this paper, we found the analytic expressions of the Green's function of fermions under the various types of symmetry breaking: vector and tensor as well as a few types of scalars.We classified the propagator according to the types of singularities: Some of them have branch cut types but some of them have pole types.By having the analytic expressions, although it is in the probe limit, we now understand why various dimensional flat bands exist and why they have finite regions of support.
Our setup refers to the order parameter field configuration with zero condensation, ⟨O Φ ⟩ = 0.For the scalar condensation in AdS 4 , an analytic study has already been made and reported [29], but in the context of AdS 5 , the presence of the condensation term gives the Dirac equations nontrivial dependence of u, making the solvability unavailable for all types of order parameter fields.This is the reason why analytic calculation for condensation was not considered in the present work.
To support the analytic results which are obtained from the probe background and also with only source type order parameters, we performed the numerical analysis to find solutions of coupled system of gravity with space-like antisymmetric 2-tensor and use it to calculate the spectral function of the fermions.Comparing the spectral functions of fermions with and without the back-reaction, we observed a qualitative agreement in the structural features.In other types of the order parameter, we do not show the results in this work, but we also found such qualitative agreement in the most cases, which will be extended to future projects.
One should also notice that our analytic results are associated with the zero temperature results, which is an extremal limit, so they should not compare with the result at high temperatures.The singularity structure significantly changes as we change the temperature.The most important remark is that our numerical spectral functions provide slight deformation and a reduction in sharpness compared to the zero-temperature limit, which is nothing but the effect of the back reaction.However, we observed the different effect of the back reaction on pole and branch-cut type singularity.
In the cases of the pole-type singularity or flat bands, we observe the negligible back reaction and high stability of the flat bands, which the structure of the singularities quanlitively remain and closely match with our analytic results.Evidently, in the cases where the rotational symmetric flat band is present: xy , the simple pole singularity structure is remarkably stable.This observation agrees with our previous work which interpreted the pole spectrum as a topological mode.However, we observed spreading out of density of state near k ⊥ = |λb| which is the back-reaction effect.
On the other hand, in the cases of the branch-cut type singularity spectrum, the singularity structure is slightly different from our analytic results.For B (SA) xy , we observe that the singularity near k ⊥ = |λb| is more fuzzy and reshapes compared to the analytic result.However, it still qualitatively remains the main feature.
We now list a few further future projects apart from removing the above limitations.First, it would be interesting to discuss the presence of branch-cut type of singularity in the propagator in view of the non-fermi liquid.Second, it should be possible to discuss the topology of the various spectral functions.This would give a precise answer to the question of what happens to the topology in the limit where the quasi-particle disappears.We hope we can come back to this issue in the near future.Finally, notice that we are in lack of chiral Γ 5 matrix to represent the chirality of the boundary.For this reason, we did not discuss the chiral dynamics in this paper.We think that it must be done by introducing another flavor of fermion to double the degrees of freedom.the structure of the gamma matrices shows us that the result of Green's functions will be the same as AdS 5 by removing complex conjugates in the expressions and eliminating the third momentum k z .The reason is that the differential equations remain the same as before.For pseudo-interaction types, however, they might be confused due to lack of the fifth gamma matrix in AdS 5 .According to Γ 5 , Γ 5x , Γ 5t , Γ 5r , they are equivalent to Γ z , Γ zx , Γ zt , Γ zr , respectively, by setting k z = 0.As a result, the duality of the Green's functions between AdS 5 and AdS 4 are obtained.See table 1.
Our results with quantization dualities of AdS 4 spectral features agree with our previous numerical study [22].However, we emphasize the trace equivalence on the table means that the trace of the dual Green's functions is the same, but each element in the Green's function might vary.So, this is a crucial clue indicating that each interaction's topological properties are different even though they share the same spectral feature.

Figure 1 :
Figure 1: Spectral Functions (SFs) of scalar source for both quantization choices.(a,b) SFs in ω-k plane.(c,d) SFs in ω = 2.5 slice, the dashed red line in figure (a), is the three dimensional object in (c,d).The figure at each plane is its projection to each plane.The blue (c) and red (d) surfaces represent the pole and the branch-cut type singularity, respectively.

Figure 2 :
Figure 2: Spectral functions (SFs) of B t source for both quantization choices.(a,d) SFs in ω-k plane.(b,e) SFs in ω = 0 plane, and (c,f) for ω = 1 plane corresponding to the dashed red line, in which the background represents the certain slices at each momentum is zero.The bare orange bulk without the surface shows the spectral function without singularity.

4. 2 . 3
Space-like polar vector, B x SS The trace of the Green's matrix (3.31), by choosing µ = x Tr G (SS)

Figure 4 :
Figure 4: Spectral functions (SFs) of B x source for both quantization choices.(a,b,e,f) SFs in ω-k x , ω-k ⊥ plane.(c,g) SFs in ω = 0, and (d,h) SFs for ω = 2.5 correspondingly to the dashed red lines.In (g,h), if k x / ∈ [−b, b] the pole type singularity disappears so that only k x / ∈ [−b, b] which the arc lines visible.The box's background represents the certain slices at each momentum is zero.

Figure 6 :
Figure 6: Spectral function (SFs) of B xy source for both quantization choices.(a,b,e,f) SFs in ω-k x , ω-k ⊥ planes.(c,g,) SFs in ω = 0, and (d,h) ω = 1 correspondingly to the dashed red lines.The spectral functions have rotational symmetry for each fixing k z .The background of the box represents the certain slices at each momentum is zero.

Figure 7 :
Figure 7: Spectral functions (SFs) for B tx source for both quantization choices.(a,b,e,f) SFs in ω-k plane.(c,d,g,h) SFs in k x -k y -k z at ω = 0, 1 slices, respectively.The spectral features are analogous to B xy .Notice that the disk appearing in (g) is just the nonsingular branch-cut.

Figure 8 :
Figure 8: The classifications of spectral features for all interaction and quantization types.The table consists of spectral functions in ω-k x,y,z and k x -k y -k z at ω ≃ 0. The spectra have identical symmetry on the horizontal alignment and have the same spectral feature on the vertical alignment.d ef f is the number of the flat band, semi-metals, nonsingular, and ω-shifting spectra appearing in each k-space section.

Figure 9 :
Figure 9: Spectral functions (SFs) which are associated with the condensation order parameter fields.(a,b) and(c,d) scalar and B tu condensation SFs, lead to the complete disappearance of the fuzzy spectra, resulting in the emergence of only the Kaluza-Klein (KK) modes.It is worth noting that even in the source case of these interactions, both simple pole and branch-cut singularities can occur.However, only the simple pole type is observed in the condensation cases.(e,f) B t condensation, the types of Green's functions appear similar to those observed in the source cases.Keep in mind that combining these SFs and introducing variations makes it possible to generate all 16 types of interaction SFs.
.61) B xy /SS: By solving the Dirac equations, and plugging-in into (3.61),we get