Ginzberg-Landau-Wilson theory for flat band, Fermi-arc and surface states of strongly correlated systems

We show that we can realize the surface state together with the bulk state of various types of topological matters in holographic context, by considering various types of Lorentz symmetry breaking. The fermion spectral functions in the presence of order show features like the gap, pseudo-gap, flat disk bands and the Fermi-arc connecting the two Dirac cones, which are familiar in Weyl and Dirac materials or Kondo lattice. Many of above features are associated with the zero modes whose presence is tied with a discrete symmetry of the interaction and these zero modes are associated with the surface states. Some of the order parameters in the bulk theory do not have an interpretation of symmetry breaking in terms of the boundary space, which opens the possibility of ‘an order without symmetry breaking’. We also pointed out that the spectrum of the symmetry broken phase mimics that of weakly interacting theory, although their critical version describe the strongly interacting system.


Holographic order parameter
The strong correlation is property of a phase of general matters not a few special materials, because even a weakly interacting material can become strongly interacting in some parameter region. It happens when the fermi surface (FS) is tuned to be small, or when conduction band is designed to be flat. The Coulomb interaction in a metal is small only because the charge is screened by the particle-hole pairs which are abundantly created when FS is large. In fact, any Dirac material is strongly correlated as far as its FS is near the tip of the Dirac cone. This was demonstrated in the clean graphene [1,2] and the surface of topological insulator [3][4][5] through the anomalous transports that could be quantitatively explained by a holographic theory [6][7][8]. In the cuprate and other transition metal oxides, hopping of the electrons in 3d shells are much slowed down because the outermost 4s-electrons are taken by the Oxygen. In disordered system electrons are slowed down by the Kondo physics [9]. In twisted bi-layered graphene [10,11] flat band appears due to the formation of larger size effective lattice system called Moire lattice. In short, strong correlation phenomena is ubiquitous, where the traditional methods are not working very well, therefore new method has been longed-for for many decades.
When the system is strongly interacting, it is hard to characterize the system in terms of its basic building blocks and one faces the question how to handle the huge degrees of freedom to make a physics, which would allow just a few number of parameters. Recently, much interest has been given to the holography as a possible tool for strongly interacting system (SIS) by applying the idea to describes the quantum critical point (QCP) describing for example the normal phase of unconventional superconductivity. Notice however that the QCP is often surrounded by an ordered phase. Physical system can be identified by the information of nearby phase as well as the QCP itself.
For the ordinary finite temperature critical point, the Ginzberg-Landau (GL) theory is introduced precisely for that purpose. As is well known, it describes the transition between the ordered and disordered states near the critical point. It works for weakly interacting theory and when it works it is a simple but powerful. The order parameter depends on the symmetry of the system and the phase transition is due to the symmetry breaking. The tantalizing question is whether there is a working GL theory for strongly interacting systems. The GL theory works also because of the universality coming from the vast amount of information loss at the critical point, which resembles a black hole. For the quantum critical point, we need one more dimension to encode the evolution of physical quantities along the probe energy scale [12,13]. Therefore it is natural to interpret AdS/CFT [14][15][16] as a GL theory for the strongly interacting system where the radial coordinate describe the dependence on the renormalization scale [17][18][19][20]. For this reason we call it as Ginzberg-Landau-Wilson theory.
The transport and the spectral function (SF) have been calculated in various gravity backgrounds using the holographic method. However, it has been less clear in general for what system such results correspond to. For this we believe that the information on the ordered phase is as important as the information on the QCP itself. Clarifying JHEP01(2021)053 this point will be the first step for more serious condensed matter physics application of the holography idea and this is the purpose of this paper. The idea is to introduce the holographic order parameters of various symmetry type and calculate the spectral function in the presence of the order. The resulting features of the fermion spectrum should be compared with the Angle Resolved photo-emission spectroscopy (ARPES) data, which is the most important finger print of the materials.
Notice that both the magnetization and the gap of superconductor can be understood as the expectation value of fermion bi-linears [21] χ † σχ and χχ of the fermion χ. In fact, the expectation value of any fermion bilinears can play the role of leading order parameters. When two or more of them are non-zero, they can compete or coexist according to details of dynamics. Then, the most natural order parameter in the holographic theory should be the bulk dual field of the fermion bilinear because it contains the usual order parameter as the coefficient of its sub-leading term in the near boundary expansion. The presence of the order parameter actually characterizes the physical system off but near the critical point. We will calculate spectral functions [22][23][24][25] in the presence of the order parameter. Our prescription for them is to add the Yukawa type interaction between the order parameter and the fermion bilinear in the bulk and see its effect on the spectrum.
To be more specific, let ψ 0 be the source field of the fermion χ at the boundary and Φ 0I be the source of the fermion bilinearχΓ I χ where I = {µ 1 µ 2 · · · µ n } represent different tensor types of Gamma matrix. The extension of source fields ψ 0 and Φ 0I to the AdS bulk is the bulk dual field ψ and the order parameter field Φ I . We calculate the fermion spectral function by considering the Yukawa type interaction of the form (1.1) For example, the complex scalar can be associated with the superconductivity, and the neutral scalar to a magnetic order. We will classify 16 types of interactions into a few class of scalars, vectors and two-tensors and calculate the spectral functions. With such tabulated results, one may identify the order parameter of a physical system by comparing the ARPES data with the spectral functions. Some of the idea has been explored for scalar [26] and tensors [27,28] to discuss the spectral gap of the superconductivity. But in our paper, we will see much more variety of spectral features like flat band, pseudo gap, surface states, split cones and nodal line etc. The most studied feature of the fermion spectral function is the gap. The authors of [29,30] considered the dipole termψF rt Γ rt ψ to discuss the Mott gap. However, if we define the gap as vanishing density of state for a finite width of energy around the fermi level, the dipole term does not generate such spectrum because the band created by the dipole interaction approaches to the Fermi level for large momentum. In [28] the author reported the observation of Fermi-arc in the sense of incomplete Fermi surface. Our Fermiarc is in the sense of surface state in topological materials. In our set up, they exists in the presence of various different types of vector order.
We found that the parity symmetry controls the presence of the zero mode or gap. For example, Φ 5ψ Γ 5 ψ generates a gap as it was discovered in [26], while ΦψΓ 5 ψ can generate a zero mode. Another interesting aspect is that some of the order parameters in holographic JHEP01(2021)053 theory, especially those of tensors with radial index do not have direct symmetry breaking interpretation in the boundary theory, and this opens the possibility of 'an order without symmetry breaking'.

Surface states as zero mode of the bulk
When we describe the surface of topological insulator (TI), having insulating bulk and conducting surface, in terms of holographic theory, one may wonder whether it is possible at all, because the essence of the material is the surface properties while there is no surface of boundary of AdS. Because the bulk is gapped and has no interesting transport coefficients, we took the surface of 3d TI as the Dirac material and modeled it in terms of the AdS 4 [7,8].
Although the transport properties of the system could be described consistently with data, this is certainly unsatisfactory, because TI can be defined consistently only by the bulk and surface together.
One of the most important discovery of this work is find an alternative and possibly the correct way to describe such topological materials. After symmetry breaking, half of the 16 possible interactions give zero modes which turns out to be the same as the surface state of topological materials of various types. The point is that, the surface mode is one of the bulk modes, because it exists as a solution of the bulk equation of motion. The zero mode is just confined or localized in one direction and propagate in other spatial directions, which is the reason why it is a surface mode. For higher modes, they penetrate more to the bulk by oscillations. In this spatial picture, it is clear why the zero mode is the surface mode. On the other hand in momentum space, the "surface mode" does not look like much different from the other bulk mode apart from that its energy or mass is zero. It actually located near the Γ point ω = k = 0.
Then, we can model the 3d TI using 'the AdS5 theory with a zero mode' instead of modeling the 2d surface of 3d TI in terms of AdS4 theory. We believe this is the correct holographic theory of the TI because the bulk modes include the massless surface mode as well.
Actually the most interesting point of this paper is that we can realize the surface modes of all the known types, in holographic context from the bulk equation of motion and we will clarify when the surface modes exist in terms of a discrete symmetry. We also see that the spectrum of the symmetry broken phase mimics that of weakly interacting theory, although their critical version describe the strongly interacting system.
Finally it is also worthwhile to notice that the chirality as the monopole charge of the Berry phase can exist in any dimension, although the Weyl fermion as a representation of SU(2) exist only for the even dimensions. Therefore the presence of the Fermi arc as a zero mode which connect the two tip of the Dirac cones with + and -monopole charges can exist in any dimension too.

Summary of main results
Before we go further, we want to give a short summary of main results to motivate the reader. Since the holographic theories rely on the universality rather than microscopic details, having the spectrum to measure the dynamical exponents of the QCP and the symmetry of interaction terms are enough to identify the physical system. Using these JHEP01(2021)053   achievable by different means of combining interlayer asymmetry, sublattice asymmetry, and doping. The mechanism allows control of the band dispersion all the way from parabolic through flat band formation to Mexican hat-like. Figure 1 shows ARPES data for a 6H-SiC sample with 1.2 monolayer graphene (MLG) coverage. [As usual, the structural graphene monolayer at the interface, "zero-layer graphene" (ZLG), which is covalently bonded and acts as buffer, is not counted.] For this coverage, the MLG Dirac cone dispersion is expected to dominate with BLG contributing just a faint intensity. However, there is an additional peculiar, very intense, very sharp, and very flat band portion at 255-meV binding energy that is not present in the case of MLG on SiC (20). The new band is marked by white arrows in Fig. 1 (A to C). On the basis of both calculations and experimental data, we attribute this band to the bottom of one of the BLG bands. For this 1.2 monolayer coverage, the photoemission intensities of the BLG bands are about four times lower than those of the MLG bands, but at the same time, the flat band intensity is about three times higher than that of the MLG bands.

Experiment
There are examples in the literature where it is possible to see this intense band in the data, but it has been ignored in discussions so far (21)(22)(23), possibly because the resolution was not sufficient for details of the band dispersion. We performed the measurements in Fig. 1A with the electron wave vector perpendicular to the GK direc-tion and through the K point, at a tem energy of 62 eV. We did not observe an and K′. The features are much better respect to energy in Fig. 1B: There is one more flat band at 150 meV (blue kink in the dispersion in the 150-to 1 To judge the photoemission intensi 3D map around the K point. The cut al reveals that, in this experimental geome Dirac cone and only half of the bilaye of a destructive interference from the tw We see the flat band on both sides from sities. This is unusual for photoemissio In Fig. 1D, two constant energy cuts are data at 235-and 255-meV binding ene 20 meV, very small for ARPES otherwis change in the constant energy cuts. At 23 there is a nearly circular ring with intensi emission interference effect, but at 255 disk, without modulation by interferen The representation in Fig. 1E as a (only every 10th spectrum is shown) emission intensity of the flat band. Figu exactly intersects the K point. The sp single-peak Gaussian fitting of the topmo peaks. The resulting dispersion is show that the scattering of peak maxima en   identification idea, we will find at least four interesting results. Since this paper is the classification of the orders and resulting spectrum, we will only give pictures to compare our result and the physical system, leaving the detailed data comparison in future work.
1. For symmetry breaking by the scalar order, there can be a massless phase as well as massive ones depending on the sign of the order parameter in one flavor or quantization rule for two flavors. See figure 1. This is rather surprising since in flat space one can never have such result. This result can also be used to describe the metal insulator transition or order-disorder transition, for which we will report in a separate paper.
2. For symmetry breaking by B 5x , the spectrum can describe the Weyl fermion and its Fermi arc. The result is consistent with the Haldane's tangential attachment [31] of the fermi arc to the fermi surface. See the figure 6.
3. For the tensor order B xy , the spectrum has flat band which is suitable for describing the bilayer graphene [32,33]. See figure 2. If we add chemical potential, the spectrum resembles that of the heavy fermion in Kondo lattice. See figure 3.
Some reader may worry about describing the topological matter in terms of the holographic theory, because in topological matter the boundary plays important role but the JHEP01(2021)053  boundary and the bulk of the matter can not be modeled in holographic method. But we do not need to worry too much thanks to the bulk-edge correspondence: although we need to combine the bulk and the surface to have a well defined physical system, we know the bulk if we know the boundary and vice versa. Therefore, we can describe the boundary only for TI. Previously we found that at least for the purpose of the transports, modeling the boundary of TI as AdS boundary, works fine [7,8] for data fitting. We assume this is also the case for the spectrum. For Weyl or Dirac semi-metals, we identify the boundary of AdS as the bulk of the material because the matter. What is surprising is that we will find "surface mode" in the spectrum of the bulk in these cases.

Flat spacetime spectrum for various Yukawa interactions
To learn the effect of the each type of interaction, we first study the spectral functions(SF) of flat space fermions and classify them. The spectral functions will be delta function sharp. This will help us by suggesting what to expect in curved space if there are correspondence, because the AdS version will be a deformed and blurred version of flat space SF by interaction effects which is transformed into the geometric effect. However, AdS 4 and its boundary has difference in the number of independent gamma matrices, threrefore there are interaction terms in the bulk which does not have analogue in its boundary fermion theory.
We now consider boundary fermion χ 1 , χ 2 whose action is given by where Φ · γ = γ µ 1 µ 2 ···µ I Φ µ 1 µ 2 ···µ I and I is the number of the indices. For one flavor case, we set p 1 = 1, p 2 = 0 and set p 1 = 0, p 2 = 1 for 2 flavor. Each two component fermion in 2+1 dimension has definite helicity and the spin is locked with the momentum. Therefore with one flavor, we can not have a Pauli paramagnetism. We list 2 × 2 gamma matrices of 2+1 dimension.
Following identity is necessary and useful to construct lagrangian.

Spectrum in flat space
Because we did not introduce a lattice structure, we do not have periodic structure in momentum space. instead we focus on the band structure near the zero momentum. If we include only one flavor, only two bands will appear in the spectrum. For the zero mass, left and right modes can be split, while it can not be for the massive case. For two flavors, the number of bands is just doubled.

One flavor case:χ
For the flat space, there is not much difference between the scalar interaction and the mass term. Gap is generated as one can see from the equation of motion. See also the figure 4(a). The mass term, if exist, violate the parity symmetry.
Antisymmetric tensor: Φ · γ = B µν γ µν . In 2+1, The role ofInt. B µν is the same as that of µνλ B λ due to the second identity of eq. (2.7). Therefore no new spectrum is generated. Comparing with figure 1 and figure 2, we can see that the spectral double of the two flavor case is manifest as doubling of the bands.

2 flavor:
Here for convenience, we consider parity symmetry invariant combination of interaction terms, • Scalar: The point is that when the order parameter fields has non-zero vacuum expectation values, the result of the operation depends on the fluctuating fields, the interactions are invariant but when In each case, two forms of the interaction are equivalent because the second form is just unitary transform of the first by χ 1 → −iχ 1 , χ 2 → χ 2 . Notice that when the equation motion does not involve the i in the interaction term, the flavors shift in opposite direction for vector and anti-symmetric tensor cases while two flavors share the same spectrum for the scalar interaction. The spectrum for the two flavor system is a double of one flavor case. For scalar, gap is generate and spectrum is degenerated because two flavors has identically gapped spectrum. For vector interaction, the spectral cone of each flavor is shifted in opposite direction. Therefore in 2+1 dimensional flat space, anti-symmetric sector can be mapped to the vectors. because the role of B µν is that of µνλ B λ . However, in anti-de Sitter space, two sectors can be different.

Dirac fermions in flat 2+1 space and in AdS 4
For massless case, the spin-orbit coupling locks the spin direction to that of the momentum so that for fixed momentum only one helicity is allowed for one flavor. In AdS 4 , half of the fermion components are projected out depending on the choice of the boundary terms [34]. The spectrum of the fermions with AdS bulk mass term is still gapless, unless interaction creates a gap, because the AdS bulk mass is a measure of the scaling dimension not a gap.

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Therefore 4-component AdS 4 fermion suffer the same problem of 2 component massless fermions in 2+1 dimension. For example such spin-momentum locked fermion system does not have a Pauli paramagnetism [35]. One way to avoid such problem is to introduce two flavor and create a gap in the spectrum by coupling with non-zero scalar field Φ as we will show later.
A Dirac fermion in real system is that of 3+1 dimension even in the case the system is arranged into a two dimensional array of atoms. Therefore it should be described by two flavor of two component fermions, which corresponds to two flavor 4-component fermions in AdS. Then the spectrum of massless Dirac fermion in condensed matter system should be described as a degenerated Dirac cones. To describe the sublattice structure of the graphene, we need another doubling of the flavor. Therefore we consider only two flavor cases in the maintext, and provide the spectrum of the one flavor in the appendix for curiosity.
Notice that in 2+1 dimension, a fermion field has two component while in AdS 4 it has 4 components, where only half of the fermion components are physical [34]. Therefore, the degrees of freedom of the bulk match with those of boundary in AdS 4 theory if the number of flavor in each side are the same. However, in AdS 5 , we need to double the number of the fields, because 4 components in the boundary corresponds to the 8 components in the AdS bulk. To avoid too many cases, we will consider only AdS 4 cases here, and treat the AdS5 separately in the future if necessary. The boundary action must be chosen such that it respect the Parity symmetry as we have done in eq. (3.2), otherwise the flat space and curved space does not have correspondence especially in scalar order.

Fermion action and equation of motion
We consider the action of bulk fermion ψ which is the dual to the boundary fermion χ. Let Φ I be the dual bulk field of the operatorχΓ I χ. The question is how the Φ I couples to the bulk fermion ψ. When Φ I is a complex field, it describe a charged order like the superconductivity that has been already studied in holographic context. [36,37]. If it is real, it describes a magnetic order like anti-ferromagnetism or gapped singlet order. The main difference is the absence or presence of the order parameter with vector field A µ which is dual to the electric current J µ . We will consider both cases simultaneously and summarize simply as "without or with chemical potential", µ = A t (r)| =∞ .
The action is given by the sum
where the horizon of the metric r H = 1 3 (2πT + 4π 2 T 2 + 3µ 2 ) and µ is a chemical potential. Following the standard dictionary of AdS/CFT for the p-form bulk field Φ dual to the operator O with dimension ∆, its mass is related to the operator dimension by and asymptotic form near the boundary is Here we used the coordinate z = 1/r which is simpler due to the homogeneity of the AdS metric in this coordinate. We can find out the expression of the fields in r coordinate by using the tensorial property.
Throughout this paper, we use the probe solution Φ which is the solution in the pure AdS background. This approximation can give qualitatively the same behavior of the fermion spectral function because for finite temperature, the horizon of the black hole cut out the black hole's inner region where the true solution of Φ deviate much from the probe solution.
Following [23], we introduce φ ± by Then the equations of motion for the one flavor, with all the possible terms turned on, can be written as where matrix U K is from the kinetic terms and U Y is from the interaction term. If all types JHEP01(2021)053 of interaction terms are turned on, they are given by x, y and f is the screening factor of the metric. For AdS Schwartzschild case, f = 1 − r 3 H /r 3 . For two flavors, the equation of motion changes minimally: with the same U K and U I given above. For the clarity of the physics we turn on just one field Φ I to calculate corresponding spectral function. Φ I is the order parameter field that couples with spinor bilinear in the bulk. In this paper, we will treat it at the probe level with AdS background. Although the probe solution for Φ I does not respect all the requirements at the horizon, the IR region where the probe solution blows up by ∼ 1/r ∆ is removed by the presence of the horizon. Therefore it is a good approximation, unless the temperature is excessively small. We will separately consider the cases where order parameter field with condensation only and the case with source only in order to understand the effect of each case.

Discrete symmetries in AdS 4
To discuss the discrete symmetry, we first list the explicit forms of the Gamma Matrices we use.

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Our convention of the tensor product is that the second factor is imbedded into each component of the first factor. Notice that the construction is based on Γ µ = σ 1 ⊗ γ µ , for µ = 0, 1, 2, and Γ r was chosen to satisfy the Clifford algebra {Γ µ , Γ µ } = 2η µν . 1 2 (1 ± Γ r ) are projections to the upper (lower) two components of the 4-component Dirac spinor. In AdS space, the bulk mass of a field is not playing the role of the gap. Therefore without interaction, fermion spectrum is basically massless, and therefore helicity is a good quantum number. The upper two components are for positive helicity while lower two components have negative helicity. Depending on the boundary term, some of the components are projected out. In this paper we will choose the upper two components of the first flavor and lower two of the second flavor.
From the boundary point of view, we have scalar and vector interaction. What happened to the correspondence between the bulk and the boundary? we can reclassify the 16 AdS4 tensors in terms of 2+1 tensors.
We will see the similarities in each classes.
Below, we discuss the three discrete symmetries. T , P, C acting on the Dirac spinors and its bilinear in our gamma matrix convention. We need to know that the hermitian form of interaction lagrangian is given by for all Γ I = i1, Γ µ , Γ 5µ , Γ µν with µ, ν = t, x, y, r.
• The time reversal operation is given by T = T K where K is complex conjugation and T is a unitary matrix. From the invariance of the Dirac equation, we have x, y, r) are all real in our gamma matrix convention, we should have Therefore the invariant Hermitian bilinears correspond to following 8 matrices: On the other hand, the other half with change sign under the time reversal operation.

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• The parity symmetry (t, x, y, z) → (t, −x, −y, −z) with z = 1/r. For this, one should imagine that two AdS spaces with z > 0 and z < 0 are patched together along the hyperplane at z = 0. Notice that vierbeins are even function of z because e µ a = δ µ a √ g µµ and the horizon of the mirror geometry is located at −z H . The operation P : ψ(t, x, y, r) → Γ 0 ψ(t, −x, −y, −z) realizes the symmetry, under which a fermion bilinear transformsψ (3.24) Then the invariant Hermitian quadratic forms correspond to following 8 gamma matrices: On the other hand, the other half with change sign under the parity operation. Later we will see that the fermions with interactions invariant under the Parity will have zero modes, that would be interpreted as a surface mode, if there were an edge of the boundary of the AdS.
• The charge conjugation in our Gamma matrix convention is given by C = CK with C = 1. This is due to the reality of the Γ a with a = t, x, y, r, 5. Under this symmetry, Therefore the bilinear term is invariant if the interaction is invariant under the 1 ↔ 2 and the order parameter is real.
• Next, we define the chiral symmetry under which we combine the time reversal and sublattice symmetry S : 1 ↔ 2, x, y, r). (3.28) It can be realized by X = Γ 0 KS, so that Therefore quadratic forms corresponding to following 8 hermitian gamma matrices change the sign while the other half which are anti-hermitian matrix does not change sign under this symmetry operation. Then the kinetic term effectively reverse the sign while the mass term is invariant and the equation of motion, hence the spectrum, is invariant as far as the order parameter is real and the Γ I is hermitian. Notice the set of spectral symmetry of X

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is precisely complements of that of the parity symmetry. Notice also that X could be a possible symmetry of the system because the bulk mass terms were chosen as −im(ψ 1 ψ 1 −ψ 2 ψ 2 ) instead of −im(ψ 1 ψ 1 +ψ 2 ψ 2 ), which explains our choice of the opposite signs in mass terms. However, such a change of mass term is a unitaty operation and can not change the spectrum. On the other hand The parity is a symmetry regardless of such sign choices. As we will see, the spectrum of our theory follows P while its dual follows X .

Classifying the spectrum by the order parameter for 2 flavours
we classify the spectrum into scalar, vector, and tensor along the line we discussed above. In all the figure below, we should keep in mind that the k x and the vertical one represents either ω mostly except the fixed ω slice, where the vertical axis is k y .

Summary of spectral features
Here, we classify, summarize and tabulate some essential spectral features.
Spectral classification. Following the discussion below (3.19), we classify the spectrum according to the 2+1 Lorentz tensor.
• There are 4 scalars. 1, Γ 5 , Γ r , Γ 5r . The first two were described above. For the gauge invariant fields B µ , we should set B r = B 5r = 0. In fact, even for non-gauge invariant case,the last two are identical to the zero Yukawa coupling in our gamma matrix representation. For scalar interactions, the roles of source and condensation are qualitatively the same.
• There are three classes of vectors: B µ , B rµ and B 5µ . The source creates the split cones and the condensation creates just asymmetry. The first two are invariant under the parity symmetry showing zero mode related features like Fermi-arc and surface states(Ribbon band).
• There are 3 rank 2-tensor terms: Γ xy , Γ tx , Γ ty ,. The first one is parity invariant and has zero modes.
Gap vs zero modes with scalar order. Out of the 16 interaction types, only parity symmetry breaking scalar interaction with Γ = Γ 5 ) creates a gap without ambiguity. Both source and the condensation create gaps. On the other hand, parity invariant scalar with Γ = i1 has a zero mode Dirac cone in spectrum, which is much sharper than the case of the non-interacting case due to the transfer of the spectral weight to the zero mode by the interaction. The genuine physical system with full gap will be described by this coupling.
Pseudo gap. When the interaction is parity non-invariant, the spectrum has pseudo gap apart from Γ 5 which produces real gap. Seven interactions corresponding to

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same energy at all momenta except at the projection of the Weyl points onto the sBZ (Fig. 6, top left). At those two points, surface states can leak into the bulk even at E F ¼ 0 and are not well defined. If one considers other energies, the momentum region occupied by bulk states grows as shown at the bottom of Fig. 6. The presence of these bulk states allows for surface states that are impossible to realize in both strictly 2D and on the nodes, the Fermi arc is displaced by virtu velocity. The surface states all disperse in the and inherit the chiral property of the Chern states. At the same time, the bulk Ferm encloses a nonvanishing volume, and their p the sBZ is now a pair of filled disks that enc node momenta. How are the Fermi arc attached to the projection of the bulk Ferm the top right of Fig. 6, a plot of both surface ( bands projected to the sBZ is shown, and s dispersion at two energies resemble the two the bottom of Fig. 6. In a conventiona dispersion traversing a band around a close contour in momentum space returns one momentum. In contrast, in a WSM system, closed contour around an end point of the moves between the valence and conduction b analogy is the Riemann surface generated by function (Fang et al., 2016). Therefore, structure, although impossible in 2D, is allow state since the surface states can be absorbed on moving away from the Weyl nodes in en Haldane (2014) argued that the Fermi arc must be tangent to the bulk Fermi surfaces pro sBZ. This follows from the fact that the surf convert seamlessly into the bulk states as they termination points. Putting this differently, depth of the surface state wave function gro point of projection onto the bulk states the merge with the bulk states. They should inheri the bulk states, which implies they must be a tially to the bulk Fermi surface projections as s  have pseudo gaps. Therefore the pseudo gap is a typical phenomena rather than an exception for general interaction in this theory, while the true gap is a rare phenomena. For vectors B µ , B 5µ and B rµ the role of source term is to generate the split Dirac cones along k µ direction, µ = 0, 1, 2, while that of the condensation is to generate an anisotropy. This is just like the flat space cases. However, there is a very interesting phenomena when the interaction term is invariant under the parity: that is, for B 5 i , B ri i = x, y, there exist a spectral line connecting the tips of two Dirac cones at ω = 0 plane. This resembles the "Fermi-arc" in the study of Dirac or Weyl-semi metal. The fact that we found a surface mode was a surprise at first, since we thought that the arc is present only if there is a boundary of the matter. However, the paper JHEP01(2021)053 by Vishwanath et al. [38,39] clearly stated that the Fermi arc at the surface Brillouine Zone(sBZ) at the figure 6(a)up-left is a projection along z-direction of a bulk mode at E = 0 in 6(a)up-right. At each slice of energy, there is a spectral line which tangentially connects the two split cones, as described by Haldane in [31]. See three figures in 6(a)down. In (ω, k) space, they form a sliding-frame like band for 1 flavor theory, and a wedge shape band for the 2-flavor theory. This is the "surface states" which exists regardless of the presence of the edge of the physical system. The Fermi-arc in the bulk is the line where this surface band crosses the Fermi level E F . What is observed in the experiment is the projected version of this to the sBZ. The edge mode is localized at the boundary in position space but it is inside the bulk of momentum space. Now looking at the figure 6(b,c,d,e) of our 1-flavor theory spectral functions, our arc lines and cones satisfy all the conditions described above so that it is clearly the same spectrum which is presented in the figure 6(a)up-right, which describes the Weyl metal. Similarly, 6(f,g,h,i) of our 2-flavor theory describe the Dirac metal where there are two fermi arcs forming a closed curve. It is crucial to notice that our zero( or zero-mass) mode in figure 6(e,i) describe the sliding frame of not the Dirac cones. This is true for any surface mode. Namely, any zero mode describes a surface mode which is localized at the surface, if actual surface exist. It is easier to think if one consider the TI, where the surface mode exists as a bulk zero mode connecting the upper and lower band which are separated by a gap.

Fermi-arc.
To make the long story short, the so called surface mode is part of the bulk modes in momentum space, therefore it is natural to discover 'the surface spectrum' from the bulk spectrum.
Finally we want to remark that although there is no Weyl fermion in 2+1 dimension, one can still define the positive and negative monopole charges of the Berry connection. Almost all construction for the Weyl fermion in momentum space continue to exist in 2+1. They include the surface states are the same apart from their dimensionality. That is why the fermi-arc, which is associated with Wey fermion appears in our AdS4 calculation. In fact, the spectrum of the boundary fermion for the AdS4 and that of AdS5 shares the same features. The so called surface states exist as bulk zero modes in both AdS5 as well as AdS4. 1 Flat band. B xy interaction introduces a flat band which is a disk like isolated band at the fermi level ω = 0. If chemical potential is applied, the disk bend like a bowl and the fermi level shifts.

Zero mode and parity symmetry.
In the presence of the background field B I with coupling B Iψ Γ I ψ, the spectrum shows the zero modes if the quadratic form is parity invariant.

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Duality. If we change the boundary term to S bdry = 1 2 bdry d 3 x i(ψ 1 ψ 1 −ψ 2 ψ 2 ), then the spectrum of dual pairs are exchanged. By the dual pair, we mean one of following set of pairs: with indices running t, x, y, r. We found that, in this case, the presence of the zero mode are protected by the chiral symmetry X we defined earlier.
Order without rotational symmetry breaking. The presence of the non-vanishing order parameter field means the breaking of the some rotational or Lorentz symmetry from the bulk point of view. However, from the boundary point of view, some order parametrs involving r-index, like B rt does not have obvious symmetry breaking interpretation and therefore they can be interpreted as 'orders without symmetry breaking'.
The table 1 summarizes all the features we found. We attributed the presence of the zero modes to the protection of the parity invariance. The zero mode is of course the key for the surface states. The presence of the zero mode results in the bright crossing of the Dirac cone with the Fermi-level. This means that the zero modes create sharp Fermi-surface, which was orginally fuzzy due to the strong interaction at the boundary. This is one of the most interesting observation made in this paper. That is, the parity invariant interaction can make a strongly interacting system be fermi-liquid like. Earlier in the figure 3 we gave comparison of the spectrum with coupling B xyψ Γ xy ψ in the presence of chemical potential and that of the heavy fermion in Kondo lattice. More explicit comparison with the experimental data is left as a future project.

Spectral Function (SF) with scalar interaction
4.2.1 Parity symmetry breaking case: L int = Φ 5 (ψ 1 Γ 5 ψ 2 +ψ 2 Γ 5 ψ 1 ) We begin with the simplest case where the order parameter field is scalar field. We choose m 2 Φ = −2 in AdS 4 for simplicity. Then [40,41] Φ 5 = M 05 z + M 5 z 2 , (4.1) in the probe limit. We consider source only and condensation only cases separately. For complex Φ with nonzero M it describe the cooper pair condensation while for real case it may describe a chiral condensation or a random spin singlet condensation where lattice spins pair up to form singlets, the dimers, in random direction so that there is no net magnetic ordering. In fact, in lattice models with antiferromagnetic coupling, the ground state is anti-ferromagneticaly ordered if frustrations and randomness are small enough.

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Order p./ Figure# Gap zero mode spectral feature possible dual system In the table of "Gap", denotes gap at the fermi-level, represents gap off the fermi level and × is gapless. SC=superconductivity, RS=Random Singlet. A(k x , ω) means we consider the spectral function A as the function of k x and ω. Under k x ↔ k y those with one spatial index are assymetric. All others are symmetric. NA=not available.
On the other hand, it has random singlet (RS) state [42][43][44][45] if there is a randomness, a distribution of next-nearest site couplings. Whether a RS like state has a gap or not depends on the details of the lattice symmetry as well as the size of the randomness [46][47][48][49][50][51][52]. Our philosophy is to bypass all such details and characterize the system only by a few order parameter, assuming this is possible at least near the critical points. From our calculation, a RS state with gap is described by a scalar order. Notice that the dipole type interaction F rtψ γ rt ψ, which was used to study the Mott physics [29,30,53], does not generate a true gap, because its density of state does not really has a gap although its spectral function has gap like features in small momentum region. This is because the spectral function shows a band that approaches to the fermi level for large momentum.  Spectrum in potential picture. More characteristic feature is the appearance of Kaluza-Klein (KK) modes in the figure 7, which is due to the effective z 2 Schrödinger potential for large z generated by the condensation part: the effective Schrödinger potential V ∼ Φ 2 5 /z 2 which goes like ∼ M 2 5 z 2 for large z [41]. Comparing the effect of the scalar condensation with that of the scalar source, the gap is generated by condensation is smaller than that generated by the source, as shown in 7 (a,b).
In the presence of chemical potential or temperature, the effect of z 2 term is suppressed because both T and µ increase the horizon size r 0 and the region 'inside' the black hole, z > z 0 = 1/r 0 is cut out. Then the rising potential z 2 also disappear, and the potential near the horizon collapses into −∞ because near the horizon, Furthermore the solution should satisfy the infalling boundary condition, so that instead of the infinitely many clean quantized eigenvalues (KK modes), only finitely many imaginary eigenvalues due to the tunneling to the horizon appears. See figure 8. This explains the fuzziness and disappearance of KK modes in 7(d) in the presence of the chemical potential. For the vector and tensor cases, there can be a pole between the horizon and the boundary. We emphasize that this case is not related to the rotational symmetry breaking. The Z 2 symmetry is not encoded in this model either. So one natural candidate is the spin JHEP01(2021)053 liquid with a gap [54,55]. This case may be also useful to describe the coupling between the localized (lattice) spin net work and the itinerant electron, namely the Kondo physics.

Parity preserving scalar interaction: L int = iΦ(ψ 1 ψ 2 +ψ 2 ψ 1 )
For the consistency with scalar model, we choose the m 2 Φ = −2 so that we still have Φ = M 0 z + M z 2 . The spectrum is gapped for both source and condensation. For the latter case zero chemical potential case shows sharp KK modes. Compared with the scalar case, spectrum is sharper. See figure 7 (c,d) for the spectral functions with pseudo scalar source and condensation respectively. The system is similar to the scalar but with Parity symmetry broken. The most famous case is the pion condensation in nuclear physics.

Vectors
From the 2+1 dimension boundary point of view there are three classes of vectors: B µ , B rµ and B 5µ , respectively. In each class, the source shifts the two degenerate Dirac cones, one to negative and the other to positive k µ directions. The last two classes are invariant under the parity showing zero mode related feature like Fermi-arc and surface states(Ribbon band). See figures 10, and 9.

Polar vector:
B µ is the extension of the source field that couples to boundary fermion currentχΓ µ χ. We can set the mass of vector and axial-vector order field to zero. Then, Notice that there are asymmetry between x and y direction. Therefore when B x → B y , it has to be followed by k x → k y at the same time.
Source. SF for B x coupling with source only is just superposition of two shifted SF's of zero coupling cases along the k x . Different flavors shift in opposite directions.
Condensation. Anisotropy instead of cone splitting is created. See figure and 9 and 10.

Pseudo vector:
Pseudo vectors mostly follows the pattern of polar vectors. One important point to emphasize is the line shaped zero mode at ω = 0 plane. At higher slice ω = 2, there are also lines connecting two cicles representing the two shifted Dirac cones. See figure 9(b)(c). So the total 3 dimensional figure is like figure 6 with double of the surface modes, which we call Ribbon bands.

Antisymmetric 2-tensor
6 anti-symmetric rank 2 tensors can be decomposed into three B µr , µ = 0, 1, 2 which should be reclassified as a boundary vector and the rests B tx , B ty , B xy . The former was already described above.
Notice the manifest zero mode Disk in B tr from the figure 11. There are rotational symmetry in B tr , B xy , but not in B xr . Spectrum of B ty and B xr are ambiguous without the views in k x -k y at various ω slices, which we provide in figure 12. Both of them have split cones and the zero modes. Both have Ribbon bands connecting the two cones.

Conclusion
We classified the Yukawa type interactions according to its Lorentz symmetry of the boundary theory and calculate their spectral functions. We met many interesting features that appear in the strongly correlated system: gap, pseudogap, flat band and Fermi-arc stuctures appear. Out of the 16 interaction types, only parity breaking scalar interaction (Γ = 1) creates a gap without ambiguity. Both source and the condensation create gaps. However, the parity conserving scalar (Γ = Γ 5 ) has a zero mode Dirac cone in spectrum, which is much sharper than the case of the non-interacting case due to the transfer of the spectral weight to the zero mode by the interaction. The genuine physical system with full gap will be described by this coupling . For vector B µ , B 5µ , B rµ show feature of split cones if the order parameter field have source part. The last two are invariant under the parity , there are the Fermi-arc. Another interesting feature is the flat disk band of B xy , which might be useful to describe the twisted bi-layered graphene. If chemical potential is applied, the disk bend like a bowl and the fermi level shifts, resembling the band of the Kondo lattice.

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There are three classes of vectors, B µ , B rµ and B 5µ , respectively. The source creates the splited cones and the condensation creates just asymmetry. The first two are invariant under the reflection showing zero mode related features like Fermi-arc and surface states(Ribbon band). There are 3 tensor types: Γ xy , Γ tx , Γ ty . These respect the parity and has zero modes.
Since the spectral data is the fingerprint of a matter, we would be able to determine the order of a strongly interacting system by comparing the calculated spectral function in the presence of these orders with the experimental data. We expect that our result will give an insight for magnetic orders of strongly interacting materials.
Final remark is that the quartic or higher terms do not contribute to the spectral functions.Therefore it is enough to discuss the effect of the Yukawa coupling to calculate the leading order effects of order parameter fields on the fermion spectrum. In fact, the Yukawa coupling terms are most relevant ones in low energy. Here we consider the gravity background of asymptotically AdS4 with Lorentz invariance.
In the future, we will study the AdS 5 version of this paper, which related to Weyl Semi-metal instead of Dirac semi-metal. It will be also interesting to extend our work to higher quantum critical points. There are ten classes of different topological insulator/superconductors depending on the discrete symmetries. It will be also interesting to realize all of such 10 folds way in terms of the explicit laglangian. One more possibility is to study the effect of combinations of the Yukawa interactions to create different types of spectral features. Studies in these directions are under progress.

B The role of the chemical potential
Let's begin by looking at the simplest scalar case. As one can see from figure 16, the main effect of the chemical potential is three folds: the first is to shift the Fermi-level and the second is to make the spectrum fuzzier. The third one is to introduce the asymmetry between the positive and negative frequency regions.      Now we consider the one flavor B xy and B tr interactions which are in a sense dual to each other. To consider the case of spontaneously broken symmetry, we set B (−1) = 0. Most important feature here is the appearance of the flat band and gated gap, which means that the gap is reachable by gating. For µ = 0, exact flat band is generated at w = 0. If we turn on the chemical potential, the central flat band is bent to give a shallow-bowl shaped band. With increasing A fuzzy flat band is created at higher energy. See figure 17(d). Notice the similarity of the spectrum with that of the heavy fermion spectrum in Kondo lattice, where an impurity spin interact with the itinerant electron with anti-ferromagnetic coupling. Indeed, our interaction term B xyψ Γ xy ψ is the form of the Kondo coupling S imp ·ψ σψ in the boundary if we interpret the order parameter field B xy as the impurity spin along z axis, because Γ xy is the corresponding spin generator matrix.
Both source and condensation seem to develop a gap in the negative energy region while Fermi level passes through the conduction band. Since gap usually means gap containing the Fermi-level to give an insulator, we need a new name. We call such gap as gated gap because we expect that we can get an insulator by gating the system, that is by applying external electric field. Therefore we studied the evolution of B rt in µ. Figure 18 shows, however, that gap is generated only in a window of negative µ around µ = −1.

B.1 B rt vs F rt
We want to make a comment on comparing this case with the work of Phillips et al. [29,30] where the dipole interaction F rtψ Γ rt ψ was considered in search of the gap. The difference with ours is that our B rt is an order parameter independent of the basic vector field A µ while F rt = ∂ r A t . Namely, rt is related to the chemical potential by B (0) rt = µ/r 0 , our model with non-zero chemical potential is exactly the same as the dipole interaction model. In fact, figure 19 shows that the overall features of two cases are similar. However, for zero chemical potential, our model still have the non-zero order parameter B    in the negative energy region while Fermi level passes through the conduction band. Since gap usually means gap containing the Fermi-level to give an insulator, we need a new name. We call such gap as gated gap because we expect that we can get an insulator by gating the system, that is by applying external electric field. Therefore we studied the evolution of B rt in µ. Figure 19 shows, however, that gap is generated only in a window of negative µ around µ = −1.
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