In this work we consider the offdiagonal coupling between two supersymmetric SYK models which preserves both supersymmetry and solvability. We found that the interaction terms of the N2 supersymmetric SYK have a holographic interpretation as a possible supersymmetric traversable wormhole. First we introduce the coupling in the trivial Homologic N1 SYK model as a simplified example. Similar couplings can be implied in N2 chiral SYK model with BPS states. We propose a special form of N4 SYK by introducing supermultiplets and which also naturally include the coupling terms. The holographic picture of N4 SYK does not have an eternal solution in the low energy limit. And the effective actions are studied in both thermal limit and low energy limit. We also investigate the SYKlike thermal field double states of the supersymmetric SYK and the transmission amplitude between singleside N2 models in Lorentz time. Additionally the multiside N24 OTOCs are also studied.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']
Offdiagonal coupling of supersymmetric SYK model
{'Off-diagonal coupling of supersymmetric SYK model': 'Chenhao Zhang a and Wenhe Cai ∗ a  \na Department of Physics, Shanghai University, Shanghai, 200444, China', 'Abstract': 'In this work, we consider the off-diagonal coupling between two supersymmetric SYK models, which preserves both supersymmetry and solvability. We found that the interaction terms of the N=2 supersymmetric SYK have a holographic interpretation as a possible supersymmetric traversable wormhole. First we introduce the coupling in the trivial Homologic N=1 SYK model as a simplified example. Similar couplings can be implied in N=2 chiral SYK model with BPS states. We propose a special form of N=4 SYK by introducing supermultiplets, and which also naturally include the coupling terms. The holographic picture of N=4 SYK does not have an eternal solution in the low energy limit. And the effective actions are studied in both thermal limit and low energy limit. We also investigate the SYK-like thermal field double states of the supersymmetric SYK and the transmission amplitude between single-side N=2 models in Lorentz time. Additionally, the multi-side N=2,4 OTOCs are also studied.', '1 Introduction': 'Recently, supersymmetric holographic wormholes have gained attention. However, these holographic wormholes still lack a simplified and solvable model. In previous works[1, 2, 3, 4], SYK model is introduced as a Gaussian random Majorana fermionic model with good holographic properties, which is dual to a NAdS spacetime. The emergent IR conformal symmetry is also demonstrated in the description of NAdS2/NCFT1 duality[5, 6]. Additionally, the supersymmetric SYK model originates from this research on the cold horizon[7, 8]. Supersymmetric SYK(SSYK) models are supersymmetric extensions of N=0 SYK model with dynamical bosons[9, 10, 11]. They also exhibit an emergent supersymmetric conformal symmetry, which is dual to NAdS spacetime with additional N=1 and N=2 supersymmetry and Grassmann variables[12, 13, 14]. These models have also been extensively studied[15, 16, 17, 18] and are characterized by valuable BPS states[19]. For instance, in double scaling limit[20, 21, 22], both non-supersymmetric and supersymmetric wormholes featuring statistical chords have been introduced[23, 24, 25]. However, the intricate structures of these entangled systems remain unexamined within the context of an adequate holographic model.  \nIn this paper, we introduce a first-order interaction into the supersymmetric SYK model, which maintains both supersymmetry and solvability. This model is similar to the approach used in the N=0 coupled SYK model(MQ)[26, 27]. And we explore its low energy effective action[28, 29], thermal phase structure and causality. Previous work has proposed a simple supersymmetrization of JT gravity, exhibiting promising holographic properties[30, 31]. Notably, the supersymmetric JT gravity closely resembles the N=0 model. Furthermore, we extend this to supergravity with N=4 supersymmetry, which also exhibits duality due to the unique form of our N=4 SYK model.  \nThe main goal of this paper is to examine the supersymmetric interaction term in both the N=2 and a streamlined N=4 SYK model. These models are formed using a special form of chiral supermultiplet combining two N=2 SYK models and have a natural representation of spinor. Currently, supersymmetric SYK models are under extensive investigation, focusing on aspects such as thermal effective actions, phase structures, and out-of-time correlators. However, they do not involve the detailed study of thermal phase interaction parts as wormholes. In this context, we aim to delve deeper into the causal properties involving excitation transmission and build upon prior research[32, 33, 34, 35, 36, 37], further enrich the understanding of supersymmetric extension.  \nThe second goal is to study the N=4 model. Unlike higher-dimensional N=1 and N=2 models[38, 39], various N=4 SYK models have been proposed[40, 41, 42]. The N=4 model includes chiral, vector, and tensor supermultiplets. In this paper, we utilize the 1D supermultiplet. The superfields here differ significantly from the bosonic case. Because of its unique structure, the holographic properties of this N=4 SYK model can be inferred from the N=2 model in superspace, although direct study presents challenges. We introduce interaction terms as in the  \nN=1 and N=2 models and examine black hole-wormhole structures.  \nThis paper is organized as follows. In Section II, we briefly review the N=1 and N=2 supersymmetric SYK models, and propose a simple form of N=4 SYK model. We then introduce the first-order interaction into these models. In Section III, we begin with the holographic duality of NAdS spacetime and the SYK model, and further examine the supersymmetric NAdS and the supersymmetric SYK model with interaction terms. The holographic properties of N=4 model are also studied. In Section IV, we propose the thermal field double states in supersymmetric SYK model. Section V verifies the causality of the N=2 wormhole through excitation transmission and out-of-time correlation (OTOC). Section VI is the conclusion and discussion.', '2.1 A brief review of N=1 and N=2 SYK': "The N=1 SYK model has been the most widely studied attempt at a supersymmetric SYK model. Subsequent research has revealed that it possesses fewer physical interactions. Nevertheless, the N=1 SYK model still remains a valuable case study for exploring supersymmetry, despite its limited significance in supersymmetric physics. Its thermal properties exhibit similarities to those of the N=2 model. Furthermore, N=1 SYK theory can be formulated using independent supercharges, analogous to the N=0 model  \nQ = i q -1 2 ∑ i 1 i 2 ...i q C i 1 i 2 ...i q ψ i 1 ψ i 2 ...ψ i q . (1)  \nψ i s represent Majorana fermions on sites ranging from 1 ...N . In the initial definition, the interaction fermions q are assigned a value of 3. The corresponding coupling C ijk is an N × N × N Gaussian random antisymmetric tensor, specified by a constant J with units of energy. The random average of the coupling C exhibits the same form as in the N=0 SYK  \nC i 1 i 2 ...i q = 0 , C 2 i 1 i 2 ...i q = 2 J N 2 . (2)  \nThe super charge of N=1 model closely resembles that of the N=0 SYK. N=1 Hamiltonian is the square of supercharge  \nH = Q 2 = E 0 + ∑ 1 ⩽ i<j<k ⩽ N J ijkl ψ i ψ j ψ k ψ l . (3)  \nIt could be considered as a special form of N=0 SYK, and  \nE 0 = ∑ 1 ⩽ i<j<k ⩽ N C 2 ijk ,  \nJ ijkl = -1 8 ∑ a C aij C kla . (4)  \nHere, E 0 is a disorder averaged constant, while J ijkl constitutes a random matrix but not Gaussian independent. Supercharge is constructed from Majorana fermions and should adhere to fermionic anticommutation relations. For simplicity, an auxiliary bosonic field can be introduced  \n{ Q,ψ i } = Qψ i = i ∑ 1 ⩽ j<k ⩽ N C ijk ψ j ψ k ≡ b i Qb i = Hψ i = ∂ τ ψ i . (5)  \nThe Lagrangian consists of original fermionic and auxiliary bosonic field operators, as well as the supercharge relations  \nL = ∑ i   1 2 ψ i ∂ τ ψ i -1 2 b i b i + i ∑ 1 ⩽ j<k ⩽ N C ijk b i ψ j ψ k   . (6)  \nFor simplicity, we can define a superfield within the corresponding superspace. This allows us to constrain the bosons and fermions, and to describe the supersymmetric theory without sacrificing generality.  \nΨ ( τ, θ ) = ψ ( τ ) + θb ( τ ) , (7)  \nTherefore, we introduce the super covariant derivative  \nD θ ≡ ∂ θ + θ∂ τ ,  \nD 2 θ = ∂ τ . (8)  \nWe can define a superfield bilinear correlation function by combining the correlation functions of f-f, f-b, b-f and b-b components  \nG ( τ, θ ; τ ' , θ ' ) = ⟨ Ψ ( τ, θ ) Ψ ( τ ' , θ ' ) ⟩ = 〈 ( ψ ( τ ) + θb ( τ )) ( ψ ( τ ' ) + θ ' b ( τ ' ))〉 (9)  \n≡ G ψψ ( τ, τ ' ) + √ 2 θG bψ ( τ, τ ' ) -√ 2 θ ' G ψb ( τ, τ ' ) +2 θθ ' G bb ( τ, τ ' ) . (10)  \nThen, we return to the original SSYK model and rewrite the effective action into superspace  \nS EFF = ∫ dθdτ ( -1 2 Ψ i D θ Ψ i ) + J 3 N 2 ∫ dθ 1 dτ 1 dθ 2 dτ 2 ( Ψ i Ψ i ) 3 . (11)  \nNote that this action is diagonalized into the N=0 SYK form using superfields and covariant derivatives. Additionally, the saddle point provides describing equations of motion, incorporating disorder averaging  \nD θ G ( τ, θ ; τ '' , θ '' ) + ∫ dτ ' dθ ' G ( τ ' , θ ' ; τ '' , θ '' ) ( J G ( τ ' , θ ' ; τ '' , θ '' ) 2 ) = ( θ -θ '' ) δ ( τ -τ '' ) . (12)  \nN=2 supersymmetric SYK can be generated from the symmetry breaking of N=1 theory. We reorganize the square of supercharges into a pair of identical, conjugated supercharges with SU(2)  \nQ = i ∑ 1 ⩽ i<j<k ⩽ N C ijk ψ i ψ j ψ k , ¯ Q = i ∑ 1 ⩽ i<j<k ⩽ N ¯ C ijk ¯ ψ i ¯ ψ j ¯ ψ k . (13)  \nC and ¯ C represent an independent Gaussian random coefficient and the fields are replaced by a pair of conjugated complex fields. Notice that fermions here only anticommute with fields in conjugated components, where the N=1 supersymmetry breaks by  \n{ ψ i , ¯ ψ j } = δ i j , { ψ i , ψ j } = 0 , { ¯ ψ i , ¯ ψ j } = 0 . (14)  \nThe N=2 Hamiltonian is determined by the anticommutator of the conjugate supercharges  \nQ 2 = 0 , ¯ Q 2 = 0 , H = { Q, ¯ Q } = | C | 2 + ∑ ijkl J kl ij ψ i ψ j ¯ ψ k ¯ ψ l . (15)  \nWe can also average the random Gaussian coefficient C using an energy-dependent parameter J  \nC ijk ¯ C ijk = 2 J N 2 , (16)  \nIt is a random parameter, independent of Gaussian distribution. Analogous to the N=1 model, auxiliary bosonic fields are introduced for both chiral fermions through the commutators of conjugate supercharges  \n{ Q,ψ i } = 0 , { ¯ Q, ¯ ψ i } = 0 , b i ≡ i { ¯ Q,ψ i } = ∑ 1 ⩽ j<k ⩽ N ¯ C ijk ¯ ψ j ¯ ψ k , ¯ b i ≡ i { Q, ¯ ψ i } = ∑ 1 ⩽ j<k ⩽ N C ijk ψ j ψ k . (17)  \nThe covariant derivative operators are expressed in terms of conjugate supersymmetric components.  \nD ≡ ∂ θ + ¯ θ∂ τ ,  \n¯ D ≡ ∂ ¯ θ + θ∂ τ . (18)  \nIn N=2 algebra, various approaches are permitted for structuring the superfield. Additionally, it is convenient to opt for a method known as the chiral superfield, which satisfies  \n¯ DΨ i = 0 , D ¯ Ψ i = 0 . (19)  \nAnd we obtain the effective actions  \nS eff = 1 2 ∫ dθd ¯ θdτ ¯ Ψ i Ψ i + ∫ dθd ¯ θdτ ¯ Ψ i 1 ¯ Ψ i 2 ¯ Ψ i 3 〈 ¯ C i 1 i 2 i 3 C j 1 j 2 j 3 〉 Ψ j 1 Ψ j 2 Ψ j 3  \n, ¯ S eff = 1 2 ∫ d ¯ θdθdτΨ i ¯ Ψ i + ∫ d ¯ θdθdτΨ j 1 Ψ j 2 Ψ j 3 〈 C j 1 j 2 j 3 ¯ C i 1 i 2 i 3 〉 ¯ Ψ i 1 ¯ Ψ i 2 ¯ Ψ i 3 . (20)  \nThen, we can define creation and annihilation operators using covariant derivatives, and subsequently construct the bi-linear Fock space from the superfield  \nΨ i ( τ, θ, ¯ θ ) = ψ i ( τ + θ ¯ θ ) + θb i , ¯ Ψ i ( τ, θ, ¯ θ ) = ¯ ψ i ( τ + ¯ θθ ) + ¯ θ ¯ b i . (21)  \nNotice that the superfields adhere to anticommute relations, and they possess conjugated chiral properties. Additionally, we can define the bi-chiral correlation function with superfields  \nG ( τ, θ, ¯ θ ; τ ' , θ ' , ¯ θ ' ) = 〈 ¯ Ψ ( τ, θ, ¯ θ ) Ψ ( τ ' , θ ' , ¯ θ ' )〉 ≡ G ψψ ( τ -θ ¯ θ, τ ' + θ ' ¯ θ ' ) + √ 2 ¯ θG bψ ( τ, τ ' + θ ' ¯ θ ' ) -√ 2 θ ' G ψb ( τ -θ ¯ θ, τ ' ) +2 ¯ θθ ' G bb ( τ, τ ' ) . ¯ G ( τ, θ, ¯ θ ; τ ' , θ ' , ¯ θ ' ) = 〈 Ψ ( τ, θ, ¯ θ ) ¯ Ψ ( τ ' , θ ' , ¯ θ ' )〉 ≡ ¯ G ψψ ( τ + θ ¯ θ, τ ' -θ ' ¯ θ ' ) + √ 2 θ ¯ G bψ ( τ, τ ' -θ ' ¯ θ ' ) -√ 2 ¯ θ ' ¯ G ψb ( τ + θ ¯ θ, τ ' ) +2 θ ¯ θ ¯ G bb ( τ, τ ) . (22)  \nWe can also formulate the coupled Schwinger-Dyson equation of chiral-anti chiral, and describe the N=2 SYK system using the conjugate components  \nD θ G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) + ∫ dτ ' dθ ' d ¯ θ ' G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) ( J G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) q ) = ( ¯ θ -¯ θ '' ) δ ( τ -τ '' -θ ¯ θ + θ '' ¯ θ '' ) , (23)  \nD ¯ θ G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) + ∫ dτ ' dθ ' d ¯ θ ' ¯ G ( τ, θ, ¯ θ ; τ ' , θ ' , ¯ θ ' ) ( J ¯ G ( τ ' , θ ' , ¯ θ ' ; τ '' , θ '' , ¯ θ '' ) q ) = ( θ -θ '' ) δ ( τ -τ '' -θ ¯ θ + θ '' ¯ θ '' ) . (24)  \nIn order to understand the supersymmetric SYK models more precisely, we can also consider the supercharges with a chord diagram. As shown in Figure 1(a), we use the blue line on the left-hand side (LHS) to represent a single SYK system with coupling C ijk (which is known as  \nFigure 1: Interpretation of the chord diagram as the supersymmetric SYK model (a) N=1 SYK model (b)N=2 SYK model with chirals  \n<!-- image -->  \na supercharge). The black line on the right-hand side (RHS) represents the same SYK system that is to be interacted with. We suppose the interaction, represented by chords that start on the LHS and end on the RHS. The pink area includes the interactions between a single fermion ψ i on the LHS and the entire SYK system on the RHS, as well as the classical SYK coupling C ijk inside the RHS. We define this pink region as the auxiliary boson b i . It is easy to prove that the dimension of fermions is equal to the dimension of those auxiliary bosons.As depicted in Figure 1(b), the N=2 supersymmetry forbids the interaction inside a single chiral. When we define the blue line as the chiral supercharge with ψ i , the red line is defined as anti-chiral with ¯ ψ i . The green line will represent the same anti-chiral supercharge that is to be interacted with, while the black line represents the anti-chiral supercharge. We also define the pink region as the chiral boson b i and the yellow region as the chiral boson ¯ b i .", 'N=4 Bilinear SYK': "The N=4 SYK model was derived from considerations of a cold horizon. The operator Φ represents a chiral multiplet that exhibits both bosonic and fermionic dependencies (which is different from the chiral supermultiplet in [40, 41]), and it is characterized by Gaussian randomness.  \nL = ∫ d 2 θΩ αβγ Φ α Φ β Φ γ + h.c, (25)  \nthe coefficients Ω αβγ as a SYK-like model are also Gaussian random.  \n⟨ Ω αβγ ⟩ = 0 , 〈 Ω 2 αβγ 〉 = J. (26)  \nNumerous models exhibit a Hamiltonian form. We opt for the bilinear scalar supermultiplet approach. Here we define  \nL = ∑ i,α [ ∂ τ ¯ ϕ i ∂ τ ϕ i + ¯ ψ α i ∂ τ ψ i α + ¯ F i F i ] +3 C ijk ( F i ϕ j ϕ k + ϵ αβ ψ i α ψ j β ϕ k ) + h.c. (27)  \nParameter α and β correspond to fermionic indices 1 and 2. And ϵ 11 = ϵ 22 = 0, and ϵ 12 = ϵ 21 = 1. We commence our construction using fermions within the framework of N=2 supersymmetry  \n{ ψ i α , ¯ ψ β j } = δ β α δ i j . (28)  \nNow we consider two identical N=2 SYK models in Fig. (1) b, which is remarked by index 1 and index 2. Since we have introduced an additional dimension as N=2 supercharges with SU(2), an additional U(1) symmetry can emerge between index 1 and index 2  \nQ α = i ∑ 1 ⩽ i<j<k ⩽ N C βγ ijk ψ ' i α ψ j ' β ψ ' k γ , ¯ Q β = i ∑ 1 ⩽ i<j<k ⩽ N ¯ C ijk αγ ¯ ψ ' α i ¯ ψ ' β j ¯ ψ ' γ k . (29)  \nSupercharges are essentially Gaussian random, and the coefficient C also involves multiple indices α , β and γ , which vanish with the random coefficient. However, these indices can disappear through transformations, and we will see that it has less physical meaning and would be vanished in integrals. For simplicity, we choose α = β = γ in single supercharge and making the supercharges identical. Thus the indices α and β associated with supercharges at positions 1 and 2 exhibit rotational symmetry. And the auxiliary boson F is pertinent to higher-order cross terms involving supercharges with identical indices.  \nb i = { Q α , ¯ ψ α i } , ¯ b i = { ¯ Q α , ψ i α } . (30)  \nWe can also generate the bosonic operator b , which has been involved in both N=1 and N=2 supersymmetry.  \nH = ψ α ψ β { ¯ Q α , ψ α } 2 ψ α ψ β + ψ α ψ β { ¯ Q α , ψ α } ψ α { ¯ Q α , ψ α } ψ β + h.c = { ¯ Q α , Q α } { ¯ Q β , Q β } + T { ¯ Q α , Q α } { ¯ Q β , Q β } + h.c = { ¯ Q α , Q α } { ¯ Q β , Q β } . (31)  \nHere we use the operator T to exchange of the sort ordering of chiral fermionic pair, which vanishes when we ignore the α -β coupling in supercharges. To establish the N=4 SYK theory, the superfields are defined by  \nΦ i = ϕ i + θ α ψ i α + θ 2 F i = ( ψ ' i α + θ α b i ) ( ψ ' i β + θ β b i ) = ψ ' i α ψ ' i β + θ α b i ψ ' i β -θ β b i ψ ' i α + θ α θ β b i b i  \n¯ Φ i = ϕ i + ¯ θ α ¯ ψ α i + ¯ θ 2 ¯ F i = ( ¯ ψ ' α i + ¯ θ α ¯ b i ) ( ¯ ψ ' β i + ¯ θ β ¯ b i ) = ¯ ψ ' α i ¯ ψ ' β i + ¯ θ α ¯ b i ¯ ψ ' β i -¯ θ β ¯ b ¯ ψ ' α i + ¯ θ α ¯ θ β ¯ b i ¯ b i , (32)  \nHere we have defined the bosonic operators ϕ and F , as well as the fermionic operators ψ α and ψ β , all of which exhibit multiple chiral aspects within the Lagrangian  \nϕ  \nϕ  \ni = ϵ αβ ψ ' i α ψ ' i β , ψ i α = ϵ β α b i ψ ' i β , ψ i β = ϵ α β b i ψ ' i α , F i = b i b i , ¯ i = ϵ αβ ¯ ψ ' α i ¯ ψ ' β i , ¯ ψ α i = ϵ α β ¯ b i ¯ ψ ' β i , ¯ ψ β i = ϵ β α ¯ b i ¯ ψ ' α i , ¯ F i = ¯ b i ¯ b i . (33)  \nHere we use ψ i α and ¯ ψ α i to represent fermions in two identical N=2 SYK models. We can also decouple the N=4 theory back to N=2. Given that the superfield theory can also be incorporated into the N=4 theory, the covariant derivative is defined as  \nD α = ∂ ∂θ α + ¯ θ α ∂ ∂τ , ¯ D β = ∂ ∂ ¯ θ β + θ β ∂ ∂τ , D 2 θ ≡ 1 4 ϵ αβ D α D β = D 1 D 2 , D 2 ¯ θ ≡ 1 ϵ αβ ¯ D α ¯ D β = ¯ D 1 ¯ D 2 . (34)  \n4  \nSince we have introduced the auxiliary parameters and superfields, the action of the N=4 SYK model can be explicitly expressed as  \nS = ∫ dτdθ 1 dθ 2 d ¯ θ 1 d ¯ θ 2 ( ¯ Φ i Φ i ) + ∫ dτdθ 1 dθ 2 C ijk Φ i Φ j Φ k + ∫ dτd ¯ θ 1 d ¯ θ 2 ¯ C ijk ¯ Φ i ¯ Φ j ¯ Φ k = ∫ dτ [ ˙ ¯ ϕ i ˙ ϕ i + ¯ ψ α i ˙ ψ i α + ¯ F i F i +3 C ijk ( F i ϕ j ϕ k + ϵ αβ ψ i α ψ j β ϕ k ) + h.c ] . (35)  \nThis action can also be constrained by the equation of motion  \nD  \n2 θ G ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) + ∫ dτ ' d 2 θ ' G AB ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) ( J G AB ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) 2 ) = ( ¯ θ -¯ θ '' ) ( ¯ θ -¯ θ '' ) δ ( τ -τ '' ) .  \n(36)  \nD  \n¯ 2 θ ¯ G ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) + ∫ dτ ' d 2 ¯ θ ' ¯ G AB ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) ( J ¯ G AB ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ; τ '' , θ '' 1 , θ '' 2 , ¯ θ '' 1 , ¯ θ '' 2 ) 2 ) = ( θ -θ '' ) ( θ -θ '' ) δ ( τ -τ '' ) .  \n(37)  \nThe N=4 correlation function is represented by  \nG ( τ, θ 1 , ¯ θ 1 , θ 2 , ¯ θ 2 ; τ ' , θ ' , ¯ θ ' , θ ' 2 , ¯ θ ' 2 ) = 〈 ¯ Φ ( τ, θ 1 , ¯ θ 1 , θ 2 , ¯ θ 2 ; τ ' , θ ' , ¯ θ ' , θ ' 2 , ¯ θ ' 2 ) Φ ( τ, θ 1 , ¯ θ 1 , θ 2 , ¯ θ 2 ; τ ' , θ ' , ¯ θ ' , θ ' 2 , ¯ θ ' 2 )〉  \n- -√ 2 ¯ θ ' α G ψαϕ ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 + θ α ¯ θ α , τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 ) + ¯ θ 1 ¯ θ 2 θ 1 θ 2 G FF ( τ, τ ' )\n- = G ϕϕ ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 , τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 ) + √ 2 θ α G ϕψα ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 , τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 -θ ' α ¯ θ ' α )\n- +2 ¯ θ α θ ' β G ψαψβ ( τ, τ ' ) + ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 + θ α ¯ θ α , τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 -θ ' β ¯ θ ' β )\n- + θ 1 θ 2 G ϕF ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 , τ ' ) + ¯ θ 1 ¯ θ 2 G Fϕ ( τ, τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 )\n- + √ 2 ¯ θ α θ 1 θ 2 G ψαF ( τ -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 + θ α ¯ θ α , τ ' ) + √ 2 ¯ θ 1 ¯ θ 2 θ α G Fψα ( τ, τ ' + θ ' 1 ¯ θ ' 1 + θ ' 2 ¯ θ ' 2 -θ ' β ¯ θ ' β ) , (38)  \n¯  \nG  \n(  \nτ, θ  \n,  \n¯  \nθ  \n, θ  \n,  \n¯  \nθ  \n;  \nτ  \n'  \n, θ  \n'  \n,  \n¯  \nθ  \n'  \n, θ  \n'  \n2  \n1  \n1  \n2  \n2  \n,  \n¯  \nθ  \n'  \n2  \n=  \n)  \n〈  \nΦ  \n(  \nτ, θ  \n,  \n¯  \nθ  \n, θ  \n,  \n¯  \nθ  \n;  \nτ  \n'  \n, θ  \n'  \n,  \n¯  \nθ  \n'  \n, θ  \n'  \n2  \n1  \n1  \n2  \n2  \n,  \n¯  \nθ  \n'  \n2  \n¯  \n)  \nΦ  \n(  \nτ, θ  \n,  \n¯  \nθ  \n, θ  \n,  \n¯  \nθ  \n;  \nτ  \n'  \n, θ  \n'  \n,  \n¯  \nθ  \n'  \n, θ  \n'  \n2  \n,  \n- = ¯ G ϕϕ ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 , τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 ) + √ 2 ¯ θ α ¯ G ϕψα ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 , τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 + θ ' α ¯ θ ' α )  \n1  \n- -√ 2 θ ' α ¯ G ψαϕ ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 -θ α ¯ θ α , τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 ) + ¯ θ 1 ¯ θ 2 θ 1 θ 2 ¯ G FF ( τ, τ ' )\n- +2 θ α ¯ θ ' β ¯ G ψαψβ ( τ, τ ' ) + ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 -θ α ¯ θ α , τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 + θ ' β ¯ θ ' β )\n- + ¯ θ 1 ¯ θ 2 ¯ G ϕF ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 , τ ' ) + ¯ θ 1 ¯ θ 2 ¯ G Fϕ ( τ, τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 )\n- + √ 2 θ α ¯ θ 1 ¯ θ 2 ¯ G ψαF ( τ + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 -θ α ¯ θ α , τ ' ) + √ 2 θ 1 θ 2 ¯ θ α ¯ G Fψα ( τ, τ ' -θ ' 1 ¯ θ ' 1 -θ ' 2 ¯ θ ' 2 + θ ' β ¯ θ ' β ) . (39)  \nSome of these components would vanish, and we will discuss this problem later.", '2.2 Introduce the coupling': "For N=1 SYK, we can propose a similar first-order interaction term to that used in MQ theory in superspace for consistent. The coupled theory encompasses two separate N=1 SYK models, along with the interaction term  \nH int = iµ∂ θ ∑ j ( Ψ j L Ψ j R -Ψ j R Ψ j L ) . (40)  \nHere, we have introduced the super derivative to constrain the theory using a delta function in a single-sided superposition. Subsequently, we obtain the action that includes the interaction  \n1  \n2  \n2  \n¯  \nθ  \n'  \n2  \n)〉  \nterm  \nS eff N = -log Pf ( D half integer ) + log det ( D integer ( iw n ) 2 ) + ∑ A,B = L,R 1 2 ∫ dτdτ ' [ Σ ψψ,AB ( τ, τ ' ) G ψψ,AB ( τ, τ '  \n) + Σ ψb,AB ( τ, τ ' ) G ψb,AB ( τ, τ ' ) + Σ bψ,AB ( τ, τ ' ) G bψ,AB ( τ, τ ' ) + Σ bb,AB ( τ, τ ' ) G bb,AB ( τ, τ ' ) -( δ AB +(1 -δ AB ) ( -1) ( q -1) / 2 ) J ( G q -1 ψψ,AB ( τ, τ ' ) G bb,AB ( τ, τ ' ) -( q -1) 2 G q -2 ψψ,AB ( τ, τ ' ) G A ψb,AB ( τ, τ ' ) G S bψ,AB ( τ, τ ' ) -( q -1) 2 G q -2 ψψ,AB ( τ, τ ' ) G S ψb,AB ( τ, τ ' ) G A bψ,AB ( τ, τ ' ) )] . (41)  \nWe can further utilize the variation in effective action to derive the coupled equation of motion  \nD θ G AB ( τ, τ ' ) -∑ C ( -iµϵ AC ∂ θ G AB ( τ, τ ' ) -∫ dτ '' dθ '' Σ AC ( τ, τ '' ) G BC ( τ, τ '' ) ) = δ AB ( θ -θ ' ) δ ( τ -τ ' ) . (42)  \nIn the generalization of the N=2 supersymmetric model, the interaction Hamiltonian represents the correlation between chiral and anti-chiral supercharges. The interactions between single-sides should vanish. When considering the two-part interaction term  \nH int = iµ∂ θ ∑ j ( Ψ j L ¯ Ψ j R -Ψ j R ¯ Ψ j L )  \n¯ H int = iµ∂ ¯ θ ∑ ( ¯ Ψ j L Ψ j R -¯ Ψ j R Ψ j L ) .  \n, j (43)  \nThe corresponding N=2 chiral and anti-chiral actions, when combined with the interaction term, lead to the equations of motion  \nD θ G AB ( Z, Z ' ) -∑ C ( -iµϵ AC ∂ θ G AB ( Z, Z ' ) -∫ dτ '' dθ '' Σ AC ( Z, Z '' ) G BC ( Z, Z '' ) ) = δ AB δ ( Z -Z ' ) ,  \nD ¯ θ ¯ G AB ( Z, Z ' ) -∑ C ( -iµϵ AC ∂ ¯ θ ¯ G AB ( Z, Z ' ) -∫ dτ '' d ¯ θ '' ¯ Σ AC ( Z, Z '' ) ¯ G BC ( Z, Z '' ) ) = δ AB δ ( Z -Z ' ) . (44)  \nWe assume that there is symmetry between the chiral and anti-chiral Green's functions  \nG ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) = ¯ G ( τ, θ, ¯ θ ; τ '' , θ '' , ¯ θ '' ) . (45)  \nWe can write down the SD equations. Notably, the N=1 equations are similar to the N=2  \nequations in terms of components  \nΣ ψψ,AB ( τ ) = ( q -1) ( -1) ( q -1) / 2 JG q -2 ψψ,AB ( τ ) G bb,AB ( τ ) -( q -1) ( q -2) ( -1) ( q -1) / 2 JG q -3 ψψ,AB ( τ ) ,  \nG A ψb,AB ( τ ) G S bψ,AB ( τ ) -( q -1) ( q -2) ( -1) ( q -1) / 2 JG q -2 ψψ,AB ( τ ) G S ψb,AB ( τ ) G A bψ,AB ( τ )  \nΣ bb,AB ( τ ) = JG ψψ,AB ( τ )  \nΣ S bψ,AB ( τ ) = -( q -1) JG A ψb,AB ( τ ) G q -2 ψψ,AB ( τ )  \nΣ A bψ,AB ( τ ) = -( -1) ( q -1) / 2 ( q -1) JG S ψb,AB ( τ ) G q -2 ψψ,AB ( τ )  \nΣ S ψb,AB ( τ ) = ( q -1) JG A bψ,AB ( τ ) G ψψ,AB ( τ )  \n2 q , 2 , 2 , 2 ,  \n( q -1)  \nΣ A ψb,AB ( τ ) = ( -1) ( q -1) / 2 2 JG S bψ,AB ( τ ) G ψψ,AB ( τ ) . (46)  \nFactor ( -1) ( q -1) / 2 originates from the fermionic sign. In the N=2 theory, we need to adjust the ψ time variable as follows: τ → τ -θ ¯ θ and τ ' → τ ' + θ ' ¯ θ ' .  \nAdditionally, it is noteworthy that the components of Green's functions G ψb,AB and G bψ,AB are neither symmetric nor anti-symmetric. Nevertheless, these components can be regularized through the exchange of the Grassmann variables θ and ¯ θ  \nG ψb,AB θ → θ ¯ θG ψb,AB ,  \n¯ θG bψ,AB → G bψ,AB . (47)  \n2 ,  \nIt is easy to check  \n-θG ψb,AB ¯ θG bψ,AB →-¯ θθG ψb,AB G bψ,AB , (48)  \nwhere we have defined the component -θ ¯ θG ψb,AB is anti-symmetric, whereas the component G bψ,AB is symmetric. Alternatively, we can also define the component G ψb,AB as symmetric and G bψ,AB is anti-symmetric without affecting the physical and numerical results, and this is due to the symmetry between these two components.  \nSimilarly, we can apply the same process to make the component G bψ,AB anti-symmetric and make G ψb,AB to symmetric  \nG bψ,AB θ → θ ¯ θG bψ,AB ,  \n¯ θG ψb,AB → G ψb,AB .  \nAnother important observation is that these components must satisfy the periodic boundary condition due to the diagonal term, and their low energy ansatz is precisely zero. This allows us to separate them into symmetric and anti-symmetric components and apply the Matsubara method without considering additional subdivisions. The multiple combination of the components  \nFigure 2: Supersymmetric Green's function in components fixed J = 3 , µ = 0 . 3 , T = 0 . 01 , q = 5 (a) Single-side fermionic Green's function G ψψ,AA , (b) Single-side bosonic Green's function G bb,AA (c) multi-side Anti-symmetric fermion-bosonic Correlation function G A ψb,LR . (d) multiside symmetric fermion-bosonic Correlation function G S ψb,LR  \n<!-- image -->  \nθ ¯ θG ψb,AB G bψ,AB can be defined as antisymmetric, similar to the combination θ ¯ θG ψψ,AB G bb,AB . And we can still split the Green function terms into anti-symmetric parts G A ψb,AB and symmetric parts G S ψb,AB , with a similar division for the anti-symmetric G A bψ,AB and symmetric G S bψ,AB .  \nThe numerical results in Figure.(2) show that supersymmetric Green's function in components fixed J = 3 , µ = 0 . 1 , T = 0 . 01. In figure.(2)a, the fermionic Green's function behaves like N=0 SYK, but no longer symmetry of Euclidean time τ . Its Retarded side ( τ > 0) decays more faster than its advanced side ( τ < 0). This is very close to the supersymmetric SYK with fermionic chemical potential results in [18]. However, this is different from cSYK[43] and Maldacena-Qi[26]. This model characterized by supersymmetric correlation is constrained on  \nboundary, implying that only non-vanishing fermion-bosonic and boson-fermionic correlations exist. And it cannot undergo renormalization similar to that described in [18]. In Figure.(2)b, the bosonic component approximates a delta function, which is an inherent consequence of the original supersymmetric SYK model. We also plot the fermion-bosonic Correlation function in both anti-symmetric figure.(2)c and symmetric figure.(2)d components. Upon including the coupling term, these components are no longer regarded as negligible in previous studies. In Figure.(2)c, the anti-symmetric fermion-bosonic correlation function closely resembles the fermionic correlator G ψψ . Meanwhile, the symmetric fermion-bosonic correlation function in Figure (2)d decreases gradually and approaches zero as we integrate over the periodic time β = 1 /T . The model also exhibits a sudden decay around the origin, time 0, which may be influenced by the delta function present in the bosonic Green's function. One can verify that setting the components G ψψ,LR , G bb,LR , G ψb,AA equal to zero yields a stable solution. Additionally, other Green's functions, such as the boson-fermion conjugate and the left-right conjugate, can be derived from the components discussed in Figure (2).  \nAnd the solutions of Green's function in frequency space are  \nG ψψ,AB ( ω ) = Det ( A ψψ,AB ( D half integer )) Det ( D half integer )  \nG S ψb,AB ( ω ) = -Det A S ψb,AB ( D integer ) Det ( D integer ) .  \n, G bb,AB ( ω ) = -Det ( A bb,AB ( D integer )) Det ( D integer ) , G A ψb,AB ( ω ) = Det ( A A ψb,AB ( D half integer ) ) Det ( D half integer ) , G S bψ,AB ( ω ) = -Det ( A S bψ,AB ( D integer ) ) Det ( D integer ) , G A bψ,AB ( ω ) = Det ( A A bψ,AB ( D half integer ) ) Det ( D half integer ) , ( ) (49)  \nIn this work, we introduce auxiliary fermions with bosonic integral frequency and bosons with fermionic half-integral frequency, which should statistically equal zero. The matrices D half integer and D integer represent the Jacobian matrix with symmetric and anti-symmetric properties, respectively, and the coefficient -1 serves as a bosonic factor in this context. We can also absorb this factor and inverse the ς  \nD half integer =       -iw n -Σ LL,ψψ -Σ LR,ψψ -Σ A LL,ψb -iµ -Σ A LR,ψb -Σ RL,ψψ -iw n -Σ RR,ψψ iµ -Σ A RL,ψb -Σ A RR,ψb -Σ A LL,bψ iµ -Σ A LR,bψ -1 -0 -0 -iµ -Σ A RL,bψ -Σ A RR,bψ -0 -1 -0       ,  \nD integer =       -iw n -0 -0 -Σ S LL,ψb iµ -Σ S LR,ψb -0 -iw n -0 -iµ -Σ S RL,ψb -Σ S RR,ψb -Σ S LL,bψ -iµ -Σ S LR,bψ -1 -Σ LL,bb -Σ LR,bb iµ -Σ S RL,bψ -Σ S RR,bψ -Σ RL,bb -1 -Σ RR,bb       .  \nZero elements in a matrix represent the possible values for Σ . We assume that the bosonic Σ s are zero, indicating that they also exhibit zero values at half-integer frequencies. In contrast, the fermionic Σ exhibit zero values at integer frequencies.  \nAs demonstrated in the equations, these algebraic cofactors of the matrix originate from the Jacobi method applied to Schwinger-Dyson equations.  \nThe energy can be evaluated in thermal limit  \nE N = ∫ d ¯ θ [ 2 q -1 D G LL + iµ ( 1 -2 q -1 ) G LR ] τ → 0 + + h.c. (50)  \nWe can precisely calculate these values within the frequency branch  \nE N = 2 q -1   T ∑ half integral ( Σ LL,ψψ ( iw n ) G LL,ψψ ( iw n ) + µ ( q -3 2 ) Im [ G A LR,ψb ( iw n ) + G A LR,bψ ( iw n ) ] )   + 2 q -1   T ∑ integral ( Σ LL,bb ( iw n ) G LL,bb ( iw n ) + µ ( q -3 2 ) Im [ G S LR,ψb ( iw n ) + G S LR,bψ ( iw n ) ] )   . (51)  \nAdditionally, the free energy is derived from the saddle point effective action  \nF N = -T log Z N = T S eff N = -T   2 log (2) + ∑ half integral ( log D half integral ( iw n ) ( iw n ) 2 + 2 q -2 q Σ LL,ψψ ( iw n ) G LL,ψψ ( iw n ) ) + 2 q -2 q Σ A LR,ψb ( iw n ) G A LR,ψb ( iw n ) + Σ A LR,ψb ( iw n ) G A LR,bψ ( iw n ) ] -∑ integral ( log D integral ( iw n ) ( iw n ) 2 -2 q -2 q Σ LL,bb ( iw n ) G LL,bb ( iw n ) -2 q -2 q Σ S LR,ψb ( iw n ) G S LR,ψb ( iw n ) -Σ S LR,ψb ( iw n ) G S LR,bψ ( iw n ) ) . (52)  \nIn this section, we have renormalized the determinant terms  \n∑ half integral (log ( iw n )) = ln 2 .  \nWe have introduced iw n to eliminate the physical influence of auxiliary fermions in D integer . As there are no natural fermions with integral frequencies, the integral part of (log ( iw n )) does not contribute any additional ln 2.  \n<!-- image -->  \nFigure 3: Free energy and energy of N=2 Supersymmetric SYK fixed J = 3 , q = 5 (a)Free energy under condition µ = 0 . 3 , 0 . 5 , 0 . 8, (b) Energy under condition µ = 0 . 3 , 0 . 5 , 0 . 8,  \n<!-- image -->  \nThe free energy as a function of temperature is plotted in Figure.(3)a. We cool the system from its initial maximum temperature and heat it again. We consider the number of coupling q = 5 in Eq. (1), which means the theory should correspond to a non-supersymmetrc q=4 SYK. We observe a phase transition in Figure.(3)a. The free energy observed is analogous to the Hawking-Page transition in standard MQ and cSYK models. However, after introducing the supersymmetric coupling, the bosonic and multistate contributions to the free energy term gradually transform the 'wormhole-phase' into a state different from the so-called eternal wormhole. Furthermore, the phase transition is also influenced by this effect(as Figure.(3)a). The energy as a function of inverse temperature β is plotted in Figure.(3)b. Similarly, we can observe a comparable ensemble result regarding the energy behaviors. For T < T c , there are two distinct solution, and consist the phase transition. We can also obtain the smooth curve outside the transition region.  \nThe second-order transition ends at a critical point ( T c = -1 . 15 , µ c = 0 . 7). We first cool the system from maximal temperature and then heat it back to its original temperature. We obtain that the phase transition exists after we introduce the super-coupling. The wormhole phase in low energy will also influenced by additional bosonic contribution, and the free energy are no longer constant when we involve the coupling.  \nFor N=4 model, we can include the first-order interaction term in the Hamiltonian and while maintaining the supersymmetry and ensuring that the model remains solvable  \nH int = ∂ 2 θ iµ ( Φ i L ¯ Φ iR + Φ i R ¯ Φ iL ) = iµF i L ¯ ϕ iR + iµF i R ¯ ϕ iL , ¯ H int = ∂ 2 ¯ iµ ( ¯ Φ iL Φ i + ¯ Φ iR Φ i ) = iµ ¯ ϕ iL F i + iµ ¯ ϕ iR F i . (53)  \nθ R L R L  \nG 3 = ( ΦΦ ) 3 = ( FF ) ( ϕϕ ) ( ϕϕ ) -( ϕϕ ) ( ψ α ψ α ) ( ψ β ψ β ) + ( ϕϕ ) ( ψ α ψ β ) ( ψ β ψ α ) + 2 ( ϕϕ ) ( Fϕ ) ( ϕF ) -( ψ α ϕ ) ( ϕψ α ) ( ψ β ψ β ) -( ψ β ϕ ) ( ϕψ β ) ( ψ α ψ α ) + ( ψ β ϕ ) ( ϕψ α ) ( ψ α ψ β ) + ( ψ α ϕ ) ( ϕψ β ) ( ψ β ψ α ) . (54)  \nThen we can write the N=4 action, including the specific interaction term between the fields  \nS eff N = -1 2 log Pf ( D half integer ) + log det ( D integer ) + ∑ A,B = L,R 1 2 ∫ dτdτ ' [ Σ ψ 1 ψ 1 ,AB ( τ, τ ' ) G ψ 1 ψ 1 ,AB ( τ, τ '  \n- ) + Σ ψ 2 ψ 2 ,AB ( τ, τ ' ) G ψ 2 ψ 2 ,AB ( τ, τ ' ) + Σ FF,AB ( τ, τ ' ) G FF,AB ( τ, τ ' ) + Σ ϕϕ,AB ( τ, τ ' ) G ϕϕ,AB ( τ, τ ' ) + Σ Fϕ,AB ( τ, τ ' ) G Fϕ,AB ( τ, τ ' ) + Σ ϕF,AB ( τ, τ ' ) G ϕF,AB ( τ, τ ' ) -J ( ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G ψ 1 ψ 1 ,AB ( τ, τ ' ) G ψ 2 ψ 2 ,AB ( τ, τ ' ) -G q -1 ϕϕ,AB ( τ, τ ' ) G FF,AB ( τ, τ ' ) -( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G Fϕ,AB ( τ, τ ' ) G ϕF,AB ( τ, τ ' ) )] . (55)  \nand  \nD 2 θ G AB ( Z, Z ' ) -∑ ( -iµϵ AC ∂ 2 θ G AB ( Z, Z ' ) -∫ dτ '' dθ '' 1 dθ '' 2 Σ AC ( Z, Z '' ) G BC ( Z, Z '' ) ) = δ AB δ ( Z -Z ' ) ,  \nC D .  \n2 ¯ θ ¯ G AB ( Z, Z ' ) -∑ C ( -iµϵ AC ∂ 2 ¯ θ ¯ G AB ( Z, Z ' ) -∫ dτ '' d ¯ θ '' 1 ¯ θ '' 2 ¯ Σ AC ( Z, Z '' ) ¯ G BC ( Z, Z '' ) ) = δ AB δ ( Z -Z ' ) (56)  \nFor each component, we have determined the solution through variations in the associated variables  \nΣ ψ 1 ψ 1 ,AB ( τ ) = J ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G ψ 2 ψ 2 ,AB ( τ, τ ' ) ,  \nΣ ψ 2 ψ 2 ,AB ( τ ) = J ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G ψ 1 ψ 1 ,AB ( τ, τ ' ) ,  \nΣ ϕϕ,AB ( τ ) = -( q -1) ( q -2) JG q -3 ϕϕ,AB ( τ, τ ' ) G ψ 1 ψ 1 ,AB ( τ, τ ' ) G ψ 2 ψ 2 ,AB ( τ, τ ' ) +( q -1) JG q -2 ϕϕ,AB ( τ, τ ' ) G FF,AB ( τ, τ ' ) +( q -1) ( q -2) JG q -3 ϕϕ,AB ( τ, τ ' ) G Fϕ,AB ( τ, τ ' ) G ϕF,AB ( τ, τ ' ) ,  \nΣ FF,AB ( τ ) = -JG q -1 ϕϕ,AB ( τ, τ ' ) ,  \nΣ ϕF,AB ( τ ) = -J ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G Fϕ,AB ( τ, τ ' ) ,  \nΣ Fϕ,AB ( τ ) = -J ( q -1) G q -2 ϕϕ,AB ( τ, τ ' ) G ϕF,AB ( τ, τ ' ) . (57)  \nSimilarly, we can write down the saddle point Green's functions in frequency space  \nG ψ 1 ψ 1 ,AB ( ω ) = Det ( A ψ 1 ψ 1 ,AB ( D half integer )) Det ( D half integer )  \nG ψ 2 ψ 2 ,AB ( ω ) = Det ( A ψ 2 ψ 2 ,AB ( D half integer )) Det ( D half integer )  \nG ϕϕ,AB ( ω ) = -Det ( A ϕϕ,AB ( D integer )) Det ( D integer )  \n, , ,  \nG FF,AB ( ω ) = -Det ( A FF,AB ( D integer )) Det ( D integer ) , G ϕF,AB ( ω ) = -Det ( A ϕF,AB ( D integer )) Det ( D integer ) , (58)  \nG Fϕ,AB ( ω ) = -Det ( A Fϕ,AB ( D integer )) Det ( D integer ) .  \nCorresponding Jacobi matrix is  \nD integer =       ω 2 n -Σ LL,ϕϕ -Σ LR,ϕϕ -Σ LL,ϕF iµ -Σ LR,ϕF -Σ RL,ϕϕ ω 2 n -Σ RR,ϕϕ iµ -Σ RL,ϕF -Σ RR,ψF -Σ LL,Fϕ iµ -Σ LR,Fϕ 1 -Σ LL,FF -Σ LR,FF iµ -Σ RL,Fϕ -Σ RR,Fϕ -Σ RL,FF 1 -Σ RR,FF       ,  \nD half integer =       -iw n -Σ LL,ψ 1 ψ 1 -Σ LL,ψ 1 ψ 2 -Σ LR,ψ 1 ψ 1 -Σ LR,ψ 1 ψ 2 -Σ LL,ψ 2 ψ 1 -iw n -Σ LL,ψ 2 ψ 2 -Σ LR,ψ 2 ψ 1 -Σ LR,ψ 2 ψ 2 -Σ RL,ψ 1 ψ 1 -Σ RL,ψ 1 ψ 2 -iw n -Σ RR,ψ 1 ψ 1 -Σ RR,ψ 1 ψ 2 -Σ RL,ψ 2 ψ 1 -Σ RL,ψ 2 ψ 2 -Σ RR,ψ 2 ψ 1 -iw n -Σ RR,ψ 2 ψ 2       .  \nIn Figure.(4), we have plotted the Green's function in N=4 SYK model. In Figure.(4)a, we consider the fermionic Green's function, and we can see that it is very similar to the Green's function in the N=2 SYK model and includes a fermionic chemical potential. In addition, we also consider the bosonic Green's function, which is related to the supersymmetric sector of the theory. In Figure.(4)c, the Green's function of auxiliary bosons approaches the delta function,exhibiting behavior similar to that of bosonic N=2 Green's functions. Furthermore, we can also generate it from the Grassmann algebra. Unlike the N=1 and N=2 models, the N=4 SYK model also contains original bosons that have no direct counterparts in lower-dimensional Grassmann theories. In Figure.(4)b these bosons decay over time. Additionally, we numerically obtain the multiside correlators, which are constrained by supersymmetry within the context of G ϕF,LR and its conjugate. In Figure.(4)d, the multiside correlators have both real and imaginary parts, which influence the phase constant of transformations. The real part in Figure.(4)c behaves like the bosonic component in Figure.(4)b, while the imagine part is also influenced by the fermionic Green' function in Figure.(4)a. One can also make the super-coupling constant between left and right side iµ to be real, and the real side and imagine parts may interact or transform under certain conditions. The components G ϕϕ,LR , G FF,LR , G ϕF,AA , G Fϕ,AA , G ψαψβ,AB , G ψαϕ,AB , G ϕ,ABψα that equal to 0 are stable solutions, while G ψαF,AB , G Fψα,AB should be integrated out in the analysis.  \n)  \n(  \nG  \n0.4  \n0.2  \n0  \n-0.2  \n-0.4  \n-0.6  \n-0.8  \n-1  \n-0.5  \n-0.4  \n-0.3  \n-0.2  \n-0.1  \n0  \n0.1  \n0.2  \n0.3  \n0.4  \n0.5  \n/  \n(a)  \nFigure 4: Supersymmetric Green's function in components fixed J = 3 , q = 5 (a) Single-side fermionic Green's function G ψψ,AA fixed µ = 0 . 4 i, T = 0 . 01, (b) Single-side bosonic Green's function G ϕϕ,AA fixed µ = 0 . 4 , T = 0 . 01 (c) Single-side auxiliary bosonic Green's function G FF,AA fixed µ = 0 . 4 , T = 0 . 01 (d) multi-side Correlator between different bosons in real Re [ G ϕF,LR ] and imagine Im [ G ϕF,LR ]  \n<!-- image -->  \n0.2  \n0.1  \n0  \n-0.1  \n-0.2  \n-0.3  \n-0.5  \n-0.4  \n-0.3  \n-0.2  \n-0.1  \n0  \n0.1  \n0.2  \n0.3  \n0.4  \n0.5  \n/  \n(d)  \nThe energy can also be calculated as  \nE N = ∫ d ¯ θ 1 d ¯ θ 2 [ 2 q -1 D 2 G LL + iµ ( 1 -2 q -1 ) G LR ] τ → 0 + + h.c = 2 q -1   T ∑ half integral ( Σ LL,ψψ ( iw n ) G LL,ψψ ( iw n ))   + 2 q -1   T ∑ integral ( Σ LL,ϕϕ ( iw n ) G LL,ϕϕ ( iw n ) + Σ LL,FF ( iw n ) G LL,FF ( iw n ) + µRe ( q -3 2 ) ( G LL,ϕF ( iw n ) + G LL,Fϕ ( iw n )) )] + h.c. (59)  \n)  \n(  \nG  \n0.5  \n0.4  \n0.3  \n0.2  \n0.1  \n0  \n-0.1  \n-0.2  \n-0.3  \n-0.4  \n-0.5  \n-0.5  \n-0.4  \n-0.3  \n-0.2  \n-0.1  \n0  \n0.1  \n0.2  \n0.3  \n0.4  \n0.5  \n/  \n(b)  \n0.3  \nRe[G  \nIm[G  \nF  \nF  \n(  \n)]  \n(  \n)]  \nHere we use real or imagine component of the correlation function, depending on how we define the interaction parameter µ and make the energy real. We obtain the free energy from saddle point solutions  \nF N = -T log Z N = T S eff N = -T   2 log (2) + ∑ half integral ( log D half integral ( iw n ) ( iw n ) 2 + 4 q -4 q Σ LL,ψψ ( iw n ) G LL,ψψ ( iw n ) ) -∑ integral ( log D integral ( iw n ) ( iw n ) 2 -2 q -2 q Σ LL,ϕϕ ( iw n ) G LL,ϕϕ ( iw n ) -2 q -2 q Σ LL,FF ( iw n G LL,FF ( iw n ) ) -2 q -2 q Σ LL,ϕF ( iw n ) G LL,ϕF ( iw n ) -Σ LL,Fϕ ( iw n ) G LL,Fϕ ( iw n ) )] . (60)  \nIn summary, we consider the cross term in N=4 theory, which is characterized by the component G ϕϕ , the N=4 theory turns to bosonic model. We have studied the free energy and energy as a function of temperature, and there still exists two identical phases. However, the wormhole-black hole picture is altered due to the contribution of G ϕϕ . Because it is mainly a bosonic model, the free energy returns to zero when the temperature is large. And the thermal structure is no longer a eternal solution via the variation of inverse temperature β . We will give a further discussion in the context of the low-energy effective action.", '3.1 Low energy effective action': "The holographic picture of N=0 is relevant to the emergent conformal symmetry in the low energy limit. In previous works, the N=1 supersymmetric algebra has been shown to contain an extra supersymmetric reparametrization. And we can define sets of those reparametrization transformations using a more general coordinates  \nτ → τ ' ( τ, θ ) , θ → θ ' ( τ, θ ) . (61)  \nUnder a more general reparameterization, the correlation function with certain dimensions should be invariant under coordinate transformation rescaling  \nG ( τ 1 , θ 1 ; τ 2 , θ 2 ) = Ber ( τ ' 1 , θ ' 1 ; τ 1 , θ 1 ) 1 q Ber ( τ ' 2 , θ ' 2 ; τ 2 , θ 2 ) 1 q G ( τ ' 1 , θ ' 1 ; τ ' 2 , θ ' 2 ) , (62)  \nwhere the derivative terms in superspace are with Berezinian  \nBer ( τ ' , θ ' ; τ, θ ) ≡ Ber ( ∂ τ τ ' ∂ τ θ ' ∂ θ τ ' ∂ θ θ ' ) . (63)  \nIn low energy limit, the corresponding Berezinian can be simplified to Grassmann super Jacobian derivative factor, and this simplification gives rise to a nearly super conformal symmetry, with conformal dimension 1/q  \nBer ( τ ' , θ ' ; τ, θ ) = D θ θ ' . (64)  \nThe correlation functions exhibit a super conformal symmetry in the IR limit which is similar to single side SSYK, and this symmetry is accompanied by the preservation of covariant derivative transformations  \nG ( τ 1 , θ 1 ; τ 2 , θ 2 ) = ( D θ 1 θ ' 1 ) 1 q ( D θ 2 θ ' 2 ) 1 q G ( τ ' 1 , θ ' 1 ; τ ' 2 , θ ' 2 ) . (65)  \nwhich is invariant under the super transformation  \nτ ' = τ + ϵ + θη, θ ' = θ + η. (66)  \nHere ϵ represents an arbitrary bosonic translation, and η is a Grassmann variable. Additionally, we perform a special reparameterization of the SDiff N=1 supersymmetry  \nτ → τ ' = f ( τ ) , θ → θ ' = √ ∂ τ f ( τ ) θ. (67)  \nWe briefly review the discussion in [9]. Furthermore, a convenient generator of transformations with super conformal symmetry is a reparameterization of inversion  \nτ → τ ' = -1 τ , θ → θ ' = θ τ . (68)  \nThis algebra is invariant under the transformations of the OSp(1,2) group and can be represented by fractional linear transformations involving three bosonic generators and two fermionic generators  \nτ  \n=  \naτ  \n+  \nαθ  \n+  \nb  \ncτ  \n+  \nγθ  \n+  \nd  \n,  \nθ ' = βτ + eθ + δ cτ + γθ + d , (69)  \nwhere we have introduced the coefficients to represent certain physical quantities  \n( βτ + eθ + δ ) ( e + θβ ) + ( aτ + αθ + b ) ( -γ + θc ) -( cτ + γθ + d ) ( -α + θa ) = 0 , (70)  \ni.e.  \n'  \neβ -aγ + αc = 0 ,  \ne 2 + βδ +2 αγ + bc -ad = 0 , eδ -γb + αd = 0 . (71)  \nIn low energy limit, this generator comprises SL(2,R), and this formalism leads to an N=0 theory with Schwarzian-like properties. This result can be extended to a similar process involving N=0 considerations, thereby deriving a super Schwarzian action. The non-supersymmetric Schwarzian action possesses a reparameterized global conformal symmetry and can be formulated in terms of derivatives, constituting a distinct functional form  \n∂ τ 1 ∂ τ 2 log τ ' 1 -τ ' 2 τ 1 -τ 2 = ∂ τ 1 τ ' 1 ∂ τ 2 τ ' 2 ( τ ' 1 -τ ' 2 ) 2 -1 ( τ ' 1 -τ ' 2 ) 2 , (72)  \nIn the limit τ 1 → τ 2 the theory gives the Schwarzian action  \nS [ f ( τ ) , τ ] = f ''' f ' -3 2 ( f '' f ' ) 2 . (73)  \nIn super space extension, since the N=1 SYK has extended the fermion field as a super composite fermionic field, we can substitute the derivative operators with their super-covariant forms  \nD τ 1 D τ 2 log τ ' 1 -τ ' 2 τ 1 -τ 2 = D τ 1 τ ' 1 D τ 2 τ ' 2 ( τ ' 1 -τ ' 2 ) 2 -1 ( τ ' 1 -τ ' 2 ) 2 . (74)  \n(74) leads to the N=1 super Schwarzian form  \nS [ τ ' , θ ' ; τ, θ ] = D 4 θ ' Dθ ' -2 D 3 θ ' D 2 θ ' ( D 4 θ ' ) 2 , (75)  \nand which can be written in components  \nS [ τ ' , θ ' ; τ, θ ] = S f ( τ ' , θ ' ; τ, θ ) + θS b ( τ ' , θ ' ; τ, θ ) . (76)  \nBy analyzing this formula, we can integrate out the Grassmann variable, thereby simplifying the N=1 super Schwarzian action to a bosonic form.  \nS A = -Nα S ∫ dτdθ S [ τ ' , θ ' ; τ, θ ] = -Nα S ∫ dτ S b ( τ ' , θ ' ; τ, θ ) . (77)  \nα S is a constant derived from the low energy symmetry breaking term and the two point function. Additionally, we can utilize an appropriate reparameterization like N=0 SYK, which simplifies the bosonic component to a function of the Schwarzian derivative supplemented by the fermionic variable η  \nS A = -Nα S 2 ∫ dτ S ( tanh h A 2 , τ ) + ηη ''' +3 η ' η '' -S ( tanh h A 2 , τ ) ηη ' . (78)  \nIn the IR limit, superconformal symmetry in SSYK allows us to construct a correlation operator between two decoupled models through super conformal reparameterizations h in singleside SSYK. The correlation function in low energy limit admits solutions with ansatz  \nG ( τ 1 , θ 1 ; τ 2 , θ 2 ) = b | τ 1 -τ 2 -θ 1 θ 2 | 2 ∆ , (79)  \nwhere the conformal dimension is restricted by ∆ = 1 /q .In addition, we can introduce the coupling action by hyperbolic reparameterizations  \nS int = µ 2 ∫ dτdθ [ bD θ ( θ ' ) L D θ ( θ ' ) R cosh 2 h L ( τ + θη ( τ )) -h R ( τ + θη ( τ )) -( θ ' ) L ( θ ' ) R 2 ] 1 q . (80)  \nThanks to the mathematical relations in SSYK superspace and the terms we have included, the interaction terms in low energy preserve super conformal symmetry and behave similarly to an N=0 theory, except that correlations and derivative have been replaced by super covariant counterparts. Moreover, the correlation form reverts to a non-supersymmetric case when the superspace prolongation vanishes.  \nAdditionally, we can introduce an infinitesimal N=2 reparameterization involving two conjugate Grassmann variables ¯ η , η , as well as one bosonic variable h  \nτ ' → h ( τ + θ ¯ η ( τ ) + ¯ θη ( τ ) ) , θ ' → exp( ia ( τ )) √ ∂ τ h ( τ ) [ θ + η ( τ ) ( τ + θ ¯ θ )] , ¯ θ ' → exp( -ia ( τ )) √ ∂ τ f ( τ ) [ ¯ θ + ¯ η ( τ ) ( τ -θ ¯ θ )] . (81)  \nThis relationship leads to N=2 super Schwarzian action with the low-energy symmetry breaking.  \nS A = -Nα S ∫ dτdθd ¯ θ S [ τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ] . (82)  \nα S is also a constant that arises from four-point and higher-order modifications. The low energy N=2 Schwarzian action in thermal bosonic sector should exhibit an additional reparameterization U(1) symmetry and denoted by  \nS b = Nα S ∫ dτ ( -Sch ( tanh h A 2 , τ ) +( ∂ τ a ) 2 ) . (83)  \nCompared to N=2 IR superconformal limit, low energy interaction terms can be classified into chiral and antichiral components, and we can reparameterize the function using a similar thermal form as in the N=1 case  \nG ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = b ∣ ∣ τ 1 -τ 2 -θ 1 ¯ θ 1 -θ 2 ¯ θ 2 +2 θ 1 ¯ θ 2 ∣ ∣ 2 ∆ ,  \nb  \n¯ G ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ∣ ∣ τ 1 -τ 2 + θ 1 ¯ θ 1 + θ 2 ¯ θ 2 -2 θ 1 ¯ θ 2 ∣ ∣ 2 ∆ , S int = µ ∫ dτdθd ¯ θ   bD ¯ θ ( ¯ θ ' ) L D θ ( θ ' ) R cosh 2 h L ( τ,θ ' , ¯ θ ' ) -h R ( τ,θ ' , ¯ θ ' ) -( θ ' ) L ( ¯ θ ' ) L -( θ ' ) R ( ¯ θ ' ) R +2( θ ' ) L ( ¯ θ ' ) R 2   ∆ , ¯ S int = µ ∫ dτdθd ¯ θ   bD θ ( θ ' ) L D ¯ θ ( ¯ θ ' ) R cosh 2 h L ( τ,θ ' , ¯ θ ' ) -h R ( τ,θ ' , ¯ θ ' ) +( θ ' ) L ( ¯ θ ' ) L +( θ ' ) R ( ¯ θ ' ) R -2( θ ' ) L ( ¯ θ ' ) R 2   ∆ . (84)  \nThe total action is given by a chiral combination  \nS int,total = S int + ¯ S int . (85)  \nAs demonstrated in [9], a similar N=2 super transformation involves one bosonic variable ϵ and two fermionic Grassmann variables, namely η and ¯ η  \nτ → τ ' = τ + ϵ + θ ¯ η + ¯ θη, θ → θ ' = θ + η, ¯ θ → ¯ θ ' = ¯ θ + ¯ η. (86)  \nIn the IR limit, chiral and anti-chiral correlation functions exhibit superconformal symmetry respectively. The N=2 coordinate transformations in the low energy limit are also constrained by the supercovariant derivative and the chiral and anti chiral Jacobians  \nG ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ( D ¯ θ 1 ¯ θ ' 1 ) 1 q ( D θ 2 θ ' 2 ) 1 q G ( τ ' 1 , θ ' 1 , ¯ θ ' 1 ; τ ' 2 , θ ' 2 , ¯ θ ' 2 ) , ¯ G ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ( D θ 1 θ ' 1 ) 1 q ( D ¯ θ 2 ¯ θ ' 2 ) 1 q ¯ G ( τ ' 1 , θ ' 1 , ¯ θ ' 1 ; τ ' 2 , θ ' 2 , ¯ θ ' 2 ) . (87)  \nFor example, we can also introduce chiral-anti chiral coordinates  \nτ ± = τ ± θ ¯ θ. (88)  \nNote the reparameterization of inversion can also yield a global conformal symmetry  \nτ → τ ' = -1 τ , θ → θ ' = θ τ , ¯ θ → ¯ θ ' = ¯ θ τ , τ ± → τ ' ± = -1 τ ± . (89)  \nA global superconformal group can be expressed in terms of chiral and anti chiral linear transformations  \nτ ' + = aτ + + αθ + b  \nτ ' -= ¯ aτ -+ ¯ αθ + b ¯  \nθ ' = βτ + + eθ + δ  \n¯ θ ' = βτ -+ ¯ eθ + δ ¯ .  \ncτ + + γθ + d , ¯ ¯ cτ -+ ¯ γθ + d , cτ + + γθ + d , ¯ ¯ ¯ cτ -+ ¯ γθ + d (90)  \nWhich has an additional restriction defined by coordinates  \nτ ' + -τ ' -= 2 θ ' ¯ θ ' , (91)  \ni.e.  \na ¯ c -¯ ac = 2 β ¯ β,  \nb ¯ d -¯ bd = 2 δ ¯ δ,  \nα ¯ c -¯ aγ = 2 e ¯ β,  \na ¯ γ -c ¯ α = 2 β ¯ e,  \nα ¯ d -¯ bγ = 2 e ¯ δ,  \nb ¯ γ -¯ αd = 2 δ ¯ e,  \na ¯ d + b ¯ c -¯ ad -¯ bc = 2 β ¯ δ +2 δ ¯ β, a ¯ d -α ¯ γ -b ¯ c +¯ ad -¯ αγ -¯ bc = 2 β ¯ δ +2 e ¯ e -2 δ ¯ β. (92)  \nCompared to N=0 and N=1, explicit symmetry breaking in low energy limit also leads to a N=2 super Schwarzian action in superderivative form  \nS [ τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ] = ∂ τ ¯ D ¯ θ ' ¯ D ¯ θ ' -∂ τ Dθ ' Dθ ' -2 ∂ τ θ ' ∂ τ ¯ θ ' ( ¯ D ¯ θ ' ) ( Dθ ' ) = S f ( τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ) + θS θ ( τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ) + ¯ θS ¯ θ ( τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ) + θ ¯ θS b ( τ ' , θ ' , ¯ θ ' ; τ, θ, ¯ θ ) . (93)  \nFor the N=4 theory, we begin with the ladder diagram in the N=2 case. The propagator involving two identical superfields should correspond to a four-point contribution, which is known as solvable ladder diagrams in SYK model. In low energy limit, we consider 0st order kernel  \nF 0 ( τ 1 , τ 2 ; τ 3 , τ 4 ) = -G ( τ 1 , τ 3 ) G ( τ 2 , τ 4 ) + G ( τ 1 , τ 4 ) G ( τ 2 , τ 3 ) .  \nit could return to two fermionic superfields, and we have obtain the correlation function for the N=4 SYK model  \nG ( τ, θ 1 , ¯ θ 1 , θ 2 , ¯ θ 2 ; τ ' , θ ' 1 , ¯ θ ' 1 , θ ' 2 , ¯ θ ' 2 ) = -b ∣ ∣ τ -τ ' -θ 1 ¯ θ 1 -θ ' 1 ¯ θ ' 1 +2 θ 1 ¯ θ ' 1 ∣ ∣ 2 ∆ ∣ ∣ τ -τ ' -θ 2 ¯ θ 2 -θ ' 2 ¯ θ ' 2 +2 θ 2 ¯ θ ' 2 ∣ ∣ 2 ∆ + b ∣ ∣ τ -τ ' -θ 1 ¯ θ 2 -θ ' 1 ¯ θ ' 2 +2 θ 1 ¯ θ ' 2 ∣ ∣ 2 ∆ ∣ ∣ τ -τ ' -θ 2 ¯ θ 1 -θ ' 2 ¯ θ ' 1 +2 θ 2 ¯ θ ' 1 ∣ ∣ 2 ∆ . (94)  \nThe conformal limit still holds for a single propagator but not for kernels. The second term arises from the general reparametrization. And the formal result reverts to a copy of N=2 theories when we trivially reparametrize the theory (they should also be integrated and lead to different physics).  \nWe can derive a simple form of the N=4 reparametrization  \nθ ' 1 → exp( ia 1 ( τ )) ( ∂ τ tanh ( τ + θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ θ 1 + η 1 ( τ + θ 1 ¯ θ 1 )] ,  \n¯ θ ' 1 → exp( -ia 1 ( τ )) ( ∂ τ tanh ( τ -θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ ¯ θ 1 + ¯ η 1 ( τ -θ 1 ¯ θ 1 )] , θ ' 2 → exp( ia 2 ( τ )) ( ∂ τ tanh ( τ + θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ θ 2 + η 2 ( τ + θ 2 ¯ θ 2 )] , ¯ θ ' 2 → exp( -ia 2 ( τ )) ( ∂ τ tanh ( τ -θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ ¯ θ 2 + ¯ η 2 ( τ -θ 2 ¯ θ 2 )] , h 1 = tanh τ + θ 1 ¯ θ 1 + θ 1 ¯ η 1 + ¯ θ 1 η 1 + θ 2 ¯ θ 2 + θ 2 ¯ η 2 + ¯ θ 2 η 2 2 , h 2 = tanh τ -θ 1 ¯ θ 1 + θ 1 ¯ η 1 + ¯ θ 1 η 1 + θ 2 ¯ θ 2 + θ 2 ¯ η 2 + ¯ θ 2 η 2 2 , (95)  \nhere we use h 1 and h 2 to denote the reparameterized initial time and final time respectively, and the U(1) variables a 1 and a 2 are also allowed.  \nFurthermore, we consider the effective reparameterized interaction action under a special constraint  \nS  \nint = µ ∫ dτdθd ¯ θ   bD ¯ θ 2 ( ¯ θ ' 1 ¯ θ ' 2 ) L D θ 2 ( θ ' 1 θ ' 2 ) R cosh 2 ( h L -h R -( θ ' 1 ) L ( ¯ θ ' 1 ) L -( θ ' 1 ) R ( ¯ θ ' 1 ) R +2 ( θ ' 1 ) L ( ¯ θ ' 1 ) R )( h L -h R -( θ ' 2 ) L ( ¯ θ ' 2 ) L -( θ ' 2 ) R ( ¯ θ ' 2 ) R +2 ( θ ' 2 ) L ( ¯ θ ' 2 ) R ) 2 -bD ¯ θ 2 ( ¯ θ ' 1 ¯ θ ' 2 ) L D θ 2 ( θ ' 1 θ ' 2 ) R cosh 2 ( h L -h R -( θ ' 1 ) L ( ¯ θ ' 2 ) L -( θ ' 1 ) R ( ¯ θ ' 2 ) R +2 ( θ ' 1 ) L ( ¯ θ ' 2 ) R )( h L -h R -( θ ' 2 ) L ( ¯ θ ' 1 ) L -( θ ' 2 ) R ( ¯ θ ' 1 ) R +2 ( θ ' 2 ) L ( ¯ θ ' 1 ) R ) 2   . (96)  \nThe N=4 model allows four independent Grassmann variables. Here, we consider an infinitesimal transformation. And for simplicity, we assume that there is no dependence between θ ' 1 and θ 2 , it is easy to check  \nD θ 1 θ ' 1 D θ 2 θ ' 2 = D 2 θ ( θ ' 1 θ ' 2 ) , θ ' → θ ' 1 θ ' 2 . (97)  \nWe consider an arbitrary physical function F , which should be invariant under a coordinate transformation  \nD 2 θ F ( τ, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) = D 2 θ τ ' ∂ τ ' F + D 2 θ θ ' 2 ∂ ' 2 θ F + D 2 ¯ θ ¯ θ ' 2 ∂ ' 2 ¯ θ F = D 2 θ θ 2 D ' 2 θ F + D 2 ¯ θ ¯ θ 2 D ' 2 ¯ θ F + ( D 2 θ τ ' -¯ θ ' 1 ∂ θ ' 2 + ¯ θ ' 2 ∂ θ ' 1 -¯ θ ' 1 ¯ θ ' 2 ∂ τ ' ) ∂ τ ' F. (98)  \nOne interesting special condition in mathematical manipulations is the process of reducing the higher order terms  \nD 2 θ τ ' = ( ¯ θ ' 1 ∂ θ ' 2 -¯ θ ' 2 ∂ θ ' 1 + ¯ θ ' 1 ¯ θ ' 2 ∂ τ ' ) D 2 θ θ ' 2 , D ¯ θ α = D 2 θ ¯ θ α = D 2 θ ¯ θ 2 = 0 . (99)  \nThis restriction also leads to a transformation invariant, which exhibits similarities to a certain aspect of super conformal theories. Furthermore, we will see that this condition is highly beneficial in discovering a holographic theory within the context of gravitational physics.  \nWe can also consider an analogous approach utilizing Berezin integrals, which is equivalent in certain respects to the previously discussed methods  \nBer ( ∂ τ τ ' ∂ τ θ ' 2 ∂ θ 2 τ ' ∂ θ 2 θ 2 ) = Ber ( ∂ τ τ ' ∂ τ θ ' 2 D 2 θ τ ' -( ¯ θ 1 ∂ θ 2 -¯ θ 2 ∂ θ 1 + ¯ θ 1 ¯ θ 2 ∂ τ ) ∂ τ τ ' D 2 θ θ ' 2 -( ¯ θ 1 ∂ θ 2 -¯ θ 2 ∂ θ 1 + ¯ θ 1 ¯ θ 2 ∂ τ ) ∂ τ θ ' 2 ) = Ber ( ∂ τ τ ' ∂ τ θ ' 2 D 2 θ τ ' D 2 θ θ ' 2 ) = ( D 2 θ θ ' 2 ) -1 Ber ( ∂ τ τ ' ∂ τ θ ' 2 ¯ θ ' 1 ∂ θ ' 2 -¯ θ ' 2 ∂ θ ' 1 + ¯ θ ' 1 ¯ θ ' 2 ∂ τ ' 1 ) = ¯ D 2 θ ¯ θ ' 2 , (100)  \nwhere we have assumed that the infinitesimal translation exhibits symmetry between the two Grassmann variables θ 1 and θ 2  \nG ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ( D 2 ¯ θ 1 ¯ θ 2 ' 1 ) 1 q ( D 2 θ 2 θ 2 ' 2 ) 1 q G ( τ ' 1 , θ ' 1 , ¯ θ ' 1 ; τ ' 2 , θ ' 2 , ¯ θ ' 2 ) , ¯ G ( τ 1 , θ 1 , ¯ θ 1 ; τ 2 , θ 2 , ¯ θ 2 ) = ( D 2 θ 2 θ 2 ' 1 ) 1 q ( D 2 ¯ θ 1 ¯ θ 2 ' 2 ) 1 q ¯ G ( τ ' 1 , θ ' 1 , ¯ θ ' 1 ; τ ' 2 , θ ' 2 , ¯ θ ' 2 ) . (101)  \nTerms with two identical Grassmann variables and derivatives of τ will break the usual super conformal limit of reparametrization in N=4 theory. But an important feature is that, when one of the Grassmann variables is ignored, the N=4 theory should reduce to the N=2 theory. Furthermore, we can construct the N=4 effective action based on the N=2 theory. The aforementioned approach of constructing the N=4 effective action from N=2 theory can also be applied  \nto the gravitational sector later. The behavior of the infinitesimal translation in the context of N=4 theory should exhibit some similarities to that in the N=2 theory, but with some notable differences, as follow  \nθ 1 → ξ -iϵρ, θ 2 → ξ -iϵρ, ¯ θ 1 → ¯ ξ -iϵ ¯ ρ, ¯ θ 2 → ¯ ξ -iϵ ¯ ρ. (102)  \nSince we have the first-order restriction specified in the equation, we can proceed to write down the infinitesimal generators ρ and ¯ ρ in a manner similar to that in the N=2 theory  \nρ = -ξ ' ¯ ρ = -¯ ξ ' . (103)  \nThen we can characterize the effective action as super Schwarzian  \nSchw ( x, ξ, ¯ ξ ; τ, θ, ¯ θ ) = ∂ τ ( D ¯ θ 1 ¯ ξ ) ( D ¯ θ 2 ¯ ξ ) ( D ¯ θ 1 ¯ ξ ) ( D ¯ θ 2 ¯ ξ ) + ( D ¯ θ 1 ¯ ξ ) ∂ τ ( D ¯ θ 2 ¯ ξ ) ( D ¯ θ 1 ¯ ξ ) ( D ¯ θ 2 ¯ ξ ) + ∂ τ ( D θ 1 ξ ) ( D θ 2 ξ ) ( D θ 1 ξ ) ( D θ 2 ξ ) + ( D θ 1 ξ ) ∂ τ ( D θ 2 ξ ) ( D θ 1 ξ ) ( D θ 2 ξ ) -2 ∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) = ( D 2 ¯ θ ∂ τ ¯ ξ 2 ) D 2 ¯ θ ¯ ξ 2 + ( D 2 θ ∂ τ ξ 2 ) D 2 θ ξ 2 -2 ∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) . (104)  \nThis effective action arises from two identical N=2 SYK models, includes a last term that arises from the first-order crossing term. It can be verified that if we eliminate one of the Grassmann variables, the theory will reduce to the N=2 theory. Furthermore, this result indicates that the effective action is mainly Schwarzian in the absence of super space, and the additional cross term would break the emergent super conformal symmetry.  \nD 2 θ ξ 2 = ( ∂ ∂θ 1 + ¯ θ 1 ∂ ∂τ ) ξ ( ∂ ∂θ 2 + ¯ θ 2 ∂ ∂τ ) ξ = ∂ 2 θ ξ 2 + D θ 1 ξ ∂ ∂θ 2 ξ + ∂ ∂θ 1 ξD θ 2 ξ + ¯ θ 1 ¯ θ 2 ∂ ∂τ ∂ ∂τ ξ 2 . (105)  \nCompared to the mean field methods and SD equations we have studied in the previous section, we can also derive the IR Green's function for the N=2 SYK model. Firstly, when we neglect the higher-order terms and concentrate on the low-energy theory, the effective propagators in the low-energy limit can be written as  \nG ψψ ( τ ) ∼ sgn ( τ ) c ψ ( τ ) , G bb ( τ ) ∼ -δ ( τ ) c b ( τ ) ∼ 1 c b ( τ ) . (106)  \nwhere the functions c are the conformal factors in propagates that and are determined by the conformal dimension, which is related to the equation G ψψ ( τ ) = -∂ τ G bb ( τ )  \nThe N=4 theory could be defined as a specific theory  \nG ψψ ( τ ) ∼ 1 2 sgn ( τ ) c ψ ( τ ) , G FF ( τ ) ∼ δ ( τ ) c F ( τ ) ∼ 1 c ' F ( τ ) , G ϕϕ ( τ ) ∼ 1 2 | τ | c ϕ ( τ ) . (107)  \nwith G ϕϕ ( τ ) = ∂ τ G ψψ ( τ ) = ∂ 2 τ G FF ( τ ). However, the component G ϕϕ is weird. Since the absolute value of the time order in the conformal factor should be less than 1, the Green's function Σ ϕϕ is not a convergent function in large τ and β limit(compared to the other component). According to the [39], the components with certain dimension will dominate the bare propagator. Considering the Replica trick, it is also a periodic function.  \nThen, we can further qualitatively analyze the thermal entropy using the mean field ansatz, when considering the zero-temperature limit  \nS ≡ 1 log ( Trρ n  \nexp( -S ) ∼ 1 Z 2 ∫ DϕDψDF exp( -S eff (Σ ϕϕ ( τ )))  \n1 -n ) (108)  \nSince the partition function conclude a first-order dependence of logarithm, it will not converge to a certain constant due to the contribution of Σ ϕϕ , and it has a dramatic change around the periodic short time τ = β/ 2(where β is large enough).", '3.2 Gravitational action on boundary': "In this subsection, we consider the holographic duality of the supersymmetric SYK model. First, we briefly review the low energy SYK model with N=0,1,2 supersymmetry and their holography duality, which can be described as JT gravity  \nS = -1 16 πG [∫ M d 2 x √ gϕ ( R +2) + 2 ∫ ∂M du √ hK ] , (109)  \nwhich is contained in a global AdS 2 metric  \nds 2 = dt 2 + dz 2 z 2 . (110)  \nWe can also apply a finite curve reparameterization to the boundary  \n1 ϵ 2 = t ' 2 + z ' 2 z 2 (111)  \nThis form leads to a Schwarzian form with NAdS boundary condition as ϵ approaches zero  \nS [ t ( u )] = ∫ duϕ r ( u ) Sch ( t, u ) . (112)  \nWe can also consider N=1 super JT gravity as in [12]  \nS = -1 16 πG [ i ∫ d 2 zd 2 θEΦ ( R + --2) + 2 ∫ ∂M dudθΦK ] , (113)  \nHere we use φ to represent supersymmetric dilatons [44, 45], and E is the supervielbein to harmonize the spinor and scalar fields. In [12], it also gives a special supercomformal gauge condition  \ndu 2 +2 θdθdu 4 ϵ 2 = dz ξ E 1 ξ dz π E ¯ 1 π . (114)  \nThe superconformal gauge condition is the most important and widely researched among those relevant to quantum gravity and supersymmetric field theories  \n( E + θ E 1 θ E + z E 1 z ) = ( e Σ 2 + θe Σ D θ e -Σ 2 -θe Σ -e Σ D θ e -Σ 2 e Σ ) .  \nwith supersymmetrization condition. In order to obtain the effective action, the vielbein on both dimensions has to satisfy  \nDz = θDθ, z = t + iϵ ( Dξ ) 2 , .  \nDt ( u, θ ) = ξ ( u, θ ) Dξ ( u, θ )  \nHere the supersymmetric gravity will contribute an additional spinor component. To convert these spinor terms back to the scalar superspace, we introduce the supervielbein. And we follow the notation E denotes vielbein components relevant to inverse density, which enables us to  \nK = T A D T n A T A T A , (115)  \nwhere T is a tangent vector along the boundary, normalized the orthogonal vector n to T A n A = 0.  \nK = 4 ϵ 2 S [ t, ξ ; u, θ ] . (116)  \nThe covariant derivative D T contains contributions from the super derivative and spinor terms, as expressed in the equation D T n A = Dn A + DΩ . Furthermore, the boundary action exhibits properties similar to those of the Schwarzian derivative  \nS bdy = ∫ dudθΦ r ( u, θ ) S [ t, ξ ; u, θ ] . (117)  \nAnd we can also introduce sets of interaction operators on both left and right boundary  \nS int = g ∑ i ∫ dudθO i L ( u, θ ) O i R ( u, θ ) . (118)  \nO is a set of N operators with super conformal dimension ∆ . The dimension of g is given by [energy] 2 ∆ -1 [46, 47]  \nS int = g ∑ i ∫ dudθd ¯ θ ( ¯ O i L ( u, θ, ¯ θ ) O i R ( u, θ, ¯ θ ) + O i L ( u, θ, ¯ θ ) ¯ O i R ( u, θ, ¯ θ )) , (119)  \nwhere O are sets of N operators with super conformal dimension ∆ , and they can be obtained by supersymmetrizing N=0 double traces. g has the dimension as [energy] 2 ∆ -1 . Then we can construct the duality of interaction part of SSYK  \n〈 O ( t 1 P ) O ( t 2 P )〉 = ∣ ∣ t 1 P -t 2 P -θ 1 P θ 2 P ∣ ∣ -2 ∆ . (120)  \nAfter applying the thermal form reparametrization  \nh = tanh ( τ + θη ) 2 ,  \nθ ' = ( ∂ τ tanh ( ( τ + θη ) 2 )) 1 2 ( θ + η + 1 2 η∂ τ η ) . (121)  \nWe can also write the N=2 JT gravity from [13]  \nS = -1 16 πG [∫ d 2 zd 2 θEΦ ( R +2) + ∫ d 2 zd 2 ¯ θE ¯ Φ ( ¯ R +2 ) +2 ∫ ∂M dudθd ¯ θ ( Φ + ¯ Φ ) K ] , (122)  \nwith the boundary external supercurvature  \n∫ ∂M dudθd ¯ θ K = ∫ ∂M dudθK + ∫ ∂M dud ¯ θ ¯ K, K = T A D T n A T A T A , ¯ K = T A ¯ D T n A T A T A . (123)  \nA special N=2 supersymmetric gauge constraint is also introduced through a natural process of supersymmetrization  \ndu 2 +2 ¯ θdθdu +2 θd ¯ θdu +2 θ ¯ θdθd ¯ θ 4 ϵ 2 = dz ξ E l ξ dz π E ¯ l π . (124)  \nHere we use the chiral density and the supervielbeins to rewrite the supergravity in super space.  \nThen we have the reparameterization, involving a super Schwarzian effective action and exhibiting holographic duality with the SSYK model  \nDz = ¯ θDθ,  \n¯ ¯ ¯  \nDz = θ D θ, z = t + iϵ ( Dξ ) ( ¯ D ¯ ξ ) , Dt ( u, θ, ¯ θ ) = ¯ ξ ( u, θ, ¯ θ ) Dξ ( u, θ, ¯ θ ) , ¯ Dt ( u, θ, ¯ θ ) = ξ ( u, θ, ¯ θ ) ¯ D ¯ ξ ( u, θ, ¯ θ ) .  \n(125)  \nThe extrinsic supercurvature is very similar to the theory with N=1 supersymmetry.  \nK = -4 ϵ 2   ξ '' Dξ -ξ ' ( Dξ ' ) ( Dξ ) 2 + ( ¯ D ¯ ξ ' ) ξ ' ( Dξ ) ( ¯ D ¯ ξ )   , K = -4 ϵ 2   ¯ ξ '' ¯ Dξ -¯ ξ ' ( ¯ D ¯ ξ ' ) ( ¯ D ¯ ξ ) 2 + ( Dξ ' ) ¯ ξ ' ( ¯ D ¯ ξ ) ( Dξ )   . (126)  \nWe can also rewrite the boundary action in terms of the super Schwarzian  \nS bdy = ∫ dudθd ¯ θ ( Φ r + ¯ Φ r ) S [ t, ξ, ¯ ξ ; u, θ, ¯ θ ] . (127)  \nWe can also built the N=2 correlation function between boundaries. Since fermions with N=2 supersymmetric interact with conjugate chiral particles, we have restricted the sets of operators O and ¯ O based on their chirality. Likewise, we can utilize the thermal reparametrization solutions  \nh 1 = tanh τ + θ ¯ θ + θ ¯ η + ¯ θη 2 ,  \nh 2 = tanh τ -θ ¯ θ + θ ¯ η + ¯ θη 2 ,  \nθ ' → exp( ia ( τ )) ( ∂ τ tanh ( τ + θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ θ + η ( τ + θ ¯ θ )] ,  \n¯ θ ' → exp( -ia ( τ )) ( ∂ τ tanh ( τ -θ ¯ θ + θ ¯ η + ¯ θη 2 )) 1 2 [ ¯ θ + ¯ η ( τ -θ ¯ θ )] ,  \nS int = g 2 2∆ ∫ dτdθd ¯ θ   D θ ( θ ' ) L D ¯ θ ( ¯ θ ' ) R cosh 2 h L ( τ + θ ¯ η ( τ )+ ¯ θη ( τ ) ) -h R ( τ + θ ¯ η ( τ )+ ¯ θη ( τ ) ) -( θ ' ) L ( ¯ θ ' ) R + ( ¯ θ ' ) L ( θ ' ) R 2   1 q ,  \n¯ S int = g 2 2∆ ∫ dτdθd ¯ θ   D ¯ θ ( ¯ θ ' ) L D θ ( θ ' ) R cosh 2 h L ( τ + θ ¯ η ( τ )+ ¯ θη ( τ ) ) -h R ( τ + θ ¯ η ( τ )+ ¯ θη ( τ ) ) -( θ ' ) L ( ¯ θ ' ) R + ( ¯ θ ' ) L ( θ ' ) R 2   1 q . (128)  \nHere we use h 1 and h 2 to denote the reparameterized initial time and final time.  \nWe also want to derive the effective action of N=4 holographic theory in the low temperature limit. First, we consider the special form of N=4 super Schwarzian as discussed in the context of the N=4 low energy limit.  \nSchw ( x, ξ, ¯ ξ ; τ, θ, ¯ θ ) = ( D 2 ¯ θ ∂ τ ¯ ξ 2 ) D 2 ¯ θ ¯ ξ 2 + ( D 2 θ ∂ τ ξ 2 ) D 2 θ ξ 2 -2 ∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) . (129)  \nThis action is not globally conformal or superconformal, unlike the N=0 and N=1,2 SYK models which exhibit special reparameterization symmetries. Unlike the 2D analogues of the SYK model, this gravitational action is also restricted to 1+1d. Since there exists only one time scale, higher order terms with cross terms between the Grassmann variables emerge. These cross terms are not invariant under conformal transformations. A special case is when SL(2,R) emerges in the low energy limit of N=0 SYK, whereas SU (1 , 1 | 1) symmetry does not emerge under N=2 conditions.  \nHowever, it can still be derived from N=2 SYK model. Since the infinitesimal translation group is similar to that in the N=2 SYK model, we can propose a generalization based on the N=2 NAdS spacetime. Similarly, in the N=4 case, we expect this N=4 NAdS spacetime to exhibit properties analogous to those of the N=4 SYK model. To extend to N=4 spacetime, we introduce supersymmetry into the relevant Grassmann variable components, building upon the N=2 NAdS spacetime framework. When we eliminate one Grassmann variable α by imposing certain constraints or symmetries  \n∂ θ α ∂ ¯ θ α ( θ α ¯ θ α L ) .  \nand the theory will return to N=2.  \nWe can easily adopt an ansatz for the specific form of the metric within the possible context of N=4 JT gravity  \nS JT = -1 16 πG N (∫ M d 2 zd 4 θ E -1 Φ ( R -2) + ∫ M d 2 zd 4 ¯ θ ¯ E -1 ¯ Φ ( ¯ R -2 ) +2 ∫ M dzd 2 θd 2 ¯ θ ( Φ b + ¯ Φ b ) K ) . (130)  \nThe parameters in this equation should also be constrained with certain conditions, which means we should restrict the N=4 theory back to N=2 when it involves first-order of the other Grassmann variable. This external curvature term can be written in Grassmann components  \n∫ M dzd 2 θd 2 ¯ θ K = ∫ M dzd 2 θK + ∫ M dzd 2 ¯ θ ¯ K. (131)  \nSince we want to obtain the formula of the effective action, it is natural to define the external curvature in terms of Grassmann components, which are relevant to or derived from the N=2 theory  \nK = T A ¯ D T 1 n A 1 ¯ D T 2 n A 2 T 2 = T A ¯ D 2 T n A T 2 , ¯ K = T A D T 1 n A 1 D T 2 n A 2 T 2 = T A D 2 T n A T 2 . (132)  \nHere we use ¯ D T 1 , ¯ D T 2 , D T 1 , D T 2 to express the scalar-spinor covariant derivatives with different Grassmann variables. And these derivatives are performed with a super connection term Ω ξ  \nD 2 T n A = D 2 T n A + ( D 2 T z ξ Ω ξ ) n A ,  \n¯ D 2 T n A = ¯ D 2 T n A + ( ¯ D 2 T z ξ Ω ξ ) n A . (133)  \nWe have preliminarily proposed the formalism of the curvature. It is easy to check that this curvature includes N=2 curvature with a single Grassmann component θ or ¯ θ .  \nWe can also write the boundary effective action up to 2nd order  \n( du 2 +2 ¯ θ 1 dθ 1 du +2 θ 1 d ¯ θ 1 du +2 θ 1 ¯ θ 1 dθ 1 d ¯ θ 1 ) ( du 2 +2 ¯ θ 2 dθ 2 du +2 θ 2 d ¯ θ 2 du +2 θ 2 ¯ θ 2 dθ 2 d ¯ θ 2 ) 4 ϵ 2 = dz ξ E l ξ dz π E ¯ l π . (134)  \nCompared to N=2 theory, this equation is generated from the N=4 supercovariant derivative. Parameter E and ¯ E as chiral density leads to supervielbein, which unifies the spinor and scalar components, which also contain N=2 theories.  \nIn order to develop the holographic theory of field theory, the boundary reparameterization should satisfy certain constraints  \nDz = ( ¯ θ 1 ∂ θ 2 -¯ θ 2 ∂ θ 1 + ¯ θ 1 ¯ θ 2 ∂ z ) D 2 θ 2  \nθ , ¯ Dz = ( θ 1 ∂ ¯ θ 2 -θ 2 ∂ ¯ θ 1 + θ 1 θ 2 ∂ z ) D 2 ¯ θ ¯ θ 2 , z = t + iϵ ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 ) ,  \nDt ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) = ( ¯ ξ ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ θ 2 -¯ ξ ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ θ 1 + ¯ ξ 2 ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ z ) D 2 θ ξ 2 , ¯ Dt ( u, θ 1 , θ 2 , , ¯ θ 1 , ¯ θ 2 ) = ( ξ ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ ¯ θ 2 -ξ ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ ¯ θ 1 + ξ 2 ( u, θ 1 , θ 2 , ¯ θ 1 , ¯ θ 2 ) ∂ z ) D 2 ¯ θ ¯ ξ 2 . (135)  \nAn important signature here is that this action should contain the N=2 gravity which is dual to the N=2 SYK model. If we eliminate terms with the Grassmann variable θ 1 and ¯ θ 1 or θ 2 and ¯ θ 2  \n∂ θ α ∂ ¯ θ α ( θ α ¯ θ α S ) ,  \nThe action now simplifies to N=2 action, which contains a bosonic part that relates to the nonsupersymmetric Schwarzian action. We can easily verify this in terms of the effective action. However, the method for verifying the N=4 construction is incomplete. Naturally, we ignore terms of first order, which do not appear in the N=2 model. Now, we proceed to examine this further. Notice that a stronger condition is required for the full consistency of the theory  \n¯ D 2 ¯ θ ¯ ξ 2 = ¯ D ¯ θ ¯ ξ ¯ D ¯ θ ¯ ξ, D 2 θ ξ 2 = D θ ξD θ ξ, (136)  \nwhen our reparametrization ξ has only first-order dependence on τ . The single reparametrization ξ should involve a Grassmann variable. Assuming the relevant equation holds, we can then proceed to consider the curvature tensor, which can be expressed in its component form  \nK = -2 ϵ 2 D 2 θ   ( D 2 θ ∂ τ ξ 2 ) D 2 θ ξ 2 -∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 )   ,  \n¯ K = -2 ϵ 2 ¯ D 2 ¯ θ   ( D 2 ¯ θ ∂ τ ¯ ξ 2 ) D 2 ¯ θ ¯ ξ 2 -∂ τ ξ 2 ∂ τ ¯ ξ 2 ( D 2 θ ξ 2 ) ( D 2 ¯ θ ¯ ξ 2 )   . (137)  \nOne can easily verify that the curvature exhibits features characteristic of N=2 curvature.  \nD 2 θ ξ 2 = ( ∂ ∂θ 1 + ¯ θ 1 ∂ ∂τ ) ξ ( ∂ ∂θ 2 + ¯ θ 2 ∂ ∂τ ) ξ = ∂ 2 θ ξ 2 + D θ 1 ξ ∂ ∂θ 2 ξ + ∂ ∂θ 1 ξD θ 2 ξ + ¯ θ 1 ¯ θ 2 ∂ ∂τ ∂ ∂τ ξ 2 . (138)  \nThe correlation term is written in ansatz  \n〈 ¯ O ( t 1 P ) O ( t 2 P )〉 = 〈 O ( t 1 P ) ¯ O ( t 2 P )〉 = ∣ ∣ t 1 P -t 2 P -θ 1 α ¯ θ 2 α -¯ θ 1 α θ 2 α -θ 1 β ¯ θ 2 β -¯ θ 1 β θ 2 β ∣ ∣ -2 ∆ . (139)  \nFor both N=2 and N=4 theories, the super reparametrization differs significantly from that of N=1 theory.", '4 Thermofield double state': "The thermofield double in the Majorana SYK model is defined as the entangled state that relates the original system to its time-reversed counterpart at a finite temperature  \n| TFD β ⟩ = 1 √ Z β ∑ n exp( -βE n / 2) | n ⟩ 1 ⊗| ˆ n ⟩ 2 . (140)  \nwhere β = 1 /T . Since N=1 SSYK supercharge has the same structure as Majorana fermions except for parity has changed from odd to even, and we can propose a projection from N=0 to N=1 by changing q to -q. The Hilbert space remains the in its original form to directly represent the supercharge, and it indirectly relates to the system's Hamiltonian.  \n| TFD β ⟩ = 1 √ Z β ∑ n exp ( -βQ 2 n / 2 ) | n ⟩ 1 ⊗| ˆ n ⟩ 2 . (141)  \nThe interaction term is proportional to Majorana fermions, which act as supercharges in eq.(6) and eq.(9).  \nΨ i = ( I + θQ ) ψ i = ( I + θQ ) ( C † i + C i ) , (142)  \nwhich exhibits anticommutative relations and behaves like fermions  \nH int = iµ∂ θ ∑ i Ψ L i Ψ R i = iµ ∑ i ( QC L † i C R i -QC R † i C L i ) . (143)  \nThen, within the N=1 framework, we can explicitly define the eigenstates of the superspace operator  \nH | Ψ r n ⟩ = Q 2 | Ψ r n ⟩ = E n | Ψ r n ⟩ ,  \nH † ∣ ∣ ∣ Ψ l m 〉 = Q 2 † | Ψ r n ⟩ = E ∗ m ∣ ∣ ∣ Ψ l m 〉 , (144)  \nwith  \n∑ n | Ψ l n ⟩ ⟨ Ψ r n | = I, ⟨ Ψ l n | Ψ r m ⟩ = δ nm . (145)  \nGround states of the interaction part are also generated from the N=0 MQ model  \n| I ⟩ = ∏ i ( | 1 ⟩ L,i | 0 ⟩ R,i + i | 0 ⟩ L,i | 1 ⟩ R,i ) , (146)  \nand  \nH † int ∣ ∣ ∣ Ψ l 0 〉 = -µNQ 0 ∣ ∣ ∣ Ψ l 0 〉 , H int | Ψ r 0 ⟩ = -µNQ 0 | Ψ r 0 ⟩ . (147)  \nThe TFD β =0 state shall be defined using an anti-unitary transformation, as described in[37, 48]  \n| TFD 0 (0) ⟩ = 1 2 N/ 2 ∑ q ∑ n q | n q ⟩ 1 ⊗ ( e -iηπΓ 4 e -iq ( π 2 ) ) P | n q ⟩ 2 , Γ = ( -1) -q + N 2 , P -1 C i P = ηP † i , ⟨ ˆ n -q ' | C R i | ˆ m -q ⟩ = -e -iπ ( -q + N 2 ) ⟨ m q | C L i | n q ' ⟩ , ⟨ ˆ n -q ' | C R † i | ˆ m -q ⟩ = -e -iπ ( -q ' + N 2 ) ⟨ m q | C L † i | n q ' ⟩ . (148)  \nWe now define the states and parity P in terms of supercharges. These supercharges in the equation act like Hamiltonian in N=0 model and give a constant without changing the states in Fock space. Since the energy diagram of the N=1 SYK model can be projected onto N=0 model[49, 50, 51, 52], the N=1 SYK model could be considered as a special form of the N=0 SYK model, and is also referred to as 'fake superspace'. The ground states of N=1 SYK model are isomorphic to the trivial homology of supercharges but not to cohomology. Furthermore, we know that the ground state of the N=1 SYK model is not a BPS state. However, we will see that the BPS ground states of the N=2 and N=4 SYK models have an infinite number of projections onto supercharges or onto the N=0 SYK model.  \n⟨ TFD β =0 | H int | TFD β =0 ⟩ = 1 2 N ∑ qq ' ∑ mn 〈 ˆ n -q ' ∣ ∣ ∣ 2 ⊗ 〈 n q ' ∣ ∣ ∣ 1 ( iµ ∑ i ( QC L † i C R i -QC R † i C L i )) | m q ⟩ 1 ⊗| ˆ m -q ⟩ 2 = 1 2 N ∑ qq ' ∑ nm ∑ i ( iµe iπ ( -q + N 2 ) 〈 n q ' ∣ ∣ ∣ QC L † i ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ C R i | ˆ m -q ⟩ -iµe iπ ( -q ' + N 2 ) e 2 α 〈 ˆ n -q ' ∣ ∣ ∣ QC R † i ∣ ∣ ∣ n ' q ''' 〉〈 n q ' ∣ ∣ ∣ C R i | m q ⟩ ) , (149)  \ngives  \n⟨ H int ⟩ = 1 2 N ∑ qq ' ∑ nm ∑ i [ -iµ 〈 n q ' ∣ ∣ ∣ QC L † i | m q ⟩ ⟨ m q | C L i ∣ ∣ ∣ n q ' 〉 + iµ ⟨ m q | QC L † i ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ C L i | m q ⟩ = ∑ qq ' ∑ nm ∑ i iµ/ 2 N 〈 n q ' ∣ ∣ ∣ QC L † i | m q ⟩ ⟨ m q | C L i ∣ ∣ ∣ n q ' 〉 = -µ 2 N ∑ n,q ∑ i ⟨ n q | 1 QC L † i QC L i | n q ⟩ 1 = Const. (150)  \nWe can utilize the same process in the N=2,4 SYK model from Appendix A, which also yield similar results.  \nAdditionally, we can calculate the Witten index to confirm the presence of fermionic ground states.  \nI ( r ) = Tr [ ( -1) F e 2 πirQ R ] .  \nIn the N=2 model, the Witten index is redefined to reflect specific properties or conditions relevant to the model  \nI ( r ) = e iπN ( r q -1 2 ) ( 1 -e 2 πir q ) N = ( 2 sin ( πr q )) N .  \nThe Witten index is very similar for N=4 SYK, which also incorporates the anti-commutation within the Hamiltonian. The N=4 ground states enforce one of the anti-commutators of the supercharges to be zero, thereby providing give  \nI ( r ) = e iπN ( 2 rq -r 2 q 2 -1 2 ) ( 1 -e 2 πi ( 2 rq -r 2 ) q 2 ) N = ( 2 sin π ( 2 rq -r 2 ) q 2 ) N .  \nThis quantity also vanishes when r = 0. Here R-charge is defined as Q R = 1 q Q", '5.1 Real time correlation and transmission amplitude': "In both N=1 and N=2 Supersymmetric SYK models, the Lorentz Wightman correlation function can be derived from the properties of superfields  \nG > AB ( t 1 , t 2 ) = -i G > AB ( it 1 , it + 2 ) = -i lim G > AB ( it 1 + ϵ, it 2 -ϵ )  \nG < AB ( t 1 , t 2 ) = -i G < AB ( it + 1 , it -2 ) = -i lim G < AB ( it 1 -ϵ, it 2 + ϵ )  \n-ϵ →-0 , ϵ → +0 ,  \nG R AB ( t 1 , t 2 ) = ϑ ( t 1 -t 2 ) ( G > AB ( t 1 , t 2 ) -G < AB ( t 1 , t 2 ) ) , G A AB ( t 1 , t 2 ) = ϑ ( t 2 -t 1 ) ( G > AB ( t 1 , t 2 ) -G < AB ( t 1 , t 2 ) ) . (151)  \nThe Schwinger-Dyson equation in Lorentzian time version is defined as  \niD t 1 G > AB ( t 1 , t 2 ) = µϵ AC ∂ θ G > CB ( t 1 , t 2 )+ ∫ dtdθ ( Σ R AC ( t 1 , t ) G > CB ( t, t 2 ) + Σ > AC ( t 1 , t ) G A CB ( t, t 2 ) ) ,  \niD t 1 G R AB ( t 1 , t 2 ) -µϵ AC ∂ θ G R CB ( t 1 , t 2 ) -∫ dtdθ ( Σ R AC ( t 1 , t ) G R CB ( t, t 2 ) + Σ A AC ( t 1 , t ) G A CB ( t, t 2 ) ) ,  \n= δ AB ( ¯ θ 1 -¯ θ 2 ) δ ( t 1 -t 2 ) , (152)  \nThe corresponding equation of motion  \nΣ > AB ( t 1 , t 2 ) = J G > AB ( t 1 , t 2 ) G > AB ( t 1 , t 2 ) Σ R AB ( t 1 , t 2 ) = ϑ ( t 1 -t 2 ) ( Σ > AB ( t 1 , t 2 ) -Σ < AB ( t 2 , t 1 ) ) , (153)  \nand written in components  \nΣ > ψψ,AB ( t ) = -( q -1) ( -1) ( q -1) / 2 JG >q -2 ψψ,AB ( t ) G > bb,AB ( t ) + ( q -1) 2 ( q -2) ( -1) ( q -1) / 2 JG >q -3 ψψ,AB ( t ) , G >A ψb,AB ( t ) G >S bψ,AB ( t ) + ( q -1) 2 ( q -2) ( -1) ( q -1) / 2 JG >q -2 ψψ,AB ( t ) G >S ψb,AB ( t ) G >A bψ,AB ( t ) , Σ > bb,AB ( t ) = -JG >q ψψ,AB ( t ) , Σ >S bψ,AB ( t ) = ( q -1) 2 JG >A ψb,AB ( t ) G >q -2 ψψ,AB ( t ) , Σ >A bψ,AB ( t ) = ( -1) ( q -1) / 2 ( q -1) 2 JG >S ψb,AB ( t ) G >q -2 ψψ,AB ( t ) , Σ >S ψb,AB ( τ ) = ( q -1) 2 JG >A bψ,AB ( τ ) G > ψψ,AB ( τ ) , Σ >A ψb,AB ( t ) = -( -1) ( q -1) / 2 ( q -1) 2 JG >S bψ,AB ( τ ) G > ψψ,AB ( τ ) . (154)  \nWe can rewrite the retarded component using both bosonic and fermionic partitions  \nG > ψψ,AB ( ω ) = G R ψψ,AB ( ω ) -( G R ψψ,AB ( ω ) ) ∗ 1 + exp ( -βω ) , G > bb,AB ( ω ) = G R bb,AB ( ω ) -( G R bb,AB ( ω ) ) ∗ -1 + exp ( -βω ) , G >S bψ,AB ( ω ) = G RS bψ,AB ( ω ) -( G RS bψ,AB ( ω ) ) ∗ -1 + exp ( -βω ) ,  \n∗  \nG >A ψb,AB ( ω ) = G RA ψb,AB ( ω ) -( G RA ψb,AB ( ω ) ) 1 + exp ( -βω ) , G >A bψ,AB ( ω ) = G RA bψ,AB ( ω ) -( G RA bψ,AB ( ω ) ) ∗ 1 + exp ( -βω ) , G >S ψb,AB ( ω ) = G RS ψb,AB ( ω ) -( G RS ψb,AB ( ω ) ) ∗ -1 + exp ( -βω ) . (155)  \nIn this context, the bosonic and fermionic factors are incorporated into the contour integrations, rendering the traditional Matsubara methods obsolete. With the frequency being discrete, we can now consider the original bosonic and fermionic factors in real time  \nD Lorentz =       w n -Σ LL,ψψ -Σ LR,ψψ -Σ A LL,ψb -iµ -Σ A LR,ψb -Σ RL,ψψ w n -Σ RR,ψψ iµ -Σ A RL,ψb -Σ A RR,ψb -Σ A LL,bψ iµ -Σ A LR,bψ -1 -0 -0 -iµ -Σ A RL,bψ -Σ A RR,bψ -0 -1 -0       , D Lorentz inverse =       w n -0 -0 -Σ S LL,ψb iµ -Σ S LR,ψb -0 w n -0 -iµ -Σ S RL,ψb -Σ S RR,ψb -Σ S LL,bψ -iµ -Σ S LR,bψ -1 -Σ LL,bb -Σ LR,bb iµ -Σ S RL,bψ -Σ S RR,bψ -Σ RL,bb -1 -Σ RR,bb       , G ψψ,AB ( τ ) = Det ( A ψψ,AB ( D Lorentz )) Det ( D Lorentz ) , G bb,AB ( τ ) = -Det ( A bb,AB ( D Lorentz inverse )) Det ( D Lorentz inverse ) , G A ψb,AB ( τ ) = Det ( A A ψb,AB ( D Lorentz ) ) Det ( D Lorentz ) , G S bψ,AB ( τ ) = -Det ( A S bψ,AB ( D Lorentz inverse ) ) Det ( D Lorentz inverse ) , G S ψb,AB ( τ ) = Det ( A S ψb,AB ( D Lorentz inverse ) ) Det ( D Lorentz inverse ) , G A bψ,AB ( τ ) = -Det ( A A bψ,AB ( D Lorentz ) ) .  \nDet ( D Lorentz ) (156)  \nHere we naturally ignore the excitation at initial time because the numerical approach around point 0 could lead to a divergence in the bosonic Retarded correlator. This implies that we must select the Lorentz time such that t ∈ ( -t max , -1) ∪ (1 , t max ).Since the frequency-dependent Green's functions encompass all the information within this system, omitting the zero point will  \nG  \nG  \n0  \n-0.05  \n-0.1  \n2.5  \n2  \n1.5  \n1  \n0.5  \n0  \n-0.5  \n-1  \n-1.5  \n-2  \n-800  \n0.1  \n0.05  \n80  \n90  \n100  \n110  \n120  \n130  \n140  \n150  \n160  \nt  \n(c)  \n30  \n40  \n50  \n60  \n70  \n80  \n90  \n100  \n110  \nt  \nFigure 5: Package solution of Real-time Green's function fixed J = 3 , q = 5 , mu = 0 . 4 , T = 0 . 01 (a) 1st package (b) 2nd package (c) 3rd package (d) 4th package  \n<!-- image -->  \nnot affect the system's properties, similar to how the Matsubara method was applied in previous works. We have the wave package solution in numerical limit λ max = 100000 and long-time cut off at t max = 40000 has been plotted in Figure.(5). This package fermionic solution gradually returns to 0, allowing us to employ the numerical long-time cutoff technique. One notable feature is that the bosonic component does not return to 0 in the long-time limit. However, it approaches a harmonic solution and becomes analytical, which we consider reliable when the number of oscillations is sufficiently large.  \nWe have plotted the 1-5 packages of the solution in Figure.(5). When we introduce an excitation at the initial Lorentz time, this signal information will periodically recover. Additionally, the excitation decays after recovery, exhibiting behavior similar to the dispersion of packages. The first revival excitation in Figure.(5)a occurs around Lorentz time t 1 = 58, and the nu-  \n-600  \n-400  \n-200  \n0  \n200  \n400  \n600  \n800  \nt  \n(a)  \nRe[G  \nRe[G  \n,AA  \n>  \nbb,AA  \n>  \nbb,AA  \nIm[G  \n>  \nIm[G  \nRe[G  \n>  \n>  \nb ,LR  \nIm[G  \n(t)]  \n(t)]  \n(t)]  \n,AA  \n(t)]  \nb,LR  \n(t)]  \n(t)]  \nRe[G  \nRe[G  \n,AA  \n>  \nbb,AA  \n>  \nbb,AA  \nIm[G  \n>  \nIm[G  \nRe[G  \n>  \n>  \nb ,LR  \nIm[G  \n(t)]  \n(t)]  \n(t)]  \n,AA  \n(t)]  \nb,LR  \n(t)]  \n(t)]  \n>  \n>  \nG  \n0.1  \n0.05  \n0  \n-0.05  \n-0.1  \nRe[G  \nRe[G  \n,AA  \n>  \nbb,AA  \n>  \nbb,AA  \nIm[G  \n>  \nIm[G  \nRe[G  \n>  \n>  \nb ,LR  \nIm[G  \n(t)]  \n(t)]  \n(t)]  \n,AA  \n(t)]  \nb,LR  \n(t)]  \n(t)]  \n>  \nmerical Green's function results between peaks are | max(Re( G ψψ )) -min(Re( G ψψ )) | = 0 . 219 and | max(Im( G ψψ )) -min(Im( G ψψ )) | = 0 . 212. The second revival excitation occurs around Lorentz time t 2 = 119 in Figure.(5)b, and we have | max(Re( G ψψ )) -min(Re( G ψψ )) | = 0 . 115 and | max(Im( G ψψ )) -min(Im( G ψψ )) | = 0 . 110. The third revival excitation occurs around Lorentz time t 3 = 178, and | max(Re( G ψψ )) -min(Re( G ψψ )) | = 0 . 080, | max(Im( G ψψ )) -min(Im( G ψψ )) | = 0 . 081. There are also several othe revival excitation t 4 = 234 , t 5 = 292 , t 6 = 349 from Figure.(5)a, and the peaks of the Green's functions decays with time.  \nFor N=4 SYK model, the real-time processes do not function effectively. We introduce a long-time cutoff following the periodic condition from 0 ∼ β . However, the bosonic ansatz G ϕϕ ∼ 0 . 5 | t | exhibits long-time divergence, rendering this saddle method unreliable directly. Consequently, we will further approximate this problem using the fully diagonal method in the next subsection.", '5.2 Out-of-time-ordered correlator': 'In this subsection, we investigate the causal and chaotic behaviors using the exact diagonal method and out-of-time-ordered correlator, which allows us to examine these characteristics in a finite system. Here, we primarily consider the correlation between different sectors with a non-zero Hamiltonian.  \nFirst, as in the previous section, we replace the fermions in the supercharges with creation and annihilation operators  \nψ i = C † i + C i .  \nThese operators is represented by 2 × 2 Pauli matrices. For the case of SYK model with N fermions on single side, we can represent the theory using a 4 N × 4 N Fock matrix.  \nHere we define the regularized out-of-time-ordered correlator. In order to compare the results with previous work[53, 54], we can shift the multi-side OTOCs by 1-F(t) and invert the phase factors  \nF ( t ) = 1 -∑ ij Z ( β ) Tr ( ρW ( t ) ρV (0) ρW ( t ) ρV (0)) Tr ( ρ 2 Wρ 2 W ) Tr ( ρ 2 V ρ 2 V ) , (157)  \nwhere the partition function Z ( β ) = Tr (exp ( -βH )) and the density of the finite temperature propagator ρ = exp( -βH/ 4). Here the operators W and V comprise the fermions. And W ( t ) represents the time-evolution of W .  \nIn N=1 theory, we can then define the fermionic operator  \nW ( t ) = exp ( iH † t )( C L † i + C L i ) exp( -iHt ) , V (0) = C R † j + C R j . (158)  \nand it can be treated as a special form of N=0 theory.  \n<!-- image -->  \nFigure 6: Averaged early-time OTOCs with 60 implement µ = 0 . 15 (a) multi-side OTOC between N=2 algebra ψ L and ¯ b R , (b) multi-side OTOC between N=4 algebra ϕ L and F R ,  \n<!-- image -->  \nAnalyzing the off-diagonal term in the Hamiltonian, we thus define the effective N=2 operators  \nW ( t ) = exp ( iH † t )( C L † i + C L i ) exp( -iHt ) , V (0) = ¯ b R j = { Q R , ¯ C R † j + ¯ C R j } = { Q R , ¯ c R † j +¯ c R j } . (159)  \nHere, we can simplify the chiral representation by the components C L i and ¯ C R † i or alternatively C L † i and ¯ C R i .  \nAnd for N=4 theory, the non-zero terms in Hamiltonian are  \nW ( t ) = exp ( iH † t ) ( ϕ L i ) exp( -iHt ) = exp ( iH † t )( c † + c ) αL i ( c † + c ) βL i exp( -iHt ) , V (0) = ¯ F Rj = ¯ b αR j ¯ b βR j = { Q, ¯ c † +¯ c } αL j { Q, ¯ c † +¯ c } βR j . (160)  \nHere we have plotted the early-time behaviors of the OTOCs in of N=2 and N=4 theory with 3 identical fermions in single-side supercharge. As shown in Figure.6, we calculate the average of 60 implementations for OTOCs. Our result shows that the early-time OTOCs with larger coupling J and inverse temperature β decay decay much slower, as indicated by the slopes. It is also suggested that a smaller Lyapunov exponent should be implied with larger coupling and lower temperature.', '6 Conclusion': "In this work, we have investigated the interaction terms between two identical supersymmetric SYK models.  \nWe first consider a special form of N=4 SYK model with a supermultiplet. This model is generated from two identical N=2 SYK models and is constrained by particle conservation, represented by Grassmann integration, which introduces additional auxiliary bosons. Then, we introduce first-order interaction terms in the N=1,2,4 SYK models using superfields, without breaking the boundary supersymmetry and solvability. In the thermal limit, we evaluate Green's function outside of super space. Subsequently, we analyze the free energy and energy. For N=1,2 theories, there also exists a Hawking-Page-like phase transition, which is also found in the N=0 SYK. However, unlike the N=0,1,2 SYK models, the N=4 SYK is primarily a bosonic model, and its phase structure does not involve a wormhole-black hole transition.  \nThe N=1,2 effective action in the low energy limit is reparameterized as superconformal and returns to the super Schwarzian. It should go back to the usual SYK when we ignore the supersymmetric part. The N=4 effective theory is also generated from two identical N=2 theories and is superconformal under certain conditions. In this work, the holographic picture of supersymmetric SYK is also discussed. We consider the supersymmetric JT gravity, which is a natural form of the supersymmetrized nAdS. One notable feature is that the great contribution of component ϕ in the IR limit makes the supersymmetric black holes in N=4 undergo an evaporation process.  \nWe also elaborated on these coupled N=1,2,4 SYKs regarding the entanglement property with thermal field double. By definition, these supersymmetric TFD states can be generated from the N=0 case. We have verified that the models with interaction are maximally entangled, and this property is not compromised after supersymmetrizing the SYK. Additionally, we verify the causality of the possible wormholes. The real-time revival dynamics are studied in N=2 theory, and we also calculate the exact diagonals of the N=2,4 theories and study the out-oftime-ordered correlator.", 'Acknowledgements': 'This work is supported by NSFC China(Grants No.12275166, No.11805117 and No.11875184)', 'Appendix A: Calculation of N=2,4 thermalfield double state': "N=2 TFD states are similar to N=1 and N=0 states. However, since we have introduced two independent supercharges but only one Hamiltonian constraint, the ground states are no longer parallel vacuum states but depend on the chiral anticommutation of supercharges. But we can still define maximal entangled states and the TFD states.  \n| TFD β ⟩ = 1 √ Z β ∑ n exp ( -β { Q, ¯ Q } n / 2 ) | n ⟩ 1 ⊗| ˆ n ⟩ 2 , (A.1)  \nwhile states that exhibit a double nature are relevant to chiral supercharges that possess maximal entanglement  \n| n ⟩ 1 = ( | Q ⟩ 1 ∣ ∣ ¯ Q 〉 1 ) n = (∣ ∣ ψ + i ¯ ψ 〉 1 ∣ ∣ ψ -i ¯ ψ 〉 1 ) n ,  \n| ˆ n ⟩ 2 = ( e -iηπΓ 4 e i ( q +¯ q ) ( ϕ -π 2 ) ) P (∣ ∣ ψ + i ¯ ψ 〉 2 ∣ ∣ ψ -i ¯ ψ 〉 2 ) , Γ = ( -1) -q -¯ q + N 2 , (A.2)  \nThe equations satisfy the anti-commute relations and we can build the corresponding Hilbert space to a specific type of entanglement  \n{ ψ i + i ¯ ψ i , ψ j + i ¯ ψ j } = i ( δ i j + δ j i ) , { ψ i -i ¯ ψ i , ψ j -i ¯ ψ j } = -i ( δ i j + δ j i ) { ψ i -i ¯ ψ i , ψ j + i ¯ ψ j } = i ( δ i j -δ j i ) = 0 , { ψ i + i ¯ ψ i , ψ j -i ¯ ψ j } = -i ( δ i j -δ j i ) = 0 . (A.3)  \nAlgebra can be written using creation and annihilation operators  \nψ i = c i † + c i ¯ ψ i = ¯ c i † +¯ c i C † = c i † + i ¯ c i , C = c i + i ¯ c † i , ¯ C = c i † -i ¯ c i , ¯ C † = c i -i ¯ c † i , (A.4)  \nAnd we can further simplify the algebra with the N=2 anti-commute relation  \nψ i = c i † ¯ ψ i = c i C † = (1 + i ) c i † , C = 0 , ¯ C † = 0 , ¯ C = (1 -i ) c i , (A.5)  \nwhich could be generated from N=1 symmetry breaking. This is a simple form of supersymmetric algebra, and we can also utilize the chiral conjugate by deleting C † and ¯ C .  \nrecalling  \nΨ i = ψ i + θ { ¯ Q,ψ i } , ¯ Ψ i = ¯ ψ i + ¯ θ { Q, ¯ ψ i } . (A.7)  \nWe can rewrite the interaction terms with complex chiral fields  \nH int = iµ∂ θ ( Ψ Li ¯ Ψ R i -Ψ Ri ¯ Ψ L i ) ,  \nand  \n¯ H int = iµ∂ ¯ θ ( ¯ Ψ L i Ψ Ri -¯ Ψ R i Ψ Li ) , (A.8)  \nIn N=2 superspace, the fermion algebra involves the anti-commutation relations between the creation and annihilation operators D or ¯ D  \nH int = iµe -iϕ ∂ θ ( ψ i + θ { ¯ Q,ψ i }) L ( ¯ ψ i + ¯ θ { Q, ¯ ψ i }) R -iµe iϕ ∂ θ ( ψ i + θ { ¯ Q,ψ i }) R ( ¯ ψ i + ¯ θ { Q, ¯ ψ i }) L , = iµe -iϕ { ¯ Q,ψ i } L ¯ ψ R i -iµe iϕ { ¯ Q,ψ i } R ¯ ψ L i = iµe -iϕ { ¯ Q, ( c i † + c i )} L (¯ c i † +¯ c i ) R -iµe iϕ { ¯ Q, ( c i † + c i )} R (¯ c i † +¯ c i ) L = iµe -iϕ { ¯ Q, ( C i † + ¯ C i † + C i + ¯ C i ) 2 } L ( C i † -¯ C i † + C i -¯ C i 2 i ) R -iµe iϕ { ¯ Q, ( C i † + ¯ C i † + C i + ¯ C i ) 2 } R ( C i † -¯ C i † + C i -¯ C i 2 i ) L ¯ H int = iµe -iϕ ∂ ¯ θ ( ¯ ψ i + ¯ θ { Q, ¯ ψ i }) L ( ψ i + θ { ¯ Q,ψ i }) R -iµe iϕ ∂ ¯ θ ( ¯ ψ i + ¯ θ { Q, ¯ ψ i }) R ( ψ i + θ { ¯ Q,ψ i }) L . = iµe -iϕ { Q, ¯ ψ i } L ψ iR -iµ { Q, ¯ ψ i } R ψ iL = iµe -iϕ { Q, (¯ c i † +¯ c i ) } L ( c i † + c i ) R -iµe iϕ { Q, (¯ c i † +¯ c i ) } R ( c i † + c i ) L = iµe -iϕ { Q, C i † -¯ C i † + C i -¯ C i 2 i } L ( ( C i † + ¯ C i † + C i + ¯ C i ) 2 ) R -iµe iϕ { Q, C i † -¯ C i † + C i -¯ C i 2 i } R ( ( C i † + ¯ C i † + C i + ¯ C i ) 2 ) L (A.9)  \n〈 ˆ n -q ' ∣ ∣ ∣ C R i ∣ ∣ ∣ n ' q ''' 〉 = -e iπ ( -q -¯ q + N 2 ) e -iϕ 〈 m q ' ∣ ∣ ∣ C L i ∣ ∣ ∣ n q ' 〉 , 〈 ˆ n -q ' ∣ ∣ ∣ ¯ C R i ∣ ∣ ∣ n ' q ''' 〉 = -e iπ ( -q -¯ q + N 2 ) e -iϕ 〈 m q ' ∣ ∣ ∣ ¯ C L i ∣ ∣ ∣ n q ' 〉 , P -1 C i P = ηP † i . (A.6)  \nAnd we can still simplify them into an algebraic form  \nH int = iµe -iϕ { ¯ Q, C i † 2 } L ( C i 2 i ) R -iµe -iϕ { ¯ Q, ¯ C i † 2 } L ( ¯ C i 2 i ) R -iµe iϕ { ¯ Q, C i † 2 } R ( C i 2 i ) L + iµe iϕ { ¯ Q, ¯ C i † 2 } R ( ¯ C i 2 i ) L ¯ H int = iµe -iϕ { Q, C i † 2 i } L ( C i 2 ) R -iµe -iϕ { Q, ¯ C i † 2 i } L ( ¯ C i 2 ) R -iµe iϕ { Q, C i † 2 i } R ( C i 2 ) L + iµe iϕ { Q, ¯ C i † 2 i } R ( ¯ C i 2 ) L (A.10)  \nSince N=2 Hamiltonian eigenstates are created by applying the appropriate creation and annihilation operators  \nH ∣ ∣ Ψ r m , ¯ Ψ r n 〉 = { Q, ¯ Q }∣ ∣ Ψ r m , ¯ Ψ r n 〉 = E mn ∣ ∣ Ψ r m , ¯ Ψ r n 〉 , H † ∣ ∣ ∣ Ψ l m , ¯ Ψ l n 〉 = { Q, ¯ Q } † ∣ ∣ Ψ r m , ¯ Ψ r n 〉 = E ∗ mn ∣ ∣ ∣ Ψ l m , ¯ Ψ l n 〉 , (A.11)  \nwith  \n∑ n,k ∣ ∣ ∣ Ψ l n , ¯ Ψ l k 〉 〈 Ψ r n , ¯ Ψ r k ∣ ∣ = I, 〈 Ψ l n , ¯ Ψ l k ∣ ∣ ∣ Ψ r m , ¯ Ψ r l 〉 = δ nm δ kl . (A.12)  \nAnd we can further propose the ground states  \n∣ ∣ Ψ 0 , ¯ Ψ 0 〉 = N ∏ j ( 1 √ 2 | 1 ⟩ L,j | 0 ⟩ R,j + i √ 2 | 0 ⟩ L,j | 1 ⟩ R,j ) = N ∏ j ∑ abcd ( 1 √ 2 | a ± 1 , a ⟩ L,j | b, b ⟩ R,j + i √ 2 | c, c ⟩ L,j | d ± 1 , d ⟩ R,j ) , (A.13)  \nwith  \nH † int ∣ ∣ ∣ Ψ l 0 , ¯ Ψ l 0 〉 = -µN ( 1 + θ ¯ Q 0 ) ( 1 + ¯ θQ 0 ) ∣ ∣ ∣ Ψ l 0 , ¯ Ψ l 0 〉 , H int ∣ ∣ Ψ r 0 , ¯ Ψ r 0 〉 = -µN ( 1 + θ ¯ Q 0 ) ( 1 + ¯ θQ 0 ) ∣ ∣ Ψ r 0 , ¯ Ψ r 0 〉 . (A.14)  \nThe Q 0 is a constant which is relevant to the ground state supercharges. And we use abcd to represent the excited states with abcd energy levels in each supercharge Q L , ¯ Q L and Q R , ¯ Q R . We denote Q = ¯ Q in the N=2 theory in vacuum, while the excitation of Q is higher than ¯ Q by one, these are the first excited states, vice versa.  \nWe have imposed the entangled states in the N=2 TFD  \n| m q ⟩ 1 ⊗| ¯ m -q ⟩ 2 = N ∏ j ∑ q ∑ kl ( | k q ± m q , k q ⟩ 1 ,j | l -q ± m -q , l -q ⟩ 2 ,j ) (A.15)  \nIndicate that  \n⟨ TFD β =0 | H int | TFD β =0 ⟩ = 1 2 N ∑ qq ' ∑ mn 〈 ˆ n -q ' ∣ ∣ ∣ 2 ⊗ 〈 n q ' ∣ ∣ ∣ 1 ( iµ ∑ i ( e -iϕ { ¯ Q, C i † 2 } L ( C i 2 i ) R -iµe -iϕ { ¯ Q, ¯ C i † 2 } L ( ¯ C i 2 i ) R -iµe iϕ { ¯ Q, C i † 2 } R ( C i 2 i ) L + iµe iϕ { ¯ Q, ¯ C i † 2 } R ( ¯ C i 2 i ) L )) | m q ⟩ 1 ⊗| ˆ m -q ⟩ 2 = 1 2 N ∑ qq ' ∑ nm ∑ i ( iµe iπ ( -q -¯ q + N 2 ) 〈 n q ' ∣ ∣ ∣ { ¯ Q, C i † 2 } L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i 2 i ) R | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ { ¯ Q, ¯ C i † 2 } L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i 2 i ) R | ˆ m -q ⟩ -iµe iπ ( -q ' -¯ q ' + N 2 ) e iϕ 〈 ˆ n -q ' ∣ ∣ ∣ { ¯ Q, C i † 2 } R ∣ ∣ ∣ n ' q ''' 〉 〈 n q ' ∣ ∣ ∣ ( C i 2 i ) L | m q ⟩ + iµe iπ ( -q ' -¯ q ' + N 2 ) e iϕ 〈 ˆ n -q ' ∣ ∣ ∣ { ¯ Q, ¯ C i † 2 } R ∣ ∣ ∣ n ' q ''' 〉〈 n q ' ∣ ∣ ∣ ( ¯ C i 2 i ) L | m q ⟩ ) (A.16)  \nand  \n⟨  \nH int ⟩ = 1 2 N ∑ qq ' ∑ nm ∑ i {-iµe -iϕ 〈 n q ' ∣ ∣ ∣ { ¯ Q, C i † 2 } L | m q ⟩ ⟨ m q | ( C i 2 i ) L ∣ ∣ ∣ n q ' 〉 + iµe -iϕ 〈 n q ' ∣ ∣ ∣ { ¯ Q, ¯ C i † 2 } L | m q ⟩ ⟨ m q | ( ¯ C i 2 i ) L ∣ ∣ ∣ n q ' 〉 + iµe iϕ ⟨ m q | { ¯ Q, C i † 2 } L ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( C i 2 i ) L | m q ⟩ -iµe iϕ ⟨ m q | { ¯ Q, ¯ C i † 2 } L ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( ¯ C i 2 i ) L | m q ⟩ = ∑ qq ' ∑ nm ∑ i iµ/ 2 N 〈 n q ' ∣ ∣ ∣ { ¯ Q, C i † 2 } L | m q ⟩ ⟨ m q | ( C i 2 i ) L ∣ ∣ ∣ n q ' 〉 -〈 n q ' ∣ ∣ ∣ { ¯ Q, ¯ C i † 2 } L | m q ⟩ ⟨ m q | ( ¯ C i 2 i ) L ∣ ∣ ∣ n q ' 〉 = -µ 2 N ∑ n,q ∑ i ⟨ n q | 1 { ¯ Q, C i † 2 } L ( C i 2 i ) L -{ ¯ Q, ¯ C i † 2 } L ( ¯ C i 2 i ) L | n q ⟩ 1 = Const.  \n(A.17)  \nThe Hamiltonian of N=4 SYK model is  \nH = ϵ αβ ψ α ψ β { ¯ Q α , ψ α } 2 ϵ αβ ψ α ψ β + ϵ αβ ψ α ψ β { ¯ Q α , ψ α } ψ α { ¯ Q α , ψ α } ψ β ϵ αβ + h.c = { ¯ Q α , Q α } { ¯ Q β , Q β } + T { ¯ Q α , Q α } { ¯ Q β , Q β } + h.c = { ¯ Q α , Q α } { ¯ Q β , Q β } + h.c (A.18)  \nTherefore, the N=4 supersymmetric SYK thermal field double is defined as  \n| TFD β ⟩ = 1 √ Z β ∑ n exp ( -β { ¯ Q α , Q α } { ¯ Q β , Q β } n / 2 ) | n ⟩ 1 ⊗| ˆ n ⟩ 2 , (A.19)  \nThe Hilbert space is also contains the eigenvectors of the supercharges  \n| n ⟩ 1 = ( | Q ⟩ α 1 ∣ ∣ ¯ Q 〉 α 1 | Q ⟩ β 1 ∣ ∣ ¯ Q 〉 β 1 ) n ∼ ( ∣ ∣ ψ + i ¯ ψ 〉 α 1 ∣ ∣ ψ -i ¯ ψ 〉 α 1 ∣ ∣ ψ + i ¯ ψ 〉 β 1 ∣ ∣ ψ -i ¯ ψ 〉 β 1 ) n , | ˆ n ⟩ 2 = ( e -iηπΓ 4 e i ( q α +¯ q α + q β +¯ q β )( ϕ -π 2 ) ) P ( ∣ ∣ ψ + i ¯ ψ 〉 α 2 ∣ ∣ ψ -i ¯ ψ 〉 α 2 ∣ ∣ ψ + i ¯ ψ 〉 β 2 ∣ ∣ ψ -i ¯ ψ 〉 β 2 ) , Γ = ( -1) -q α -¯ q α -q β -¯ q β + N 2 , (A.20)  \nHere we have defined sets of new eigenvectors, which are also the eigenvectors of Hamiltonian. It is straightforward to verify that these new eigenvectors satisfy the properties of being eigenvectors of the Hamiltonian  \n{ ψ i α + i ¯ ψ αi , ψ j β + i ¯ ψ βj } = i ( δ i j + δ j i ) δ αβ , { ψ i α -i ¯ ψ αi , ψ j β -i ¯ ψ βj } = -i ( δ i j + δ j i ) δ αβ { ψ i α -i ¯ ψ αi , ψ j β + i ¯ ψ βj } = i ( δ i j -δ j i ) δ αβ = 0 , { ψ i α + i ¯ ψ αi , ψ j β -i ¯ ψ βj } = -i ( δ i j -δ j i ) δ αβ = 0 . (A.21)  \nWe can also rewrite these operators in the Hilbert space  \nψ i α = c i † α + c i α ¯ ψ αi = ¯ c † αi +¯ c αi C i α † = c i † α + i ¯ c αi , C αi = c i α + i ¯ c † αi , ¯ C i † α = c i † α -i ¯ c αi , ¯ C i α = c i α -i ¯ c † αi , (A.22)  \nwith  \n〈 ˆ n -q ' ∣ ∣ ∣ C R αi ∣ ∣ ∣ n ' q ''' 〉 = -e iπ ( -q α -¯ q α -q β -¯ q β + N 2 ) e -iϕ 〈 m q ' ∣ ∣ ∣ C L αi ∣ ∣ ∣ n q ' 〉 , 〈 ˆ n -q ' ∣ ∣ ∣ ¯ C R αi ∣ ∣ ∣ n ' q ''' 〉 = -e iπ ( -q α -¯ q α -q β -¯ q β + N 2 ) e -iϕ 〈 m q ' ∣ ∣ ∣ ¯ C L αi ∣ ∣ ∣ n q ' 〉 , P -1 C i P = ηP † i . (A.23)  \nWe can further introduce the interaction Hamiltonian into the superspace  \nΦ = ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β } ¯ Φ = ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β } (A.24)  \nThe interaction Hamiltonian is defined  \nH int = ∂ 2 θ ( Φ Li ¯ Φ R i + Φ Ri ¯ Φ L i ) ¯ H int = ∂ 2 ¯ θ ( ¯ Φ L i Φ Ri + ¯ Φ R i Φ Li ) , (A.25)  \nand  \nH int = e -iϕ ∂ 2 θ ( ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β }) L ( ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β }) R + e iϕ ∂ 2 θ ( ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β }) R ( ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β }) L , = iµe -iϕ ( { ¯ Q α , ψ α } { ¯ Q β , ψ β }) L ( ¯ ψ α ¯ ψ β ) R + iµe iϕ ( { ¯ Q α , ψ α } { ¯ Q β , ψ β }) R ( ¯ ψ α ¯ ψ β ) L = iµe -iϕ ({ ¯ Q α , ( c i † α + c i α )}{ ¯ Q β , ( c i † β + c i β )}) L (( ¯ c † αi +¯ c αi )( ¯ c † βi +¯ c βi )) R + iµe iϕ ({ ¯ Q α , ( c i † α + c i α )}{ ¯ Q β , ( c i † β + c i β )}) R (( ¯ c † αi +¯ c αi )( ¯ c † βi +¯ c βi )) L = iµe -iϕ ({ ¯ Q α , ( C i † α + ¯ C i † α + C i α + ¯ C i α 2 )}{ ¯ Q β , ( C i † β + ¯ C i † β + C i β + ¯ C i β 2 )}) L (( C i † α -¯ C i † α + C i α -¯ C i α 2 i )( C i † β -¯ C i † β + C i β -¯ C i β 2 i )) R + iµe iϕ ({ ¯ Q α , ( C i † α + ¯ C i † α + C i α + ¯ C i α 2 )}{ ¯ Q β , ( C i † β + ¯ C i † β + C i β + ¯ C i β 2 )}) R (( C i † α -¯ C i † α + C i α -¯ C i α 2 i )( C i † β -¯ C i † β + C i β -¯ C i β 2 i )) L (A.26)  \nAnd  \n¯ H int = e -iϕ ∂ 2 θ ( ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β }) L ( ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β }) R + e iϕ ∂ 2 θ ( ¯ ψ α ¯ ψ β -¯ θ α ¯ ψ β { Q α , ¯ ψ α } -¯ θ β ¯ ψ α { Q β , ¯ ψ β } + ¯ θ α ¯ θ β { Q α , ¯ ψ α } { Q β , ¯ ψ β }) R ( ψ α ψ β + θ α ψ β { ¯ Q α , ψ α } + θ β ψ α { ¯ Q β , ψ β } + θ α θ β { ¯ Q α , ψ α } { ¯ Q β , ψ β }) L , = iµe -iϕ ( { Q α , ¯ ψ α } α { Q β , ¯ ψ β }) L ( ψ α ψ β ) R + iµe iϕ ( { Q α , ¯ ψ α } { Q β , ¯ ψ β }) R ( ψ α ψ β ) L = iµe -iϕ ({ ¯ Q α , ( ¯ c † αi +¯ c αi )}{ ¯ Q β , ( ¯ c † βi +¯ c βi )}) L (( c i † α + c i α )( c i † β + c i β )) R + iµe iϕ ({ ¯ Q α , ( ¯ c † αi +¯ c αi )}{ ¯ Q β , ( ¯ c † βi +¯ c βi )}) R (( c i † α + c i α )( c i † β + c i β )) L = iµe -iϕ ({ ¯ Q α , ( C i † α -¯ C i † α + C i α -¯ C i α 2 i )}{ ¯ Q β , ( C i † β -¯ C i † β + C i β -¯ C i β 2 i )}) L (( C i † α + ¯ C i † α + C i α + ¯ C i α 2 )( C i † β + ¯ C i † β + C i β + ¯ C i β 2 )) R + iµe iϕ ({ ¯ Q α , ( C i † α -¯ C i † α + C i α -¯ C i α 2 i )}{ ¯ Q β , ( C i † β -¯ C i † β + C i β -¯ C i β 2 i )}) R (( C i † α + ¯ C i † α + C i α + ¯ C i α 2 )( C i † β + ¯ C i † β + C i β + ¯ C i β 2 )) L (A.27)  \nIn this section, we aim to investigate whether the interaction Hamiltonian affects the supersymmetry thermal field double states, and we should concentrate on the diagonal terms within the Hilbert space  \nH int = iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( C i α 2 i C i β 2 i ) R + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( ¯ C i α 2 i ¯ C i β 2 i ) R -iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( C i α 2 i ¯ C i β 2 i ) R -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( ¯ C i α 2 i C i β 2 i ) R + iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) R ( C i α 2 i C i β 2 i ) L + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ( ¯ C i α 2 i ¯ C i β 2 i ) L -iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ( C i α 2 i ¯ C i β 2 i ) L -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) R ( ¯ C i α 2 i C i β 2 i ) L (A.28)  \n¯ H int = iµe -iϕ ({ ¯ Q α , C i † α 2 i }{ ¯ Q β , C i † β 2 i }) L ( C i α 2 C i β 2 ) R + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 i }{ ¯ Q β , ¯ C i † β 2 i }) L ( ¯ C i α 2 ¯ C i β 2 ) R -iµe -iϕ ({ ¯ Q α , C i † α 2 i }{ ¯ Q β , ¯ C i † β 2 i }) L ( C i α 2 ¯ C i β 2 ) R -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 i }{ ¯ Q β , C i † β 2 i }) L ( ¯ C i α 2 C i β 2 ) R + iµe -iϕ ({ ¯ Q α , C i † α 2 i }{ ¯ Q β , C i † β 2 i }) R ( C i α 2 C i β 2 ) L + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 i }{ ¯ Q β , ¯ C i † β 2 i }) R ( ¯ C i α 2 ¯ C i β 2 ) L -iµe -iϕ ({ ¯ Q α , C i † α 2 i }{ ¯ Q β ¯ C i † β 2 i }) R ( C i α 2 ¯ C i β 2 ) L -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 i }{ ¯ Q β , C i † β 2 i }) R ( ¯ C i α 2 C i β 2 ) L (A.29)  \nThen the turning terms give  \n⟨ TFD β =0 | H int | TFD β =0 ⟩ = 1 2 N ∑ qq ' ∑ mn 〈 ˆ n -q ' ∣ ∣ ∣ 2 ⊗ 〈 n q ' ∣ ∣ ∣ 1   iµ ∑ i   e -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( C i α 2 i C i β 2 i ) + iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( ¯ C i α 2 i ¯ C i β 2 i ) R -iµe -iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( C i α 2 i ¯ C i β 2 i ) R -iµe -iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( ¯ C i α 2 i C i β 2 i ) R + iµe iϕ ({ ¯ Q α C i † α 2 }{ ¯ Q β , C i † β 2 }) R ( C i α 2 i C i β 2 i ) L + iµe iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ( ¯ C i α 2 i ¯ C i β 2 i ) L -iµe iϕ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ( C i α 2 i ¯ C i β 2 i ) L -iµe iϕ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) R ( ¯ C i α 2 i C i β 2 i ) L     | m q ⟩ 1 ⊗| ˆ m -q ⟩ 2 = 1 2 N ∑ qq ' ∑ nm ∑ i ( iµe iπ ( -q α -¯ q α -q β -¯ q β + N 2 ) 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i α 2 i C i β 2 i ) R | ˆ m -q ⟩ + 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i α 2 i ¯ C i β 2 i ) R | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i α 2 i ¯ C i β 2 i ) R | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i α 2 i C i β 2 i ) R | ˆ m -q ⟩ + iµe iπ ( -q α -¯ q α -q β -¯ q β + N 2 ) e iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) R ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i α 2 i C i β 2 i ) L | ˆ m -q ⟩ + 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i α 2 i ¯ C i β 2 i ) L | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( C i α 2 i ¯ C i β 2 i ) L | ˆ m -q ⟩ -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) R ∣ ∣ ∣ m ' q '' 〉〈 ˆ n -q ' ∣ ∣ ∣ ( ¯ C i α 2 i C i β 2 i ) L | ˆ m -q ⟩ (A.30)  \n⟨ H int ⟩ = 1 2 N ∑ qq ' ∑ nm ∑ i [ -iµe -iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L | m q ⟩ ⟨ m q | ( C i α 2 i C i β 2 i ) R ∣ ∣ ∣ n q ' 〉 -iµe -iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L | m q ⟩ ⟨ m q | ( ¯ C i α 2 i ¯ C i β 2 i ) R ∣ ∣ ∣ n q ' 〉 + iµe -iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L | m q ⟩ ⟨ m q | ( C i α 2 i ¯ C i β 2 i ) R ∣ ∣ ∣ n q ' 〉 + iµe -iϕ 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L | m q ⟩ ⟨ m q | ( ¯ C i α 2 i C i β 2 i ) R ∣ ∣ ∣ n q ' 〉 -iµe iϕ ⟨ m q | ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) R ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( C i α 2 i C i β 2 i ) L | m q ⟩ -iµe iϕ ⟨ m q | ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( ¯ C i α 2 i ¯ C i β 2 i ) L | m q ⟩ + iµe -iϕ ⟨ m q | ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) R ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( C i α 2 i ¯ C i β 2 i ) L | m q ⟩ + iµe -iϕ ⟨ m q | ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) R ∣ ∣ ∣ n q ' 〉〈 n q ' ∣ ∣ ∣ ( ¯ C i α 2 i C i β 2 i ) L | m q ⟩ = ∑ qq ' ∑ nm ∑ i iµ/ 2 N 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L | m q ⟩ ⟨ m q | ( C i α 2 i C i β 2 i ) L ∣ ∣ ∣ n q ' 〉 + 〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L | m q ⟩ ⟨ m q | ( ¯ C i α 2 i ¯ C i β 2 i ) L ∣ ∣ ∣ n q ' 〉 -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L | m q ⟩ ⟨ m q | ( C i α 2 i ¯ C i β 2 i ) L ∣ ∣ ∣ n q ' 〉 -〈 n q ' ∣ ∣ ∣ ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L | m q ⟩ ⟨ m q | ( ¯ C i α 2 i C i β 2 i ) L ∣ ∣ ∣ n q ' 〉 = -µ 2 N ∑ n,q ∑ i ⟨ n q | 1 ({ ¯ Q α , C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( C i α 2 i C i β 2 i ) L + ({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( ¯ C i α 2 i ¯ C i β 2 i ) L | n q ⟩ 1 -({ ¯ Q α , C i † α 2 }{ ¯ Q β , ¯ C i † β 2 }) L ( C i α 2 i ¯ C i β 2 i ) L | n q ⟩ 1 -({ ¯ Q α , ¯ C i † α 2 }{ ¯ Q β , C i † β 2 }) L ( ¯ C i α 2 i C i β 2 i ) L | n q ⟩ 1 = Const. 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