Schwarzian correction to quantum correlation in SYK model

We study a class of SYK-type models in large N limit from the gravity dual side in terms of Schwarzian action analytically. The quantum correction to two point correlation function due to the Schwarzian action produces transfer of degree of freedom from the quasiparticle peak to Hubbard band in density of states (DOS), a signature strong correlation. In Schwinger-Keldysh (SK) formalism, we calculate higher point thermal out-of-time order correlation (OTOC) functions, which indicate quantum chaos by having Lyapunov exponent. Higher order local spin-spin correlations are also calculated, which can be related to the dynamical local susceptibility of quantum liquids such as spin glasses, disordered metals.

There are three novel features of SYK model. The first one is the solvability at large N in the strong coupling limit. The second one is the emergence of conformal symmetry at IR limit, as well as its spontaneous breaking which results in soft modes as the pseudo Nambu-Goldstone bosons (pNGBs) [7,23,32]: we know it is dynamically broken since the Virasoro symmetry of the boundary is broken to SL(2,R) in its bulk dual. The third one is the quantum chaos behavior in 4-point correlation functions.
The Schwarzian action is determined by the pattern of spontaneous breaking of reparametrization symmetry [7]. It has been conjectured that the gravity dual of the SYK model can be described by a 2-D dilaton gravity [6]: one example is the Jakiew-Teitelboim (JT) model [36,37] [6,7]. Another example is the Almheiri-Polchinsky (AP) model [38] [8]. The low energy quantum description of SYK model is proposed to be holographically dual to a (1 + 1)-dimensional model of black hole [5,23], although it is not completely conventional AdS/CFT.
Higher dimensional generalization of SYK model -There are several way to generalize the SYK model, one is its generalization in higher dimensional spacetime [11,13,16,22,25]. It has been argued that the SYK model has a 3-D interpretation of a bulk massive scalar, which is subject to a delta function potential at the center of the interval along the extra/third dimension being a dilaton [25]. After standard S 1 /Z 2 compactification, an infinite series of Kaluza-Klein (KK) tower [39] mass spectrum of a bulk massive scalar exactly matches the strong coupling (J → ∞) spectrum of the SYK model [25], while higher point correlation functions might still not be matching [31]. To be specific, it has been shown that the asymptotic three-point function of SYK model has a string-like bulk interaction [21]. It turns out that the interaction grows much faster than that of massive KK scalar with cubic coupling, which is overlaps of the wavefunctions along the S 1 in a AdS 3 ≃ AdS 2 × S 1 bulk [31].
As a result, the KK description from bulk might be broken down in dealing with strongly correlated system.
It has also been claimed that the bulk dual to the soft mode sector of SYK model might be realized through the KK reduction from 3-D Maxwell-Einstein gravity [10,35], which leads to the 2-D JT model, where the dilaton is proportional to the KK radius. However, it is still insufficient to confirm whether there is a local bulk dual for the full SYK model.
Flavor symmetry generalization of SYK model -An alternative generalization is one in flavor symmetry spacetime, which generalize the four-majorana fermion interactions of SYK-like model to fermion with p-body interactions [12]. Since in (0+1) dimensional SYK model, the Majorana field ψ is dimensionless ([ψ] = 0), the coupling J is always dimensionful ([J] = 1). Thus, the p ≥ 4-fermion interaction is always relevant at UV. While in general this is not true in D ≥ 2, since the UV relevance of the p-fermions interactions depends on the dimension of the spacetime. For example, in a generalization of (1 + 1)-dimensional SYK model, in analogy to the 2-D Gross-Neveu model, it can be obtained by integrating out the tensor field that coupled with a new vector field [24]. In this case, the fermion field is dimensionful ([ψ] = 1/2), meanwhile the coupling is dimensionless ([J] = 0). Thus, the four fermion interaction term is marginal. Generally speaking, the generalization to more higher dimension (D ≥ 3), will inevitably lead to irrelevant p-fermion interactions.
Therefore, it might not be necessary to stick to the original SYK model [34], but instead focus on a general class of SYK-like models in (0 + 1)-dimensional spacetime with three most notable universal features: solvability in the IR and large N limit, maximal chaotic behavior and emergent conformal theory spontaneously broken at IR. Few studies, however, have examined the effects of 1-D Schwarzian correction to the quantum correlations functions and none, to our knowledge, have compared the DOS of quantum liquid such as spin glasses, non-Fermi liquid.
In this paper, motivated by the novel features of a class of quantum spin glass or disordered metals depicted by SYK-like models [43][44][45], we study a strongly interacting (0 + 1)-dimensional quantum mechanical model at large N limit, whose effective action is Schwarzian one. As it will be shown, the model owns a Hubbard band in the spectral function by transferring the degree of freedoms from quasi particle peak to side band, a Hall mark of the strongly interacting systems. Such features are observed in the spectral functions of several fcc A 3 C 60 system as well as in the transition metal Oxides. We compare with the density functional theory (DFT) [40] DOS results as Fig.4B in Ref. [53]. The interesting point is that our analytical results on the 0 dimensional system quantitatively reproduces the density of state of the dynamical mean-field theory (DMFT) [41] approach.
The paper is organized as follow. In Sec.II, we study the correlation functions from Schwarzian action at both zero and finite temperature. In Sec.III, we study the zero temperature and thermal retarded Green's functions with loop correction from pNGBs, and local dynamical susceptibility of quantum liquid. The higher order local spin-spin correlation function beyond local susceptibility are also investigated. Generalization of AdS 2 spacetime as near IR horizon of RN black hole in AdS d+1 spacetime is studied in Appendix.C. In Sec.IV, we study higher point correlation functions, the thermal OTOCs functions in SK formalism.

II. CORRELATION FUNCTIONS FROM SCHWARZIAN
In this section, based on the low energy effective action of the Schwarzian theory of time reparametrization from 2-D gravity, we calculate the correlation functions, especially the 2-point one [58].

A. The action of the model
The prototypical SYK model is described by the partition function Z(J) =´Dψ i e −S , with an action S =ˆdτ 1 2 S eff = −C gˆd tφ r (t)Sch(f (t), t), (II.2) C g is a constant depending on the bulk gravity parameter, t is the boundary time coordinate, f (t) is the field variable, φ r (t) is the normalizable part of the dilaton, which is a constant on the cutoff boundary and plays a role of external coupling, while the divergent part that blows up at boundary is absorbed by a counter term, and can be identified as the source in NAdS 2 /NCFT 1 description, Sch(f, t) is the Schwarzian derivative defined as where the prime ′ denotes the derivative with respect to t. The zero modes is described by the Schwarzian action.
By doing variation with respect to f (t), the action becomes δS eff ∼ −C gˆφr (t)dt [Sch(f (t), t)] ′ f ′ δf, (II. 4) and by using the property as and that δ(f ′ ) −1 = −(f ′ ) −2 δf ′ , one obtains the field equation of motion with respect to t(s) turns out to be which becomes [Sch(f, t)] ′ /t ′ = 0 when φ r is a constant. One of most simplest but non-trivial solution might be a non-constant functions with constant Schwarzian.

Zero temperature soft mode propagator
Consider a linear transformation f (τ ) = τ (t), then according to the composition rule of Schwarzian derivative as Sch(g(f ), t) = f ′2 Sch(g(f ), f ) + Sch(f, t), (II. 7) where g(f ) = g(f (t)) and f = f (t), one has Sch(f, t) = τ ′2 Sch(f, τ ) + Sch(τ, t), where Sch(f, τ ) = 1/2, when τ is a linear function of t, Sch(t, s) is constant and satisfies the equation of motion of the Schwarzian action, i.e., Sch(τ, t) ′ /f ′ = 0. In the perturbative approach, one can set τ (t) ≡ t + ǫk(t), (II. 8) where t = it is imaginary time and ǫ ≪ 1 is the expansion parameter, which can be chosen as the bulk gravitational interaction coupling, i.e., ǫ = κ N ∼ √ G N , which is proportional to G 1/2 N in gravity or large N −1/2 as in the SYK model. By expanding the Schwarzian action, one has By dropping total derivative term, the leading order action at ǫ 2 order reads as where C ∝φ r . By doing a Fourier transformation k(t) = n k n e int , the action becomes (II.11) where hence and forth, the Einstein summation notation convention is applied for repeated n ∈ Z. Thus, the soft mode propagator can be obtained where t ≡ t 1 − t 2 and we have used that where z = e it . Li n (x) is the the polylogarithmic function defined by the series Li n (x) = n k=1 z k /k n for |z| < 1 and B n (x) is the Bernoulli polynomial.

Thermal soft mode propagator
Consider a thermal circle transformation f (τ ) = tan(πτ /β) satisfying f (τ + β) = f (τ ) with period length β = 2π, in other words, by imposing a mapping from then the composition rule of Schwarzian derivative in Eq.(II.7) leads to an action results as Sch(f, t) = τ ′2 /2 + Sch(τ, t), when τ is a linear function of t, Sch(t, s) is constant and satisfies the equation of motion of the Schwarzian action, i.e., Sch(f, t) ′ /f ′ = 0. This can be traced back to the bulk equation of motion, which gives the dilaton solution.
In the perturbative approach as in Eq.(II.8), after expanding the effective Schwarzian action, higher order selfinteraction terms for k(τ (t)) are present, which is suppressed by factor of ǫ. By expanding the Schwarzian action, one has By dropping total derivative term, the leading order action at ǫ 2 order reads as where C ∝ C gφr . By doing a Fourier transformation as k(t) = n k n e int where t ∈ [0, 2π], the action becomes where for repeated n the Einstein summation convention is applied as stated before. Thus, the soft mode propagator can be obtained as where t ≡ t 1 − t 2 and we have used that C. Effective action of matter and two point correlation functions The n-point function of a matter field, e.g., a scalar Φ in NAdS 2 , can be computed by coupling the matter field to the bulk gravity in AdS 2 , and then rewriting the action by using f (t), and rescaling by a factor f ′ (t) ∆ at the insertion of each operator.
For a massive scalar Φ in AdS 2 spacetime in Poincaré coordinate, since all gravitational configurations in 2dimentional spacetime can be described by the metric, and the effective action is [7] and we have used the asymptotic behavior of Φ at boundary where Φ 0 (t) can be viewed as a source for a scalar operator with conformal dimension ∆, e.g., for free scalar in pure AdS 2 /CFT 1 case, ∆ = 1/2 + 1/4 + m 2 ≥ 1. Consider that the trajectory of the boundary curve is f (t), which can be transformed to the desired boundary conditions as Then, the effective action can be re-parameterized as The two point function of the dual field O(t) to the source Φ 0 can be read as

D. Thermal correlation functions
Consider expand the boundary t around the saddle of a thermal circle, according to Eqs.(II.14) and (II.8) as where we have dropped a common factor 2π/β, so to recover one has to rescale t → 2π/βt. The two point function of the dual operators can be expanded as where C n (t ij ) ≡ C n (t i , t j ). By neglecting the perturbation expansion term, ǫ → 0, the AdS 2 thermal two point functions is recovered as 26) where in the last equality, we have recovered the thermal factor t 12 → (2π)/βt 12 . The leading correction to the thermal two point-Green's function can be expressed more explicitly as where for the brief ness, we do not recover the thermal factor. It will be obvious in the following that the higher order C n (t 1 , t 2 ) for n ≥ 3 will not contributes to the leading order of all point functions. The generating functional of connected correlators can be expanded as 2∆ is the same as Eq.(II.22) with f (t) = tan(t/2) without soft mode ǫk(t) correction, and we have dropped the odd-leg source term considering that C 1 (t 12 ) = 0 or k(t) = 0. : · · · : means the time ordering. From the functional Z, one can reads 2n-point functions as  [5]. The Feynman diagrams of the loop corrections from pNGBs are depicted in Fig.1. In the large Lorentzian time with t → it and in the contour chosen in Eq.(IV.6), one has the two point functions as which is a constant and independent oft.

Zero temperature case
By using the correlation function B 2 (t 1 , t 2 ) in Eqs.(II.31) and the soft mode propagators in Eq.(II.12), one obtains the correction to the two point function as In the large Lorentzian time with t → it and in the contour chosen in Eq.(IV.6), the loop correction to the two point function turns out to be Lorentzian time independent as

III. QUANTUM LIQUID WITH SCHWARZIAN CORRECTIONS
In this section, we study the retarded Green's functions as well as local spin-spin correlation functions of quantum liquid with Schwarzian correlation in terms of "Schwarzian liquid", which can be related to the spectral functions and local dynamical susceptibility of strongly interacting quantum liquid including not only spin glass phase but also NFL phase. We also generalize the AdS 2 vacuum in 2-D gravity to higher dimensional RN-AdS d+1 vacuum in Einstein gravity with Maxwell action as in Appendix.C.
A. Quantum liquid from AdS2

Poincaré AdS2:zero temperature CFT1
In the AdS 2 spacetime in the energy coordinate z, the metric and the gauge field is linear in 1 + 1-dimensional spacetime as which is a AdS 2 spacetime in the Poincare coordinate, c µ contains UV information from a 2-D gravity. For the convenience, one may define an effective AdS 2 radius as In the momentum spacetime, the Klein-Gordon equation of a charge scalar in out-wave e −iωt+ikx (∂ t → −iω, ∂ x → ik) becomes which leads to the wave functions whereω = ω + qµ and the conformal dimension is In the special case with c µ = −1/4, it just recovers the original one.
In the near horizon limit, one obtains the asymptotic behavior of the boson wave function as The out-going wave is e −iωt+iωz−iqµz⋆ ln z , which implies that the in-falling boundary condition to be c 1 = 0. On the other hand, in the infinite boundary z → 0, one has where A and B are identified as source and response, respectively, and can be expressed more explicitly as (III.13) The two point Green's function ban be read as (III.14) By doing an inverse Fourier transformation, one haŝ where e −iπa = (−1) a = (−i) 2a and sgn(t) = t/ |t|. Thus, in the coordinate spacetime, assuming t ≫ 0, then one obtains the retarded Green's function in real coordinate spacetime, which just recovers the ansatz of the form the two point correlation function at strong coupling at zero temperature [5,28], It is worthy of noticing that the result reproduces the SYK uniform saddle point solution, by making a match as below where ∆ ≡ p −1 = 1/2 − ν q = ∆ − . In particularly, in the case that ν q = 1/4, m 2 ℓ 2 /(4c µ ) = q 2 µ 2 z 2 ⋆ , the Green's function just recovers the two-point function of SYK model with conformal dimension ∆ IR ± = 1/4 due to an emergent conformal symmetry at low energies and large N at zero temperature, is dropped. It is useful to use the Fourier transforms for symmetric and antisymmetric function aŝ Consider the AdS 2 metric in global coordinates as in (III.5) for hyperbolic case in the vacuum b = 1 as where z ∈ (0, ∞). It is worthy of noticing that it can be transformed into by making a replacement For the simplicity, let's consider the z 0 = 1 case at the beginning, we can obtain a general results by making an inverse rescaling the parameter z 0 is related to the temperature, according to Eq.(III.3). The Klein-Gordan equation are For neutral scalar case (µ = 0), the wave functions are For the charged scalar case, the Klein-Gordan equation are The wave functions are The wave function can also be re-expressed as where ρ ≡ tanh z. In the near horizon limit (ρ → 1), the wave function can be re-expressed as where ∼ means that we have dropped a common factor −i2 ν1− 1 2 (−1) ν1+iµq+ iω 2 in front of the wave function and the coefficients are Since e −iωt−i 1 2 ln (1−ρ) is the infalling wave, will impose the in-falling wave condition that a(ω) = 0, from which the relation between c 2 and c 1 can be determined. In the UV limit, one has where for ∼, we have dropped a common factor −i(−1) ν1+iµq+ iω 2 . The conformal dimension is defined as ∆ ± = 1/2 ± ν 1 . Thus, the retarded Green's functions are (III.32) By using the rescaling relation in Eq.(III.23), one obtains the Green's function as where the pre-factors z 0 are due to the rescaling of the coordinates in Eq.(III.22). For neutral case, i.e., µ = 0 and q = 0, one has where ∆ + = 1/2 + ν 1 . The equation shows that the dimension ∆ + sets the quasi-normal mode frequencies as iω n β = 2π(∆ − + n). Therefore, by doing an inverse Fourier transformation and according to the integral identity as in Eq.(D.13), one obtains the retarded Green's function in real coordinate spacetime which just recovers the ansatz of the form the two point correlation function at strong coupling at finite temperature [5,28], where sgn(t) ≡ t/ |t| is a step function. At this step, by making a comparison with that of SYK model as in Eq.(A.9), one has (III.36) The finite temperature retarded Green's function can be expanded as (III. 37) In frequency space, it can be re-expressed as G R (ω) = −iG(−iω + ǫ). As expected, at low temperature, i.e., in the large β limit, the retarded Green's function recovers the zero temperature one as The retarded Green's function obtained above describes a general class of strongly interacting system such as NFL with ∆ = 1/4 [19,34,50,54,55] and spin fluid or quantum spin glass with ∆ = 1/2, or random/disordered paramagnet [2,[43][44][45], which describes the quantum fluctuations near a critical quantum Heisenberg spin glass. The retarded Green's functions of CFT 1 or NCFT 1 in AdS 2 and NAdS 2 spacetime, can be generalized to be those in higher dimensional spacetime, as explored in Appendix.C.

B. Schwarzian retarded Green's functions
The two point correlation function of quantum liquid without Schwarzian correction in real time is (III. 38) from which, one obtains the retarded Green's function, or the susceptibility of quantum liquid as defined in Eq.(D.3) as where 0 < ∆ < 1/2, β > 0, Im ω > 0 and we have used the integral in Eq.(D.9). One can restore the temperature by multiplying each ω with factor β/(2π). The temperature dependent factor in front origins from thermal correlation function in Eq.(III.38), so that it recovers the quantum correlation function in the zero temperature limit. For zero temperature case, one has The pNGBs loop corrections to the imaginary time thermal two point functions in Eq.(II.34) can be re-expressed as real time one, by replacing t 12 with real time it as where we will assigned every t with a factor multiplying factor 2π/β. It can also be separated as two parts, one has even symmetry for time, while the other has odd symmetry as By doing Fourier transformation, the second and third part of the odd sector will be vanishing unless −1 < ∆ < 0, thus the non-vanishing part within 0 < ∆ < 1/2, comes form the even part, which turns out to bē where where we have used the integrals as in Eqs.(D.13) and (D.8).

Finite temperature case
Therefore, for finite temperature case, the pNGBs loop corrected two point thermal retarded Green's function becomes where G R (ω) is defined in Eq.(III.39), and we have used that the iteration relation 10

Zero temperature case
For zero temperature case, one can do the Fourier transformation upon Eq.(II.36), which leads tō is given as in Eq.(III.40). The dynamical local susceptibility of Shcwarzian liquid becomes where ∆ < 1/4, and χ as given in Eq.(III.50). After obtains loops correction from the Schwarzian effective action, the local suscpetibility becomes more singular at zero frequency limit ω = 0. While these terms is vanishing when ∆ = 1/2, 1/4, the physical consequence of which can be observed at finite temperature.

C. Retarded Green's function
It turns out that the loop correction from pNGBs to the thermal correlation functions, or the retarded Green's functions in Eq.(III.45), leads to a dynamically generated high energy Hubbard band in spectral function, which corresponds to the destruction of quasi-particle states in the spectral function/DOS of quantum liquid, as shown in  The C = ∞ case (dotted red curves) corresponds to conventional NFL with fragile quasi-particle picture. With the increasing of the coupling strength (or decreasing of C) up to C −1 = 3π (purple solid line), the DOS accumulates more in the ω = 0 region as the metallic phase with Fermi liquid behavior, meanwhile it develops a "slope-dig-ramp" shoulder structure, i.e., a Hubbard band at ω ≈ 0.22, which is dynamically generated DOS at finite frequency. The Hubbard band is a smoking gun indicating the presence of a bad metal phase. Among the intermediate range, there is a temperature dependent crossover between the Fermi liquid regime and bad metal regime in the strongly correlating regime, in which the quasi-particle picture is still fragile or even broken down. This signature of NFL phase with Schwarzian correction, i.e., a DOS with Hubbard band in strongly correlated region, is significant different from the conventional NFL phase, where there is no Hubbard band structure present at all. In particular, this signature has been observed in experiments in strongly correlated system, which can be calculated by DFT approach. While to fit the experimental data in the quantum liquid with Schwarizain correction, one needs only three input parameters: the temperature β, the conformal dimension ∆ and coupling strength of low energy effective Schwarzian action ∼ C −1 g that comprise the UV information of various 2-D gravity. Moreover, our exact analytical results quantitatively reproduce the DOS obtained from DMFT approach with a state-of the art numerical calculation from first principles of many-body theory [53].
In Fig.2(b), we also show the temperature evolution of retarded Green's functions of Schwarzian NFL with ∆ = 1/4 by decreasing temperature (or by increasing β) untill T = 1/(200π), which approximately corresponds to zero temperature case (dotted lines). The decreasing of the temperature from T = 1/(2π) to T = 1/(20π), the dig of DOS moves from ω ≈ 0.22 to more lower frequency region at ω ≈ 0.02, and so does the location of Hubbard band, which indicates that the dynamics is due to the pNGBs from spontaneous and explicit symmetry breaking. Meanwhile, the DOS accumulates rapidly and results in a peak at ω = 0, which implies that the quantum liquid becomes more metallic like in zero temperature limit.
In Fig.3(a-b), we show the retarded Green's function at finite temperature with β = 2π and coupling C = 1/(2π) for ∆ = 1/4 case. It is worthy of noticing that the real part of χ (2) (ω) owns a peak at ω = 0 and decays with the increasing of |ω|, and is expected to be a delta function δ(ω) at ω = 0 in the zero temperature limit as shown in Fig.3(c). While the imaginary part of local dynamical susceptibility, i.e., χ ′′ (ω) ∝Imχ (2) (ω) behaviors like a smoothness function ∼ tanh ω as given in Eq.(III.61) and shown in Fig.3(d), which is expected to be a step function jumping at ω = 0 in the zero temperature limit.
Boson and fermions-In this paper, we mainly focus on the bosonic retarded Green's function of quantum liquid with Schwarzian correction, a similar procedure might be imposed to fermion's case, which leads to NFL underlying fundamental Dirac or Weyl fermions [19,34,54,55]. For ∆ = 1/4 case, one just recovers the fractionalized Fermi liquid of lattice Anderson model [52]. To obtain thermal fermionic retarded Green's function of quantum liquid, one needs to solve the wave functions of Dirac fermions in (1 + 1)-dimensional spacetime in global AdS 2 coordinate. The exact solutions to the 2-D Dirac fermion wave functions are      shown in Appendix. B 2.

∆ = 1/3: quantum liquid
In this section, we study the spectral functions of a specific quantum liquid with Schwarzian correction with a conformal dimension ∆ = 1/3. This is an intriguing phase between Schwarzian NFL phase (∆ = 1/4) and Schwarzian spin glass (∆ = 1/2) as will be discussed in more detail in the following section.
In Fig.4.(a-b), we show the retarded Green's functions of quantum liquid with or without Schwarzian correction for ∆ = 1/3 case, and we also plot the corresponding local dynamical susceptibility in Fig.4.(c-d), which characters the local spin-spin correlation of disordered state. loc (ω), (d) Im χ (2) loc (ω) at high temperature with β = 2π (solid green/pink line) or at low temperature with β = 20π (dashed cyan/magenta line) The Fig.5(a) shows the evolution of retarded Green's function with respect to the coupling coefficient C −1 (where C ∼ C gφr ), for spin glass (with ∆ = 1/2) with Schwarzian correction, in terms of "Schwarzian spin glass". The coupling coefficient C −1 (where C ∼ C gφr ) characterizes the coupling strength of pNGBs loop corrections to matter two point correlation functions, according to Eq.(II.34). The C = ∞ case (dotted red curves) corresponds to conventional spin glass, and the local susceptibility becomes exact step function θ(ω) in the limit ǫ → 0. With the increasing of the coupling strength (or decreasing of C) up to C −1 = 3π (solid purple line), the DOS accumulates more in the ω = 0 region and develops a small dig at the ω ≈ 0.5.
In Fig.5(b), we also show the evolution of retarded Green's functions of Schwarzian spin glass with respect to the temperature. By decreasing temperature (or by increasing β) from T = 1/(2π) down to T = 1/(20π) as well as T = 1/(200π) (black/red dotted line). As expected, the DOS spread out among the frequency space at finite temperature, but there is still a peak at ω = 0, and a plateau in the ω > 0 region at low temperature limit as T → 0.

Large p or small ∆ behavior
In this section, we consider the physical consequence when ∆ becomes smaller as shown in Fig.6. This is equivalent to increasing the number of interacting particles, i.e., p ≡ 1/∆ [5]). As stated before in the introduction section, the p-fermion interacting vertex with p ≥ 4 is only UV relevant in (0 + 1)-dimensional spacetime. Without loss of generality, we chose some specific value for conformal dimension as ∆ = 1/8, /16, 1/32, 1/64, respectively. For larger p or smaller ∆, the spectral functions ImG R (ω) becomes more sharper at ω = 0. Consequently, the life time of the quasi-particle becomes longer as shown in Fig.6. Conversely, for smaller p or larger ∆, the life time becomes shorter and the DOS shows non-quasi-particle behavior at low frequency.
With the increasing of p = ∆ −1 , or the decreasing of conformal dimension ∆ from ∆ = 1/8 to ∆ = 1/64, the spectral functions ImG R (ω) become more and more centralized at ω = 0. It shows more metallic behavior at low frequency region, meanwhile the depth of Hubbard band increases towards low frequency region, and so does the location of Hubbard band.  (2) (ω); (f) Imχ (2) (ω). We have chosen input parameters as β = 2π and C = 1/(2π). With the increase of p, the density spectral function becomes more central localized at low frequency region, i.e., ω ∼ 0.

D. High order local spin-spin correlation
By using the tree level retarded (real time) Green's function in Eq.(III.35), it is straightforward to calculate the local spin-spin correlation function, namely the dynamical local spin susceptibility χ loc (ω) as defined in Eq.(D.5) (III.49) By using retarded Green's function defined in Eq.(III. 35), and according to Eq.(D.13), we are able to calculate the local spin susceptibility at zero temperature as where the conformal dimension is limited as 0 < ∆ < 1/4 and the frequency must be in the upper complex plane Imω > 0. We have also used (−i) = e −i π 2 in the last equality of above equation. For finite temperature case, by making a rescaling t → (2π/β)t or ω → β/(2π)ω, the dynamical local spin susceptibility at finite temperature becomes By using Eq.(III.35), one has Based upon which, the higher order local spin susceptibility with respect to the frequency becomes where x ≡ 2∆ − iωβ/(2π), the prime is with respect to the frequency and we have used the definition of functions defined in Eq.(D.10).

Static local susceptibility
The leading order low frequency behavior of local spin susceptibility is a constant, i.e., χ(ω) = const. + O(ω), in which, the constant term is inverse proportional to the temperature as While it turns out that for the special ∆ = 1/4 case, the imaginary sector of χ(0) is divergent Consider first derivative of χ ′′ loc (ω) with respect to frequency ω, one obtains that the static local spin susceptibility χ loc (0) is inversely proportional to the square of temperature, i.e., β 2 as from which, it turns out that for the special ∆ = 1/2 case, the imaginary part of χ While at the 2-nd order derivative of χ ′′ loc (ω) with respect to ω, the static local spin susceptibility χ ′′′′ (0) for both ∆ = 1/2 and ∆ = 1/4 case, becomes convergent and is inversely proportional to the cubic of temperature, i.e., β 3 as where ζ(3) ≈ 1.20206 is the Riemann zeta function. The effective bath for the local spin is given by the local spin-spin correlation function itself, which have nontrivial low frequency behavior, which appears only as a subdominant correction to the leading low frequency behavior χ (n) (0) ∼ const given β.

Marginal NFL
(III.59) the result just recovers the retarded Greens'f function of fractionalized Fermi liquid phase of the lattice Anderson model [52], which can be obtained in an analogy procedure for fermion case, by solving 2-D Dirac equation as shown in Appendix. B 2. More generally, it is a special case of NFL in a doped Mott insulator [19,34,50,54,55]. For the ∆ = 1/4 case, the low frequency behavior of local spin susceptibility χ (2)T loc (ω) is given by Eq.(III.60).
where we have used that ψ(−1/2) = ψ(3/2) = 2 − γ E − 2 ln(2) and ψ(1/2) = −γ E − 2 ln(2). Thus, one obtains the universal form for low frequency behavior of the dynamical local spin-spin correlation susceptibility [50] Imχ (2)T loc (ω) = −Imψ which is simply a smoothed-out version of the step function at zero temperature (Imχ (2) loc (ω) = π tanh(πω)) or in large frequency limit, i.e., T → 0, or T ≪ ω. The local dynamical susceptibility implies that [47] which is precisely of the form for spin and charge fluctuations in the phenomenological "marginal non-Fermi liquid" (mNFL) description of High-T c cuprates in the strange metal region. The marginal critical point can be viewed as a concrete realization of the bosonic fluctuation spectrum needed to support a mNFL. In this case which just recovers the spin-polarization correlation function of "marginal NFL" [47,48,50]. For higher order local spin-spin correlation functions, at finite temperature case, one has Thus, one obtains where in the last equality, we have used reflection principle in Eq.(III.66) as

Spin glass
For ∆ = 1/2 case, the result also recovers the retarded Greens's function of spin glass [2,[43][44][45], which describes the quantum fluctuations near a critical quantum Heisenberg spin glass. The low frequency behavior of local spin susceptibility χ ′′ loc (ω) is given by Eq.(III.71) as from which, one obtains where in the last equality, we have used reflection principle in Eq.(III.66).
The higher order susceptibility is In this case, one obtains where we have used the refection principle in Eq.(III.70).

IV. HIGH POINT CORRELATION FUNCTIONS
In this section, based on the low energy effective action of the Schwarzian theory of time reparametrization from 2-D gravity, we calculate the high point functions, especially the 4-point correlations function [59,60]. The physical consequence of the four point functions can be detected in the system of quantum chaos [11,15,27,33,64,66,69], which can be characterized by an exponential growth of the thermal out-of-time-order correlating (OTOC) [63,[65][66][67][68][69] four point function with a scrambling timet.
A. Four-point function
According to the relative ordering of the time, there are several possibility, one is t 1 > t 2 > t 3 > t 4 , in this case one obtains which can be viewed as arising from energy fluctuations. After recovering the thermal factor t → (2π/β)t, one just recovers the connected four point function in Eq.(3.131) in Ref. [5]. Each two point function C 1 (t 12 ) generates an energy fluctuation, which affects each other. This result does not depend on the relative distance between the pair of points. In the double limit of t 12 → 0 and t 34 → 0, one has The other results is obtained with the time order t 1 > t 3 > t 2 > t 4 , in this case, one has F (4) which depends on the overall separation of the two pair.
In the absence of cross distance t 23 , i.e., when t 2 = t 3 , the result F

OTOCs
A simple diagnostic of quantum chaos is consider a square of the commutator by taking an expectation value in some thermal state, by considering a quantity, i.e., the commutator of operators separated in time as [1,65] where W(t) and V(t) are two different operators dual to the source Φ 0 (t), and · · · β = Z −1 Tr[e −βH · · · ], where the subscript β is introduced to denote the thermal expectation value at temperature T = β −1 . The behavior of C(t) in a chaotic system is By expanding it, there are four point functions in C, two of them consists of C 1 (t) in terms of Lorentzian time ordered correlators (TOCs), i.e., G With the OTOCs four point function G VWVW , by making the parameterization with SK four-contour [61,62], which is depicted in Fig.8 ( where i = 1, 3, 2, 4 and β = 2π.t is the separation of the early V operator and the later W operators. The contour goes from some initial timet 1 within Euclidean domain, along the imaginary time axis to some timet 2 , then turns to the Euclidean domain timet 3 again, and again runs along the imaginary time axis tot 4 . In thet ≫ β limit, one obtains the TOCs and OTOCs four point functions, in the SK contour as depicted in Fig.8 G (4) where G VVWW (t 1 ,t 2 ,t 3 ,t 4 ) etc., and for the last equality of OTOCs, we have transferred the Euclidean time τ (t) in Eq.(II.8) to Minkowski time, i.e., t i → it and recover the temperature by rescaling t i → 2πt i /β, while the β in front comes from recovering of the thermal factor, i.e., by multiplying a factor 1 → β/2π. The behavior of thermal OTOCs at later time shows a exponential expansion with a Lyapnov exponent λ L = 2π/β, which indicates the growth rate of chaos in thermal quantum systems with large number of degrees of freedom, and is bounded in a universal system [66], as λ ≤ λ L = 2π/β. The Eq.(IV.7) is valid under the condition that t r ≪ t ≪ t s , where t r is relaxation time, t s is the scrambling time and t s ∼ λ −1 L ln C when C(t) becomes of O(1) under time long time evolution. By selecting the exponential increasing mode and doing Fourier transformation, in the choice of SK contour 16 as in Eq.(IV.6), or depicted in Fig.8. The chaotic mode of OTOCs in frequency space becomes G (4) , Imω > λ L , (IV. 8) while the normal mode of TOC becomes which is singular at ω = 0. The chaotic behavior of four-point OTOCs functions in frequency space is shown in Fig.9. For maximal chaotic behavior with λ L = 1 (β = 2π), it results in a non-zero frequency bump in the in low frequency region at large Lorentizian time. While at low temperature limit for (β = 4π), the peak of the bulk moves more closer to low frequency range, or equivalently, a much more larger Lorentizian time to saturate the chaos, which corresponds to the non-maximal chaotic behavior with λ L = 1/2. In the zero temperature limit, λ L → 0 (β → ∞), as expected, the peak of the bump moves to the ω = 0, and the mass spectrum of pNGB becomes NGB like.  VWVW (ω, β)/π of quantum liquid with Schwarzian correction: maximal chaotic behavior with λL = 1 (β = 2π) (cyan/purple thick lines) or non-maximal chaotic behavior with λL = 1/2 (β = 4π) (green/magnet dashed lines). We have chosen a set of input parameters as ∆ = 1/4, C = 1/(2π).

Zero temperature case
By using the correlation function B 1 (t 1 , t 2 ) in Eqs.(II.31) and the soft mode propagators in Eq.(II.18), the correlation to the connected four-point function turns out to be G (4) (t 1 , t 2 , t 3 , t 4 ) = ǫ 2 B 1 (t 12 )B 1 (t 34 ) /[t 12 t 34 ] 2∆ . For the normal ordering of the time case, i.e., t 1 > t 2 > t 3 > t 4 , the correlation function turns out to be vanishing which means although the two point function B 1 (t 12 ) generates an energy fluctuation, they do not affects each other. While for the crossing time ordering case, i.e., t 1 > t 3 > t 2 > t 4 , the corresponding four-point function turns out to be non-vanishing.
which is proportional to the overall separation of the two pair t 23 . In the absence of cross distance t 23 , i.e., when t 2 = t 3 , the result G VWVW just recovers G (4) VVWW = 0. In the case t 3 → t 1 and t 2 → t 4 , one has G (4)

B. Six-point function
It is straightforward to calculate the higher point functions as in Eq.(II.28) such as the six-point functions obtained in Eq.(II.29). The typical Feynman diagrams of six point TOCs and OTOCs functions are depicted in Fig.10, respectively. For the convenience of viewing physical consequence of six-point functions, one may generalize the SK four-contour in Eq.(IV.6) to be six-contour as shown in Fig.11, by increasing the real time with equal pace and imaginary time separately with for OTOCs as where the time order is i = 1, 3, 2, 5, 4, 6. For example, in the SK six contour chosen in Fig.11, the six-point correlation function can be expressed more elegantly asG 6 (t 1 , · · · , t 6 ) = F 6 /4 3∆ , where F 6 are where F VWVX WX = ǫ 3 C 1 (t 1,2 )C 1 (t 3,4 )C 2 (t 5,6 )θ(t 32 )θ(t 54 ) , F VWVWX X = ǫ 3 C 1 (t 1,2 )C 1 (t 3,4 )C 2 (t 5,6 )θ(t 32 ) , One may entail an infinitesimal Im t52 = δ > 0 to entail that t2 is earlier than t5 along imaginary time line, and in the end set it to be zero, which does not affect the results.
In a similar manner as in calculating the six-point function, in the chosen SK eight-contour as in Fig.13 with the OTOCs time ast i with i = (1,3,2,5,4,7,6,8), the thermal eight-point functions turns out to be where we have introduced the new notation C 4 t 7,8 ). It is easy to check that it satisfy the relation For the higher point OTCs, one would expect that the thermal system will approach the chaos much faster with time less than τ L ≡ 1/λ L .

V. DISCUSSIONS AND CONCLUSION
The SYK model is an intriguing quantum mechanical model displaying both a spontaneous and explicit breaking of an emergent reparametrization symmetry Diff 1 . The breaking patten of this symmetry determines many feature of the low energy dynamical property of the model and some are expected to be universal in strongly interacting IR fixed point at large N limit.
Features of SYK like model -The most fabulous features of the SYK model is the solvability in the strongly interacting IR fixed point at large N limit. The mass spectrum of the SYK model is obtained by solving Schwinger-Dyson equation and the spectrum of two-point and fourpoint function, as well as more higher-point functions are computed [3,5,31].
The other interesting features of the model is that in the strong coupling limit (βJ ≫ 1), the four point function saturates the maximal chaotic bound since it is dominated by the universal sector of gravity [66]，which is characteristic of a gravity theory with black hole solutions [64]. The saturation means it achieves the maximally allowed chaos quantified by the Lyapunov exponent λ = 2πβ, the growing rate of a thermal four-point OTOCs functions, as defined on the Keldysh contour [1,6], V i (0)W j (t)V i (0)W j (t) β ∼ e λt /N , which is true at a time range between the dissipation time and the scrambling time, i.e., t ∈ (λ −1 , λ −1 ln N ). The exponential growing manner reflects a underlying chaotic dynamics.
Another novel feature of the model is the emergent conformal symmetry, i.e., the time reparametrizations diffeomorphism symmetry Diff 1 , or Virasoro symmetry, at low energy and its spontaneous and explicit breaking [5,7].
Spontaneous breaking of Diff 1 -In the SYK model, the emergent Diff 1 symmetry is spontaneously broken down to SL(2, R) symmetry [56,57], which is kept in the Schwarzian action (the Lagrangian) at finite frequency. From the gravity viewpoint, the Diff 1 symmetry is an approximate asymptotic boundary symmetry of the perfect AdS 2 at IR conformal fixed point (ω = 0 or J = ∞), and is spontaneously broken down to a one dimensional global conformal group SO(2, 1) ∼SL(2, R) symmetry, or large diffeomorphism Diff owned by the AdS 2 symmetry.
Explicit breaking of Diff 1 -In the SYK model, the emergent Diff 1 symmetry is also explicitly broken, since the symmetry is not kept by the Lagrangian any more as one slightly moves away from the IR conformal fixed point, where the kinetic term ∂ τ becomes relevant at low frequency or strong coupling region (ω ≪ 1 or J ≫ 1). From the gravity viewpoint, the bulk spacetime is slightly deviated from AdS 2 vacuum to near-AdS 2 (NAdS 2 ) by taking account of the backreaction due to arbitrary tiny energy excitation.
Diff 1 symmetry breaking pattern-The pattern of spontaneous breaking of the Diff 1 results in an infinite number of zero mode, namely, the NGBs characterized by the coset Diff 1 /SL(2, R). As the Diff 1 symmetry is explicitly broken, the leading order dynamical correction is described by a SL(2, R) invariant Schwarzian derivative of the reparameterization f (t) in terms of effective Schwarzian action in Eq.(A.10) as described in Appendix. A, which determines many aspects of the theory. As will be seen, the dynamics of the Schwarzian correction to the quantum correlations of SYK model is characterized by Schwarizian action with an SL(2, R) unbroken symmetry. As the Diff 1 symmetry is explicitly breaking due to a small but non-vanishing derivation ǫ ∼ κ ∼ J −1 (this is equivalent to a small ω ≪ 1 or the presence of a relevant kinetic term ∂ τ ), associated with an infinitesimal fluctuation field k(t) in a dynamical reparameterization function f = t + ǫk(t), as in Eqs.(II.8) and (II.14) for zero and finite temperature cases, respectively. To be brief, the Diff 1 symmetry is parameterized by f and is explicitly broken by the small fluctuation ǫk(t), which is parameterized by the coset Diff 1 /SL(2, R).
Quantum and thermal correlations-For zero temperature case as in Eq.(II.11), the zero modes of the fluctuation field, i.e., NGBs, leads to a zero action in the IR conformal fixed point (J = ∞ or ǫ = 0), while the soft modes of the fluctuation field, i.e., pNGBs, leads to a non-vanishing action when the classic solution is deviated away from the conformal limit (a small but finite ǫ ∼ J −1 ). For finite temperature case as in Eq.(II.17), the first exciting state of pNGBs, i.e., the n = ±1 soft modes, also leads to a zero action. In both cases, the 2-point function is singular and needs to be regularized. Consequently, the 2-point and 4-point correlation functions of matter field, obtains loop corrections from the pNGBs as the reminiscent effect of the broken of time reparamterization invariance Diff 1 .
In conclusion, we study the retarded Green's function of quantum liquid with Schwarzian correction, which can be depicted by a (0+1)-dimensional strongly interacting quantum mechanical/statistics model dual to a general (1 + 1)dimensional classical dilaton gravity model. Based upon the two point correlation functions of matter, which get loop corrections from pNGBs in coset Diff 1 /SL(2, R), we obtain the bosonic retarded Green's functions as well as local dynamical susceptibility, i.e., the 2-nd order local spinspin correlation functions for quantum liquid. We also calculate the four point as well as higher point thermal OTOCs functions in SK formalism, which cultivate the quantum chaos at large real time.
To manifest our results, we show the spectral functions of not only Schwarzian spin-glass described with conformal dimension ∆ = 1/4 (p = 4) but also Schwarzian NFL with ∆ = 1/2 (p = 2), as well as a specific quantum liquid phase with ∆ = 1/3 (p = 3). Large p-body behavior of quantum liquid with Schwarzian correction are studied too. Moreover we make comparison with the leading order retarded Green's functions, which just recovers the results of quantum liquid from AdS 2 /CFT 1 approach.
In the infrared (IR) conformal fixed point with zero frequency (ω = 0) where the Diff 1 symmetry is emergent, the spectral functions owns Fermi liquid [42] peak in DOS at ω = 0 and leads to typical metallic behavior. The symmetry is spontaneously broken to SL(2, R) and leads to zero modes on the boundary in terms of "boundary graviton" [7], which are the Fourier modes of the Diff 1 symmetry. At finite frequency (ω = 0), the Diff 1 symmetry is explicit broken. As its physical consequence, the system develops a feature which is interpreted as bad metalic behavior with a high energy Hubbard band dynamically generated. In the intermediate region, there is a temperature dependent crossover between Fermi liquid phase and bad metal phase in the strongly correlation regime, in which the quasi-particle picture is fragile or even broken down.
We make generalization of 4-point correlation to higher point correlation functions. As non-inclusive demos, we show concise analytic results on 6-point as well as 8-point thermal OTOCs functions in SK contour, which exhibit exponential growth until progressively a longer timescale and thus sensitive to more fine grained quantum chaos. We also obtain analytic expression for 3-order and 4-th order local spin-spin correlation functions.
The quantum liquid with Schwarzian correction can be related not only to quantum spin glass or disordered metals without quasi-particles scenery depicted by SYK like model, but also NFL phase [47,49,50,52,54,55]. We study the matter retarded Green's function by taking account of the loop corrections from pNGBs to the matter two point correlation function, and unexpectedly find a Hubbard band or dynamically generated DOS in the spectral functions, which is due to the spontaneous and explicit breaking of time reparameterization symmetry and is a distinct signature of quantum liquid with Schwarzian correction, comparing with the conventional strongly interacting quantum liquid. The existence of pNGBs mode in the quantum liquid with Schwarzian correction also provides a dynamical mechanism for explaining the commonly observations of bad metal in strongly correlated system.
where χ i are N Majorana fermions, satisfying {χ i , χ j } = δ ij , interacting with random interactions involving 4 fermions at a time. J ijkl is a Gaussian random infinite-range exchange interaction of all-to-all quartic coupling, which are mutually uncorrelated and satisfies the Gaussian's probability distribution function P (J ijkl ) ∼ exp (−N 3 J 2 ijkl /12J 2 ), which leads to zero mean E[J ijkl ] = 0 and variance E[J 2 ijkl ] = 3!J 2 /N 3 with width of order J/N 3/2 , respectively. The E[· · · ] denotes an average over disorder. The J is the only one effective coupling after the disorder averaging for the random coupling J ijkl . The random couplings J ijkl represents disorder, and does not correspond to a unitary quantum mechanics [14,32]. For euclidean time τ = it, the model can be viewed alternatively as a 1-dimensional statistical model of Majorana fermions. For finite temperature case, the quantum mechanical model can be alternatively depicted in a quantum statistics. By using a Hubbard-Stratonovich transformation, it is possible to rewrite the original partition function of SYK model as a functional integral of the form [1,[43][44][45] as whereS = S/N is a disorder-averaged non-local effective action by doing Gaussian integral over the disorder and integrating out fermions after introducing a bilocal field G(τ, τ ′ ) and a Lagrange multiplier field Σ(τ, τ ′ ). Pf denotes the Pfaffian, and the first term of the action can also be reexpressed as ln[det(∂ τ − Σ)]/2, τ, τ ′ are Matsubara times and p = 4 denotes the number of Majorana fermion in the vertex. At large N limit, i.e, a model with the number of Majorana fermion N ≫ 1, by doing variation with respect to G and Σ, or equivalently by counting the resummed Feynman diagrams, the solution of SYK model is described by the Schwinger-Dyson (SD) equations in real spacetime as where G = G(τ, τ ′ ) is the two-point Green's function, Σ = Σ(τ, τ ′ ) is one particle irreducible (1PI) self energy. The first kinetic term in the G represents a conformal breaking term as will be clear in the following. Substituting the full solutions of above classical equations of motion back into the effective action in the partition functions, one obtains the leading large-N saddle point free energy F in low temperature expansion as [5,11] where e 0 is the ground state energy density, s 0 is te zero temperature entropy density and c v ≡ γβ −1 is the specific heat density. · · · denote terms with higher order in β −1 .
In the ultraviolet (UV) limit at short distance, ω ≫ J, the kinetic term dominates and the four-fermion interactions term is irrelevant so that the theory has N weakly interacting massless Majorana fermions. The fermions have a two point function given by G 0 (τ ) = sgn(τ )/2 regardless of temperature, or G(ω) = iω −1 in frequency space, assuming the time translation symmetry is kept. The action is invariant under arbitrary time reparametrizations and consequently the Hamiltonian is zero. While in the low energy IR limit at large distance, the frequency (in the momentum spacetime ∂ τ ∼ iω) is much smaller than the UV coupling J, i.e., ω ≪ J, means the model becomes strongly interacting at low energies. Consequently, the kinetic term ∂ τ can be dropped, so that the SD equations in the IR limit is modified to be conformal invariant ones as, In this case, the SD equation in the conformal limit are reparameterization invariant, which means that under an infinitesimal transformation of the time reparameterization τ → τ + ǫ(τ ), the two point function transforms as G → G + δ ǫ G with In this case, G + δ ǫ G still solve the conformal SD equations in Eq.(A.5).
As a physical consequence, an extra general reparametrization symmetry with an arbitrary function f (t) and conformal invariance is emergent in the IR limit as long as ω ≪ J or equivalently, J → ∞, namely, in strong interactions, as τ → f (τ ), where ∆ = 1/p is the conformal dimension of CFT 1 , which is explicitly broken by the kinetic term ∂ τ in medium energy range, i.e., ω ∼ J, which is the explicit symmetry breaking parameter. To be brief, both two point function G and self energy Σ are conformal invariant in the IR background. Therefore, βJ ≫ 1 can also be viewed as the conformal limit of the model. In the IR limit, the action is reparametrization invariant by dropping the kinetic term ∂ τ inside the action, while the solution G is only SL(2, R) invariant. Thus one can view reparametrization invariance as an emergent symmetry of the IR theory, which is spontaneously broken by the conformal solution G. The emergent full reparameterization symmetry, i.e., the Virasoro group SL(2), is presented by the generators The zero modes in the effective action can be viewed as Nambu-Goldstone (NG) modes for the spontaneous breaking of the full SL(2) conformal symmetry down to SL(2, R).
Since the action is SL(2, R) gauge invariant, in the path integral, one need to dived the the integral by a volume of SL(2, R).
At zero and finite temperature, the two point function has a conformal ansatz form at zero and at finite temperature, respectively, as where ∆ is the IR conformal dimension, which turns out to be inversely proportional to the d.o.f p of the disordered interaction, i.e., ∆ = 1/p and b p = (2πJ 2 ) −1 (1 − 2∆) tan (π∆) at leading order in 1/N . The Green's function represent the low frequency behavior of the retarded Green's function for the SYK model in the strong coupling limit. The ansatz form above can be obtained by applying the reparameterization at saddle point f (τ ) = τ in zero temperature case, while in finite temperature case, the time direction is considered Euclidean and compactified into a thermal circle f (t) = e 2πit/β or f (τ ) = tan (πτ /β) satisfying f (τ + β) = f (τ ). To be brief, the thermal quantum mechanics or quantum statistics can be achieved through the reparameterization of a zero temperature quantum mechanics, by mapping a straight line of imaginary time τ to a thermal circle, i.e.,τ = tan (πτ /β) with periodic boundary conditions over a periodic lattice length β.
Effective Schwarzian action-In the low energy limit, the model can be described by a local effective action proportional to the Schwarzian derivative [1] in terms of Schwarzian theory [23], which can be understood as the dynamics of a Goldstone bosons f (τ ), a near-zero mode for the breaking of reprarametrization invariance [5], with a coefficient of order (βJ) −1 as [1] where Sch(f, τ ) is the Schwarzian derivative in Eq.(II.3), which is invariant under SL(2) symmetry f → (af + b)/(cf +d) and it is an exact symmetry at zero temperature since f (τ ) = τ . The prime indicates the derivative with respect to the τ . f (τ ) is the Nambu-Goldstone bosons, or the zero modes involving large diffeomorphisms, which are nontrivial on the boundary. When one move away from the IR fixed point (ω ≪ J → ∞), the NG bosons cease to be zero mode and leads to a non-zero action, i.e., Sch(f (τ ), τ ) = 0, e.g, f (τ ) = tan(πτ /β), a black hole with finite temperature as a deformed parameter from AdS 2 . At the finite temperature, the effective action becomes S eff = −2π 2 N α/(Jβ) The form of the action itself implies an SL(2, R) invariant solution f → (af + b)/(cf + d) with a, b, c, d ∈ R and ad − bc = 1, which is the same as SL(2, C) . For instance, at finite temperature, f (τ ) = tan(πτ /β), the Schwarzian is Sch(f, τ ) = 2π 2 /β 2 . Since the effect coupling of the theory 1/g 2 ∝ N α/(Jβ 2 ), at large N and fixed temperature, the theory is weakly coupled, dominated by fluctuations around the saddle point τ , but is strongly coupled at ultra low temperature, i.e., g ∝ β. It also interesting to consider a reparameterization f (τ ) → tan (πǫ(τ )/β), the Schwarzian becomes Sch(f (τ ), τ ) = Sch(f (τ ), τ ) + 2(π/β) 2 f ′2 . Considering a small reparameterization τ → τ + ǫ(τ ), by using the equivalent form of Schwarzian with terms of the total derivatives, the action can be re-epxressed as a local one as where we have dropped the total derivative terms (ǫ ′′ /ǫ) ′ , the action has an expression of lowest order in derivatives that vanishes for global SL(2) transformation. Consider a small fluctuation on the fixed parametrization ǫ(τ ) = τ +ǫ(τ ), and expanded up to quadratic order, one obtains the quantum action in terms of Pseudo Nambu-Goldsonte (PNG) boson fieldǫ. In this case, not only the SL(2) symmetry is broken by the SL(2, R) invariant IR G solution, but also is explicitly broken, which gives a small Schwarzian action forǫ(τ ), which is vanishing in the strong interacting limit J → ∞ at order of large N . The effective action is a potential term for the zero mode, thus, the Schwarzian action can be viewed as a mass term for PNG boson. Therefore, the low energy effective Schwarzian action above makes reparameterization modesǫ PNG bosons, in terms of soft modes [9,32].

Self energy
With the two point correlation functions G(t), it is also possible to consider the scattering rate of quantum liquid with Schwarzian correction, by doing fourier transformation the imaginary time self-energy. For example, as in SYK model, according to the Schwinger Dyson equations in Eqs.(A.3) or (A.5) with p = 1/∆, the self energy can be calculated as where we have used Eq.(III.38).
where the prime denotes the derivative with respect to the frequency ω.

Partition functions and free energy
According to Eq.(A.4), by using the thermal parameterization f (t) = tan (πt/2), the free energy in low temperature expansion i.e., β ≫ 1, can be obtained from the effective action.
From the effective action of the gravity sector in Eq.(II.2), one obtains the free energy which leads to the zero temperature entropy S = N s 0 = βF = 2π 2 β −1 and the specific heat C V = N c v = γβ −1 = 4π 2 Cβ −1 , which are both linear in temperature. While from the effective action of the scalar matter given in Eq.(II.22), where the first term is a finite one as leading IR correction under the case that ∆ < 3/2, since this free energy due to matter F χ ∝ β 1−2∆ dominates over the free energy due to the gravity F ∝ β −2 at low temperature limit. While the second term is a UV divergent term, since it is a constant, thus contributes to the ground state energy density e 0 as obvious in Eq.(A.4). One can also include the one-loop exact Schwarzian partition function by direct functional path integration of the Schwarzian theoryas [23] from which, the one loop corrections to the free energy is obtained from Therefore, the loop corrections of graviton soft mode to the scalar matter field turns to contributes a finite logarithmic temperature term for the free energy.
Appendix B: Wave function in global AdS2

Boson in global AdS2
Consider the AdS 2 metric in global coordinates as where ρ ∈ (−∞, +∞) and τ ∈ (−∞, +∞), c µ contains UV information of 2-D gravity. The global coordinate are within the range ρ ∈ (∞, 0) for z ∈ (0, ∞). Thus, it is expected that the theory is dual to copies of conformal qantum mechanics CQM 1 and CQM 2 on two boundaries via AdS 2 /CFT 1 . The Klein-Gordon equation in this coordinates becomes which gives the wave functions as where z ≡ tanh ρ and
In the representation, the chirality operator is γ 3 = γ 0 γ 1 = σ 3 , thus the wave function contains two Weyl fermion with opposite chirality as φ = (ψ + , ψ − ) T . The Dirac equation (✚ ✚ D − m)ψ = 0 in the global AdS 2 coordinate in Eq.(B.1), becomes which can be combined to be two decoupled equations of motion for ψ ± , respectively, as Alternatively, the Dirac equation in the global AdS 2 coordinate in Eq.(B.6) are which can be combined to be two decoupled equations of motion for ψ ± , respectively, as The solutions to Dirac equation in Eqs.(B.9) or (B.11) tuns out to be where (C. 35) In the IR limit at large distance, we have f (r) ≈ 1, or energy scale is much larger than the chemical potential µ (but still much less than the UV scale) is simply a vacuum with conformal symmetry AdS d+1 . The near horizon geometry is given by AdS 2 × R d−1 , i.e., ds 2 = ds 2 AdS2 + (r 2 ⋆ /ℓ 2 )dx 2 , which indicates the boundary system should develop an enhanced symmetry group including scaling invariance. For T = 1 2πζ0 and T = 0 case, respectively, one has AdS 2 in global and local/Poincaré coorindates, respectively, as where ℓ 2 is the curvature radius of AdS 2 given by For the second equality, we have used the coordinate transformation as In the finite temperature limit ζ → ζ 0 , the metric and the vector potential of Maxwell field are dominated by The time direction also shrinks to zero, and the the spatial direction approaches a constant, the Maxwell field approaches zero. The RG scales is flowing from AdS d+1 with scale z to near horizon boundary with scale z 0 , a single scale of the boundary theory. In the boundary field theory aspect, the theory flows from CFT d in the UV to the IR-CFT 1 , which is a conformal symmetries of a (0 + 1)dimensional conformal quantum mechanics (CQM) [72][73][74], including the scaling symmetry in the time direction. Therefore, the corresponding IR fixed point is just a IR-CFT 1 , which is a conformal symmetry only in the time direction. The new conformal symmetry is emergent and has relation with the collective motion of the large number of charged excitation. Of course one has to take a notice that the spatial direction can also have important physical consequence.
In the zero temperature limit ζ → 0, the metric and the gauge field in Eq.(C.36) reduce to the T = 0 case in Eq.(C.37). It is worthy of emphasizing here that the central charge is infinite, since it is proportional to the volume of the d-dimensional transverse space R d−1 . To have a finite central charge one could replace R d−1 by other manifold, i.e., a torus.
The time direction shrinks to zero, and the the spatial direction approaches a constant, the Maxwell field approaches zero. The AdS 2 symmetry is emergent in the nearhorizon region. The AdS 2 is isomorphic to a full SL(2, R) symmetry, which own the scaling isometry: where only the time sector scales. The finite t corresponds to the long time limit of the original time coordinate, meanwhile the short distance limit of the original spatial coordinates. Thus, the metric obtained above should apply to the low frequency limit, since ω is the frequency conjugate to t ω ∼ T ≪ µ. (C.42) In the low frequency limit, the d-simensional boundary theory at finite charge density should be described by a CFT 1 , in terms of IR CFT of the boundary theory, which is an emergent conformal symmetry due to collective behavior of a large number of degrees of freedom. It is not related to the microscopic conformal symmetry in the UV, which is broken by finite charge density.
The metric is the AdS 2 slice of the high dimensional RN-AdS d+1 , the letters with bar above is associated witht, T andT are Hawking temperature with respect to the coordinates t andt, respectively. In this case, the frequency conjugated to the rescaled timet becomes

a. Near horizon field equations of motion
Consider a massive scalar field φ with mass in the bulk action as where the covariant derivative is defined as D N ≡ ∇ N − iqA N . From the action, one can obtain the equations of motion for the complex scalar, Let's consider a charged scalar field in the background metric and gauge field at zero temperature T = 0, which is given by Eq.(C.37), which recovers the asymptotic AdS 2 spacetime in Eq.(III.5), by setting ℓ 2 = 1 and e d = µ. To be explicit, by considering that g xx = 1/r 2 ⋆ , A x = 0, the field equations become, where φ = φ(t, ζ). Alternatively, one can expand φ in momentum space by doing Fourier transformation, where D a = ∇ a − iqA a and The indices a, b only run over t and ζ. From the action, one can obtain the equations of motion for the complex scalar, ) which can be written as where the ± sign corresponds to out/in going waves (∂ t → ∓iω and ∂ x → ±ik), respectively,

b. IR correlation functions of scalars
Firstly, let's consider the case in the infinite boundary conditions, The EOM is dominated by the singularity at z ⋆ → 0, The asymptotic behavior of the solution are where the two exponents are respectively For pure AdS 2 case in (1 + 1)-dimensional spacetime, one has ℓ 2 = 1, k = 0, and e d = µ. Thus, It is worthy of emphasizing that it is possible that the ν k become pure imaginary, once the electric charge q becomes sufficiently large. Then, ν k becomes imaginary for sufficiently small k 2 < k 2 0 (for a given m, this always occurs for a sufficiently large q) with where z ⋆ ≡ ℓ 2 /r ⋆ and the local chemical potential µ L is defined as in Eq.(C.63). By observing the conformal dimension in neutral background as in Eq.(C.21). It can be viewed that the background electric field acts through the charge as an effective negative mass square, which make it possible that the total mass square m 2 − q 2 µ 2 becomes negative and resulting in an imaginary conformal dimension. For a neutral scalar operator with q = 0 in AdS d+1 spacetime , one can obtain the oscillatory region mass window, Where the lower limit comes form the stability of vacuum theory, i.e., Breitenlohner-Freedman (BF) bound of AdS d+1 and the upper limit comes from the condition k 2 0 > 0. For AdS 2 case, one has  at which, the NFL phase (since ν k < 1/2) of CFT 1 is separated into steady and oscillatory due to the charge instability.
It is worthy of noticing that at ν k = 0, or equivalently which is impossible for neutral black brane with q = 0, since m k > 0 is always positive unless m = 0 and k = 0(no scalar at all). As a result, the original two independent solutions are degenerate. In this case, the equation of motion of charged scalar in the infinite boundary conditions at IR fixed point in Eq.(C.55) become where φ = φ(ω, ζ). The asymptotic behavior of the solution are where ζ > 0. Secondly, let's consider a more generic case of the solution. In frequency space φ(t, ζ) = e −iωt φ(ω, ζ), the equation of the motion for a charged scalar φ can be written as where the plus and minus sign corresponds to the out/in going waves (∂ t → ∓iω). Note that ω can be scaled away by redefining ζ, reflecting the scaling symmetry of the background solution. the potential can be expressed as a function depending on the dimension of operators in the infinite boundary conditions, where in the last identity, we have used the Eq.(C.57). According to the asymptotic behavior of the solution in Eq.(C.56) and the EOMs above, it is obviously that the frequency ω dependence in the potential V (ζ), can be scaled away from by redefining the ζ → qe d /ω so that the EOMs recovers Eq.(C.67). Thus, the solution to the generic case will be of form φ = B ζ ω Thus after imposing the infalling boundary condition on φ, the correlated functions are expected to be of form (C.73) This implies a coordinate space correlation function by doing an inverse Fourier transformation 1 , with the conformal dimension ∆ of the boson operator O B given by ∆ = 1 2 + ν k , which is nothing but ∆ IR + defined in Eq.(C.57). It is worthy of noticing that the dimension ∆ depends on the charge q through ν k . In particular, it is possible for ν k to become imaginary when the charge q becomes sufficiently large. This implies that in the constant electric field A t = e d /ζ with sufficiently large charge q can be pair produced, which cause an instability for scalars. When ν k is imaginary, there is an ambiguity in specifying G R since one can choose either A or B as the source term.
1. For real ν k : The ratios G R (ω)/G A (ω) become a pure phase and one find that The factor e iγ k and thus c(k) always lies in the upperhalf complex plane, while for scalars e iγ k +2πν k i always lies in the lower-half complex plane. Namely, for ν k ∈ (0, 1/2), γ k + 2πν k > π, ⇒ π − γ k < 2πν k . (C.98) 2. For pure imaginary ν k = −iλ k (λ k > 0), the ratio becomes real and give the modulus of c(k).
e −4πλ k < e −2πλ k + e −2πqe d e 2πλ k + e −2πqe d = e 2iγ k < 1. (C.99) 3. For generic ν k , G k (ω) and G R (ω, k) have a logarithmic branch point at ω = 0. One can choose the branch cut along the negative imaginary axis, i.e., the physical sheet to be θ ∈ (−π/2, 3π/2), which resolves into a line of poles along the branch cut when going to finite temperature.

d. Finite Temperature case
For a charged scalar field at finite temperature, the near horizon region is a charged black brane in NAdS 2 × R d−1 space-time at finite charge density, the background metric and gauge field are given by Eq.(C.36) where the physical coordinates are listed as below IR horizon r ⋆ ≤ r 0 ≤ r < ∞, UV boundary z ⋆ ≥ z 0 ≥ z > 0, ζ ⋆ ≥ ζ 0 ≥ ζ > 0, Thus, one needs to choose the proper coefficients so that the expansion wave factors are completely canceled and only the in-falling wave solution are present in the near horizon region. It turns out that one has to choose where we have used that Γ[1 + α] = αΓ[α] and (−1) 2ν k = 1. In this case, the final solution of bulk equation in Eq.(C.105) with pure in-falling wave near the horizon becomes, In the infinite boundary condition(z → 0, ζ → z ⋆ /(d(d − 1)) ≪ 1), the EOM is dominated by where ∆ IR ± = 1/2 ± ν k . On the other hand, in the infinite boundary condition, Eq.(C.105) becomes from which, we have C 3 ∼ (−2) −2ν k C 2 , (−2) 2ν k C 4 ∼ C 1 . Thus, one can read the correlator functions (C.114) By using Eq.(C.109), we obtain the retarded Green's function as . (C.115) By using that finite temperature definition T as in Eq.(C.101), we have ), (C.116) where g b is a scaling function given by .

(C.117)
It is worthy of noticing that at zero temperature the original branch point at ω = 0 disappears and the branch cut is replaced at finite temperature by a line of poles parallel to the next imaginary axis. In the zero temperature limit (T → 0), these pole line emerges as a branch cut. At finite low temperature, the near horizon geometry is a black brane in AdS 2 . This IR geometry results in the Green's functions at the finite temperature. The fermion self energy at finite temperature becomes ) 2ν k ∼ c k ω 2ν k . (C.118) In the zero temperature limit, i.e., T → 0, the line of discrete poles of the Gamma function at finite temperature emerges as a branch cut for ω 2ν k at T = 0.