Spectral Form Factor for Time-dependent Matrix model

Time-dependent matrix model can be converted to a two matrix model with coupling via path integral formulation. Two point correlation function for such a two matrix model made of $M_1$ and $M_2$, each over Gaussian unitary ensemble has been studied. Fourier transform of it gives the Spectral Form Factor, which comes with rounding off near Heisenberg Time for this case. This rounding off has been obtained for same matrix interaction ($M_1 -M_1$ correlation) and different matrix interaction ($M_1 - M_2$ correlation function) with first order correction. Rounding off behavior has a (1/N) expansion behavior and can be related to instanton effect near Heisenberg time. With duality relation being considered this two matrix model correlation has been related to open intersection number which connect this to Riemann surface with boundary removed.


Introduction
Matrix models are proven tools for the study of random geometry, strings, twodimensional quantum gravity, chaos, M-theory(non-perturbative string theory), spins on a random lattice, etc. The time-dependent matrix model is considered with explicit time dependence for eigenvalues of the matrix. Two matrix model from the time-dependent matrix model can be constructed following [1,2]. Two matrix model was discussed in the context of the Ising model coupled to gravity in [3]. It has been subsequently generalized for 2-d quantum gravity coupled to (p,q) conformal fields. Time-dependent matrix model and two matrix model has been studied by [2,[4][5][6].Two matrix model and 2-d quantum gravity relation has been established and extensively studied in [7][8][9][10][11]. With a specific potential of two matrices, critical regime for various (p,q)-rational string theories has been evaluated in [12]. Matrix models are also an important part for describing black holes and chaos within it. This has a huge section of literature [13][14][15]. In recent times random tensor model has been related to the random matrix model and spectral form factor.They are extensively discussed in [16][17][18][19]. Spectral form factor for the Ising chain has been discussed in [20][21][22]. Ramp region of SFF in late time has a universal nature due to chaotic behavior. It has been solved numerically and analytically for the SYK model in [23][24][25][26]. The same has been discussed for the gravity model in [27,28].Two-point correlation function and spectral form factor for two Matrix model has been studied in [1,29,30] previously. Here we have followed the model of Gaussian matrix with an external source ,previously considered in [1]. We have computed two types of two-point correlation function:-1. Between the same matrices( kind of auto correlation function or diagonal correlation function) defined as - We call it same matrix interaction 2. In other case two different matrix gives the two point correlation function( off-diagonal correlation function) We call it different matrix interaction Starting with a time dependent matrix M which has a correlation function varying with time, we choose its Hamiltonian to be of type Here M is a time dependent hermitian matrix of size N × N and p is defined by p =Ṁ and M is N × N . Following the same procedure as in [1] with application of path integral method the time-dependent model can be readily generalized to a two matrix model with time dependent coupling in between the matrices. Reducing this time varying matrix model to two matrix model gives wide application. Spectral Form Factor(SFF) is defined as Fourier-transform of the two point correlation function and it readily gives information about basic properties of the system like integrability or time-reversal symmetry.Two point correlation function for a system with Hamiltonian "H" defined as in [31] ρ (2) For a two matrix case with interaction there can be two type of correlation, one between same matrices and other one between different matrices. We have studied both of them in finite large N limit with analytical solutions.
In section 3 we have studied the two-point correlation function and its Fourier transform, Spectral form factor. Using Kernel method it can be given as a exact solution of Hermite polynomial summation. We then choose Kazakov contour-integral representation [32] of correlation function for analytical solution and use saddle point method to calculate its exact form for arbitrary order of N.Then we evaluate SFF and average it following same path as in [1,31]. We then compare both the solutions for specific N cases. The universality of the ramp in SFF is related to the universal Dyson kernel,which is obtained in the limit N → ∞, with a fixed N ∆E, where ∆E is a scaled short energy distance. To obtain this universal nature asymptotics of Hermite polynomials can be used. Hermite polynomials are the solution of the GUE kernel which in the previously described limit gives universal Dyson sine kernel as solution.Near Heisenberg time, the rounding behavior, which we will discuss may not be universal. However, the study of the rounding behavior may shed a light on important physics, such as non perturbative effect. We have studied the rounding off behavior of SFF near Heisenberg time. This gives a smooth evolution from classical to quantum nature, and also most possibly can be connected with phase transition.
In section 4 we have studied the correlation function for same matrix interaction and again use contour integral and saddle point method to evaluate its analytical form. Its Fourier transform gives the spectral form factor and by studying both cases(Same matrix and different matrix interaction) we get complementary behavior between them. In section 5 we looked into the 1/N expansion of correlation function which is important to study its connection with single/multi instantons type cases. After analytically deriving exact form of saddle point for different order we use it on different matrix interaction case. We get the perturbative nature of correction by this and compare with the solution by Hermite polynomial summation.

Time dependent matrix model
Consider a N×N Hermitian matrix M, with time dependence. [1] Time dependent correlation function for such a matrix is defined by:- where t = t 1 − t 2 and t 1 ,t 2 are different times. Correlation function of this form is Fourier transform of a quantity U (α, β) α,β are the Fourier transform variables.This time dependent matrix model correlation function is equivalent to two matrix model on Gaussian ensemble correlation function.This can be shown by Path integral method and it holds for any finite N.
Hamiltonian for time dependent matrix model:- M is dependent on t and p =Ṁ Therefore Now by Path integral formulation ,we define Then: Now ground state energy of free independent N 2 fermions is N 2 2 ,so (2.10) Then U (α, β) becomes; (2.12) Where Z = N 2 and λ, µ are scaled down by a factor {e −t sinh(t)} to compensate the previous change in variable. where c = e −t . Change in notation in two matrix model formulation. Let the matrix be A → M 1 , B → M 2 .There is a coupling between this matrix which time dependent and denoted by c = e −t . The Gaussian distribution is given by

Two Matrix model correlation function and dynamical form factor
Density of state ρ(λ) derived by Fourier transform of U A (z) We have considered an external matrix A coupled to matrix M 1 acting as a source.At last step we will reduce it to zero to get our desired result.The eigenvalue of M 1 and M 2 are r i and ξ i .
Here we have used HarishChandra-Itzykson-Zuber formula to change the measure from integration over matrix to integration over eigenvalues of the matrix.∆(r) = i<j (r i − r j ) is the Vandermonde determinant.
Now using the above expression in Eq:-(3.2) we first do the gaussian integral over dr i and get the form as:- If we take the external source term to zero(a i → 0) Correlation function is defined as the Fourier transform of this function:- Two level correlation function and its Fourier transform -dynamical form factor(Spectral form factor) are important measure for a model.This two level correlation function has two parts.From two interacting matrix and between the same matrix. For two different matrix it can be written as:- U 0 (z 1 , z 2 ) computed in the same with introduction of external source (A) and then letting it to zero.
We perform the gaussian integral with linear term on dr and simplify the expression as:- Changing the integral by contour integral with α 1 = α 2 and taking the external matrix tends to zero.

Disconnected Spectral Form Factor
Disconnected coorelation function from the disconnected part of Eq:-(3.14) is decomposed to two parts then by Fourier transform we change to two point correlation function:- This gives the disconnected two point correlation function:- Now we move to one matrix model ρ 1 ,the density of state for matrix model,following [1] :- Therefore two matrix two point correlation function Eq:-(3.16) is very similar to one matrix model density of states.
After a Fourier transform and setting the values λ = 0 and µ = ω we get the spectral form factor.  We have averaged this dynamical form factor over an interval [0,t] and plot this average value:-

Connected Spectral Form Factor
Connected part has a form:- there are poles at u=0,v=0. Then there is one more pole at v = u − iz 1 N gives same expression as for So this expression is written :- then the kernel equation can be solved in two ways:-1. Hermite polynomial summation 2. Saddle point approximation for the integrals

Kernel solution from hermite polynomial summation
In [1] authors has solved this via expressing the kernel as hermite polynomial in contour integral representation:- with an auxilary variable introduction this can be written as:- The kernel K N (λ, µ) can be represented as The summation of the series ( Therefore the kernel in hermite polynomial term We solve this Hermite polynomial representation and after Fourier transform we get the SFF which is again averaged over an interval; Saddle points are solution of the equations df 1 du = 0, df 2 dv = 0 Saddle point approximation of the integral around this points

For Finite N
Now we consider the finite N limit . First kernel for this case:- Saddle point equation for first kernel:- So solving the saddle point equation ∂f ∂u = 0, ∂f These gives the saddle points as :- Fluctuation around saddle points to get the solution of kernel:- So solving the saddle point equation ∂f ∂u = 0, ∂f So this gives the saddle points Fluctuation around saddle points to get the solution of kernel:- Two Point Correlation Function then represented as :- (a) SFF for c=0.9, N=100, at whole range, Figure 3. Correlation function behavior for two matrix model using Eq. (3.48) Transforming the equation in terms of ω = 2 sin φ This two point correlation function has a similar behavior of Bessel function but it has a exponential decay within it as shown in Fig:-3. Now we need to compute Fourier transform of this two point correlation function to get the dynamical form factor.
We choose the singularities of the above equation to find the integral by residue theorem.The singularities are given by the following equation:- This gives the saddle points:- First two are branch points so we choose our contour avoiding this two point and compute residue w.r.t other two points.

Two point correlation function between same matrices
Following the same change of measure and distribution function we solve this by external source matrix A.
After letting a i → 0 (4.5) This is same as one point correlation function or level density. Now we take the case for α 1 = α 2 disconnected part of correlation function has the form:- Now we use one transformation for connected part :- now we do the Fourier transform two get the two point correlation function:- (4.10)

Solving this Integral by four-variable saddle point method.
Here we consider four variable saddle point solution discussed in [33,34] Equation for saddle point evaluation:- So our saddle points are the simultaneous solution of four equations.
Which takes the form:- Solving this equation gives sixteen set of solution as the saddle points. Now using saddle point method for four variables with the transformation:- (4.14) Now changing the transformation to its previous form and setting λ = 0 by Then this gives us the correlation function for same matrix model.
, ω → 0 (4.17) Finding residue w.r.t this poles gives us the Spectral Form Factor.We plotted its time average defined as:- x 0 is the biggest maximum of g(x).So, g (x 0 ) = 0 Now we change the integration variable to So the taylor expansion of Ag(x) and f (x) around x 0 can be given as So the saddle point expression gives For two variable saddle point method Now we repeat this same method for two variable saddle point approximation For saddle point approximation we choose the main contributing points of the integral and this set of point is given by:- Now we change the integration variable to Now we make Taylor expansions of AH(x, y) and F (x, y) around x 0 and y 0 and choose up to certain terms to get the 1 A n terms completely for n = 1 2 , 1 in e AH(w,z) F (w, z) e AH(w,z) F (w, z) = Exp AH(x 0 , y 0 ) + 1 2 w 2 H (2,0) (x 0 , y 0 )

Second order contribution of SFF
Now we evaluate nest order contribution for correlation function using second term of the (Eq:-(5.9)). We use this relation for Expression of both the kernels in Eq:-(3.36) and Eq:-(3.41). We follow the exactly same procedure thereafter and at first evaluate the two-point correlation function. Now we evaluate spectral form factor by Fourier transform exactly as Eq:-(3.49).We choose the singularities of this equation to find the inegral by residue theorem.The singularities are same as previous case. Then we compute residue w.r.t these points3.4.2.
Now we use Grassmann variable ψ α and χ β for this integral. We know for Grassmann variable integration has determinant form given by:- Where c and c † are Grassmann variables So we can write the correlation function in integral form as So writing the averaging in integral form:- Now we change the measure by HarishChandra-Itzykson-Zuber formula and get this matrix integral in eigen value integration format with use of Vandermonde determinant like previous case:- on integrating over q i this gives four fermionic term which can be simplified by auxiliary matrices. We choose B 1 B 2 to be hermitian matrix of size k 1 × k 1 ,k 2 × k 2 . D is complex rectangular matrix of size k 1 × k 2 .
Solving the integral we the integral over grassman variables as With this simplification we solve the Eq:-(6.6) and get it in much simplified form det(X i ) (6.10) Now we use a transformation This simplifies the integral :- Tr(log(1−K i )) (6.11) Here we have used det(A) = e Tr(Log(A)) and the matrix K is reduced from X We set A=aI with constraint a = √ 1 − c 2 . Now we can expand Log(1-K) in taylor series upto 3rd term, Considering upto K 3 gives the term in power of exponential [Eq:-(6.11)] as:- Now at the edge of the spectrum for the matrix M 1 edge scaling limit at large N gives:- Dropping the negiligable terms ( Q is the decoupled part generated after integration over B 2 Integrating out D † and D gives logarithmic term:- This has been related to Airy Matrix model coupled with a logarithmic potential (Kontsevich -Penner model ) in [35] Derivation for B 4 1 term So, Tr(Log(1-K)) has terms from four contribution, as trace is there we can consider only the diagonal terms in each of Tr[K n ]. So for Tr[ 1 Although Tr(B 4 ) term is absent in the edge scaling, this term can be derived as [36][37][38]. Two converging saddle points gives rise to fold singularity as in the B 3 1 expression.This is related to Airy kernel discussed in [39]. For extended Airy Kernel Eq:-(6.21) cubic singularity becomes quartic term.This is expressed in terms of Pearcey function and showed in [36,37] on the level spacing distribution for hermitian random matrices with an external field. If H=H 0 +V where H 0 is a fixed matrix and V is an N × N random GUE matrix. H 0 has eigenvalues ±a each with multiplicity N 2 . Spectrum of H 0 is such that there is a gap in the average density of eigenvalues of H which is thus split into two pieces. With N → ∞ density of eigenvalues supported on single or double interval depending on size of a. At the closing of gap the limiting eigenvalue distribution has Pearcey kernel structure.When the spectrum of H 0 is tuned so that the gap closes limiting eigenvalue distribution have the same structure as Pearcey kernel.   [40][41][42][43][44][45]. In [30] two matrix model correlation function has been related to Kontsevich-Penner Matrix model near Heisenberg time.Using Replica method they have studied the intersection number discussion in this context. In our previous calculation we have obtained a rounding off behavior near Heisenberg time. The universal behavior of SFF ramp region Dyson sine kernel is now changed. It suggests that some new kind of description is needed in this region. Kontesevich [46] and Penner [47] Matrix models gives the edge behavior and open boundaries for the punctured open Riemann surfaces. This has been explained in [30,42,45,[48][49][50][51].Universal Dyson sine kernel gives one important feature of underlying Guassian Unitary Ensemble , its stationary nature under Dyson Brownian motion.But now universality of sine kernel are no more available. To explain the rounding off behavior we need to consider Brownian motion near edges.This brownian motion effect is related to time dependence of the model.

Discussion
In this paper starting from time dependent Gaussian Unitary Ensemble(GUE) matrix model we converted it in two matrix model and with contour integral representation for correlation function, SFF and average of SFF has been calculated and discussed. We have considered both type of correlation function and also the next order contribution of 1/N expansion, for saddle point integral. SFF for different matrix correlation has been shown to have a rounding off near Hisenberg time τ = τ c , a crossover in this point. In [1] it has been discussed as breakdown of one-matrix model and singularity at this point and also referred to the case of mesoscopic dirty metals discussed in [52]. For our same matrix correlation function and spectral form factor it gives a decaying average spectral form factor which is consistent with GUE behavior of SFF.Second order contribution calculated here from the 1 N expansion of saddle point integral gives same rounding off behavior and appear as correction to the first order solution. Here the rounding off behavior is different with increasing dimension of matrix(N).