On large $q$ expansion in the Sachdev-Ye-Kitaev model

We consider the Sachdev-Ye-Kitaev (SYK) model where interaction involves $q$ fermions at a time. We find the next order correction to the thermal two-point function in the large $q$ expansion. Using this result we find the next order correction to the SYK free energy.


INTRODUCTION
The Sachdev-Ye-Kitaev (SYK) model is a quantum mechanical model of N interacting Majorana fermions χ i , i = 1, . . . , N with the Hamiltonian [1,2]: where {χ i , χ j } = δ ij and J i1...iq are random couplings drawn from a Gaussian distribution with zero mean and a width J 2 i1...iq = (q − 1)!J 2 /N q−1 . One is usually interested in computing correlation functions, and particularly two-point function at temperature T = 1/β: At the large N limit only melonic Feynman diagrams contribute to the two-point function in the SYK model. These diagrams can be resummed and one obtains a nonperturbative Schwinger-Dyson equation: where G(iω n ) = β 0 dτ e iωnτ G(τ ) and ω n = 2πβ −1 (n + 1/2). It is not possible to solve this equation analytically, but one can find solution in the infrared limit, where ω is small and the bare −iω n -term in (3) can be neglected [1][2][3][4]: where J 2 b q π = (1/2 − 1/q) tan(π/q). Nevertheless it is still interesting to obtain some analytic approximation for G(τ ) which interpolates both UV and IR regions. One way to proceed is to use the large q expansion. The first order in 1/q was found in [3]. In this note we compute the next 1/q 2 correction and argue that it improves the approximation significantly, such that it agrees with numerical results quite well. At the next section we compute 1/q 2 correction to the two-point function. Next we compare the large q results and numerics. At the end we compute the large q free energy and the coefficient of the Schwarzian action.

LARGE q TWO-POINT FUNCTION
We consider the large q ansatz for the two-point function [3]: For the self-energy (3) we find (we assume that q is even) where a new coupling constant J 2 = 2 1−q qJ 2 is introduced. From now on we work on the interval τ ∈ [0, β] and we can omit sgn(τ ) in all formulas. Expanding G(iω n ) −1 in 1/q series up to 1/q 2 term using (5) we obtain where g * g(iω n ) ≡ β 0 dτ e iωnτ g 2 (τ ). Then using the equations (3) and (6) and going back to the coordinate space we find differential equations for each order of 1/q: and the functions g(τ ) and h(τ ) satisfy the boundary conditions g(0) = g(β) = 0 and h(0) = h(β) = 0. Now we introduce a convenient variable x = πv 2 − πvτ β . Then the first equation has the solution Using this solution the second equation can be represented as The solution to this equation can be written as where the Green's function G(x, y) obeys the equation with the boundary conditions G(− πv 2 , y) = G( πv 2 , y) = 0. One can solve this equation and obtain an explicit formula for the Green's function where V ≡ πv 2 + cot πv 2 and x > ≡ max(x, y) and x < ≡ min(x, y). Computing the convolution πv β and using the explicit formula for the Green's function (13) we obtain from (11) h ) and g(x) is given in (9). One can compute explicitly the integral (formulas from [5] are useful)

COMPARISON WITH NUMERICAL RESULTS
In this section we compare the large q result with the numerical solution of the Schwinger-Dyson equation (3). In general we expect the large q formula to work well when |g(τ )| ≪ q and |h(τ )| ≪ q 2 . These inequalities are fulfilled when βJ ≪ πe q/2 .
Looking at the explicit formula (15) it is tempting to exponentiate the result and to introduce an exponentiated large q two-point function which is equivalent to (5) up to order 1/q 2 . We plot numerical and the large q results for q = 4 and different values of βJ in figure 1. We can see that the exponentiated result works very precisely even for large βJ, whereas the large q answer (5) deviates significantly from numerics at large βJ. . The blue dash-dotted line is the large q approximation (5) with 1/q 2 term. The blue dashed line is the exponentiated two-point function (17).

LARGE q FREE ENERGY
The leading large N approximation to the free energy in the SYK model is [2,6] To avoid evaluating the Pfaffian it is convenient to differentiate the free energy by J∂ J [3] where from (9) and (15) we find Next, using (16) and we can integrate back and obtain −βF/N = 1 2 log 2 + Expanding the free energy at strong coupling by using that we find where the first three terms are the ground state energy, the zero-temperature entropy and the temperature dependent correction to the entropy. The zero temperature entropy coincides with the large q expansion of the formula [2,7] The last term in (24) agrees with the formula reported in [8][9][10].
We note that one can improve approximation by using more terms near q = 2 [9]. We plotted Pade approximation and numerical results adapted from [3] in figure  2. We see that the Pade approximation is very close to numerics. The black circles correspond to numerical results adapted from [3]. The blue solid line corresponds to the two-sided Pade approximation (29).