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Crossposted from math.stackexchange. (Link: https://math.stackexchange.com/questions/2848558/is-this-function-meromorphic)

Question

Let $$e^{g\left(\tau,T,J_1,J_2\right)}=\frac{2}{\left(\frac{J_1}{vT}\right)^2+\cos\left(v\left(1-2T\tau\right)\right)\sqrt{\left(\frac{J_1}{vT}\right)^4+\left(\frac{J_2}{vT}\right)^2}}$$

where $v$ is given by $g\left(0,T,J_1,J_2\right)=0$. That is,

$$2=\left(\frac{J_1}{vT}\right)^2+\cos\left(v\right)\sqrt{\left(\frac{J_1}{vT}\right)^4+\left(\frac{J_2}{vT}\right)^2}$$

Suppose, $F\left(T,J_1,J_2,q\right)$ solves

$$\frac{\partial F}{\partial J_1}=-\frac{J_1}{2q^2}\int_0^\beta e^g\mathrm{d}\tau$$ $$\frac{\partial F}{\partial J_2}=-\frac{J_2}{8q^2}\int_0^\beta e^{2g}\mathrm{d}\tau$$ $$F\left(T,0,0,q\right)=-\frac{T}{2}\ln2$$

Then does $F$ diverge and what are its asymptotic divergence exponents?

Background

$$G=\frac{1}{2}\mathrm{sgn}\left(\tau\right)\left(1+\frac{g\left(\tau\right)}{q}+O\left(\frac{1}{q^2}\right)\right)$$ is the solution for the two point function a generalised SYK model with Hamiltonian $$\mathscr{H}=-i^{\frac{q}{2}}\sum\limits_{1\leq i_1<\cdots<i_q\leq N}J^{\left(1\right)}_{i_1,\cdots,i_q}\psi_{i_1}\cdots\psi_{i_q}-i^{q}\sum\limits_{1\leq i_1<\cdots<i_{2q}\leq N}J^{\left(2\right)}_{i_1,\cdots,i_{2q}}\psi_{i_1}\cdots\psi_{i_{2q}}$$

where $q$ is large. From the Schwinger–Dyson equations we get,

$$\frac{\partial^2g}{{\partial\tau}^2}=2\left(J_1^2e^g+\frac{J_2^2}{2}e^{2g}\right)$$

where $J_1$ and $J_2$ are the standard deviations of $J^{\left(1\right)}$ and $J^{\left(2\right)}$. The above given $g$ is the only solution of this equation that doesn't diverge and satisfies $g\left(0\right)=g\left(\beta\right)=0$. Then $F$ is the free energy, and the question is basically looking for phase transitions.

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  • $\begingroup$ diverge where? as $J_i\to\infty$? as $J_i\to0$? $\endgroup$ – AccidentalFourierTransform Jul 12 '18 at 15:49
  • $\begingroup$ Anywhere. Want to find a phase transition at some values of $J_i$ $\endgroup$ – Chetan Vuppulury Jul 12 '18 at 15:51

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