## Time correlation function

In this calculation, the bath spectrum used include Ohmic bath with exponential cutoff funciton and the Debye bath.

### Temperature

high temperature, $\beta=0.25$
low temperature,$\beta=10,50$
$\beta=1/k_BT=0.25,1,5,10,50$

### Scaling limit

$\omega_c/\Delta=0.25,2,2.5,5$

### Debye Spectrum

The imaginary part of C(t) can be replaced by a single exponential function

Meanwhile, this need us to fit the real part more carefully.