  We consider the limit behavior of partition function of directed polymers in
random environment represented by linear model instead of a family of
i.i.d.variables in $1+1$ dimensions. Under the assumption that the correlation
decays algebraically, using the method developed in [Ann. Probab.,
42(3):1212-1256, 2014], under a new scaling we show the scaled partition
function as a process defined on $[0,1]\times\RR$, converges weakly to the
solution to some stochastic heat equations driven by fractional Brownian field.
The Hurst parameter is determined by the correlation exponent of the random
environment. Here multiple It\^{o} integral with respect to fractional Gaussian
field and spectral representation of stationary process are heavily involved.
