  In this paper, we conduct rigorous error analysis of the Lie-Totter
time-splitting Fourier spectral scheme for the nonlinear Schr\"odinger equation
with a logarithmic nonlinear term $f(u)=u\ln|u|^2$ (LogSE) and periodic
boundary conditions on a $d$-dimensional torus $\mathbb T^d$. Different from
existing works based on regularisation of the nonlinear term $ f(u)\approx
f^\varepsilon(u)=u\ln (|u| + \varepsilon )^2,$ we directly discretize the LogSE
with the understanding $f(0)=0.$ Remarkably, in the time-splitting scheme, the
solution flow map of the nonlinear part: $g(u)= u {\rm e}^{-{\rm} i t
\ln|u|^{2}}$ has a higher regularity than $f(u)$ (which is not differentiable
at $u=0$ but H\"older continuous), where $g(u)$ is Lipschitz continuous and
possesses a certain fractional Sobolev regularity with index $0<s<1$.
Accordingly, we can derive the $L^2$-error estimate: $O\big((\tau^{s/2} +
N^{-s})\ln\! N\big)$ of the proposed scheme for the LogSE with low regularity
solution $u\in C((0,T]; H^s( \mathbb{T}^d)\cap L^\infty( \mathbb{T}^d)).$
Moreover, we can show that the estimate holds for $s=1$ with more delicate
analysis of the nonlinear term and the associated solution flow maps.
Furthermore, we provide ample numerical results to demonstrate such a
fractional-order convergence for initial data with low regularity. This work is
the first one devoted to the analysis of splitting scheme for the LogSE without
regularisation in the low regularity setting, as far as we can tell.
