Trotterization
Furthermore, since spins can flip as a function of (imaginary) time, they also need to be added.
This is easily seen from the trotterization of the Hamiltonian,
$$ \begin{aligned} H &= -J \sum_i \sigma^z_i \sigma^z_{i+1} - h \sum_i \sigma_i^x \ &= H_z + H_x, \end{aligned} $$
and then the partition function
$$ \begin{aligned} Z &= \mathrm{Tr}; e^{-\beta H}, \ &= \mathrm{Tr}; e^{-\beta (H_z + H_x)}, \ &= \lim_{M \rightarrow \infty} \left(e^{\beta H_z/M} e^{\beta H_x/M}\right)^M, \end{aligned} $$
where $\beta = \Delta \tau M$ evolves in imaginary time.
This is the result of Path-Integral QMC: the 1D quantum Ising model is mapped to a 2D classical Ising model.
$$ S = -K_s \sum_{i,n} \sigma_{i,n} \sigma_{i+1,n} - K_t $$
This allows for more statistically independent configurations (because of detailed balance?) to be generated in fewer steps, compared to single spin-flip methods.
The magnetization $|m|$ can distinguish between a ferromagnetic (FM) and a paramagnetic phase.
Chi
The Binder cumulant is defined as
$$ U_L = 1 - \frac{\langle m^4 \rangle}{3 \langle m^2 \rangle^2} $$
Binder parameter