Transverse Ising model

Transfer matrix methods

Transfer Matrix Method

Quantum Monte Carlo

We can emply the Wolff algorithm to accelerate the flipping of spins by instead flipping a cluster of spins. In this algorithm (there are other cluster algorithms), a randomly selected spin (uniform sampling) is connected to its neighbours iff they share the same sign. The probability that both neighbors have the same sign is \(1-e^{-2\beta J}\), and is used for calculating adding links to the cluster (since they are assumed frozen).

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

\[ \begin{aligned} Z&=\mathrm{Tr}\;e^{-\beta H},\\ &= \mathrm{Tr}\;e^{-\beta(H_z+H_x)},\\ &=\lim_{M\rightarrow \infty}\qty(e^{\beta H_z/M}e^{\beta H_x/M})^M,\\ \end{aligned} \]

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 step, compared to single spin-flip methods.