# What are the state-of-the-art methods for simulating a time-dependent transverse-field Ising model?

Consider a spin-1/2 Ising model with time-dependent transverse field:

$$H = - \sum_{i

Given the initial state $$|\psi(0)\rangle$$ (usually $$|\psi(0)\rangle = \left( \frac{|0\rangle + |1\rangle}{\sqrt{2}} \right)^{\otimes N}$$), what are the state-of-the-art methods for numerically calculating the expectation value of the final state with respect to an observable $$\sigma^z_i$$: $$\langle\psi(t')|\sigma^z_i|\psi(t')\rangle$$?

### My current method

I believe this method is called discrete-time path-integral Monte Carlo method.

Referring to chapter 5.3 of (Binder and Heermann, 2019), by dividing the Hamiltonian into two non-commuting parts and apply Trotterization, we can obtain a corresponding classical Hamiltonian ($$z_i^k = \pm 1$$) that is one-dimensional higher: $$H_c = - \sum_{k=1}^P \left( \sum_{i $$J^\perp(t) = -\frac{PT}{2} \ln \tanh \frac{\Gamma(t)}{PT}$$ and then use a classical Monte Carlo method (e.g. simulated annealing) to sample the classical states at each time step. The expectation value $$\langle\psi(t')|\sigma^z_i|\psi(t')\rangle$$ is estimated by $$\frac{1}{P} \sum_{k=1}^P z_i^k$$.

• In how many dimensions are you mainly interested? I believe Matrix-Product States (aka DMRG) are excellent in 1D by now, but much harder for larger dimensions Oct 27, 2021 at 10:50
• @Wouter Do you mean the number of spins or the connectivity? The number of spins is usually around a few hundreds, sometimes up to ~2,000, with full connectivity between spins (i.e. generally $J_{i, j} \neq 0 \forall i, j$). Simulating 2,000 spins isn't necessary, but I would like to simulate at least 100 spins with full connectivity. Oct 27, 2021 at 14:38
• @Wouter I looked into DMRG, unfortunately I need to calculate problems that have denser interactions between spins than even a 3D lattice model. Oct 27, 2021 at 14:51
• What about simple free fermion simulation? You can turn this problem into a time-dependent Majorana fermion chain and evolve in time by keeping track of the correlation matrix alone.
– Zack
Oct 27, 2021 at 15:07
• – Zack
Oct 27, 2021 at 18:11