# Difference between classical, semi-classical and quantum Ising model

I have a question I have been struggling about for two weeks now and would be very happy for any advice or direct help here in this forum.

The question is: What is the difference between classical, semi-classical and quantum Ising model? I believe a good answer here helps many people since literature or the internet does not tell too much or just gives you vague answers. I found one answer What is the difference between classical and quantum Ising model? that does not really answer my question.

For simplicity, let us restrict our reflections on the case of the one-dimensional case (d=1).

The Hamiltonian of the Ising model with longitudinal field $h$ looks like $$H=-\sum_{\langle ij \rangle} J_{ij} s_i^z s_j^z-\sum_i h_i s_i^z$$ where the summation runs over all nearest neighbours spins.

How I understand these three cases is the following:

1. Classical Ising model: fields and spins are classical (the spin has continuous values between -1 and +1. The dynamics of the Hamiltonian (for example in simulated annealing) is described by the classical equation of motion.
2. Quantum Ising model: Here we can rewrite the classical spins as $\sigma_i^z$, thus they have just distinct values ($\pm 1$). The fields however are continuous (I've never seen a case where the fields were quantised like in QED for example). The dynamics of the Hamiltonian is described by the Schroedinger equation.
3. Semi-classical Ising model: The fields are classical and spins quantised, the dynamics of the system should be described by the classical equation of motion, although I really do not know if that makes sense at all.

I would be very happy for direct help and literature and thank you a lot in advance!

• The quantum Ising model has a $\sigma_x$ field, not a $\sigma_z$ field! You should really make clear why the other question doesn't answer yours. -- Also, do you have any link to the "semi-classical Ising model", or did you just come up with that yourself? (3+5 hits on google!) – Norbert Schuch Aug 23 '18 at 16:04
• And no, in the classical ising model the s_i also only have values $\pm1$. – Norbert Schuch Aug 23 '18 at 16:10
• To add to what @NorbertSchuch says, in the classical case the dynamics is not given by "the classical equation of motion" (the spins being discrete, what would those be?). One usually use, in this case, stochastic evolutions, such as Glauber or Kawasaki dynamics. – Yvan Velenik Aug 23 '18 at 16:13
• Thank you for your answers. Yes, that is a problem Hamiltonian, one wants to find the ground state in, for example in combinatorial optimisation problems. In the case of an transverse field, of course there is a $\sigma_i^x$ and thus the spins do not commute, so mathematically classical and quantum Ising spin models are different. I read the word semi-classical sometimes when it comes to approximations, so I just wanted to know what that exactly means. – A. Hartmann Aug 24 '18 at 12:06
• @A.Hartmann The problem I see is that the question is phrased as a non-canonical question yet you want a canonical answer. – Norbert Schuch Aug 24 '18 at 13:47