# Is there angular momentum conservation in models like the Ising model?

In Quantum Mechanics conservation laws are fundamental, I was thinking about spin altering models of interaction such as the Ising Model and realized that it isn't at all clear how angular momentum conservation occurs in such a situation. If there is spin flipping occurring where does the contrary compensating angular momentum go as to have global angular momentum conservation? The interaction fields absorb this angular momentum?

• There is nothing fundamental about conservation laws in either classical mechanics or quantum mechanics. A simple rigid wall does, for instance, not conserve momentum. Neither does the Kepler problem with a fixed center mass conserve momentum. In QM none of the potentials where V(x) is a function of just one coordinate is momentum conserving and a system with magnetic field does not conserve angular momentum. WE have to chose potential functions that are. Sometimes we do and sometimes we don't. It's a choice and we have to be aware of its consequences. Commented May 31, 2023 at 1:27
• @FlatterMann you should write this as answer, so it does not disappear! Commented May 31, 2023 at 3:49
• @hyportnex It doesn't really answer the question about the Ising spin model, though. It's just a general remark why having non-conservative models is a perfectly fine approximation in appropriate situations (nobody is going to complain about the planet speeding up or slowing down a bit just because a kid bounces a ball of a wall). The two answers given, so far, are addressing the concerns of the OP much better and I really don't know enough about the Ising spin model to give an appropriate answer. Thanks for the support, though. Commented May 31, 2023 at 5:41
• Although only tangentially related to your question, note that one can (and one does!) study the Ising model in a fixed-magnetization ensemble. (It is actually quite a natural thing to do in the lattice gas interpretation of the model, since it then corresponds to a canonical constraint.) Commented May 31, 2023 at 14:29

It is the bath degrees of freedoms which absorb or provide the compensating forces.

It depends on the ensemble you are using:

1. Microcanonical Ensemble: The number of + spin and number of - spin is conserved since energy is conserved.

2. Canonical Ensemble: The system in question can exchange energy and spin with a bath at temperature T. So No conservation.

3. Grand Canonical Ensemble: No conservation.

in 2 and 3, it is the bath degrees of freedoms which absorb or provide the compensating forces. But all three will agree in relations between average quantities as the fluctuations die down as some power of N(Number of Particles).

• Regarding 1., note that fixing the energy does not fix the magnetization (think of a 2d grid with one spin having two + neighbors and two - neighbors, for instance; in this case, you can flip the spin at 0 energetic cost). Commented May 31, 2023 at 15:55
• Yeah, true. Energy conservation leads to number conservation only with some magnetic field bias. Commented May 31, 2023 at 17:55

Angular momentum is conserved, but sometimes that isn't the important thing. An Ising model might be used to study magnetic properties of a solid, or phase transitions. Details of how angular momentum is conserved are not the primary interest, and don't have a large effect on these properties.

If spins flip and primarily align in one direction, a tiny rotation of the solid can make angular momentum be conserved.

Sometimes a benefit of a model is that it ignores things that are not of immediate interest. This simplifies calculations, and makes more processor time available for the things that do matter.

The Hamiltonian of the Ising model is usually assumed to be $$H=-J\sum_{i,j} s_is_j$$ There is no momentum in this expression and no kinetic energy! In Statistical physics, it is common that you can integrate part of the partition function. Think to the ideal gas: \eqalign{ {\cal Z}&=\int e^{-\beta\sum_i[{p_i^2\over 2m}+V(\vec r_i)]}d^{3N}\vec p d^{3N}\vec r\cr &=(2\pi mk_BT)^{3N/2} \int e^{-\beta\sum_iV(\vec r_i)}d^{3N}\vec r } The same has been done with the Ising model. Indeed, the partition function of the Ising model is usually written as $${\cal Z}=\sum_{\{s_i=\pm 1\}} e^{-\beta H}$$ One can notice that there is no integration (or summation) over the momentum of the spins. It is because this integration has already been performed, as in the above example of the ideal gas! What is known as the Ising Hamiltonian is actually not an Hamiltonian but only the potential energy. There is no kinetic energy anymore. Therefore, it is not possible to derive conservation laws for momentum or angular momentum.

• In the continuum (scaling) limit, where the ising model approaches a scalar field theory, can't you write down momentum operators? Commented May 31, 2023 at 18:53