Suppose we have a 3d Ising Model(NN interaction) in simple cubic lattice, if we define a subsystem of it to be a 2d plane of spins(for example all sites with z = L/2, L being the linear system size) and also define this subsystem energy just like a 2d Ising model(only 4 neighbor).

What is the probability distribution of this subsystem states?

For the 3d system itself this probability distribution is proportional to exp(-E/T) but I do not think for the 2d subsystem this holds.

  • $\begingroup$ 3d Ising model is a hard problem. I think it is possible to find an explicit probability distribution of a subsystem states for the simple 1d Insing model and for the not so simple 2d Ising model. $\endgroup$ – Gec Nov 22 '18 at 17:41

The total density operator for the 3D system is: $$\rho = \frac{1}{\mathcal Z} e^{-\beta H}$$ where $H \equiv - J \sum_{\langle i,j\rangle} \sigma_i^z \sigma_j^z$ is the Ising Hamiltonian with nearest neighbor interactions. Now if you want to describe the density operator of the 2D subsystem (and hence the probability distribution), which I'll call $A$, you need to take a partial trace over all other degrees of freedom ($\bar A$), i.e. all spins outside that 2D plane. So the density operator of the subsystem is: $$\rho_A \equiv \mathrm{Tr}_{\bar A} (\rho) = \frac{1}{\mathcal Z} \mathrm {Tr}_{\bar A}(e^{\beta J \sum_{\langle i,j\rangle} \sigma_i^z \sigma_j^z})$$ I don't think there is an analytic way to exactly evaluate such an expression; so you would probably need to resort to numerical methods for it.

For reference, even calculating simple quantities like just the partition function of the Ising model is a hard problem (except in 1D and 2D where exact expressions exist). People usually resort to approximation methods such as mean field theory to calculate such simple quantities. So actually calculating stuff in the Ising model by hand is usually pretty hard.

  • $\begingroup$ thank you @Sahand Tabatabaei, I don't want analytic solution, I just want to know that if in a simulation I have the total Energy(of the 3d system) and also the energy of the subsystem(treated like a 2d Ising model) what is the probability of the state? for example in 3d Ising model each configuration probability is proportional to exp(-E/T) $\endgroup$ – mehrdad Nov 22 '18 at 18:35
  • $\begingroup$ No problem. The probability distribution of states for the subsystem is not necessarily just a function of its energy anymore. In other words, states of the subsystem with the exact same energy may have a different probability of occurrence. I think you'll need to calculate the probability distribution of the 3D system, and take a partial trace over the spins not in your 2D subsystem to calculate its probability distribution. $\endgroup$ – Sahand Tabatabaei Nov 22 '18 at 19:45
  • $\begingroup$ unfortunately I am not familiar with partial trace , The probability distribution of states for the subsystem can not even be a function of both 3D system Energy and subsystem Energy?! $\endgroup$ – mehrdad Nov 22 '18 at 20:07
  • $\begingroup$ The probability distribution of the subsystem definitely is a function of the energy levels of the full 3D system (because the probability distribution of the 3D system is a function of its energy levels by the canonical distribution). However, the form of that function is non-trivial (you need to solve the partial trace to get it). If you're not familiar with partial traces or density operators, check out any graduate level QM book (e.g. Cohen Tannoudji). $\endgroup$ – Sahand Tabatabaei Nov 23 '18 at 2:42
  • $\begingroup$ Also check out these lecture notes: info.phys.unm.edu/~caves/courses/qinfo-f14/lectures/… $\endgroup$ – Sahand Tabatabaei Nov 23 '18 at 2:45

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