# Mean-Field Theory in Second Quantization Formalism

Consider the Ising model in statistical physics

$$H=-J\sum_{\left}s_{i}s_{j}-\mu h\sum_{i}s_{i}$$

In this case mean-field approximation is done by replacing the surrounding spins with their averages $$s_{i}\rightarrow\left\equiv s$$ in such a way to get an effective Hamiltonian of decoupled spins

$$H_{\rm MF}=-\left(Jzs+\mu h\right)\sum_{i}s_{i}$$

On the other hand, consider the Bose-Hubbard model

$$H-\mu N=-t\sum_{\left}b_{i}^{\dagger}b_{j}+\frac{U}{2}\sum_{i}n_{i}\left(n_{i}-1\right)-\mu\sum_{i}n_{i}$$

People usually assume that in the mean-field regime the destruction operator acquires an expectation value $$\psi\equiv \left$$. What justifies this assumption? Why is this mean-field - what is the analogy to the Ising model mean-field procedure? In addition, people continue to state that the operators decouple as

$$b^{\dagger}_{i}b_{j}=\leftb_{j}+b^{\dagger}_{i}\left-\left\left$$

How is this equation derived? I find in many books claims such as $$b_{i}=\left+\delta b_{i}$$ trying to explain the above equation, but without much justification. To me this seems like an assumption on the matrix elements of $$b_{i}-\left$$ being small, but I fail to connect this to any mean-field argument.

Let me respond to your points in a different order.

In addition, people continue to state that the operators decouple as

$$b^{\dagger}_{i}b_{j}=\leftb_{j}+b^{\dagger}_{i}\left-\left\left$$

How is this equation derived?

Writing $$b_i=\langle b_i\rangle + \delta b_i = \langle b_i \rangle + \left(b_i-\langle b_i\rangle\right),$$ we have $$b_i^\dagger b_j = \langle b_i^\dagger\rangle b_j + b_i^\dagger \langle b_j\rangle - \langle b_i^\dagger\rangle\langle b_j\rangle + \delta b_i^\dagger \delta b_j$$ Note that $$\delta b_j$$ is to be interpreted as a deviation away from the mean-field value. For mean-field theory to be accurate, this fluctuation should be relatively small, which leads us to neglect the $$\delta b_i^\dagger \delta b_j \approx 0$$ term. This really is the central assumption of mean-field theory - that some parameter doesn't deviate too much away from its average value.

Consider the Ising model in statistical physics $$H=-J\sum_{\langle i,j\rangle} s_is_j -\mu h\sum_is_i$$ In this case mean-field approximation is done by replacing the surrounding spins with their averages si→⟨si⟩≡s in such a way to get an effective Hamiltonian of decoupled spins...

Actually, let's try the same mean-field procedure as before: $$s_i = \langle s_i\rangle + \left( s_i - \langle s_i\rangle\right)$$. Then you get $$H=-J\sum_{\langle i,j\rangle} \left( \langle s_i\rangle s_j + s_i\langle s_j\rangle - \langle s_i\rangle\langle s_j\rangle + \delta s_i \delta s_j \right) -\mu h \sum_i \left( \langle s_i\rangle + \left( s_i - \langle s_i\rangle \right) \right)$$ Now we can first neglect the $$\delta s_i \delta s_j$$ term. Then we can recognize that $$\sum \langle s_i\rangle \langle s_j\rangle$$ only shifts the energy by a constant, and can neglect also this part. This yields your $$H_{MF}$$. [Note that neglecting the constant part is a standard simplification, but one that may fail sometimes, for example if we want to compare the energy of different phases in order to construct a phase diagram.]

Now, back to the Bose-Hubbard model:

People usually assume that in the mean-field regime the destruction operator acquires an expectation value $$\psi \equiv \langle b_i\rangle$$. What justifies this assumption? Why is this mean-field - what is the analogy to the Ising model mean-field procedure?

This is a rather subtle question. In a state with a fixed number of particles you'll have $$\langle b_i\rangle \equiv 0$$. That is, vanishing vacuum expectation values for operators changing the particle number. In a superfluid or Bose-Einstein-condensate (BEC) phase this changes, because the state can be (to good approximation) described by coherent states. This makes $$\langle b\rangle$$ a good order parameter - it's non-zero in the BEC phase, and zero in the normal phase. Similarly, the magnetization $$\langle s\rangle$$ is a good order parameter for the Ising model. The mean-field procedure then proceeds (in both cases) by a mean-field approximation around the presumed finite value of the order parameter.

In the case of a magnet you can literally interpret the mean-field as a background magnetic field coming from the other spins. More generally, the mean-field is the average effect from the surrounding sites/particles. In the BEC case, the effect is due to the surrounding condensate.