My question is related to Chapter 3 of Prof. Yvan Velenik's book "Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction".
For an Ising model defined on a finite volume $\Lambda \subset {Z^d}$, $Z^d$ is d-dimensional cubic lattice, the magnetization density $m_\Lambda ^\# (\beta ,h)$ is $$m_\Lambda ^\# (\beta ,h)\mathop \equiv \limits^{{\rm{ def }}} \left\langle {\frac{1}{{|\Lambda |}}\sum\limits_{i \in \Lambda } {{\sigma _i}} } \right\rangle _{\Lambda ;\beta ,h}^\#$$
Here '#' denotes the boundary condition, $h$ is the external magnetic field, $|\Lambda |$ is the number of lattice points in $\Lambda$ and ${\sigma _i}$ is the spin on point $i$, $\left\langle {} \right\rangle _{\Lambda ;\beta ,h}^\#$ is the ensemble average.
In finite system, $m_\Lambda ^\# (\beta ,h)$ should be a real analytic function and it is an odd function with respect to $h$. Therefore, no net manetization density at $h=0$.
My question is dose the above statement also holds for $m_\Lambda ^ + (\beta ,h)$, that is with the + boundary condition. As can be proved (page 106),
$$\left\langle {{\sigma _0}} \right\rangle _{\beta ,h}^ + \le m_\Lambda ^ + (\beta ,h)$$
where $\left\langle {} \right\rangle _{\beta ,h}^ + $ is an infinite-volume Gibbs state with + boundary conddition. When $\beta$ is large, such that we have spontaneous symmetry breaking, $\left\langle {{\sigma _0}} \right\rangle _{\beta ,0}^ + > 0$. Does that mean we have
$$m_\Lambda ^ + (\beta ,0)>0$$
that is spontaneous magnetization density in finite system with + boundary condition at $h=0$?
From another angle, $m_\Lambda ^\# (\beta ,0)=0$ can be seen from $m_\Lambda ^\# (\beta ,h) = - \frac{1}{{|\Lambda |}}\frac{{\partial F_\Lambda ^\# (\beta ,h)}}{{\partial h}}$, where $F$ is the Helmholtz free energy. (In the book, $\psi$ page 83 is used instead of $F$. I think they give the same result) Since, $F_\Lambda ^\# (\beta ,h)$ is an even function of $h$, $m_\Lambda ^\# (\beta ,h)$ is odd which certainly gives $m_\Lambda ^\# (\beta ,0)=0$.
But for $F_\Lambda ^ + (\beta ,h)$, what we have is $F_\Lambda ^ + (\beta ,h) = F_\Lambda ^ - (\beta , - h)$. $F_\Lambda ^ + (\beta ,h)$ is not an even function itself.
In my opinion, does that because the system with + boundary condition does not corresponds to real system. We use + boundary condition just to mimic the effect of an external magnetic field which will reduce the probability of microstates with - spins. And that is the basic idea of symmetry breaking. Actually, in finite system with + bondary condition, we truely have the fact that probability of + spins is larger than that of - spin.
I wonder if I got something wrong.