This Hamiltonian has what is known as a "spin-flip" symmetry. It means that due to the term ∑SziSzj, we can simultaneously change the sign of all the Szi operators and we still have the same Hamiltonian (the operator that commutes with the Hamiltonian is G=∏iSxi, which produces a global spin-flip over states in the z-basis).
The eigenstates (and the ground state) of this Hamiltonian depend on the magnitude of B:
-If B>>1, the dominant term of the Hamiltonian is B∑iSxi, so the eigenstates are close to product states in the x basis: |n⟩≃|←→→...⟩. This phase is called the paramagnetic phase.
-If B<<1, the dominant term of the Hamiltonian is −∑SziSzj so the eigenstates are close to cat states of the form |n±⟩≃=↑↓↓...⟩±|↓↑↑...⟩, oriented in the z-axis. Due to the negative sign in −∑SziSzj, this phase would be called antiferromagnetic.
This change of order in the eigenstates (and the ground state) is a phase transition that goes from eigenstates with a defined parity (B<<1) to states without a well defined parity B>>1. This phase transition is said to have a spontaneous symmetry-brake and I think that in quantum mechanics, although the eigenstates of B<<1 do not explicitly break the symmetry (because the cat states are a superposition of the two possibilities of the symmetry-break), the name of "spontaneous..." is kept because of the similarities with the classical case.
Going now to the question itself, when Xiao-Gang Wen says says that small B does not break the symmetry, I think he is meaning respect to the order of the ground state, saying that samll B would not be enough to produce the phase transition.
And when it is said that a finite B does not brake the symmetry neither, I think he is meaning that you cannot explicitly break/destroy the symmetry of the Hamiltonian with the term B∑iSxi, i.e., removing the spin-flip property, that is always present in our case. However, you could introduce a term in the z direction like Bz∑iSzi that explicitly breaks the symmetry of the Hamiltonian, and you lose the spin-flip property.
I hope this can help you!
This post imported from StackExchange Physics at 2020-12-03 17:31 (UTC), posted by SE-user RMPsp