Almost in every textbook of condensed matter physics, the standard description of SSB could be formulated as follows:
Consider the lattice Heisenberg model in an external magnetic field H=∑ijJijSi⋅Sj+hSz, where h is the magnitude of magnetic field and Sz=∑iSzi. Now the average magnetization per site is a function of both magnetic field h and number of lattice sites N, say m≡∑i⟨Szi⟩/N=m(N,h), where ⟨Szi⟩≡tr(ˆρSzi) with ˆρ=e−βH/tr(e−βH) the density operator. Then if limh→0limN→∞m(N,h)≠0, we say the system has SSB at temperature T. Now I get some questions:
(1)We know at finite N and zero h, m(N,h=0)=0 due to spin-rotation symmetry. But there is no reason for that limh→0m(N,h)=m(N,h=0)—[1], right? Since the function m(N,h) may not be continuous at h=0, from the math viewpoint.
(2)If Eq.[1] is correct, and hence limh→0m(N,h)=0, then limN→∞limh→0m(N,h)=0, right?
(3)If Eq.[1] is wrong, say limh→0m(N,h)≠m(N,h=0) and hence limh→0m(N,h)≠0, then what about limN→∞limh→0m(N,h)? And why don't we use this identity to define SSB?
Thank you very much.
This post imported from StackExchange Physics at 2014-03-09 08:42 (UCT), posted by SE-user K-boy