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11.22主要内容
Goldstone mode, Goldstone theorem
- Background:Heisenberg model. H=J∑⟨i,j⟩→Si⋅→Sj————Magnon(类似phonon), gapless & εk=v|k|
- Before SSB
L=12∂μϕ∂μϕ−V(ϕ),
其中V(ϕ)=m22ϕ2−λ4!ϕ4
After SSB
L=L0+cubic terms+quartic terms,
其中L0=12∂μπ∂μπ−12∂μ→σ⋅∂μ→σ−m2π2
(N−1) massless fields,1 massive field
- Application:
(1)Magnetism(Goldstone mode=magnon); (2)superfluid; (3)phonon.
- Goldstone Theorem(数学表达):
{G⟷L before SSBH⟷L after SSB⟹dim(G/H)=Goldstone mode number
Higgs mechanism(massive gauge field/boson)
- Background
Superconductivity + Maxwell equation.(1) London equation; (2) Ginzberg-Landau equation; (3) BCS theory
- N=2, ϕ=(ϕ1ϕ2)=ϕ1+iϕ2
Before SSB,
L=−14FμνFμν+(Dμϕ)∗(Dμϕ)−V(ϕ),
其中V(ϕ)=m22ϕ†ϕ−λ4!(ϕ†ϕ)2
eat Goldstone mode,
L=12∂μϕ1∂μϕ1−m2ϕ21−14FμνFμν−√2ve(Aμ−gμ)∂μϕ2+e2v2AμAμ,
其中gμ=12√2ev∂μϕ2
产生massive boson
- massive gauge boson ⇔ proca equation(只能写成A的形式,而不能写成→E,→B的形式)
break gauge invariant(Superfluid(Boson), Superconductivity(Fermion))
- 自由度不变
before SSB: ψ=(ϕ1,ϕ2,Boson(2))共4个自由度
after SSB: ψ=(ϕ1,0,Boson(2+1))共4个自由度
London equation & Ginzberg-Landau equation
其他补充
USTC|
BBS