The Nambu-Goldstone boson is a massless boson that appears when a global continuous symmetry is
broken spontaneously. Let us examine the representation of the Nambu-Goldstone boson here.
Let G be a continuous group that represents the symmetry of the system.
We adopt that the symmetry G is spontaneously broken to a subgroup H.
The Lagrangian is written as
L = iψ†∂ψ/∂t+iψ†Γk∂kψ + V(ψ),
where ψ represents a fermion field and V is the interaction term.
Let g be the Lie algebra of the group G and we write the basis of g as {Ta} (a = 1, …, dim G).
We adopt that Ta are hermitian. G are generated by {Ta}.
The symmetry breaking term is written as
LSB = λψ†Mψ,
where M ∈ {Ta} and λ is an infinitesimal parameter.
This can be regarded as a generalization of the Zeeman term in a ferromagnet. We define the order parameter as
Δ = 〈 ψ†Mψ 〉.
Here 〈 〉 denotes the expectation value in the ground state.
Then the spontaneous symmetry breaking is defined as follows:
We say that symmetry is spontaneously broken if Δ ≠ 0 in the limit λ→ 0.
For an infinitesimal parameter θ, ψ is transformed as ψ→ ψ−iθTaψ. Then,
LSB → LSB + δLSB and we have
δLSB = iθλψ†[Ta, M]ψ.
When [Ta, M] ≠ 0 and Δ ≠ 0 for λ→ 0,
the symmetry corresponding to Ta is spontaneously broken and the Nambu-Goldstone boson
(zero mode) exists.
Here the Nambu-Goldstone boson is given by the formula
πa = iψ†[Ta, M]ψ.
In fact we can show that πa is a zero energy mode[1].
The number of πa is given by NBS = dim(G/H): a =1, …, dim(G/H).
This formulation also applies to boson systems.
Thus we can give explicit formulae for the Nambu-Goldstone bosons when symmetry (G, H, and M) is given.
Please note that all the πa are not always independent each other, which means that
there may be πbs that represent the same mode each other.
In this case the number of Nambu-Goldstone modes is less than NBS [2, 3].
This occurs, for example, in a ferromagnet.
References:
[1] T. Yanagisawa: J. Phys. Soc. Jpn. 86, 104711 (2017)
Nambu-Goldstone bosons characterized by the order parameter in spontaneous symmetry breaking
[2] H. Watanabe and H. Murayama: Phys. Rev. Lett. 75, 251602 (2012).
[3] Y. Hidaka: Phys. Rev. Lett. 110, 091601 (2013).
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