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I have read that in several statistical models exhibiting spontaneous symmetry breaking, the resulting Goldstone bosons do not interact with each other via $\theta^{2n}$ terms — only via derivative terms like $(\nabla\theta)^2$.

For instance, in the XY model, the free energy has terms like $F\sim\frac\gamma2 \int (M_0^2+2M_0\delta M)(\nabla\theta)^2$. Or in the Heisenberg model, there are terms like $F\sim\frac\gamma2 \int M_0^2[(\nabla\theta)^2+\sin^2\theta(\nabla\phi)^2$, for the two Goldstone modes $\phi$ and $\theta$.

My question is, what is the physical implication of the absence of $\theta^{2n}$ interactions, and only derivative ones?

I have read that in several statistical models exhibiting spontaneous symmetry breaking, the resulting Goldstone bosons do not interact with each other via $\theta^{2n}$ terms — only via derivative terms like $(\nabla\theta)^2$.

For instance, in the XY model, the free energy has terms like $$F\sim\frac\gamma2 \int (M_0^2+2M_0\delta M)(\nabla\theta)^2.$$ Or in the Heisenberg model, there are terms like $$F\sim\frac\gamma2 \int M_0^2[(\nabla\theta)^2+\sin^2\theta(\nabla\phi)^2],$$ for the two Goldstone modes $\phi$ and $\theta$.

My question is, what is the physical interpretation of the absence of $\theta^{2n}$ interactions, and only derivative ones?

It have read that in several statistical models exhibiting spontaneous symmetry breaking, the resulting Goldstone bosons do not interact with each other via $\theta^{2n}$ terms -- only via derivative terms like $(\nabla\theta)^2$.

For instance, in the XY model, the free energy has terms like $F\sim\frac\gamma2\int (M_0^2+2M_0\delta M)(\nabla\theta)^2$. Or in the Heisenberg model, there are terms like $F\sim\frac\gamma2\int M_0^2[(\nabla\theta)^2+\sin^2\theta(\nabla\phi)^2$, for the two Goldstone modes $\phi$ and $\theta$.

My question is, what is the physical implication of the absence of $\theta^{2n}$ interactions, and only derivative ones?

I have read that in several statistical models exhibiting spontaneous symmetry breaking, the resulting Goldstone bosons do not interact with each other via $\theta^{2n}$ terms — only via derivative terms like $(\nabla\theta)^2$.

For instance, in the XY model, the free energy has terms like $F\sim\frac\gamma2 \int (M_0^2+2M_0\delta M)(\nabla\theta)^2$. Or in the Heisenberg model, there are terms like $F\sim\frac\gamma2 \int M_0^2[(\nabla\theta)^2+\sin^2\theta(\nabla\phi)^2$, for the two Goldstone modes $\phi$ and $\theta$.

My question is, what is the physical implication of the absence of $\theta^{2n}$ interactions, and only derivative ones?