Abstract
The effective Lagrangian for Nambu-Goldstone bosons (NGBs) in systems without Lorentz invariance has a novel feature that some of the NGBs are canonically conjugate to each other, hence describing $1$ dynamical degree of freedom by two NGB fields. We develop explicit forms of their effective Lagrangian up to the quadratic order in derivatives. We clarify the counting rules of NGB degrees of freedom and completely classify possibilities of such canonically conjugate pairs based on the topology of the coset spaces. Its consequence on the dispersion relations of the NGBs is clarified. We also present simple scaling arguments to see whether interactions among NGBs are marginal or irrelevant, which justifies a lore in the literature about the possibility of symmetry breaking in $1+1$ dimensions.
Highlights
In studies of any macroscopic physical systems, the behavior of the system at low temperatures, small energies, and long distances is determined predominantly by microscopic excitations with small or zero gap
We present simple scaling arguments to see whether interactions among Nambu-Goldstone bosons (NGBs) are marginal or irrelevant, which justifies a lore in the literature about the possibility of symmetry breaking in 1 þ 1 dimensions
We demonstrate how spacetime symmetries can be discussed within our effective Lagrangian formalism and see how they provide additional constraints on the parameters in the theory
Summary
In studies of any macroscopic physical systems, the behavior of the system at low temperatures, small energies, and long distances is determined predominantly by microscopic excitations with small or zero gap. The Nambu-Goldstone theorem says that there must be one gapless excitation for every broken-symmetry generator, assuming Lorentz invariance. Nambu [23,24] was probably the first to obtain the correct insight into this problem He observed that the nonzero expectation value h1⁄2Qa; Qbi makes zero modes associated with these generators canonically conjugate to each other, and the number of NGBs is reduced by 1 per such a pair. We find a set of terms that have not been taken into account in the literature This fully nonlinear effective Lagrangian contains only a few parameters that play the role of coupling constants between NGBs. By analyzing the scaling law of the dominant interaction, we discuss the stability of the symmetry-broken ground state. We clarify a confusion in the existing literature on the relation between type-B NGBs and the time-reversal symmetry in Appendix C
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