# At most $N$ gapless charge/spin modes in a system of $N$ coupled 1D chains?

Leon Balents and Matthew P. A. Fisher claimed the following without any further explanation ($N$ is the number of chains)

For a system of $N$ coupled 1D chains, the number of gapless charge modes can vary from $0$ to $N$, and likewise for spin.

in the introduction part of their highly-cited article Weak-coupling phase diagram of the two-chain Hubbard model, Phys. Rev. B 53, 12133 (1996).
This article studied two Hubbard chains (1D) coupled by simplest single-particle hopping between chains without altering site/spin index. The possible phases therein can be characterized by the number of gapless charge and spin modes. The form of coupling doesn't look to be the reason. From the context, it sounds valid in general.

Can anyone shed some light on this very general conclusion?

• Reminds me of the 1D chain of classical harmonic oscillators where you get $N$ gapless modes for $N$ coupled identical masses. Maybe it is linked to your problem somehow ? – Dimitri Jun 14 '16 at 9:11
• @Dimitri Sorry, I think $N$ means the number of chains rather than the number of masses/sites in a chain. – xiaohuamao Jun 15 '16 at 1:13
• If each chain has one gapless mode, the coupling between different chains can cause backscattering from one chain to the other which destroys the gapless modes. If you want, you could easily solve this system without the Hubbard interaction to get some insight. Just diagonalize a matrix with cos(k) on the diagonal and t (the interchain hopping) on the first off diagonals. Btw, is there no parity effect for even/odd number of chains; gapless modes should vanish in pairs? – Praan Jun 17 '16 at 16:54