# Questions on degenerate ground states and the thermodynamic limit?

For example, let's consider a $N$ spin-1/2 system on a lattice described by the Hamiltonian $H$. My questions are:

(1) If $H$ has either global $SU(2)$ spin-rotation symmetry or time-reversal symmetry, then $\forall$ finite $N=$odd, there must be exact degenerate ground states (GS). From this point of view, can we infer that in the thermodynamic limit ($N\rightarrow \infty$), the GSs of the above $H$ must be degenerate ??

(2) This is a question about the definition of GS degeneracy. Assuming $H$ is gapped, such that the GS degeneracy can be well defined. My question is: No matter how the sites number $N$ approaches $\infty$, for instance, either $N(=\text{odd})\rightarrow \infty$ or $N(=\text{even})\rightarrow \infty$, they should both give the same (exact or approximate) GS degeneracy in the thermodynamic limit, right ?? Since, at least from the math side, otherwise the GS degeneracy would depend on the way that $N\rightarrow \infty$ and it is thus not well defined.

Referring to Q.(2), let's consider the spin-1/2 Kitaev honeycomb model in a gapped phase (i.e., $\left | J_z \right |>\left | J_x \right |+\left | J_y \right |$ and $J_xJ_yJ_z\neq0$). If we take a sequence of sizes of the system $N$=odd (odd number of total spins or total lattice sites) in the thermodynamic limit, such that there always exist exact Kramer's degeneracy (due to time-reversal symmetry) as $N$ approaches $\infty$. Therefore, this kind of sequence would give rise to a ground state degeneracy (Kramer's degeneracy) in the thermodynamic limit, right? On the other hand, as pointed out by the author, both Lieb's theorem and the author's numerical study suggest that the zero-flux conﬁguration is the unique sector minimizing the ground state (GS) energy, indicating that the GS of Kitaev model is nondegenerate or unique. This is in contrary to the above Kramer's degeneracy. While it seems that the original paper didn't mention that what kind of sequence of sizes of the system is chosen as taking the thermodynamic limit. So I guess that a 'reasonable' sequence should be at least ensuring the number of total lattice sites $N=$even (e.g., we may take the number of unit cells $\rightarrow \infty$ for the thermodynamic limit), which make the GS uniqueness possible. Is my understanding correct?

Thank you very much.

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I just found this very crucial sentence "we have to specify that the system have a certain sequence of sizes as we take the thermodynamic limit" in Prof.Wen's answer, which I think could answer the above questions. –  Kai Li Apr 25 '14 at 13:48