# One-dimensional $SU(3)$ Heisenberg Model, the non-linear sigma model, $\theta$-term

Let's consider a one dimensional $$SU(N)$$ antiferromagnetic Heisenberg Model with an irreducible representation and its conjugate on alternating sites, such that they correspond to a Young tableaux with $$n$$ columns and 1 row, $$n$$ columns and (N-1) rows respectively. In the "semiclassical limit" $$n\to\infty$$ this model can be mapped onto the non linear sigma model with an additional topological term in which $$\theta=\pi n$$.

If we consider the case N=2, we get the familiar SU(2) Heisenberg chain of spin $$S$$, with $$S=\frac{n}{2}$$. This model is massive for $$S$$ integer given that the topological term vanish $$(\theta=0)$$. For $$S$$ half-integer ($$\theta=\pi$$) the Lieb-Schultz-Mattis (LSM) theorem assure us that the model has to be either gapless or have degenerate ground states. We know that in this case our model has to be gapless.

If now we consider the case N=3, we find that for $$n$$ even ($$\theta=0$$) the model is massive. What can be said in general for the case $$n$$ odd? Is there an equivalent LSM theorem for SU(3)? Can we still say that the corresponding model has to be either gapless or have degenerate ground states? I know that if $$n=1$$, the model spontaneously breaks the parity symmetry and has two degenerate ground states and is massive. Does this result follow from the semiclassical limit above even though $$n$$ is not large?

• Should the coefficient of the topological term really be $\theta = (2\pi/N)n$ ? – Dominic Else Feb 10 at 22:34
• This is my reference on the subject : qpt.physics.harvard.edu/c6.pdf – Alessandro Feb 11 at 9:05

Let me just answer the LSM part. To find the potential LSM theorem you treat the system as having symmetry $$\mathbb{Z} \times PSU(N)$$, where the generator of $$\mathbb{Z}$$ is a lattice translation times a complex conjugation (the complex conjugation is necessary for $$N > 2$$ because of the alternating of representations and their conjugates). I don't know if any papers in the literature have considered this case specifically, but adapting a standard argument, one finds that there is an LSM theorem whenever the representation of $$PSU(N)$$ at each site is projective. I believe this happens whenever $$n$$ is not an integer multiple of $$N$$.

• I am not much familiar with projective representations, so I am asking you: are the fundamental/antifundamental representations of SU(3) projective representation of PSU(3)? – Alessandro Feb 11 at 9:03
• @Alessandro Yes. $SU(N)$ is the universal cover of $PSU(N) = SU(N)/Z_N$, so you diagnose the projective representation of $PSU(N)$ just by looking at how the center $Z_N \leq SU(N)$ is represented in the rep of $SU(N)$. And indeed this is non-trivial in the fundamental representation. – Dominic Else Feb 12 at 1:00
• Nice, thanks. Can you give me some references? – Alessandro Feb 12 at 8:32
• Another couple of questions: 1) if the representation of PSU(3) is linear then the theorem tells us nothing, right? 2) A linear representation of PSU(3) would be ,for instance, 3⊗3⊗3 or $\bar{3}\otimes 3 \otimes \bar{3}$? – Alessandro Feb 12 at 15:55
• @Alessandro You can start with: en.wikipedia.org/wiki/… – Dominic Else Feb 12 at 20:17