# Even-odd effects in the $J_1-J_2$ SU($N$) Heisenberg spin chain

## TL;DR

This paper explores how the phase diagram of the SU(N) J1-J2 Heisenberg spin chain varies with N, revealing even-odd effects in the formation of N-merized phases and their properties.

## Contribution

It uncovers the parity-dependent behavior of N-merized phases and the emergence of a gapless SU(N)1 phase for odd N, combining field theory and numerical methods.

## Key findings

- N-merized phase forms above critical J2/J1 for all N.
- Even N phases smoothly connect to a zigzag two-leg ladder regime.
- Odd N phases have finite extent with no incommensuration, and a gapless SU(N)1 phase appears for larger J2.

## Abstract

The zero-temperature phase diagram of the $J_1-J_2$ SU($N$) antiferromagnetic Heisenberg spin chain is investigated by means of complementary field theory and numerical approaches for general $N$. A fully gapped SU($N$) valence bond solid made of $N$ sites is formed above a critical value of $J_2/J_1$ for all $N$. We find that the extension of this $N$-merized phase for larger values of $J_2$ strongly depends on the parity of $N$. For even $N$, the phase smoothly interpolates to the large $J_2$ regime where the model can be viewed as a zigzag SU($N$) two-leg spin ladder. The phase exhibits both a $N$-merized ground state and incommensurate spin-spin correlations. In stark contrast to the even case, we show that the $N$-merized phase with odd $N$ only has a finite extent with no incommensuration. A gapless phase in the SU($N$)$_1$ universality class is stabilized for larger $J_2$ that stems from the existence of a massless renormalization group flow from SU($N$)$_2$ to SU($N$)$_1$ conformal field theories when $N$ is odd.