Entropy Dependence of Correlations in One-Dimensional $\mathrm{SU}(N)$ Antiferromagnets

Motivated by the possibility to load multicolor fermionic atoms in optical lattices, we study the entropy dependence of the properties of the one-dimensional antiferromagnetic $\mathrm{SU}(N)$ Heisenberg model, the effective model of the $\mathrm{SU}(N)$ Hubbard model with one particle per site (filling $1/N$) in the large $U/t$ limit. Using continuous-time world-line Monte Carlo simulations for $N=2–5$, we show that characteristic short-range correlations develop at low temperature as a precursor of the ground state algebraic correlations. We also calculate the entropy as a function of temperature, and we show that the first sign of short-range order appears at an entropy per particle that increases with $N$ and already reaches $0.8k_B$ at $N=4$, in the range of experimentally accessible values.

Figure 1

Evolution of the energy per site $E$ and of the entropy per site $S$ as a function of the temperature $T$ for different $N$ on a $L=60$ chain. The inset shows the slope of the entropy at $T=0$, given in Eq. (5). The curvature being positive at $T=0$, the curves go higher than the tangent (dashed line).

Figure 2

Evolution of the structure factor $\tilde{C}(k)$ as a function of the entropy per site $S$ for different $N$ on a $L=60$ chain.

Figure 3

Structures factor $\tilde{C}(k)$ at low temperature ($k_{B}T=0.01$) for different $N$ on a $L=60$ chain, with $n={10}^7$ Monte Carlo steps. Small peaks are clearly visible at $k=\pi$ for $N=4$ and $k=4\pi /5$ and $6\pi /5$ for $N=5$. The data for $N=4$ are in perfect agreement with those of Ref. 31.