Correlations and order parameter at a Coulomb-crystal phase transition in a three-dimensional dimer model

The three-dimensional classical dimer model with interactions shows an unexpected continuous phase transition between an ordered dimer crystal and a Coulomb liquid. A detailed analysis of the critical dimer and monomer correlation functions points to a subtle interplay between the fluctuations of the crystal order parameter and the “magnetic” degrees of freedom present in the Coulomb phase. The distribution probability of the crystal order parameter suggests an emerging continuous $O\left(3\right)$ symmetry at the critical point.

Figure 1

(Color online) (a) Dimer correlations $⟨d_0^x{d}_{\mathbf{r}}^y⟩$ as a function of $\left|\mathbf{r}\right|$ and multiplied by ${\left|\mathbf{r}\right|}^3$ for a few selected $T$ above $T_c$ (system size $L=64$). For this orientation, the spin contribution vanishes. (b) Current correlations $⟨J_0^x{J}_{\mathbf{r}}^x⟩$ of the ICLM as a function of $\left|\mathbf{r}\right|$ and multiplied by ${\left|\mathbf{r}\right|}^3$ at $T=0.5$ (in the Coulomb phase) and $T_c=0.33305$ (system size $L=64$). Correlators at $T_c$ have been multiplied by 50. Simulations of the dimer model (ICLM) were performed with the algorithm of Ref. 12 (Ref. 14).

Figure 2

(Color online) Dimer correlations ${⟨d_0^x{d}_{\mathbf{r}}^x⟩}^c$. (a) $T\simeq T_c$ and just above, for different orientations of $\mathbf{r}$ $\left(L=128\right)$. (b) A few selected $T$ above $T_c$ $\left(L=64\right)$. The lattice direction $\mathbf{r}=\left[i,i,i\right]$ is such that the dipolar contribution vanishes.

Figure 3

(Color online) Monomer correlation as a function of $r/\xi$ in log-log scale for a few $T$ just above $T_c$, assuming $\xi ={\left(T-T_c\right)}^{-1/2}$ and $T_c\simeq 1.6743$ $\left(L=128\right)$.

Figure 4

(Color online) (a) Spherical character of the order parameter $\vec{N}$ distribution $\left(C_4\right)$ as a function of $1/L$ for different $T$. The data for $T=1.6$ have been divided by 2. (b) Probability distribution $P\left(N^x,N^y,\left|N^z\right|\le 0.03\right)$ as a function of $N^x$ and $N^y$ for several temperature and system sizes. The range is $N^{x,y}∊\left[-0.3,0.3\right]$ for $L<128$ and $\left[-0.2,0.2\right]$ for $L=128$.