Intermediate and spin-liquid phase of the half-filled honeycomb Hubbard model

We obtain the phase diagram of the half-filled honeycomb Hubbard model with density matrix embedding theory, to address recent controversy at intermediate couplings. We use clusters from 2–12 sites and lattices at the thermodynamic limit. We identify a paramagnetic insulating state, with possible hexagonal cluster order, competitive with the antiferromagnetic phase at intermediate coupling. However, its stability is strongly cluster and lattice size dependent, explaining controversies in earlier work. Our results support the paramagnetic insulator as being a metastable, rather than a true, intermediate phase, in the thermodynamic limit.

Figure 1

(a) DMET embeds clusters of $N_c$ sites in the underlying honeycomb lattice of $2L^2$ sites. (b)–(f) Cluster shapes in our study, for $N_c=2\text{--}12$. Also shown is the nearest-neighbor spin-spin correlation, $\langle S_i^z{S}_j^z\rangle$. Blue numbers are from $U=1.5t$ in the semimetal (SM) phase; red numbers are from $U=8.4t$ in the paramagnetic insulator (PMI) phase.

Figure 2

DMET calculations allowing spontaneous AFM order. Top left: single-particle gap against $U$ for cluster sizes $N_c=2\text{--}12$. As $U$ increases, a gap spontaneously opens at $U_{\mathrm{AFM}1}$, just after $U=3$. Top right: local density of states when $N_c=6$, calculated from spectral DMET [40], showing the Dirac cone in the SM phase. Bottom left: staggered magnetization against $U$ for $N_c=2\text{--}12$. The staggered magnetization appears at the same point as the opening of the gap. Bottom right: $U_{\mathrm{AFM}1}$ as a function of lattice size $L=12\text{--}216$ ($288\text{--}93312$ sites) and cluster size ($N_c=2\text{--}12$).

Figure 3

Top: DMET energy and Sorella et al.'s finite lattice AFQMC energy at intermediate $U$ for $L=12$ (288 sites) as a function of cluster size $N_c=2\text{--}12$. This is the most challenging regime for DMET, but the energy comparison is very favorable. PM denotes paramagnetic solution. Bottom: derivative of kinetic energy as a function of $U/t$. Overlaid are results of Meng et al. and Sorella et al. for $L=12$ [4]. Inset: kinetic energy density as a function of $U$, showing good agreement between the DMET and numerically exact AFQMC [4]. DMET qualitatively reproduces two changes in curvature (near $U=3.5$ and $U=4.3$) cited as evidence of an intermediate phase by Meng et al., although no intermediate phase is observed.

Figure 4

Top: phase transition points for cluster sizes 2–12, and two lattice sizes $L=12,216$. $U_{\mathrm{AFM}1}$ corresponds to opening of the gap. $U_{\mathrm{AFM}2}$ corresponds to the thermodynamic transition to the AFM phase. $[U_{\mathrm{AFM}1},U_{\mathrm{AFM}2}]$ is a coexistence region for the AFM phase and SM phase, although this region appears to vanish with increasing cluster size. $U_{\mathrm{PM}}$ is the transition to the PMI if AFM order is not allowed to develop. The transition is first order for $N_c=2,4,8$ and second order for $N_c=6,12$. Note that for $N_c=6,12$, $U_{\mathrm{PM}}$ is very close to $U_{\mathrm{AFM}2}$ indicating that the PMI is very competitive with the AFM phase for these cluster shapes. Bottom: ground state energy difference of PM and AFM solutions. The positive region is the coexistence region.

Figure 5

Single particle gap of paramagnetic solution with cluster sizes for $L=216$. For $N_c=2,4,8$ the transition is first order; for $N_c=6,12$ the transition is continuous. Inset: second derivative of the gap for $N_c=6,12$; we identify $U_{\mathrm{PM}}$ from the peak. Yellow squares: six-site CDMFT calculations from Ref. [1].