The extended Hubbard model on a honeycomb lattice

TL;DR

This work probes the extended Hubbard model on a half-filled honeycomb lattice using determinant quantum Monte Carlo to map the full $U$-$V$ phase diagram. By employing complex Hubbard-Stratonovich fields, it accesses the sign-free region $|V|\,\leq\,|U|/3$ and uses structure factors, double occupancy, and correlation ratios with finite-size scaling to locate SM, AFM, CDW, SC, and PS phases. The key findings are that Dirac cones remove nesting and van Hove-driven instabilities, leading to an SM exclusion near the origin, stabilization of $s$-wave SC only in the $U<0,V<0$ quadrant, and a phase diagram where AFM and CDW compete for $U>0$, $V>0$ with PS at large negative $V$. These results illuminate how lattice geometry governs many-body phases and provide a benchmark for honeycomb-based correlated systems relevant to graphene-like materials and ultracold-atom emulations.

Abstract

The lack of both nesting and a van Hove singularity at half filling, together with the presence of Dirac cones makes the honeycomb lattice a special laboratory to explore strongly correlated phenomena. For instance, at zero temperature the repulsive [attractive] Hubbard model only undergoes a transition to an antiferromagnetic [$s$-wave superconducting degenerate with charge density wave (SC-CDW)] for sufficiently strong on-site coupling, $U/t\gtrsim 3.85$ [$U/t\lesssim -3.85$]; in between these, the system is a semi-metal, by virtue of the Dirac cones. The addition of an additional interaction, $V>0$ or $V<0$, between fermions in nearest neighbor orbitals should break the SC-CDW degeneracy giving rise to a phase diagram quite distinct from the one for the square lattice. Here we perform determinant quantum Monte Carlo simulations to investigate the whole phase diagram, covering the four combinations of signs of $U$ and $V$; the use of complex Hubbard-Stratonovich fields renders the region $|V|\leq |U|/3$ free from the `minus sign problem'. We calculate structure factors associated with different orderings, which, together with the double occupancy and the average sign allows us to map out the whole phase diagram. We have found that the SM phase forms a zone from which ordered phases are excluded, preventing the stabilization of a $d$-wave SC phase, i.e., only $s$-wave pairing is allowed.

Figures (6)

• Figure 1: (a) AFM correlation ratio, $R_\text{afm}^{(0,1)}$, at fixed $V$ as a function of $U/t$, for different linear system sizes, $L$. (b) Contour plot of the cost function, $C$, used to fit $R_\text{afm}^{(0,1)}$ to a FSS form, Eq. \ref{['eq:RFSS']}. The white star locates its minimum, which in turn determines the best fit to the correlation length exponent, $\nu$, and $U_c$. (c) The resulting FSS data collapse, with fitted $\nu$ and $U_c$; the latter is indicated in (a) by the dashed vertical line.
• Figure 2: (a)-(c) Charge-density wave correlation ratio for different system linear sizes, $L$, as a function of $V/t$, for fixed $U$. (d) Double occupation for $U/t=2$, where the dashed horizontal line indicates the non-interacting value, $\left \langle D \right \rangle=0.25$; see text. (e) Average sign of the product of fermionic determinants for $U/t=2$. (f) FSS plots of the crossing points (filled squares) in (a)-(c) using sizes $L$ and $L-\Delta L$, with $\Delta L = 6$, leading to the extrapolated values (empty squares) $V_c/t = 0.50\pm 0.02$, $V_c/t = 0.67\pm 0.02$ and $V_{c}/t = 0.92\pm0.02$, respectively.
• Figure 3: Double occupancy as a function of $V$ for (a) $U/t = 4$, (b) $U/t=5$ and (c) $U/t=6$. The dashed horizontal lines correspond to $\left \langle D \right \rangle=0.25$. Average determinant sign and a function of $V$ for (d) $U/t = 4$, (e) $U/t=5$ and (f) $U/t=6$. As the system enters the CDW phase, $\langle \text{sign}\rangle$ rises sharply.
• Figure 4: (a) SC correlation ratio as a function of $U$ for $V/t=-0.2$; (b) same as (a), but for $V/t=-0.4$; (c) FSS plot of the crossing points (filled squares) in (a) using sizes $L$ and $L-\Delta$, with $\Delta L=6$, leading to the extrapolated ($L\to\infty$) value $U_{c}/t = -3.67\pm 0.05$ (empty square), indicated by the vertical dashed line in (a); (d) same as (c), but for the data in (b), leading to the extrapolated value $U_{c}/t = -3.39\pm 0.02$ respectively.
• Figure 5: (a) Average density, (b) CDW structure factor (c) double occupancy, and (d) average sign of fermionic determinants, as functions of $V/t$. All data shown are for $U=0$.