Mott physics in the half-filled Hubbard model on a family of vortex-full square lattices

We study the half-filled Hubbard model on a one-parameter family of vortex-full square lattices ranging from the isotropic case to weakly coupled Hubbard dimers. The ground-state phase diagram consists of four phases: A semimetal and a band insulator which are connected to the weak-coupling limit, and a magnetically ordered Néel phase and a valence bond solid (VBS) which are linked to the strong-coupling Mott limit. The phase diagram is obtained by quantum Monte Carlo (QMC) and continuous unitary transformations (CUTs). The CUT is performed in a two-step process: Nonperturbative graph-based CUTs are used in the Mott insulating phase to integrate out charge fluctuations. The resulting effective spin model is tackled by perturbative CUTs about the isolated dimer limit yielding the breakdown of the VBS by triplon condensation. We find three scenarios when varying the interaction for a fixed anisotropy of hopping amplitudes: (i) one direct phase transition from Néel to semimetal, (ii) two phase transitions VBS to Néel and Néel to semimetal, or (iii) a smooth crossover from VBS to the band insulator. Our results are consistent with the absence of spin-liquid phases in the whole phase diagram.

Figure 1

Ground-state phased diagram as a function of $t^{\prime }/t$ and $U/t$ as obtained from QMC (red circles) and CUT (blue circles). Solid lines are guides to the eye for the expected quantum critical lines between the Néel-ordered antiferromagnet (AF), the semimetal (SM), and the valence bond solid (VBS). Dashed vertical lines denote the specific cases $t^{\prime }/t=-1$ and $t^{\prime }/t=-2$ which play an important role in this work. The QMC data at $t^{\prime }/t=0$ stem from Ref. [13]. The data at $t^{\prime }/t=-0.5$ and $t^{\prime }/t=-1.5$ result from extrapolation to infinite size of the spin-spin correlation function using lattices up to 784 sites.

Figure 2

Illustration of the $t\text{−}{t}^{\prime }$ Hubbard model. Filled circles denote sites and dark (gray) lines refer to the coupling $t^{\prime }$ $\left(t\right)$.

Figure 3

Illustration of graphs $g_{\nu }^{\left(n\right)}$ used in the gCUT plaquette expansion. The index $\nu$ classifies the graphs by the number of Hubbard sites which are then further counted by $n=1,2,3,...$  . Red filled circles denote Hubbard sites which are connected via hopping amplitudes $t^{\prime }$ $\left(t\right)$ by black (cyan) lines.

Figure 4

Illustration of (a) two-spin and (b) four-spin interactions in the original lattice. (c) The effective lattice resulting from replacing $t^{\prime }$ bonds by effective supersites (filled black squares) together with the corresponding two-spin interactions as links.

Figure 5

Relative values for the effective spin-couplings $J_{\alpha }/J^{\prime }$ with $\alpha \in \{1,2,3,\parallel ,=,\times \}$ as a function of $t/U$ in the thermodynamic limit for (a) $t^{\prime }/t=-1$, (b) $t^{\prime }/t=-2$, and (c) $t^{\prime }/t=-3$. Empty (filled) circles are $\mathrm{gCUT}\left(4\right)$ $\mathrm{[}\mathrm{gCUT}\left(6\right)\text{]}$, filled squares are $\mathrm{gCUT}\left(8\right)$. Vertical dashed lines signal the inverse bandwidth $t/W$ according to Eq. (6). Insets: Exchange coupling $J^{\prime }/U$ as a function of $t/U$. Note that in (a) $J^{\prime }=J_1=J_2$.

Figure 6

Pinning field QMC data for the induced magnetic moment $m$ as a function of $1/L$ at $t^{\prime }/t=-1$. Here we have use a projection parameter $\Theta t=320$. Both values of the pining field $h_0=1$ (top panel) and $h_0=5$ (bottom panel) lead to the same extrapolated value of the magnetization thus providing an internal check.

Figure 7

Single-particle gap ${\Delta }_{\mathrm{sp}}$ in units of $t$ as a function of $1/L$ for different ratios $U/t$ at $t^{\prime }/t=-1$ obtained by QMC. Lines are linear extrapolations up to the thermodynamic limit.

Figure 8

Spin gap at $t^{\prime }/t=-2$ from QMC simulations. The extrapolation to the thermodynamic limit is delicate. In the gapped phase we have used the fitting form $a+b{\mathrm{e}}^{-\xi /L}$. This is certainly an appropriate choice when the correlation length $\xi$ is smaller than the system size. If the data does not support this point of view, we have used a polynomial fit to extract the infinite volume value of the spin gap.

Figure 9

One-triplon gap $\Delta$ in units of $4t^{\prime 2}/U$ as a function of $t/U$ at $t^{\prime }/t=-2$ obtained by CUTs (circles) and QMC (squares). For the CUT, we display different truncations of the $\text{g}\text{CUT}\left(\nu \right)$ with $\nu \in \{4,6,8\}$ and different maximal orders $n\in \{5,6\}$ of the pCUT series expansion. For the latter we have used dlogPadé extrapolants $[2,2]$ and $[2,3]$, respectively. The estimated quantum critical point is located at the vertical dashed line.

Figure 10

Real space spin-spin correlations at $t^{\prime }/t=-2$ from QMC simulations for (a) $U/t\ge 7$ and (b) $U/t\le 7$. Here we consider the largest distance along the $x$ axis on a $L\times L$ lattice. We fit the data to the form $a+b/L+c/L^2$.

Figure 11

One-triplon gap $\Delta$ in units of $4t^{\prime 2}/U$ as a function of $t^{\prime }/t$ and $t/U$ as obtained from CUT calculations (red points). Solid green line denotes the characteristic energy scale $t/W$. Symbols: (i) Circle denotes quantum phase transition between Néel order and VBS in the strong-coupling limit $t/U\ll 1$ and (ii) squares represent QMC results for $t^{\prime }/t=-1$ and $t^{\prime }/t=-2$ from this work.