Quantum Disordered Phase near the Mott Transition in the Staggered-Flux Hubbard Model on a Square Lattice

We investigate ground state properties of the half-filled staggered-flux Hubbard model on a square lattice. Energy gaps to charge and spin excitations and magnetic as well as dimer orders are calculated as a function of interaction strength $U/t$ by means of a constrained-path quantum Monte Carlo method. It is found that the system is a semimetal at $U/t\lesssim 5.6$ and a Mott insulator, with long-range antiferromagnetic order, at $U/t\gtrsim 6.6$. In the range $5.6\lesssim U/t\lesssim 6.6$, the ground state is a correlated insulator where both magnetic and dimer orders are absent. Furthermore, spin excitation in the intermediate phase appears to be gapless, and the measured spin-spin correlation function exhibits power-law decaying behavior. The data suggest that the nonmagnetic ground state is a possible candidate for the putative algebraic spin liquid.

Figure 1

(a) Arrangement of hopping amplitudes on a square lattice. The phase $\Phi =\pi$ is distributed equally so that $t_1=t_2=t_3=t_4=e^{i\Phi /4}$. (b) Band structure of the tight-binding Hamiltonian. (c) Top panel: power law exponent $\eta$ extracted from fits to staggered spin-spin correlation functions. Bottom panel: thermodynamic limit charge gap ${\Delta }_C$ and magnetic moment $m^2$ (see text for definition). The shaded region indicates the nonmagnetic insulating phase.

Figure 2

(a) Extrapolation of the charge gap ${\Delta }_C(L)$. Solid lines represent second-order polynomial fits to the QMC data. Inset shows the charge gap ${\Delta }_C(L)$ as a function of $U$ obtained for $L=4$, 6, 8, 10, 12, 14, and extrapolated (empty circle) values. Lines are guides to the eye. (b) Finite size extrapolation of the spin structure factor $m^2(L)$. Solid lines are second-order polynomial fits to the QMC data. Inset: $m^2(L)$ versus $U$ on finite lattices and extrapolated (empty circle) values. Lines are guides to the eye.

Figure 3

(a) Finite size extrapolation of the spin gap ${\Delta }_S(L)$. Solid lines are second-order polynomial fits to the Monte Carlo data. The inset illustrates ${\Delta }_S(L)$ and its extrapolated value. Lines are guides to the eye. (b) Long-range behavior of the staggered spin-spin correlation function $C(\mathbf{r})=(-1{)}^{\mathbf{r}}⟨S_{\mathbf{r}}^x{S}_0^x+S_{\mathbf{r}}^y{S}_0^y+S_{\mathbf{r}}^z{S}_0^z⟩$ obtained on $L=16$, 20, and 24 ($U=0$ only). Straight lines are representative power-law fits to the data: green (dot-dashed) line $L=24$ at $U=0$, black (dashed) line $L=20$ at $U=4$, and blue (solid) line $L=20$ at $U=6$.

Figure 4

Thermodynamic limit extrapolation of the columnar VBS order. In both figures, solid lines are three representative third-order polynomial fits to the Monte Carlo data. Insets show the normalized dimer structure for single bonds in $x$ or $y$ directions on finite size lattices. Lines are guides to the eye. The extrapolated value in the thermodynamic limit is indicated by orange (empty) circles in the insets.