Current carrying ground state in a bilayer model of strongly correlated systems

Strongly interacting systems have been conjectured to spontaneously develop current carrying ground states under certain conditions. We demonstrate the existence of a commensurate staggered interlayer current phase in a bilayer model by using the recently discovered quantum Monte Carlo algorithm without the sign problem for fermionic systems. A pseudospin $\mathit{SU}\left(2\right)$ algebra and the corresponding anisotropic spin-1 Heisenberg model are constructed to show the competition among the staggered interlayer current, rung singlet, and charge density wave phases.

Figure 1

(a) Sketch of a SIC phase. For clarity, we do not show the bottom layer current. By conservation, each site acts as a source or drain for the current within the bilayers. (b) The top view of the the bilayer. (c) A sketch of the SF or the DDW current pattern for comparison.

Figure 2

The double occupancy states $a$ and $c$ and the rung singlet $b$ (a), (b) and (c) are spin $\mathit{SU}\left(2\right)$ singlets and form the triplet representation of the pseudospin $\mathit{SU}\left(2\right)$ group.

Figure 3

(Color online) Phase diagram in the strong coupling limit. Two $\mathit{SO}\left(5\right)$ lines are shown as well as QMC region with no minus sign problem for any filling (hatched area): $g>0,g^{\prime }>0$, and $U_c>0$. There is also another region with $V<0$ (not shown). In the yellow region, the low-energy bosonic states are $a,b$, and $c$ as shown in Fig. 2. This is where we expect the competition between the SIC and the rung-singlet phase. Black dots correspond to where the QMC simulations are performed.

Figure 4

(Color online) Parameters are $t_{\perp }=0.1,U=0,V=0.5$, and $J=2.0$ and correspond to $g=0.25,g^{\prime }=0$, and $U_c=0$. The scaling of $\mathcal{J}\left(\vec{Q}\right)∕N$ and $J(L∕2,L∕2)$ vs $1∕L$ shows almost no finite-size effects and proves long-range order in the thermodynamic limit. (The inset shows the convergence of $\mathcal{J}\left(\vec{Q}\right)∕N$ with the projection parameter $\theta$. Typically the GS value is obtained for $\theta =20$.)

Figure 5

Finite-size scaling of the current correlations $\mathcal{J}\left(\vec{Q}\right)∕N$ showing no long-range order in the thermodynamic limit. The parameters are (i) the same as Fig. 4 except for $t_{\perp }=0.5$; (ii) $U=V=0.3,J=1.6$, and $t_{\perp }=0.5$ at half-filling; (iii) $U=0.75,V=0,J=1$, and $t_{\perp }=0$ at $1∕8$ doping. Typically the GS value is obtained for $\theta =20$.