Nonlocal Parity Order in the Two-Dimensional Mott Insulator

The Mott insulator is characterized by having small deviations around the (integer) average particle density $n$, with pairs with $n-1$ and $n+1$ particles forming bound states. In one dimension, the effect is captured by a nonzero value of a nonlocal “string” of parities, which instead vanishes in the superfluid phase where density fluctuations are large. Here, we investigate the interaction induced transition from the superfluid to the Mott insulator, in the paradigmatic Bose Hubbard model at $n=1$. By means of quantum Monte Carlo simulations and finite size scaling analysis on $L\times M$ ladders, we explore the behavior of “brane” parity operators from one dimension (i.e., $M=1$ and $L\to \infty$) to two dimensions (i.e., $M\to \infty$ and $L\to \infty$). We confirm the conjecture that, adopting a standard definition, their average value decays to zero in two dimensions also in the insulating phase, evaluating the scaling factor of the “perimeter law” [S. P. Rath et al., Ann. Phys. (Berlin) 334, 256 (2013)]. Upon introducing a further phase in the brane parity, we show that its expectation value becomes nonzero in the insulator, while still vanishing at the transition to the superfluid phase. These quantities are directly accessible to experimental measures, thus providing an insightful signature of the Mott insulator.

Figure 1

Brane parity correlator $C_0(r,M)$, evaluated at $r=L/2$, as a function of $U/t$ for ladders with $M=2$ and $L=120$.

Figure 2

Phase diagram of the bosonic Hubbard model for $n=1$: the critical interaction strength $U_c$ at which the superfluid-Mott transition occurs is reported as a function of the number of legs $M$ of the ladder.

Figure 3

Size scaling of brane parity $C_0(M)$ (i.e., $\theta =\pi$) with $M$ for $U/t=12$. The fit is performed by using Eq. (8) with $b=t^2/(2U^2)$. The results have been obtained for ladders with $L=30$, after having verified that the calculations do not change sensibly for larger values of $L$.

Figure 4

Brane parity correlator $C_1(r,M)$ (i.e., $\theta =\pi /M$), evaluated at $r=L/2$, as a function of $U/t$ for ladders with $M=2$ and various lengths $L$.

Figure 5

Left panel: Finite-size scaling of $C_1(L/2,M)$ for $M=2$ with increasing $L$, deep inside the SF regime (i.e., $U/t=2$). Right panel: Finite size-scaling of $C_1(M)$ with increasing the number of legs $M$ of the ladder, deep inside the MI phase (i.e., $U/t=12$). Here, the results have been obtained for ladders with $L=30$, after having verified that the calculations do not change sensibly for larger values of $L$.

Figure 6

Two-dimensional brane parity $C_{\alpha }$ as a function of $U/t$ for $\alpha =1$ (i.e., $\theta =\pi /M$) and $\alpha =1/2$ (i.e., $\theta =\pi /\sqrt{M}$).