Systematic Derivation of Order Parameters through Reduced Density Matrices

A systematic method for determining order parameters for quantum many-body systems on lattices is developed by utilizing reduced density matrices. This method allows one to extract the order parameter directly from the wave functions of the degenerate ground states without the aid of empirical knowledge, and thus opens a way to explore unknown exotic orders. The applicability of this method is demonstrated numerically or rigorously in models that are considered to exhibit dimer, scalar chiral, and topological orders.

Figure 1

Numbering of the sites on the two-leg ladder.

Figure 2

(a) Phase diagram obtained from earlier studies [3, 4]. (b) Values of diff1 and diff2 for the fixed areas, $\{1,2,1^{\prime }\}$ and $\{1,2,1^{\prime },2^{\prime }\}$, versus $\theta$. The values of diff1 (filled symbols) for the two areas are exactly the same. As for diff2 (open symbols), the upper and the lower points refer to $\{1,2,1^{\prime },2^{\prime }\}$ and $\{1,2,1^{\prime }\}$, respectively. Lines are guides to eyes. The vertical dotted straight line represents the self-dual point ${\theta }_c$. Our calculation assumes a quasidegeneracy of the GSs and is therefore invalid for $\theta \lesssim 0.07\pi$ and $0.39\pi \lesssim \theta$.

Figure 3

(a) Dimer covering and arrow representation. (b) Local area $\Omega$ on the cylinder. The cut $\Delta$ and the loop ${\Delta }^{*}$ can be drawn so as not to touch $\Omega$.