Detection of Exotic Order Parameters of Quantum Antiferromagnets through Reduced Density Matrices

Abstract

We propose a new method based on reduced density matrices for determining order parameters of quantum spin systems. Our method can extract the order parameter directly from ground-state wave functions without prior knowledge, and thus has a potential for detecting unknown exotic orders. We numerically apply our method to the multiple-spin exchange model on the ladder and detect the staggered dimer and the scalar chiral orders which have been found in previous studies. We also consider the resonating valence bond liquid in a solvable quantum dimer model and demonstrate through reduced density matrices that its ground states cannot be characterized by any local order parameter. Frustrated quantum antiferromagnets often exhibit exotic orders which are difficult to preconceive from the type of interactions or lattice structures. To determine the order parameters characterizing them, usually, we examine correlation functions of candidate order parameters or look for characteristic spectral properties. Such methods, however, rely on our knowledge about previously-found orders and do not work if the order is beyond our experiences. To overcome this difficulty, in this paper, we propose a new method which can determine the order parameter without prior knowledge. Suppose that we have obtained the low-energy spectrum and eigenstates of the finite-size systems by exact diagonalization, for instance. In a phase which breaks a discrete symmetry in the thermodynamic limit, we find a finite number of nearly-degenerate ground states which become asymptotically degenerate as the size of the system increases. The symmetry-breaking ground states jΨii (i = 1, 2,¢¢¢ ) in the thermodynamic limit can be constructed as the linear combinations of these nearly-degenerate ground states. An order parameter can be identified with an operator which distinguishes the symmetry-breaking ground states. The order parameter is local, if the corresponding operator can be defined on a finite local area in the entire system, even when the system size is taken to infinity. The central idea of our method is to search the order parameter by comparing the reduced density matrices (RDMs) ρ i = Tr ¯ A jΨiihΨij (i = 1, 2,¢¢¢ ), where A is an area of the system and ¯ A is its complement. If these RDMs are different (identical) in an area, an order parameter can (cannot) be defined in that area. In this way, we can elucidate in what area order parameters can be defined. In x2, we give the detailed formulation of our method. In x3, we demonstrate our method in a ladder model with the dimer and scalar chiral orders. In x4, we exploit the relation between RDMs and order parameters to prove that the resonating