Abstract
We study numerically the geometric entanglement in the Laughlin wave function, which is of great importance in condensed matter physics. The Slater determinant having the largest overlap with the Laughlin wave function is constructed by an iterative algorithm. The logarithm of the overlap, which is a geometric quantity, is then taken as a geometric measure of entanglement. It is found that the geometric entanglement is a linear function of the number of electrons to a good extent. This is especially the case for the lowest Laughlin wave function, namely the one with filling factor of 1/3. Surprisingly, the linear behavior extends well down to the smallest possible value of the electron number, namely, N = 2. The constant term does not agree with the expected topological entropy. In view of previous works, our result indicates that the relation between geometric entanglement and topological entropy is very subtle.
Highlights
The Laughlin wave function [1] is a paradigm as a variational wave function [2]
To the best of our knowledge, this is the first time that the geometric entanglement in a strongly interacting fermionic system, which is of great physical interest, is calculated
We find that the geometric entanglement EG scales linearly with the electron number N
Summary
The Laughlin wave function [1] is a paradigm as a variational wave function [2]. simple in construction, its implication is rich. The larger Omax is, the closer the original wave function Φ is to a Slater determinant, and it is reasonable to say that the less the fermions are entangled with each other. If Omax = 1, the wave function is a Slater determinant and the entanglement is zero. In [15], an efficient algorithm to construct the optimal Slater approximation of an arbitrary fermionic wave function was brought up. It is this algorithm that enables us to study the geometric entanglement in the Laughlin wave function. The Schmidt decomposition based bipartite entanglement depends on the partition of the system, which is inevitably arbitrary
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