Diverging scaling with converging multisite entanglement in odd and even quantum Heisenberg ladders

Abstract

We investigate finite-size scaling of genuine multisite entanglement in the ground state of quantum spin-1/2 Heisenberg ladders. We obtain the ground states of odd- and even-legged Heisenberg ladder Hamiltonians and compute genuine multisite entanglement, the generalized geometric measure (GGM), which shows that for even rungs, GGM increases for odd-legged ladder while it decreases for even ones. Interestingly, the ground state obtained by short-range dimer coverings, under the resonating valence bond ansatz, encapsulates the qualitative features of GGM for both the ladders. We find that while the quantity converges to a single value for higher legged odd- and even-ladders, in the asymptotic limit of a large number of rungs, the finite-size scaling exponents of the same tend to diverge. The scaling exponent of GGM is therefore capable to distinguish the odd–even dichotomy in Heisenberg ladders, even when the corresponding multisite entanglements merge. Introduction and background . The sequences of odd and even numbers mathematically reach the same infinity. The question is whether limits of functions of odd/even numbers reach the same limiting function as the respective sequences tend to infinity. In many-body physics, the dichotomy between physical properties of odd and even quantum ladders is known, and a typical result is exemplified by the seminal work of White, Noack, and Scalapino (PRL 73,886 (1994)), wherein the spin gap was shown to exhibit different behaviors in odd and even ladders. The values of the spin gaps were shown as converging to a single value with the increase in the number of legs, irrespective of whether we are with odd or even ladders. It may now be asked whether one can identify a physical quantity that would have different limiting values depending on whether odd or even legged ladders are followed to reach the infinite 2D square lattice. We answer this query in the affirmative by identifying the scaling of multisite entanglement as the physical quantity that is capable of doing the job. Main results . We evaluate the asymptotic genuine multisite entanglement of states of quantum spin models on ladder lattices with odd and even legs, and perform finite size scaling analysis of the same. We find that scaling exponents of the multisite entanglement for large lattices tend to diverge for odd and even ladders, as the number of legs are increased, even though the amount of the multisite entanglement converges in the asymptotic limit. Wider implications . We believe that the results presented in the manuscript will be of interest to the broad physics community working on stronglycorrelated systems, quantum information in many-body physics, and related quantum technologies. The methodology presented in the manuscript will, we believe, also appeal to the general theoretical physics community.