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Quantum many-body problems for identical particles:
The (anti-)symmetrized n-fold product states.

Zhang Jiang, University of New Mexico

(Session 7c : Friday from 4:30 - 5:00)

Abstract. Quantum many-body problems are notorious hard. This is partly because the Hilbert space becomes exponentially big with the particle number N. As a consequence, one needs an exponentially large number of parameters merely to record an arbitrary state, not to say calculating its time evolution. While exact solutions are often considered intractable, numerous approaches have been proposed using approximations. A common trait of these approaches is to use an ansatz such that the number of parameters either does not depend on N or is proportional to N, e.g., the matrix-product state for spin lattices, the BCS wave function for superconductivity, the Laughlin wave function for fractional quantum Hall effects, and the Gross-Pitaecskii theory for BECs. Among them the product ansatz for BECs has precisely predicted many useful properties of Bose gases at ultra-low temperature. As particle-particle correlation becomes important, however, it begins to fail. To capture the quantum correlations, we propose a new set of states, which constitute a natural generalization of the product-state ansatz. Our state of N=d× n identical particles is derived by symmetrizing the n-fold product of a d-particle quantum state. The quantum correlations of that d-particle state thus spread out to any two of the N particles. For fixed d, the parameter space of our states does not grow with N. Several properties of our states will be discussed, including the one- and two-particle reduced density matrices, multi-particle correlations, and advantages over the product ansatz. Although our states were initially proposed for bosons, we also briefly discuss its fermion correspondence and applications.