Jordan-Wigner Transformation in Higher Dimensions

Abstract

Particle statistics is one of the most fascinating aspects of quantum mechanics. While the Bose-Einstein statistics is already counter-intuitive, bosons may be understood as elementary excitations of quantized classical fields. Fermions, on the other hand, do not seem to have any classical counterpart. Formally they arise by quantizing Grassmann fields, but this may not be too helpful for intuitive understanding. Yet, fermions are allowed within principles of quantum mechanics, and indeed they are important building blocks of our universe. Not only the electrons, but also the "matter fields" in the standard model of elementary particles are fermionic. Fermi-Dirac statistics is indeed important for the stability of the matter. On the other hand, Fermi-Dirac statistics also poses computational challenge. The notorious negative sign problem in Quantum Monte Carlo simulations is a consequence of the Fermi-Dirac statistics (although the sign problem can appear also in bosonic systems). The computational difficulty of fermions has also become a renewed issue [1], as quantum computers will be based on bosonic qubits.Given the fundamental difference between bosons and fermions, it is remarkable that a mapping between a fermionic system and a bosonic system, known as Jordan-Wigner transformation [2], exists in one spatial dimension. The mapping can be defined exactly for a one-dimensional lattice, as discussed in Ref. [3]. The exact solution of the quantum S = 1/2 chain, which follows from the exact Jordan-Wigner transformation, is an important basis to understand quantum magnetism in one dimension. It is then a natural desire to extend the