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arXiv:2206.10393

Local spin description of fermions on a lattice

Adam Wyrzykowski

Published Jun 17, 2022• 0 citations
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Abstract

A local transformation from fermionic operators to spin matrices is proposed and studied in this work. For this purpose, a system of fermions on a lattice is considered and one applies the scheme to replace the fermionic variables with spin matrices, while the transformation relates only those fermionic/spin operators which are assigned to nearby lattice sites. In one dimension, this proposal yields the same result as the well-known Jordan-Wigner transformation, while not being restricted to $d=1$ dimension. To obtain the equivalent description in the spin picture, one needs to impose constraints on the spin space. Since finding the reduced spin Hilbert space constitutes a substantial stage of the whole procedure, the constraints are paid particular attention. The full set of necessary constraints is determined in both representations. To approach the task to solve the constraints, a suitable basis is constructed. The introduction of the basis in the spin representation along with the construction of the constraints and the Hamiltonian in this basis show how the transformation proposed in this work can be applied to obtain observables in the spin picture. Explicit construction of the constraints in the basis allows one to solve them and, once the basis vectors of the reduced spin Hilbert space are found, the spin Hamiltonian is expressed in this basis and diagonalized. The constraints are constructed in the basis as discussed above and analyzed with the Wolfram Mathematica programs for lattice sizes $3\times3$, $4\times3$ and $4\times4$. Their mutual relations are determined and the reduced spin Hilbert space is specified. The Hamiltonian is constructed in this representation and diagonalized. It is verified that the eigenenergies obtained in the spin picture agree with the analytic formulas from the fermionic representation.

DOI
10.48550/arxiv.2206.10393

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A local transformation from fermionic operators to spin matrices is proposed and studied in this work. For this purpose, a system of fermions on a lattice is considered and one applies the scheme to replace the fermionic variables with spin matrices, while the transformation relates only those fermionic/spin operators which are assigned to nearby lattice sites. In one dimension, this proposal yields the same result as the well-known Jordan-Wigner transformation, while not being restricted to d=1d=1d=1 dimension. To obtain the equivalent description in the spin picture, one needs to impose constraints on the spin space. Since finding the reduced spin Hilbert space constitutes a substantial stage of the whole procedure, the constraints are paid particular attention. The full set of necessary constraints is determined in both representations. To approach the task to solve the constraints, a suitable basis is constructed. The introduction of the basis in the spin representation along with the construction of the constraints and the Hamiltonian in this basis show how the transformation proposed in this work can be applied to obtain observables in the spin picture. Explicit construction of the constraints in the basis allows one to solve them and, once the basis vectors of the reduced spin Hilbert space are found, the spin Hamiltonian is expressed in this basis and diagonalized. The constraints are constructed in the basis as discussed above and analyzed with the Wolfram Mathematica programs for lattice sizes 3×33\times33×3, 4×34\times34×3 and 4×44\times44×4. Their mutual relations are determined and the reduced spin Hilbert space is specified. The Hamiltonian is constructed in this representation and diagonalized. It is verified that the eigenenergies obtained in the spin picture agree with the analytic formulas from the fermionic representation.
We revisit the Jordan-Wigner transformation, showing that-rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators-it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan-Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies
We revisit the Jordan–Wigner transformation, showing that—rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators—it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan–Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies.
We revisit the Jordan–Wigner transformation, showing that—rather than a non-local isomorphism between different fermionic and spin Hamiltonian operators—it can be viewed in terms of local identities relating different realizations of projection operators. The construction works for arbitrary dimension of the ambient lattice, as well as of the on-site vector space, generalizing Jordan–Wigner's result. It provides a direct mapping of local quantum spin problems into local fermionic problems (and vice versa), under the (rather physical) requirement that the latter are described by Hamiltonians which are even products of fermionic operators. As an application, we specialize to mappings between constrained-fermion models and spin-1 models on chains, obtaining in particular some new integrable spin Hamiltonian, and the corresponding ground-state energies.
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We explore the properties of the SO(3) Majorana representation of spin. Based on its nonlocal nature, it is shown that there is an equivalence between the SO(3) Majorana representation and the Jordan-Wigner transformation in one and two dimensions. From the relation between the SO(3) Majorana representation and one-dimensional Jordan-Wigner transformation, we show that application of the SO(3) Majorana representation usually results in Z2{Z}_{2}Z2​ gauge structure. Based on lattice Chern-Simons gauge theory, it is shown that the anticommuting link variables in the SO(3) Majorana representation make it equivalent to an operator form of compact U(1)1{\text{U(1)}}_{1}U(1)1​ Chern-Simons Jordan-Wigner transformation in 2D. As examples of its application, we discuss two spin models, namely the quantum XY model on a honeycomb lattice and the ${90}^{\ensuremath{\circ}}$ compass model on a square lattice. It is shown that under the SO(3) Majorana representation both spin models can be exactly mapped into Z2{Z}_{2}Z2​ gauge theory of spinons, with the standard form of Z2{Z}_{2}Z2​ Gauss law constraint.