© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2101.10119 · Journal of Physics A Mathematical and Theoretical

A mapping between the spin and fermion algebra

Felix Meier, Daniel Waltner, Petr A. Braun, Thomas Guhr

Published Jul 13, 2021• 0 citations

Abstract

Abstract We derive a formalism to express the spin algebra <?CDATA $\mathfrak{s}\mathfrak{u}(2)$?> s u ( 2 ) in a spin s representation in terms of the algebra of L fermionic operators that obey the canonical anti-commutation relations. We also give the reverse direction of expressing the fermionic operators as polynomials in the spin operators of a single spin. We extend here to further spin values the previous investigations by Dobrov (2003 J. Phys. A: Math. Gen. 36 L503) who in turn clarified on an inconsistency within a similar formalism in the works of Batista and Ortiz (2001 Phys. Rev. Lett. 86 1082). We then consider a system of L fermion flavors and apply our mapping in order to express it in terms of the spin algebra. Furthermore we investigate a possibility to simplify certain Hamiltonian operators by means of the mapping.

DOI: 10.1088/1751-8121/ac13dc

Related papers

On the spin–fermion connection

Stanislav V. Dobrov

Jordan–Wigner-type transformations connecting the spin-3/2 operators and two types of fermion are derived. The general condition of fermionizability of spins is obtained. Discordances in the results of a previous attempt to generalize the Jordan–Wigner transformation for all spins (Batista and Ortiz 2001 Phys. Rev. Lett. 86 1082) are pointed out. After introducing clarity, a new interpretation is given for these results.

Journal of Physics A Mathematical and General•Sep 15, 2003•10 citations

Something about spin-fermion connection

Stanislav V. Dobrov

Jordan-Wigner-type transformations connecting the spin-3/2 operators and two kinds of fermions are derived. A general condition of fermionizability of spins is obtained and a theorem establishing connection between half integer spins and fermions is proved. The fallibility of a previous attempt to generalize the Jordan-Wigner transformation for all spins (C.D. Batista and G. Ortiz, Phys. Rev. Lett. 86, 1082 (2001)) is pointed out.

arXiv:cond-mat/0306495•Jun 19, 2003•0 citations

PT -symmetric representations of fermionic algebras

Carl M. Bender, S. P. Klevansky

A recent paper by Jones-Smith and Mathur, Phys. Rev. A 82, 042101 (2010) extends $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics from bosonic systems (systems for which ${\mathcal{T}}^{2}=\mathbf{1}$) to fermionic systems (systems for which ${\mathcal{T}}^{2}=\ensuremath{-}\mathbf{1}$). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct $\mathcal{P}\mathcal{T}$-symmetric matrix representations for operator algebras of the form ${\ensuremath{\eta}}^{2}=0$, ${\overline{\ensuremath{\eta}}}^{2}=0$, $\ensuremath{\eta}\overline{\ensuremath{\eta}}+\overline{\ensuremath{\eta}}\ensuremath{\eta}=\ensuremath{\alpha}\mathbf{1}$, where $\overline{\ensuremath{\eta}}={\ensuremath{\eta}}^{\mathcal{P}\mathcal{T}}=\mathcal{P}\mathcal{T}\ensuremath{\eta}{\mathcal{T}}^{\ensuremath{-}1}{\mathcal{P}}^{\ensuremath{-}1}$. It is easy to construct matrix representations for the Grassmann algebra ($\ensuremath{\alpha}=0$). However, one can only construct matrix representations for the fermionic operator algebra ($\ensuremath{\alpha}\ensuremath{\ne}0$) if $\ensuremath{\alpha}=\ensuremath{-}1$; a matrix representation does not exist for the conventional value $\ensuremath{\alpha}=1$.

arXiv:1104.4156•Physical Review A•Aug 25, 2011•23 citations

The Group Theoretical Structure of Fermion Many-Body Systems Arising from the Canonical Anticommutation Relation. I: Lie Algebras of Fermion Operators and Exact Generator Coordinate Representations of State Vectors

Hideo Fukutome

We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.

Progress of Theoretical Physics•Mar 1, 1981•46 citations

Computational Many-Body Physics via $\mathcal M_{2^q}$ Algebra

Emil V. Prodan

The many-body Hamiltonians and other fermionic physical observables are expressed in terms of fermionic creation and annihilation operators, which form the algebra of canonical anti-commutation relations (CAR). In this work we use a canonical isomorphism between CAR and $\mathcal M_{2^\infty}$ algebras to derive analytic matrix representations of many-fermion operators. Code-lines implementing these matrix representations are supplied and Hubbard-type Hamiltonians are worked out explicitly.

arXiv:1906.07309•Jun 17, 2019•1 citation

Spin-Spin Correlators in the Majorana Representation

Alexander Shnirman, Yuriy Makhlin

In the Majorana representation of a spin 1/2 we find an identity which relates spin-spin correlators to one-particle fermionic correlators. This should be contrasted with the straightforward approach in which two-particle (four-fermion) correlators need to be calculated. We discuss applications to the analysis of the dynamics of a spin coupled to a dissipative environment and of a quantum detector performing a continuous measurement of a qubit's state.

arXiv:cond-mat/0305064•Physical Review Letters•Nov 10, 2003•58 citations

Abstract We derive a formalism to express the spin algebra <?CDATA $\mathfrak{s}\mathfrak{u}(2)$?> s u ( 2 ) in a spin s representation in terms of the algebra of L fermionic operators that obey the canonical anti-commutation relations. We also give the reverse direction of expressing the fermionic operators as polynomials in the spin operators of a single spin. We extend here to further spin values the previous investigations by Dobrov (2003 J. Phys. A: Math. Gen. 36 L503) who in turn clarified on an inconsistency within a similar formalism in the works of Batista and Ortiz (2001 Phys. Rev. Lett. 86 1082). We then consider a system of L fermion flavors and apply our mapping in order to express it in terms of the spin algebra. Furthermore we investigate a possibility to simplify certain Hamiltonian operators by means of the mapping.

A recent paper by Jones-Smith and Mathur, Phys. Rev. A 82, 042101 (2010) extends

\mathcal{P}\mathcal{T}

-symmetric quantum mechanics from bosonic systems (systems for which

{\mathcal{T}}^{2}=\mathbf{1}

) to fermionic systems (systems for which ${\mathcal{T}}^{2}=\ensuremath{-}\mathbf{1}$). The current paper shows how the formalism developed by Jones-Smith and Mathur can be used to construct

\mathcal{P}\mathcal{T}

-symmetric matrix representations for operator algebras of the form ${\ensuremath{\eta}}^{2}=0$, ${\overline{\ensuremath{\eta}}}^{2}=0$, $\ensuremath{\eta}\overline{\ensuremath{\eta}}+\overline{\ensuremath{\eta}}\ensuremath{\eta}=\ensuremath{\alpha}\mathbf{1}$, where $\overline{\ensuremath{\eta}}={\ensuremath{\eta}}^{\mathcal{P}\mathcal{T}}=\mathcal{P}\mathcal{T}\ensuremath{\eta}{\mathcal{T}}^{\ensuremath{-}1}{\mathcal{P}}^{\ensuremath{-}1}$. It is easy to construct matrix representations for the Grassmann algebra ($\ensuremath{\alpha}=0$). However, one can only construct matrix representations for the fermionic operator algebra ($\ensuremath{\alpha}\ensuremath{\ne}0$) if $\ensuremath{\alpha}=\ensuremath{-}1$; a matrix representation does not exist for the conventional value $\ensuremath{\alpha}=1$.

We study in this series the group theoretical structure of Fermion many-body systems arising from the canonical anticommutation relation of the annihilation-creation operators. Owing to the canonical anticommutation relation, a Fermion system with N single particle states has at least six Lie algebras of Fermion operators, a U(N), an SO(2N), an SO(2N+1), an SO(2N+2) and two U(N+1) Lie algebras. There are also two Clifford algebras of 2N and 2N+1 dimensions. The Fermion space is shown to belong to the spinor representations of the SO(2N), SO(2N+1) and SO(2N+2) groups. The canonical transformations generated by the U(N), SO(2N) and SO(2N+1) Lie algebras are characterized as the transformations to induce the linear U(N), SO(2N) and SO(2N+1) transformations for the Clifford algebras. The independent (quasi-) particle type wave functions of three kinds including the Hartree-Fock and Hartree-Bogoliubov wave functions are constructed by means of the canonical transformations and their relationship is studied. We derive three exact generator coordinate representations for state vectors in which the generator coordinates are the U(N), SO(2N) or SO(2N+1) group and the generating functions are the independent (quasi-) particle type wave functions. We characterize the structures of state vectors in the generator coordinate representations and study the relationship of the three representations.

Computational Many-Body Physics via

\mathcal M_{2^q}

Algebra

The many-body Hamiltonians and other fermionic physical observables are expressed in terms of fermionic creation and annihilation operators, which form the algebra of canonical anti-commutation relations (CAR). In this work we use a canonical isomorphism between CAR and

\mathcal M_{2^\infty}

algebras to derive analytic matrix representations of many-fermion operators. Code-lines implementing these matrix representations are supplied and Hubbard-type Hamiltonians are worked out explicitly.