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Generalized Jordan-Wigner transformations.

C D Batista1, G Ortiz

  • 1Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.

Physical Review Letters
|February 15, 2001
PubMed
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We developed a novel spin-fermion mapping applicable to arbitrary spin values, generalizing the Jordan-Wigner transformation. This method reveals hidden symmetries and solves previously intractable lattice models, including demonstrating the Haldane gap in spin chains.

Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Statistical Mechanics

Background:

  • The Jordan-Wigner transformation is a fundamental tool for mapping spin systems to fermionic ones, primarily for spin-1/2 systems.
  • Solving complex lattice models in condensed matter physics often requires advanced analytical or numerical techniques.
  • Understanding hidden symmetries and emergent phenomena like the Haldane gap is crucial for characterizing quantum materials.

Purpose of the Study:

  • To introduce a generalized spin-fermion mapping for arbitrary spin S, extending the capabilities of the Jordan-Wigner transformation.
  • To demonstrate the utility of this new mapping in solving previously intractable lattice models and uncovering hidden symmetries.
  • To explore the application of this mapping to specific physical systems, such as Heisenberg spin chains and strongly correlated electrons.

Related Experiment Videos

Main Methods:

  • Development of a new spin-fermion mapping valid for arbitrary spin S and regular lattices in any spatial dimension d.
  • Application of the mapping to find exact solutions for specific lattice models.
  • Analysis of the mapping's relevance to the Haldane gap phenomenon and strongly correlated electron systems.
  • Presentation of a general spin-anyon mapping for dimensions d <= 2.

Main Results:

  • A novel spin-fermion mapping for arbitrary spin S has been successfully formulated, generalizing the Jordan-Wigner transformation.
  • The mapping effectively unravels hidden symmetries in lattice models.
  • Exact solutions were obtained for lattice models that were previously unsolvable by standard techniques.
  • The existence of the Haldane gap in spin-1 Heisenberg spin chains was demonstrated using this mapping.
  • The relevance of the mapping to models of strongly correlated electrons was discussed.
  • A general spin-anyon mapping for d <= 2 was presented.

Conclusions:

  • The new spin-fermion mapping provides a powerful and versatile tool for analyzing quantum spin systems with arbitrary spin values.
  • This generalized approach opens new avenues for finding exact solutions to complex lattice models and understanding their underlying physics.
  • The mapping has significant implications for the study of quantum magnetism, topological phases, and strongly correlated electron systems.