Communications in Mathematical Physics

, Volume 131, Issue 3, pp 465–494 | Cite as

A Feynman-Kac formula for the quantum Heisenberg ferromagnet. I

  • H. Hogreve
  • W. Müller
  • J. Potthoff
  • R. Schrader


The Hamiltonian of the (anisotropic) quantum Heisenberg (anti-) ferromagnet on an arbitrary finite lattice is lifted to a Hamiltonian acting on sections of the bundle obtained by twisting a certain line bundle over the classical spin configuration space (which is a Kähler manifold) with the Dolbeault complex. This procedure is extended fromSU(2) to arbitrary compact semi-simple Lie groups and arbitrary irreducible representations. The Bott-Borel-Weil theorem gives a heat kernel representation for the original partition function in an external magnetic field. TheU(1)-gauged local Hamiltonian is the sum of the free, supersymmetric, twisted Dolbeault Laplace operator (multiplied by the inverse of an arbitrary small mass parameter) plus the lifted Hamiltonian.

The resulting (Euclidean) Lagrangian is nonlocal and describes bosons which do and fermions which do not propagate through the lattice. All fields couple to the external magnetic field. The Lagrangian contains Yukawa and Luttinger type interactions.


Partition Function Line Bundle External Magnetic Field Irreducible Representation Heat Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • H. Hogreve
    • 1
  • W. Müller
    • 2
  • J. Potthoff
    • 3
  • R. Schrader
    • 4
  1. 1.Hahn-Meitner Institut BerlinBerlinFederal Republic of Germany
  2. 2.Akademie der Wissenschaften der DDRBerlinGerman Democratic Republic
  3. 3.Universität Bielefeld and LSU at Baton RougeUSA
  4. 4.Freie Universität BerlinBerlinFederal Republic of Germany

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