Communications in Mathematical Physics

, Volume 181, Issue 2, pp 409–446

Low temperature phase diagrams for quantum perturbations of classical spin systems

  • C. Borgs
  • R. Kotecký
  • D. Ueltschi

DOI: 10.1007/BF02101010

Cite this article as:
Borgs, C., Kotecký, R. & Ueltschi, D. Commun.Math. Phys. (1996) 181: 409. doi:10.1007/BF02101010


We consider a quantum spin system with Hamiltonian
$$H = H^{(0)} + \lambda V,$$
whereH(0) is diagonal in a basis ∣s〉=⊗xsx〉 which may be labeled by the configurationss={sx} of a suitable classical spin system on ℤd,
$$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$
We assume thatH(0)(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH(0), provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • C. Borgs
    • 1
  • R. Kotecký
    • 2
  • D. Ueltschi
    • 3
  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Centre de Physique ThéoriqueCNRSMarseilleFrance
  3. 3.Institut de Physique ThéoriqueEPFLausanneSwitzerland
  4. 4.Institut für Theoretische PhysikFreie Universität BerlinBerlinDeutschland
  5. 5.Center for Theoretical StudyCharles UniversityPraha