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Quantum Mechanics

Ising Chain Hamiltonian

Hamiltonian

Section titled “Hamiltonian”

A common quantum transverse-field Ising chain is

H=−J∑iσizσi+1z−h∑iσix.H = -J\sum_i \sigma_i^z\sigma_{i+1}^z - h\sum_i\sigma_i^x.

The classical Ising chain omits the transverse-field term and uses commuting spin variables or diagonal σiz\sigma_i^z operators:

Hclassical=−J∑isisi+1.H_{\mathrm{classical}} = -J\sum_i s_i s_{i+1}.

Assumptions

Section titled “Assumptions”
  • Boundary conditions are specified: open, periodic, or another choice.
  • The sign convention for JJ is stated.
  • Pauli matrices act on sites in a tensor-product chain.
  • The transverse-field term makes the model quantum because it does not commute with the interaction term.

Common Mistakes

Section titled “Common Mistakes”
  • Confusing the classical and transverse-field Ising models.
  • Forgetting boundary-condition effects in small chains.
  • Comparing conventions where the transverse field is called hh, Γ\Gamma, or gJgJ.
  • Treating all spin-chain Hamiltonians as Ising models.
Section titled “Canonical Links”

References

Section titled “References”
  • S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, 2011.
  • P. Pfeuty, “The one-dimensional Ising model with a transverse field”, Annals of Physics 57, 79-90, 1970.