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5.1.3 Physical Hamiltonians

Since real materials may not realize the ideal Heisenberg model, more general spin Hamiltonians have been investigated, mainly with numerical simulations [65,66,67,68]. From these works, a phase diagram has been obtained for the Hamiltonian with the XXZ-type interaction and a uniaxial single ion anisotropy D:

The phase diagram is shown in Fig.36 [68]. It has been found that the Haldane phase (H) exist in a relatively large parameter region, which includes the pure Heisenberg model ($\lambda=1, D=0$).
  
Figure: Phase diagram of the Hamiltonian eq.42 is shown (cite from [68]). The symbols are: (H) Haldane phase, (N) Néel ordered phase, (XY) XY phase, (F) Ferromagnetic phase and (D) large-D phase.
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\vspace{5cm}\end{figure}

The effect of the inter-chain interactions has also been investigated [69,70,71]. Fig.37 shows the phase diagram [71] obtained from a numerical simulation of the Hamiltonian:

where (i,j) denotes the nearest inter-chain neighboring pair. It has been found that the Haldane phase (H) survives in the presence of the inter-chain interaction (J'), if the single ion anisotropy (D) is small.
  
Figure: Phase diagram of the Hamiltonian (eq.43) is shown (cite from [71]). The symbols are: (H) Haldane phase, (N) Néel ordered phase, (XY) XY phase, and (D) large-D phase.
\begin{figure}
\vspace{4.5cm}\end{figure}

The above introduced theoretical works all supported Haldane's conjecture. The next section introduces previous experimental results of several S=1 spin systems, which were studied to test Haldane's conjecture in real materials.


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Next: 5.1.4 Experimental evidence for Up: 5.1 Introduction Previous: 5.1.2 The Valence-Bond-Solid Hamiltonian