The Zero- and Higher-Order Effective Hamiltonians

To obtain the effective Hamiltonian we need to diagonalise the SMFT Hamiltonian in Floquet space. When this is not practical we should consider perturbation expansions. The van Vleck transformation [96] will be the most convenient approach in this case. The result will be an expansion of the effective Hamiltonian Heff in terms of higher-order terms with 

At this point of the discussion the zero- and higher-order terms will not be derived explicitly, but we will return to the van Vleck approach at a later stage, where we will treat the BMFT case. Following Goldman s derivation [98,99] and Mehring s secular averaging theory [14] the result of the van Vleck transformation yields 

If Hi luJt and we approximate Li(0) 1, the effective Hamiltonian to first-order becomes 

The zero-order contribution to Hef / is thus just equal to the diagonal term Ho, and depends on the matrix elements H k as in Eq. 56. We will discuss briefly which bimodal Floquet elements Hnk can contribute to the diagonal block Hq in the single-mode representation. According to Eq. 55, in order to obtain I = 0 the integer k must be equal to -nn/v. There is always a possibility that n = 0 and k = 0 and hence Hqo will always contribute to Ho- 

For spinning frequencies slower than or equal to the characteristic RF frequency (ujr ujc with 1/ k), k/v is less then 1. Since n = 1, 2 and k is an integer (it is an index), only when k/v equals 1 or 1/2 elements iT -uk/v exist that can contribute to Ho. This corresponds to the synchronisation conditions with Tr/Tc = = 1,2. For all other t /tc ratios the only element contributing 

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