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Hopfield–Bogoliubov transformation

TM Takuma Makihara
KH Kenji Hayashida
GI G. Timothy Noe II
XL Xinwei Li
NP Nicolas Marquez Peraca
XM Xiaoxuan Ma
ZJ Zuanming Jin
WR Wei Ren
GM Guohong Ma
IK Ikufumi Katayama
JT Jun Takeda
HN Hiroyuki Nojiri
DT Dmitry Turchinovich
SC Shixun Cao
MB Motoaki Bamba
JK Junichiro Kono
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We perform a Hopfield–Bogoliubov transformation to diagonalize our Hamiltonian, Eq. (4). We introduce coupled magnon annihilation operators B^L(B^U) describing the LM (UM), which are expressed in terms of the generalized qFM (qAFM) operators a^ (b^) by:

for j = L, U. The coefficients are solutions to an eigenvalue problem discussed in Supplementary Note 4. The Hamiltonian can be rewritten as:

and the ground state 0 of our coupled magnon system must satisfy:

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