Higher order spin interactions mediated by the substrate

Raw data for the plots used in the paper. Every file has the x axis as the first column and the appropriate interaction as the second column. To obtain the same values as in the paper, normalize the second column by the hopping (t = 0.5). The expression for the fourth order expansion term was obtained using the SNEG package (http://nrgljubljana.ijs.si/sneg/) but can be handled by Mathematica. nc is the non-commutative multiplication Electron field operators are expressed with the following subscripts: c[type, site, time, spin] type: 0 - creation; 1 - annihilation site: from i_1 to i_4 time: imaginary time from t_1 to t_4 spin: 1,2,3 representing x,y,z Spin operators are expressed with the following subscripts: S[site,spin] site: from i_1 to i_4 spin: 1,2,3 representing x,y,z