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I have 1-d electron gas in tight banding model with included interaction between electrons of same spinsV[itex]_{\uparrow}[/itex]=-N[itex]_{\uparrow}[/itex]Uwhere U > 0,[itex]\sigma[/itex]=[itex]\pm1[/itex]is spin up or down, and pauli interaction with outside field B is included. I have to calculate the total energy of gas at T=0K as function of of polarizationx=[itex]\frac{N_{\uparrow}-N_{\downarrow}}{N}[/itex]expand it to series up to 6th order, minimize it and find the nontrivial solution for U

[itex]N=N_{\uparrow}+N_{\downarrow}, N_{\uparrow}=\frac{N}{2}(1+x)[/itex]

and [itex]N_{\downarrow}=\frac{N}{2}(1-x)[/itex]

[itex]N_{\sigma}=\frac{N}{2}(1+{\sigma}x)[/itex]

my major problem is calculating the total energy

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# Band correlative ferromagnetism-magnetism of conductive electrons in magnetic field

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