In the article "Mean-field theory for the t-J model" (Physical Review B 40, 2610 (1989)), the authors derived a mean field theory in Slave fermion and Schwinger boson formalism for the t-J model. In the second page, it is said that "we derive self-consistent equations for A, B, D, λ, and μ, and solve them numerically for the lowest free-energy solution for various values of t/J, δ, and T".
I am now able to derive those self-consistency equations, but having great problems in solving it. I tried to minimize a cost function which is defined as the squared sum of the difference between both sides of each self-consistency equation, but the cost function landscape seems to be very ugly for the parameters A and B. I addition, in the derivation of those equations I discarded lots of number terms (not involving holon/spinon operators) in the mean field Hamiltonian, so I currently do not have an expression for the free energy. I hope that experienced researchers can provide me some suggestions and details on how the numerical solution can be found.
Dear Zhengyuan Yue
It will depend on the routine you use for it, 2 examples:
- A non linear minimization Levenberg-Marquardt routine/algorithm (Levenberg, 1944 & Marquardt, 1963).
- A self-consistent method based on a fixed point algorithm.
Usually is very slow to obtain the convergence from one step to another if the expression is a non-linear one.
Best Regards.
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