Physical mechanism underlying the relation between t, J and U, then the questions are as follows:

For the 1/2 filling: When ε_{d} is in between -U and 0, we are in the local moment (intermediate) regime in Coulomb blockade staircase, If we don't have any hopping, we have double degeneracy. The ground state -which is characterized by occupancy- is not only the single state but two states anymore.

In T/|t|-U/|t| planes: At very high temperature we have some sort of classical phase which is the mixture of those corresponding four states (LM, Mixed valance regime and the other two: |up,up>, |0>). Now, in the large repulsion limit, let's get temperature gradually down. We cooldown the system, thus we first hit U. It means that some of the states (zero and double occupied) will become eliminated from our configuration space. So, in the low energy Hilbert space will only contain up and down. In the related diagram, automatically we have a boundary (crossover) where T~U. Below this crossover, charge fluctuations are frozen out and only up and down remain in the Hilbert space. In each sites are occupied by only up and down electrons and these guys have really hard time hopping. Because, it's really difficult. You need to create a charge fluctuation. So, the only hopping process that you have to go from the case which the each site is occupied by a single electron to the case which a single site is double occupied with opposite spins due to the Pauli exclusion principle. It means that a low energy scale will be generated in this regime which is the order of square of the matrix element for the hopping devided by the energy that you have to pay for the hopping. Some low energy scale will be generated which is order of the square of t devided by U. But this is a very low scale. And again we have a boundary in low temperature and high repulsion regime order of t^2/U. In between these two boundaries, we have a local moment paramagnet state with frozen charge fluctuations. Charge fluctuations are protected by a gap (can be seen in Coulomb blockade staircase) which is order of repulsion as Δq~U. In this state the material is a Mott insulator. We still have a very large configuration space of the order of 2^n. Above the crossover line we had 4^n possible configurations. So, in the low temperature something must happen to this 2^n. We can not keep the degerate GS of this 2^n. To avoid this, usually spins are magnetically ordered except some special cases. The ~t^2/U boundary -which is the line we broke in low temperature and high repulsion regime- is the characteristic energy scale for magnetic ordering.

Why this phenomena implies this energy scale? This related to the degenerate perturbation theory. For simplicity consider a dimer with no hopping. We have two site. We can organize these two site for four states of |sigma, sigma'> by using the quantum number of total spin. The manifold of these four states is actually made of the one singlet state which S=0 (opposite spins) and the 3 (degenerate) triplet states which S=1. When hopping is zero these 3 three states are degenerate. When hopping is nonzero these four states will split. They will obviously split by favoring the singlet as the lowest energy state. Because this guys allows is a nonzero matrix element. In the presence of a nonzero hopping these two states in the expectation value of hopping are both singlet so automatically we have nonzero matrix element. If you do this for triplet state we can not get nonzero matrix element. If we compare this two expectation value we see that the singlet state is lower by an energy which is the square of the perturbation matrix element t^2 divided by the energy state of the intermediate high energy state, hence we get t^2/U. If you actually carry out the calculation properly for a dimer the energy difference of expectation values getting from singlet and triplet states will be 4t^2/U. This is called antiferromagnetic superexchange J_{AF}. This essentially is the scale of magnetically ordering and for one-band Hubbard model this is AF order. That’s my understanding.

Q1: From above explanation, Should we catch why the Hubbard model turn into the t-J model in infinite U limit? Are there enough evidence for this inference?

Q2: Does Increasing or decreasing temperature directly effect the hopping ability of electron? For example, when the system is heated, the kinetic energy of electrons will become larger and will this cause any changes of the hopping constant strenght or just shows an effect on fluctuations?

Q3: To detect the different phase islands in T/|t|-U/|t| plane, |t| should be selected as independent system parameter. That’s why we scale the other Hamiltonian parameters with |t| for comparison in different energy scales. If the answer of Q2 is “Yes…”, how can we investigate these islands for a fixed value of hopping?

Q4: Q3 is indirectly related to determine the Mott transition point for each different U by using the spectral weight. We need a fixed hopping for this purpose. Again, If the answer of Q2 is “Yes…”, I guess, there should be another justification. Are there any instructions in Luttinger theorem for this?

Q5: It has been claimed that The t-J model was first derived in 1977 from the Hubbard model by Spalek [1, 2]. Or, is this just reflects the spirit of t-J model? Contrary to this, according to some other experts, this model totally has been propounded first by Zhang and Rice to derive explicitly a single-band effective Hamiltonian for the high-T_{c} Cu-oxide superconductors [3]. Which one is duly admitted as the first study?

Q6: Relatedly with Q1; How about a unified t-J-U? Are there any other challenging work in literature except [4]? If there is an exact proof of canonical transformation between Hubbard and t-J model, is this kind of attempt still possible? If the answer is “Yes…”, How do we avoid double counting? How can be made a fine tunning on the ranges of two different interaction (Hubbard U and superexchange) at the same time?

1. K. A. Chao, J. Spalek, A. M. Oles, J. Phys. C 10, L 271 (1977), Phys. Rev. B 18, 3453 (1978).

2. The method has been originally proposed by J. Spalek, A. M. Oles, preprint of Jagiellonian University, SSPJU-6/76 (1976) and Physica B, 86-88, 375 (1977).

3. F. C. Zhang, T. M. Rice, Phys. Rev. B 37, 3759 (1988). arXiv:cond-mat/0303501