1d Hubbard model metal insulator transition

[jsfeng-fudan]

For zero temperature chain (half filling), I use the Hphi to study the MIT of Hubbard model with FullDiag method, but it is found that
the gap (E1-E0) is always zero.
t = -0.5
U=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
gaps = [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]

it is apprieated that someone can help me solve this error!

[tmisawa]

Could you tell me the definition of the "gap" that you calculated?

Usually, the charge gap Delta is defined from the chemical potential mu(Ne), i.e.,
Delta= mu (Ne+1)- mu(Ne-1) and mu(Ne+1)=(E(Ne+2)-E(Ne))/2
where Ne is the number of electrons and E(Ne) means the total energy
with Ne.
Thus, to calculate the charge gap, it is necessary to perform the calculations with the different number of electrons.

[jsfeng-fudan]

Here, for many electrons state |psi> = |0,1,0,1.....>, a binary encoded state, the ground state energy is E0, the first excited state energy is E1, gap is defined as delta = E1-E0.

[jsfeng-fudan]

the charge gap is as you said, I follow your advice. The gap vs U/t diagram is plotted as below. Here, We can't even see the MIT phase transition. the size of chain N = 8.

[jsfeng-fudan]

By the way, in HPhi, the standard mode does not provide the onsite energy epsilon, i.e., the single site occupied energy. If the onsite energy is needed, how can we deal with ?

[tmisawa]

On the MIT transition, it is known that MIT does not occur at finite U (in other words, MIT occurs at U=0) in the one-dimensional Hubbard model (see, "Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension" by Lieb an Wu ,Phys. Rev. Lett. 20, 1445). Thus, I think that your analysis is consistent with the exact solution.

[tmisawa]

You can specify the onsite energy (i.e. chemical potential) by the keyword mu in the standard mode.
Note that, this term just gives the constant shift in the total energy (-mu*Nelec) in total-particle conserved systems.

[jsfeng-fudan]

@tmisawa Thank you very much, for 1d Hubbard model, the critical point appears at U/t=0. I try to deal with 2d square hubbard model with one orbital. But the charge gap vs U/t diagram shows that MIT can not take place. Here, I use the CG method, not FullDiag. the square lattice is 4x4.

[tmisawa]

It is also known that there is no metal-transition transition in the square lattice model, i.e., MIT occurs at U/t=0.
Intuitively, we can understand that the antiferromagnetic order of (π,π) occurs in the square lattice at infinitesimal U because of the perfect nesting, which means that the metal-insulator transition occurs at U/t=0.
Thus, I think your calculation is correct.
In the Hubbard model on the cubic lattice, we expect that the MIT also occurs at U/t=0.

To realize the MIT at finite U, we should introduce some additional terms in the Hubbard model, such as the
next-nearest hopping t'. For example, in a previous study, it is shown that MIT occurs at finite U by using the exact diagonalization. However, note that the limitation of the available system size is severe in the exact diagonalization.
To reduce the finite size effects, they use the averaged twisted boundary condition in actual calculations.

[jsfeng-fudan]

@tmisawa Thank you, I think you are right . The reason of open charge gap is the perfect fermi surface nesting, the nesting vector is Q = (π,π). I try to add the next hopping in the Hamiltonian.

[jsfeng-fudan]

@tmisawa Hi, when I add the next nearest hopping t' into the Hamiltonian, the input file is :
model = "Hubbard"
method = "CG"
lattice = "square"
L = 4
W = 4
t = 1
t'= 0.5
U = 1
2Sz = 0
nelec = 14
but , the HPhi does not perform the calculations, the error message is :
Open Standard-Mode Inputfile stan.in

Skipping a line.
KEYWORD : model | VALUE : Hubbard
KEYWORD : method | VALUE : CG
KEYWORD : lattice | VALUE : square
KEYWORD : l | VALUE : 4
KEYWORD : w | VALUE : 4
KEYWORD : t | VALUE : 1
KEYWORD : t' | VALUE : 0.5
ERROR ! Unsupported Keyword in Standard mode!

####### You DO NOT have to WORRY about the following MPI-ERROR MESSAGE. #######

can you help me deal with the problem?

[tmisawa]

In my PC, I can generate the files using the attached input file (stan.txt).
Could you try to use this file?

stan.txt

The origin of the error may be related to the font of prime in t'

[jsfeng-fudan]

@tmisawa Hi, After I added the hopping of the next nearest neighbor, a non-zero phase transition point appeared in the two-dimensional square lattice. However, this phase transition point should be slightly smaller than what is mentioned in the literature, which may be caused by boundary effects. How to eliminate this effect without changing the size of the grid. After the grid became larger, the memory increased exponentially, and my computer was unable to calculate.

[jsfeng-fudan]

@tmisawa Thank you again. By the way, besides using charge gap to determine phase transition, can we use density of states and doublon to determine it? If so, how can we achieve it in HPhi?

[tmisawa]

As mentioned above, to reduce the finite size effects, the averaged boundary conditions are often used.
For details, please see "Exact diagonalization study of Mott transition in the Hubbard model on an anisotropic triangular lattice" by T. Koretune et al., and
"Ab initio derivation and exact diagonalization analysis of low-energy effective Hamiltonians for 𝛽′−X⁢[Pd⁢(dmit)2]2" by K. Yoshimi et al. By changing the hoppings manually, you can perform the averaged boundary conditions.

As shown in the paper by T. Koretsune et al., the changes in the doublon at the MIT are small. Thus, it is difficult to detect the MIT only from the doublon. And, I note that it is proposed that the quantum spin liquid exists
between the antiferromagnetic phase and the paramagnetic phase in the t-t' Hubbard model. It is difficult to detect such exotic quantum spin liquid phase by the exact diagonalization since the available system sizes are very small.

[jsfeng-fudan]

@tmisawa Thank you very much. I try to measure the spin liquid state proximity to AFM state with t-t' square lattice Hubbard model.