Exercise: Dimensional crossover

Here we will modify the code so as to allow for different hopping matrix elements along the $x$ and $y$ directions of a square lattice.

8.2.1Modifying the hopping¶

To do so we start from the submodule Hamiltonian_Hubbard_Plain_Vanilla_smod.F90, found in $ALF_DIR/Prog/Hamiltonians/, proceeding as follows:

Declare a new variable, Ham_Ty, in the module’s specification on a new line between !#PARAMETERS START# VAR_Hubbard_Plain_Vanilla and the corresponding !#PARAMETERS END#.
Modify the hopping matrix in the subroutine Ham_Hop in Vanilla:

Do I = 1,Latt%N
   Ix = Latt%nnlist(I,1,0)
   Op_T(1,nf)%O(I,  Ix) = cmplx(-Ham_T,    0.d0, kind(0.D0))
   Op_T(1,nf)%O(Ix, I ) = cmplx(-Ham_T,    0.d0, kind(0.D0))
   If ( L2 > 1 ) then
      Iy = Latt%nnlist(I,0,1)
!!!!!!! Modifications for Exercise 1a
      !Op_T(1,nf)%O(I,  Iy) = cmplx(-Ham_T,    0.d0, kind(0.D0))
      !Op_T(1,nf)%O(Iy, I ) = cmplx(-Ham_T,    0.d0, kind(0.D0))
      Op_T(1,nf)%O(I,  Iy) = cmplx(-Ham_Ty,    0.d0, kind(0.D0))
      Op_T(1,nf)%O(Iy, I ) = cmplx(-Ham_Ty,    0.d0, kind(0.D0))
!!!!!!!
   endif
   Op_T(1,nf)%O(I,  I ) = cmplx(-Ham_chem, 0.d0, kind(0.D0))
   Op_T(1,nf)%P(i) = i 
Enddo

It is a good idea as well to get the new simulation parameter written into the file info, also a change in Ham_Set.
Add Ham_Ty to the VAR_Hubbard_Plain_Vanilla name space in the parameter file parameters. This can be found in Scripts_and_Parameters_files/Start. Also set ham_name to Hubbard_Plain_Vanilla. The variable Model can be left unchanged as it is irrelevant.

After these changes, one can proceed with the workflow introduced in Downloading and using the code and tutorial.

The modified Hamiltonian_Hubbard_Plain_Vanilla_smod.F90 file can be downloaded from solution. The corresponding parameters can be downloaded from parameters. In case something went horribly wrong the original can be downloaded from original.

As an application of this code, we can once again consider a ladder system (e.g, a 2-leg ladder with L1=14 and L2=2), for different values of Ham_Ty. The results you should obtain for the total spin correlation function (file SpinT_eqJR) are summarized in Figure 1.

Spin correlation functions along one leg for the Hubbard ladder. As t_y grows the spin gap becomes large enough so as to detect the exponential decay of the spin correlation function on this small lattice size. The underlying physics of odd-even ladder systems is introduced in the article: Elbio Dagotto and T. M. Rice, “Surprises on the way from one- to two-dimensional quantum magnets: The ladder materials”, Science 271, 5249 (1996) pp. 618–623. — Figure 1:Spin correlation functions along one leg for the Hubbard ladder. As $t_y$ grows the spin gap becomes large enough so as to detect the exponential decay of the spin correlation function on this small lattice size. The underlying physics of odd-even ladder systems is introduced in the article: Elbio Dagotto and T. M. Rice, “Surprises on the way from one- to two-dimensional quantum magnets: The ladder materials”, Science **271**, 5249 (1996) pp. 618–623.

8.2 Exercise: Dimensional crossover

8.2.1Modifying the hopping¶