module shared !every function and routine which
implicit none !includes the module shared can use its
real*8, allocatable :: Hdiag(:,:),eigenvalues(:) !variables
end module shared
program diag
use shared !now the program diag knows about Hdiag and eigenvalues
implicit none !this makes sure one has do define everything explicitly
integer :: nsize !size of the matrix to be diagonalized
integer :: i,j,icount,jcount,norb
integer :: i1,i2,j1,j2
real*8 :: v
norb = 5 !size of system
nsize = norb*norb !size of hamiltonian
allocate(Hdiag(nsize,nsize)) !This is the matrix to be diagonalized
allocate(eigenvalues(nsize)) !after diagonalization, this vector
!contains the eigenvalues
v = -1.0 !this is the hopping parameter
!Now we construct the hamiltonian, and we do that by first looping over
!j, and then i. This are non-periodic boundary conditions.
icount = 0
jcount = 0
do i1=1,norb
do j1=1,norb
jcount = 0
icount = icount + 1
do i2=1,norb
do j2=1,norb
jcount = jcount + 1
if (abs(i1-i2) + abs(j1-j2) == 1)Hdiag(icount,jcount) = v !if distance == 1, put hopping parameter = v
write(11,*)'hop(',icount,jcount,')=',Hdiag(icount,jcount) !this prints the hamiltonian to disc
end do
end do
end do
end do
call diagonalize(nsize) !now Hdiag will be diagonalized.
!After diagonalization, Hdiag(:,:) will contain the eigenvectors,
!Where Hdiag(:,1) is the first eigenvector, Hdiag(:,2) the second and so on
!and eigenvalues(1) will contain the eigenvalue corresponding to
!eigenvector 1, and so on.
do i=1,nsize
write(*,*)'eigenvalue:',i,eigenvalues(i) !writing eigenvalues to screen
end do
end program diag !program is finished
subroutine diagonalize(ndiag) !Diagonalizes the matrix Hdiag
use shared
implicit none
real*8, allocatable :: work(:)
integer, allocatable :: iwork(:)
character job, uplo
external dsyevd
integer nmax, lda, lwork, liwork, info,ndiag
nmax=ndiag
lda=nmax
lwork=4*nmax*nmax+100
liwork=10*nmax
allocate(work(lwork),iwork(liwork))
job='v'
uplo='l'
call dsyevd(job,uplo,ndiag,Hdiag,lda,eigenvalues,work,lwork,iwork,liwork,info)
if (info > 0) then
write (*,*) 'failure to converge.'
stop
endif
deallocate(work,iwork)
return
endA 2D square Tight-binding model is a mathematical model used to describe the electronic structure of a 2D crystal lattice. It is commonly used in solid state physics to study the behavior of electrons in materials such as graphene or transition metal dichalcogenides.
The Hamiltonian of a 2D square Tight-binding model is a mathematical operator that represents the total energy of the system. It takes into account the kinetic energy of the electrons as well as the interactions between them, which are described by the crystal lattice structure.
The matrix for the Hamiltonian of a 2D square Tight-binding model can be constructed by considering the basis functions of the crystal lattice, which are usually chosen to be the atomic orbitals of the constituent atoms. The elements of the matrix correspond to the hopping integrals between these orbitals, which represent the strength of the electron interactions at different lattice sites.
The matrix for the Hamiltonian of a 2D square Tight-binding model allows us to solve for the energy levels and wavefunctions of the electrons in the system. This information is crucial in understanding the electronic properties and behavior of the material, such as its conductivity and band structure.
Yes, there are several simplifications and approximations involved in constructing the matrix for the Hamiltonian of a 2D square Tight-binding model. These include assuming a small number of nearest neighbor interactions, neglecting higher order interactions, and using a finite basis set to represent the wavefunctions of the electrons. These approximations are necessary to reduce the complexity of the problem and make it solvable using computational methods.