Negative sign in hopping amplitude

  • Thread starter hbaromega
  • Start date
  • #1
21
0

Main Question or Discussion Point

Might be a naive question.

In many places, specially in the Hubbard model, people use a minus sign before the tight-binding hopping amplitude. What does the negative sign signify for? Any special intention?


Thanks.
 

Answers and Replies

  • #2
DrDu
Science Advisor
6,032
759
The hopping amplitude is something like [tex] \langle\phi_j| T_\mathrm{kin}|\phi_i\rangle [/tex], hence it's sign is well defined although there is a certain arbitrariness in the choice of the phase of the phi_i.
 
  • #3
21
0
The hopping amplitude is something like [tex] \langle\phi_j| T_\mathrm{kin}|\phi_i\rangle [/tex], hence it's sign is well defined although there is a certain arbitrariness in the choice of the phase of the phi_i.
That's true. But there must be a meaning for keeping this minus sign. I have seen Hubbard's original paper. Surprisingly that paper does not have this minus sign!

If the sign is just a matter of convention, then why unnecessarily an extra minus was chosen as a convention?


Thanks.
 
  • #4
DrDu
Science Advisor
6,032
759
No, it is not just a convention, it makes a real physical difference, at least in lattices where you cannot consistently number neighbouring atoms as 1 or 2. A simple example is azulene in the Hueckel model (which is basically the Hubbard model in quantum chemistry)
 
  • #5
21
0
No, it is not just a convention, it makes a real physical difference, at least in lattices where you cannot consistently number neighbouring atoms as 1 or 2. A simple example is azulene in the Hueckel model (which is basically the Hubbard model in quantum chemistry)
Sorry. Still not clear to me. What if I write

$t_{ij}$ instead of $-t_{ij}$ in the kinetic part of the Hubbard Hamiltonian?

Does it make any difference? If yes, then what difference?
 
  • #6
21
0
@DrDu

Sorry, the latex didn't work! I guess, you could follow the notation.

And I think, the Huckel model and the tight-binding model are equivalent. Hopping from one site to other site is similar to hopping from one orbital to another orbital.
 
  • #7
27
0
Consider the infinite U limit. Here the electrons are frozen at the lattice sites because the U prohibits double occupancy, even as a virtual state. Now if you tune U to a finite value then electrons start hopping in the process getting delocalized, thus lowering its energy.

The sign of t captures this lowering of energy.

P.S. : I can't seem to construct an argument as to why delocalization would lower the energy :(
 
  • #8
DrDu
Science Advisor
6,032
759
At large U, t can be taken into account perturbationally. As t is non-diagonal, it can make a contribution beginning with second order which will be proportional to t^2/(-U). Hence it is negative (stabilizing) and does not depend on the sign of t.
However, when going around a ring with an odd number of atoms, the sign does matter in higher orders, e.g. for a ring of three atoms the you get a contribution t_12*t_23*t_31/(-U)^2. For simple orbitals, like hydrogen 1s, t is negative, typically, whence these terms lead to a stabilization.
 
  • #9
M@2
67
0
Might be a naive question.

In many places, specially in the Hubbard model, people use a minus sign before the tight-binding hopping amplitude. What does the negative sign signify for? Any special intention?


Thanks.
The sign of t is significant. For example in tight binding model it determines where is the minimum energy orbital: at k=0 or at k=Pi/2

Let us for example consider polymer of atoms with s atomic valent orbital. It is evident,that k=0 (LCAO) orbital has minimum energy.
If we take polymer of atoms with p atomic orbital we get minimum energy orbital with |k|=Pi/2

LCAO: Linear Combination of Atomic Orbitals

I can recommend the book of Nobel Prize winner in chemistry:
Roald Hoffmann, Solids and surfaces: A Chemist's View of bonding in extended structures, 1988
pages: 4-9
 
Last edited by a moderator:

Related Threads on Negative sign in hopping amplitude

  • Last Post
Replies
2
Views
11K
  • Last Post
Replies
2
Views
4K
Replies
3
Views
4K
  • Last Post
Replies
2
Views
2K
Replies
1
Views
964
Replies
0
Views
2K
  • Last Post
Replies
2
Views
12K
Replies
2
Views
1K
  • Last Post
Replies
1
Views
655
  • Last Post
Replies
2
Views
2K
Top