Bloch electrons in a magnetic field: Rigorous justification of the Peierls-Onsager effective Hamiltonian

1989 ◽  
Vol 17 (3) ◽  
pp. 247-252 ◽  
Author(s):  
G. Nenciu
Keyword(s):  
Magnetic Field ◽  
Effective Hamiltonian ◽  
Bloch Electrons

Von Neumann lattices of Wannier functions for Bloch electrons in a magnetic field

1986 ◽  
Vol 403 (1824) ◽  
pp. 135-166 ◽  
Keyword(s):  
Magnetic Field ◽  
Phase Space ◽  
Periodic Function ◽  
Degrees Of Freedom ◽  
Two Dimensions ◽  
Effective Hamiltonian ◽  
Hall Conductance ◽  
Von Neumann ◽  
Wannier Functions ◽  
Bloch Electrons

The problem of Bloch electrons in a magnetic field in two dimensions can be reduced to a one-dimensional problem with a Hamiltonian Ĥ that is a periodic function of x ^ and p ^ . Wannier functions can be defined for the sub-bands of the spectrum of this effective Hamiltonian. When the Chern class (quantized Hall conductance integer) of the sub-band is zero, the Weyl-Wigner formalism can be used to represent these Wannier functions by a von Neumann lattice. It is shown how this von Neumann lattice of Wannier functions can be defined for irrational as well as rational magnetic fields. An important benefit from using the Weyl-Wigner formalism is that symmetries of the periodic potential are reflected by symmetries of the effective Hamiltonian in phase space. It is shown how the Wannier functions can be defined so that their Wigner functions have the point symmetries of the effective Hamiltonian. An example of how these results can prove useful is given: if we take matrix elements of the Hamiltonian between the Wannier states of a sub-band, we derive a new effective Hamiltonian describing this sub-band, which is again a periodic function of coordinate and momentum operators. Since, by projecting onto a sub-band, we have also reduced the number of degrees of freedom, this operation is a renormalization group transformation. It is shown that the symmetry of the new effective Hamil­tonian in phase space is the same as that of the original one. This preservation of symmetry helps to explain some unusual properties of the spectrum when the Hamiltonian has fourfold symmetry.

Nonlinear electromagnetic response of bloch electrons in a magnetic field

1968 ◽  
Vol 58 (1) ◽  
pp. 93-106 ◽  
Author(s):  
K. C. Rustagi ◽  
C. S. Warke ◽  
S. S. Jha
Keyword(s):  
Magnetic Field ◽  
Electromagnetic Response ◽  
Bloch Electrons

scholarly journals Effective Hamiltonian for a half-filled asymmetric ionic Hubbard chain with alternating on-site interaction

2016 ◽  
Vol 30 (03) ◽  
pp. 1550260 ◽  
Author(s):  
I. Grusha ◽  
M. Menteshashvili ◽  
G. I. Japaridze
Keyword(s):  
Magnetic Field ◽  
Nearest Neighbor ◽  
Effective Hamiltonian ◽  
Heisenberg Chain ◽  
Spin Hamiltonian ◽  
Effective Spin ◽  
One Dimensional ◽  
The One ◽  
Strong Repulsion ◽  
Ionic Hubbard Model

We derive an effective spin Hamiltonian for the one-dimensional half-filled asymmetric ionic Hubbard model (IHM) with alternating on-site interaction in the limit of strong repulsion. It is shown that the effective Hamiltonian is that of a spin S = 1/2 anisotropic XXZ Heisenberg chain with alternating next-nearest-neighbor (NNN) and three-spin couplings in the presence of a uniform and a staggered magnetic field.

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