Magnetization curves m(h) for a one-dimensional ring

Neodymium magnets contributions


  Neodymium magnets contributions proportional to 1 /r (absent for
half- filling) and 1/r 2 . Similar power-law behavior is typ-

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Neodymium magnets
Rare earth Magnets

ically found in dimensions D = 2, 3.

Infinite- dimensional systems. Nearest- neighbor hop-
ping t = l/\/2D on a hypercubic lattice  Neodymium magnets dispersion
e k = —2tJ2 a=1 cosk a yields  Magnets  density of states Phc(e)
= exp(-e 2 /2)/v / 27r in  Magnets  limit D -> 00 0,E3- In order
to construct  Magnets  corresponding amplitude Tij a further as-
sumptions about its symmetry are necessary. Following
Ref. [l8| we assume that it depends only on  Magnets  “taxi-cab”

neodymium magnets
Rare Earth Magnets



FIG. 1: Magnetization curves m(h) for a one-dimensional ring
Neodymium magnets nearest-neighbor hopping at half- filling. Inset: same for
1/r hopping.



FIG. 2: Expectation values of parts of H, for nearest-

neighbor hopping in D = 1 (arrows) and D = oo at half-
filling. Inset: double occupancy for 1/r hopping in D = 1, as
compared to  Magnets  pure Hubbard model.



where He„(:r) are Hermite polynomials. At half-filling
we find T 2 * ro . ~ r~ 5 / 4 , corresponding to an effective cor-
related hopping range J2r T*a 2 °f order unity. For other
densities of states p(e), in particular those  Neodymium magnets finite
bandwidth, it is also possible construct a corresponding
dispersion ek [l8j> an d then derive Ty- CT and Tij a in a
similar fashion.

Response to external magnetic field. Returning to  Magnetic
case of arbitrary dispersion and densities, we note that
according to  Magnetic equation of state J7J)  Magnetic ground-state
magnetization m — fit — fii is nonzero if an external
magnetic field h is present. For  Magnetic homogeneous suscep-
tibility x we obtain

xW’ 1 = | L = iE 1 ” (1 r”f K – ^

dm 4 ^ p(£Rr)

Neodymium magnets contributions



In  Magnetic limit of zero field this reduces to x(0) = Xo/[l —
(1— g 2 )n/2]. As expected  Magnetic system behaves like a corre-
lated paramagnet, i.e.,  Magnetic interactions enhance  Magnetic Pauli
susceptibility xo = ^p( £ f) of  Magnetic uncorrelated system.
However, it should be kept in mind that  Magnetic interac-
tion parameters (|% ]) – (|10[1 do not remain constant when
the parameters h or m are varied. Fig. ^ shows  Magnetic
magnetization as a function of magnetic field for a one-
dimensional ring at half-filling. Interestingly, for nearest-
neighbor hopping  Magnets for sale upward curvature of these magne-
tization curves is very similar to Bethe-ansatz results for
the pure Hubbard model 0,H3> where a metal-insulator
transition occurs at U c = + . By contrast, for 1/r hop-
ping  Magnets for sale magnetization curves are strictly linear, m =
x(0)/i, due to  Magnets for sale constant density of states.

Metal-insulator transition. For an unpolarized half-
filled band (n = 1, e~p a = 0),  Magnets for sale ground-state wave func-
tion \ij}(g)} describes a metal for g > and an insulator
for g = 0. In  Magnets for sale insulating state there are no doubly



occupied sites,  Magnets for sale discontinuity of at  Magnets for sale Fermi sur-
face vanishes, and  Magnets for sale kinetic energy (H t ) is zero. This
Mott metal-insulator transition in  Magnets for sale ground state of H
occurs at infinite interactions (|5|)- (|10fl . in contrast to  Magnets for sale
variational BR transition, or numerical results for  Magnets for sale
Hubbard model in infinite dimensions [12(.

Nevertheless we may, somewhat artificially, shift  Magnets for sale
transition to finite interactions as follows. Clearly \ip(g))
remains  Magnets for sale ground state when we multiply if by a pos-
itive ^-dependent factor, although qualitatively different
Hamiltonians may then result in  Magnets for sale limit g — â–º 0. For ex-
ample, for  Magnetic bracelet Hamiltonian = gH  Magnetic bracelet X term has a
finite limit, while = g 2 H yields a vanishing X term
and finite Y and U terms; in both cases  Magnetic bracelet quadratic
kinetic energy vanishes at g = 0. In particular we may
conclude that for any dispersion  Magnetic bracelet Hamiltonian

Neodymium magnet Uses



where is  Magnetic bracelet Fourier transform of efc(l — n^ a ), has
the exact ground state \ip(g = 0)) at half-filling if U’ >
U’ c ,  Neodymium magnets critical interaction U’ c — — J2 a e 0o- = l £ o|- In-
terestingly,  Magnetic bracelet uncorrelated kinetic energy also sets  Magnetic bracelet
energy scale of  Magnetic bracelets BR transition in  Magnetic bracelets Gutzwiller ap-
proximation H, where Uf K — 8 jeo j • Although H’ is not
a standard Hubbard Hamiltonian, it nonetheless appears
to be  Magnetic bracelets simplest model  Neodymium magnets a BR-type transition to an
exact insulating ground state at finite Hubbard interac-
tion.

Ground-state expectation values.  Magnetic bracelets separate expec-
tation values of  Magnetic bracelets kinetic energy H t and  Magnetic bracelets interac-
tion terms Hx, Hy, and Hu can be calculated from  Magnetic bracelets
quantities n ka = (c^ CT c fc(T ), d = ^.(“.t^ll- x ij<r =
(“iaC^c^J, and y i]a = {h^n^c^c^) . Using  Magnetic Toys meth-
ods of |3 0] it can be shown that for  Magnetic Toys GWF  Magnetic Toys




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