Theory of quantum metal to superconductor transitions in highly conducting systems Page: 22 of 46
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H , there must still be a sharp, continuous quantum phase transition at a shifted critical
field, Hel.
Specifically, in the present case, the superconducting domains are regions with magneti-
zation m near 0, and with the local magnitude of the order-parameter, AI, ~ A0, while the
metallic regions have m ~ xHg and miniscule magnitude of the superconducting order.
The volume fraction of the two phases is a function of Hg; it is roughly a 50-50 mixture
when Hg ~ H (0), and the superconducting fraction decreases monotonically with increasing
Hg. However, global phase coherence is not lost at Hg ~ H(0), where on the mean field
level the superconducting fraction first fails to percolate. Rather, as in the other problems
we have examined, it occurs when the Josephson coupling between superconducting regions
becomes sufficiently weak, which in turn occurs when the superconducting fraction is small
and the superconducting regions far separated.
Because the superconducting regions have a characteristic size large compared to 0,
and the magnitude of the order parameter is large, the dynamics of phase fluctuations
is determined by electric field fluctuations, and consequently (according to Eq. 10) x2 ~
AO exp LZ' G2D]-
To determine the distribution of Josephson couplings, we note that in an SNS junction,
when the normal part of the junction is partially spin polarized, [40] the Josephson coupling
oscillates in sign as a function the coordinates. Specifically, at T = 0,
G2DDtr r - r' r -r'
J(r, r') ~ G r exp(- -r ) cos( ),
Ir - r' 2 LHI LHI
J(rr') 2 r 2 (47)
, , 2DIr r - r'
J (r, r') ~ F (r, r') C2 cos ( ) ,-
Ir - r'I ( LHII)
where LHI = Dt r/pH , and F(r,r') is a sample specific function (IFI - 1) which has
random variations both in modulus, and in sign.
The mesoscopic fluctuations of J again dominate the average at distances large compared
to LH . Thus we can estimate the critical magnetic field Hcl at which the zero temperature
phase transition to the metallic phase takes place. For H > H.c, the probability of finding a
superconducting puddle is ~ exp [- (Hg - Hc )2 /272H2 ]. As a result, following the same22
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Spivak, B. Theory of quantum metal to superconductor transitions in highly conducting systems, article, April 6, 2010; United States. (https://digital.library.unt.edu/ark:/67531/metadc930638/m1/22/: accessed May 1, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.