Vortex pinning by a columnar defect in planar superconductors with point disorder

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Vortex pinning by a columnar defect in planar
superconductors with point disorder
Anatoli Polkovnikov Yariv Kafri, David Nelson
Department of Physics, Harvard University.
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Plan of the talk

1. Vortex physics in 11 dimension. Mapping to a
   Luttinger liquid.
2. Effects of point disorder. Vortex glass phase.
   Response to a columnar pin.
3. Unzipping of a single vortex from a columnar pin
   and a twin plane with point disorder.
4. Unzipping a single vortex from a two-dimensional
   Luttinger liquid. Revealing the Luttinger liquid
   parameter.

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A single vortex line in a planar superconductor
L?
Free energy
Partition function
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Many vortex lines in a planar superconductor
L?
a
u is the coarse-grained phonon displacement field
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Luttinger liquid parameter
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Point disorder
Random phase,? ??0,2?
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Correlation functions
Vortex liquid phase
1
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Add a columnar pin
Contribution to free energy
Kane-Fisher problem with no disorder
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High-temperature weakly interacting (liquid) phase
Both columnar defect and point disorder are
irrelevant. Thermal fluctuations dominate pinning
and disorder.
Low-temperature strongly interacting (glassy)
phase
Columnar pin and point disorder become relevant.
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Flow diagram.
Columnar pin is always irrelevant !!!
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Friedel oscillations around a columnar pin
(linear response in V)
Slowest asymptotic decay at the vortex glass
transition (g1).
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Free fermion limit, g1
Partition function
The ground state of the N-particle system is the
Slatter determinant of N-highest eigenstates of
the evolution operator
Find eigenstates numerically for a given
realization of disorder by discretizing space and
time.
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Free fermion limit, g1
201 sites, filling factor 0.1
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Extract exponent ? (average over 65536
realizations of point disorder)
RG result
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Response to a weak transverse field.
h
Traffic jam scenario
Np is the number of vortices prevented from
tilting by a columnar pin (pinning number)
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No disorder
I. Affleck, W. Hofstetter, D.R. Nelson, U.
Schollwöck (2004)
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With point disorder
In an infinite sample g1 corresponds to the
strongest divergence of Np with either L or 1/h.
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Unzipping of a single vortex line
f plays a role of a local transverse magnetic
field acting on a vortex
Unzipping transition at the critical force ffc
What are the critical properties of this
transition?
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No disorder, unzipping from a columnar pin
?
N. Hatano, D. Nelson (1997)
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Add point disorder
?
bulk
defect
Relation to anomalous diffusion
D. Huse, C. Henley (1985)
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Dominant disorder on the defect
Fragmented Columnar pin ?1/2
Disordered twin plane ? 1/3, ?2/3
?x
?
Dominant disorder in the bulk
2D ? 1/3
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Disordered columnar pin (?1/2)
D. Lubensky and D. Nelson (2000).
x
Replica calculation
?
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Replica and numerical calculations for a
disordered columnar pin
Replica derivation gives exact result!
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Unzipping from a twin plane (? 1/3)
Agrees with exact numerical simulations.
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General case. Bulk randomness
Effective disorder on the defect due to finite
extent of the localized state.
Asymptotically the main contribution comes from
disorder generated on the defect!!!
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Unzipping from a columnar pin in 2D with bulk
disorder
Finite size scaling
Extract exponent ?1/(1-?) from numerics
Anticipate ?1.5 from bulk part (?1/3), ?2 from
columnar pin part (?1/2).
Effectively have unzipping from a disordered pin
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Critical force versus point disorder in 11d
As expected, there is no unbinding transition in
11d due to point disorder
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Pulling a vortex from a twin plane with an array
of flux lines
Create a dislocation (magnetic monopole) in the
twin plane
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Method of images energy of a dislocation
distance ? from the boundary is equal to the
energy of a dislocation pair of opposite signs.
Schulz, Halperin, Henley (1982)
Compute boson-boson correlation function using
Luttinger liquid formalism.
I. Affleck, W. Hofstetter, D.R. Nelson, U.
Schollwöck (2004)
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Conclusions

1. Columnar pin is always irrelevant in the presence
   of point disorder.
2. The columnar pin is least irrelevant at the
   vortex glass transition (g1).
3. The number of vortices prevented from tilting by
   a columnar pin in a weak transverse magnetic
   field has a maximum at g?1.
4. Point disorder changes critical properties of an
   unzipping transition of a single vortex line from
   an extended defect.
5. Unbinding transition properties from a twin plane
   in the presence of many flux lines drastically
   depends on the Luttinger parameter g.

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Finite size scaling
x
absorbing boundary conditions
Clean case ?1
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