(1/2m) is bending energy of vortices

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Classical (London) free energy

• (1/2m) is bending energy of vortices
• U is vortex-vortex (classical) interaction
• V is interaction with columnar pins at positions
  Rn
• h is proportional to transverse field

-this is Feynman path integral for N particles
moving in 2-dimensions (transverse to field
direction) in imaginary time, t -when h0 this is
standard action for interacting bosons - the h
term tends to drive them in a particular
direction corresponding to tilting field relative
to t (pin) direction -this is a rich problem
with many open questions -in particular there is
an interesting competition between tendencies of
vortex lines to be parallel to field and to
pins -the case of a single vortex in a random
pinning potential gives A non-Hermitian version
of the localization problem -increasing h favours
extended states
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• Vortices parallel to a thin film with a single
  pin
• already shows interesting behaviour related to
• well-known (Kane-Fisher) Luttinger liquid
  physics
• electrons in quantum wire with single defect
• New twists
• bosons not fermions
• Non-Hermitian term in action

Quantum many body Hamiltonian is
Here n(x) is the density operator, n?? -there
is a uniform pure imaginary vector potential,
h -we can treat this problem using a phonon
representation for the bosons (Haldane) -du/dx
is density and conjugate field f is phase of ?
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bosonized Lagrangian is
-same bosonized Lagrangian arises starting from
interacting fermions -c is phonon velocity and g
is Luttinger liquid parameter which depends on
density and interactions -in dilute limit g?1
correspondling to non-interacting fermions -with
increasing density g increases or decreases
depending on details of interactions (which must
be repulsive) -we think g may increase then
decrease with increasing density for realistic
vortex-vortex interactions -ignoring h, e0 is
relevant for glt1 and irrelevant for ggt1 -so in
long distance limit we expect e0 to renormalize
to infinity for glt1 corresponding to a very
strongly pinned vortex -we get Friedel
oscillations in density
-this is a quasi-long ranged vortex lattice
caused by a single pin -true vortex lattice cant
form in 2 dimensions without pins
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what is effect of tilting pin? -g1 (free
fermion) case is simplest to study just solve
Non-Hermitean single particle Schroedinger
equation with impurity -Friedel oscillations
arise from interference of incident and
reflected waves -non-Hermitian scattering
wave-functions have exponentially decaying
reflected waves
-leads to exponentially decaying Friedel
oscillations
effectively, the renormalization of the pinning
strength to Infinity is cut off at a length
scale 1/h -this is the spacing of vortices along
the pin direction
1/2h
-similar to commensurate-incommensurate
transition but Involves pinning at a single
location only
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• Imaginary Current
• -tilted field drives bosons in one direction
• The current is J(-i/L)dE0/dh where E0 is
  groundstate energy
• -this is pure imaginary and corresponds
  physically to
• transverse magnetization and also torque required
  to rotate
• field away from pin direction
• -with no pin Jihn0/m
• -the pin reduces this current by temporarily
  delaying vortices
• We define the number of pinned vortices, NP by
• Jih(N-NP)/mL
• NP diverges as h?0 like (n0/h)(1/g-1)ln h for
  relevant pin
• this can be understood from a traffic jam
  picture

particles are stuck in a line-up, like cars at a
toll-booth, waiting to pass by pin (at maxima of
density oscillations) -line-up extends out to
distance 1/2h -interactions just slighly enhance
NP for a relevant pin and decrease it for
irrelevant pin NP?1/h3-2g, ggt1 -torque
measurements on a thin film superconductor with
dilute columnar pins could observe Luttinger
liquid physics!