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Detecting quantum duality in experiments: how superfluids become solids
in two dimensionsPhysical Review B 71, 144508 and 144509 (2005),
cond-mat/0502002, cond-mat/0511298
Leon Balents (UCSB) Lorenz Bartosch (Harvard) Anton Burkov (Harvard)
Predrag Nikolic (Harvard) Subir Sachdev (Harvard)
Krishnendu Sengupta (Saha Institute, India)
Talk online at http://sachdev.physics.harvard.edu
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The cuprate superconductors
Cu
O
LaLa2CuO4
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The cuprate superconductors
La2CuO4
Mott insulator: square lattice antiferromagnet
jiij
ij SSJH ⋅= ∑><
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The cuprate superconductors
La2-δSrδCuO4
Superfluid: condensate of paired holes
0S =
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Experiments on the cuprate superconductors show:
• Proximity to insulating ground states with density wave order at carrier density δ=1/8
• Vortex/anti-vortex fluctuations for a wide temperature range in the normal state
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The cuprate superconductor Ca2-xNaxCuO2Cl2
Multiple order parameters: superfluidity and density wave.Phases: Superconductors, Mott insulators, and/or supersolids
T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).
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Distinct experimental charcteristics of underdoped cuprates at T > Tc
Measurements of Nernst effect are well explained by a model of a liquid of vortices and anti-vortices
N. P. Ong, Y. Wang, S. Ono, Y. Ando, and S. Uchida, Annalender Physik 13, 9 (2004).
Y. Wang, S. Ono, Y. Onose, G. Gu, Y. Ando, Y. Tokura, S. Uchida, and N. P. Ong, Science299, 86 (2003).
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STM measurements observe “density” modulations with a period of ≈ 4 lattice spacings
LDOS of Bi2Sr2CaCu2O8+δ at 100 K.M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science, 303, 1995 (2004).
Distinct experimental charcteristics of underdoped cuprates at T > Tc
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Experiments on the cuprate superconductors show:
• Proximity to insulating ground states with density wave order at carrier density δ=1/8
• Vortex/anti-vortex fluctuations for a wide temperature range in the normal state
Needed: A quantum theory of transitions between superfluid/supersolid/insulatingphases at fractional filling, and a deeper
understanding of the role of vortices
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Superfluids near Mott insulators
• Vortices with flux h/(2e) come in multiple (usually q) “flavors”
• The lattice space group acts in a projective representation on the vortex flavor space.
• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.
• These modulations may be viewed as strong-coupling analogs of Friedel oscillations in a Fermi liquid.
Superfluids near Mott insulators
• Vortices with flux h/(2e) come in multiple (usually q) “flavors”
• The lattice space group acts in a projective representation on the vortex flavor space.
• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.
• These modulations may be viewed as strong-coupling analogs of Friedel oscillations in a Fermi liquid.
The Mott insulator has average Cooper pair density, f = p/qper site, while the density of the superfluid is close (but need
not be identical) to this value
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Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integrated from 1meV to 12meV at 4K
100Å
b7 pA
0 pA
Vortices have halos with LDOS modulations at a period ≈ 4 lattice spacings
J. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).
Prediction of VBS order near vortices: K. Park and S. Sachdev, Phys. Rev. B 64, 184510 (2001).
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OutlineOutline
I. Superfluid-insulator transition of bosons
II. Vortices as elementary quasiparticle excitations of the superfluid
III. The vortex “flavor” space
IV. Theory of doped antiferromagnets: doping a VBS insulator
V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor
VI. Predictions for experiments on the cuprates
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I. Superfluid-insulator transition of bosons
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Bose condensationVelocity distribution function of ultracold 87Rb atoms
M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemanand E. A. Cornell, Science 269, 198 (1995)
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Apply a periodic potential (standing laser beams) to trapped ultracold bosons (87Rb)
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Momentum distribution function of bosons
Bragg reflections of condensate at reciprocal lattice vectors
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
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Superfluid-insulator quantum phase transition at T=0
V0=0Er V0=7Er V0=10Er
V0=13Er V0=14Er V0=16Er V0=20Er
V0=3Er
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Bosons at filling fraction f = 1Weak interactions:
superfluidity
Strong interactions: Mott insulator which preserves all lattice
symmetries
M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).
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Bosons at filling fraction f = 1
0Ψ ≠
Weak interactions: superfluidity
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Bosons at filling fraction f = 1
0Ψ ≠
Weak interactions: superfluidity
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Bosons at filling fraction f = 1
0Ψ ≠
Weak interactions: superfluidity
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Bosons at filling fraction f = 1
0Ψ ≠
Weak interactions: superfluidity
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Bosons at filling fraction f = 1
0Ψ =
Strong interactions: insulator
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Bosons at filling fraction f = 1/2
0Ψ ≠
Weak interactions: superfluidity
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Bosons at filling fraction f = 1/2
0Ψ ≠
Weak interactions: superfluidity
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Bosons at filling fraction f = 1/2
0Ψ ≠
Weak interactions: superfluidity
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Bosons at filling fraction f = 1/2
0Ψ ≠
Weak interactions: superfluidity
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Bosons at filling fraction f = 1/2
0Ψ ≠
Weak interactions: superfluidity
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Bosons at filling fraction f = 1/2
0Ψ =
Strong interactions: insulator
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Bosons at filling fraction f = 1/2
0Ψ =
Strong interactions: insulator
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Bosons at filling fraction f = 1/2
0Ψ =
Strong interactions: insulator
Insulator has “density wave” order
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Superfluid-insulator transition of bosons at generic filling fraction f
The transition is characterized by multiple distinct order parameters (boson condensate, density-wave order……..)
Traditional (Landau-Ginzburg-Wilson) view:Such a transition is first order, and there are no precursor fluctuations of the order of the insulator in the superfluid.
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Superfluid-insulator transition of bosons at generic filling fraction f
The transition is characterized by multiple distinct order parameters (boson condensate, density-wave order..........)
Traditional (Landau-Ginzburg-Wilson) view:Such a transition is first order, and there are no precursor fluctuations of the order of the insulator in the superfluid.
Recent theories:Quantum interference effects can render such transitions second order, and the superfluid does contain precursor CDW fluctuations.
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
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OutlineOutline
I. Superfluid-insulator transition of bosons
II. Vortices as elementary quasiparticle excitations of the superfluid
III. The vortex “flavor” space
IV. Theory of doped antiferromagnets: doping a VBS insulator
V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor
VI. Predictions for experiments on the cuprates
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II. Vortices as elementary quasiparticleexcitations of the superfluid
Magnus forces, duality, and point vortices as dual “electric” charges
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Excitations of the superfluid: Vortices
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Excitations of the superfluid: Vortices
Central question:In two dimensions, we can view the vortices as
point particle excitations of the superfluid. What is the quantum mechanics of these “particles” ?
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In ordinary fluids, vortices experience the Magnus Force
FM
( ) ( ) ( )mass density of air velocity of ball circulationMF = i i
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Dual picture:The vortex is a quantum particle with dual “electric”
charge n, moving in a dual “magnetic” field of strength = h×(number density of Bose particles)
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);
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OutlineOutline
I. Superfluid-insulator transition of bosons
II. Vortices as elementary quasiparticle excitations of the superfluid
III. The vortex “flavor” space
IV. Theory of doped antiferromagnets: doping a VBS insulator
V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor
VI. Predictions for experiments on the cuprates
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III. The vortex “flavor” space
Vortices carry a “flavor” index which encodes the density-wave order of the
proximate insulator
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Bosons at rational filling fraction f=p/q
Quantum mechanics of the vortex “particle” in a periodic potential with f flux quanta per unit cell
Space group symmetries of vortex Hamiltonian:
2
1 1 1 4
Magnetic space group: ;
; ; 1
ifx y y x
y x x y
T T e T T
R T R T R T R T R
π
− − −
=
= = =
The low energy vortex states must form a representation of this algebra
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At filling = / , there are species of vortices, (with =1 ), associated with degenerate minima inthe vortex spectrum. These vortices realizethe smallest, -dimensional, representation of the
f p q qq
q
q
ϕ …
magnetic algebra.
Simplest representation of magnetic space group by the quantum vortex “particle” with field operator ϕ
Vortices in a superfluid near a Mott insulator at filling f=p/q
21
2
1
: ; :
1 :
i fx y
qi mf
mm
T T e
R eq
π
π
ϕ ϕ ϕ ϕ
ϕ ϕ
+
=
→ →
→ ∑
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Vortices in a superfluid near a Mott insulator at filling f=p/q
The wavefunction of the vortices in flavor space characterizes the density-wave order
ϕ
( )
ˆ
* 2
1
Density-wave order:
Status of space group symmetry determined by 2density operators at wavevectors ,
: ;
:
qi mnf i mf
mn
i xx
n
y
mn
p m n
e
T
e
q
e T
π π
πρ
ρ ρ
ρ ϕ ϕ +=
=
→
= ∑i
Q
QQ Q
Q
( ) ( )
ˆ
:
i ye
R R
ρ ρ
ρ ρ
→
→
iQQ Q
Q Q
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Vortices in a superfluid near a Mott insulator at filling f=p/q
The excitations of the superfluid are described by the quantum mechanics of flavors of low energy vortices moving in zero dual "magnetic" field.
The orientation of the vortex in flavor space imp
qi
i lies a particular configuration of density-wave order in its vicinity.
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12( + )=
Mott insulators obtained by condensing vortices at f = 1/2
Valence bond Valence bond solid (VBS) ordersolid (VBS) order
Valence bond Valence bond solid (VBS) ordersolid (VBS) order
Charge density Charge density wave (CDW) orderwave (CDW) order
( ) .Can define a common CDW/VBS order using a generalized "density" ieρ ρ= ∑ Q rQ
Q
r
All insulators have 0 and 0 for certain ρΨ = ≠Q Q
C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001) S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
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Mott insulators obtained by condensing vortices at f = 1/4, 3/4
unit cells;
, , ,
all integers
a bq q ab
a b q
×
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Vortices in a superfluid near a Mott insulator at filling f=p/q
The excitations of the superfluid are described by the quantum mechanics of flavors of low energy vortices moving in zero dual "magnetic" field.
The orientation of the vortex in flavor space imp
qi
i lies a particular configuration of VBS order in its vicinity.
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Vortices in a superfluid near a Mott insulator at filling f=p/q
The excitations of the superfluid are described by the quantum mechanics of flavors of low energy vortices moving in zero dual "magnetic" field.
The orientation of the vortex in flavor space imp
qi
i lies a particular configuration of VBS order in its vicinity.
Any pinned vortex must pick an orientation in flavor space: this induces a halo of VBS order in its vicinityi
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OutlineOutline
I. Superfluid-insulator transition of bosons
II. Vortices as elementary quasiparticle excitations of the superfluid
III. The vortex “flavor” space
IV. Theory of doped antiferromagnets: doping a VBS insulator
V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor
VI. Predictions for experiments on the cuprates
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IV. Theory of doped antiferromagnets: doping a VBS insulator
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g = parameter controlling strength of quantum fluctuations in a semiclassical theory of the destruction of Neel order
(B.1) Phase diagram of doped antiferromagnets
La2CuO4
Neel order
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g
La2CuO4
Neel order
VBS order
or
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
(B.1) Phase diagram of doped antiferromagnets
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
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g
Neel order
La2CuO4
VBS order
or
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
(B.1) Phase diagram of doped antiferromagnets
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
Upon viewing down spins as hard-core
bosons, these phases map onto “superfluid”
and VBS phases of bosons at f=1/2
considered earlier
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g
La2CuO4
Neel order
VBS order
or
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
(B.1) Phase diagram of doped antiferromagnets
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
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VBS order
or
g
(B.1) Phase diagram of doped antiferromagnets
La2CuO4
Dual vortex theory of for interplay between VBS order and d-wave superconductivity
Dual vortex theory of for interplay between VBS order and d-wave superconductivity
δHole densityNeel order
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g
(B.1) Phase diagram of doped antiferromagnets
Neel order
La2CuO4
VBS order
δHole density
132δ =
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g
(B.1) Phase diagram of doped antiferromagnets
Neel order
La2CuO4
δHole density
116δ =
VBS order
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g
(B.1) Phase diagram of doped antiferromagnets
Neel order
La2CuO4
δHole density
18δ =
VBS order
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g
(B.1) Phase diagram of doped antiferromagnets
Neel order
La2CuO4
δHole density
d-wave superconductivity above a critical δ
VBS order
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OutlineOutline
I. Superfluid-insulator transition of bosons
II. Vortices as elementary quasiparticle excitations of the superfluid
III. The vortex “flavor” space
IV. Theory of doped antiferromagnets: doping a VBS insulator
V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor
VI. Predictions for experiments on the cuprates
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V. Influence of nodal quasiparticles on vortex dynamics in a d-wave
superconductor
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2
A effective mass
~
where ~ is a high energy cutoff
vF
finite
m vΛ
Λ ∆
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21
sub-Ohmic damping with
Universal function of FF
vC v v− ∆⎛ ⎞= ×⎜ ⎟
⎝ ⎠
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22
Bardeen-Stephen viscous drag with
Universal function of FF
vC v v− ∆⎛ ⎞= ×⎜ ⎟
⎝ ⎠
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22
Bardeen-Stephen viscous drag with
Universal function of FF
vC v v− ∆⎛ ⎞= ×⎜ ⎟
⎝ ⎠
Effect of nodal quasiparticles on vortex dynamics is relatively innocuous.
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OutlineOutline
I. Superfluid-insulator transition of bosons
II. Vortices as elementary quasiparticle excitations of the superfluid
III. The vortex “flavor” space
IV. Theory of doped antiferromagnets: doping a VBS insulator
V. Influence of nodal quasiparticles on vortex dynamics in a d-wave superconductor
VI. Predictions for experiments on the cuprates
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VI. Predictions for experiments on the cuprates
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Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integrated from 1meV to 12meV at 4K
100Å
b7 pA
0 pA
Vortices have halos with LDOS modulations at a period ≈ 4 lattice spacings
Prediction of VBS order near vortices: K. Park and S. Sachdev, Phys.
Rev. B 64, 184510 (2001).
J. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).
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Measuring the inertial mass of a vortex
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Measuring the inertial mass of a vortex
p
estimates for the BSCCO experiment:
Inertial vortex mass Vortex magnetoplasmon frequency
Future experiments can directly d
101 THz = 4 meV
etect vortex zero point motionby
v e
Preliminar
m m
y
ν≈
≈
looking for resonant absorption at this frequency.
Vortex oscillations can also modify the electronic density of states.
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Superfluids near Mott insulators
• Vortices with flux h/(2e) come in multiple (usually q) “flavors”
• The lattice space group acts in a projective representation on the vortex flavor space.
• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.
• These modulations may be viewed as strong-coupling analogs of Friedel oscillations in a Fermi liquid.
Superfluids near Mott insulators
• Vortices with flux h/(2e) come in multiple (usually q) “flavors”
• The lattice space group acts in a projective representation on the vortex flavor space.
• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.
• These modulations may be viewed as strong-coupling analogs of Friedel oscillations in a Fermi liquid.
The Mott insulator has average Cooper pair density, f = p/qper site, while the density of the superfluid is close (but need
not be identical) to this value