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8/3/2019 Michele Correggi- Vortices in Rotating Bose-Einstein Condensates: A Review of (Recent) Mathematical Results
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Vortices in Rotating Bose-Einstein CondensatesA Review of (Recent) Mathematical Results
Michele Correggi
CIRMFBK, Trento
15/09/2009Mathematical Models of Quantum FluidsUniversita di Verona
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 1 / 35
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Outline
1 General Setting and Background: the Gross-Pitaevskii (GP) Theory
for Rotating Bose-Einstein (BE) Condensates [A].
2 Vortices and Rotational Symmetry Breaking.3 The Thomas-Fermi (TF) Limit of the GP Theory:
Harmonic trapping potentials [IM].
(Strongly) Anharmonic trapping potentials [CY].
Main References
[A] A. Aftalion, Vortices in Bose-Einstein Condensates, 2006.
[CY] M.C., J. Yngvason, J. Phys. A 41 (2008), 445002.
[IM] R. Ignat, V. Millot, J. Funct. Anal. 233 (2006), 260306.
Physics: A.L. Fetter, Rev. Mod. Phys. 81 (2009), 647691.
Numerics: W. Bao, in Dynamics in Models of Coarsening,Coagulation, Condensation and Quantization, 2007.
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General Setting and Background
The GP Theory of a Rotating BE Condensate in a Trap
2d BE condensate rotating along the zaxis with angular velocity .External trap given by a potential V(r) (V(r) as r ).The stationary ground state properties of a rotating BE condensatecan be described through minimizers of the GP energy functional (inthe non-inertial rotating frame).
L = i(xy yx) is the zcomponent of the angular momentum.2 is the coupling parameter ( scattering length).The TF limit is 0 (large coupling and/or fast rotation).
The GP Energy Functional
EGP [] =R2
dr
||2 L + V(r) ||2 + ||
4
2
.
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 3 / 35
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General Setting and Background
Minimization of the GP Functional
EGP
[] =R2
dr iA
2
+
V(r) 2r2
4||2 + |
|4
2
The vector potential is A = ez r/2.GP ground state energy EGP = inf2=1 EGP[].GP stands for any corresponding minimizer.
Boundedness from Below of EGPAssume that V L2loc(R3) and either V(r) r2, < 2, or V(r) rs,
s > 2, as r = EGP
is bounded from below. Usually one considersthe harmonic potential V(r) = r2 (upper bound on !),
(strongly) anharmonic (homogeneous) potentials V(r) rs, s > 2(no upper bound on !): the simplest example is a confinement tothe unitary disc
B1 with Neumann boundary conditions (s =
).
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 4 / 35
G l S i d B k d
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General Setting and Background
Existence of a Minimizer GP
Under the same hypothesis on V, (at least) one minimizer GP.
GP
solves the GP time-independent equation
GP LGP + VGP + 22GP2 GP = GPGP.
The chemical potential GP is fixed by the L2 normalization:
GP
= EGP
+ 2
GP
44.
If V is smooth, the same is GP.
Uniqueness of GP
The minimizer is not necessarily unique.Non-uniqueness is due to the presence of the angular momentumterm. If = 0, EGP is strictly convex in = ||2.Non-uniqueness is strictly related to the occurence of isolated vortices
rotational symmetry breaking.
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 5 / 35
G l S tti d B k d
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General Setting and Background
Ginzburg-Landau vs. Gross-Pitaevskii
[GL] Minimize EGL
w.r.t. u andA with h = curl(
A), hex = (uniform magnetic field),
EGL[u, A] =Ddr
iA u2 + |h hex|2 + 2(1 |u|2)2
.
[GP] MinimizeEGP w.r.t. L2 normalized (r) = (r)u(r),
EGP [] = ETF[] +Sdr
iA
u2 + 2 1 |u|2
.
Main Differences:
In GL there is an additional minimization w.r.t. A, whereas in GP A is
given = major differences in the minimizers but the ground stateenergies can be close in certain regimes.No L2 normalization in GL = no chemical potential in GL = nodensity in GL. Not only a technicality since there is some physicsbehind it!
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Vortices and Rotational Symmetry Breaking
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Vortices and Rotational Symmetry Breaking
Vortices and Rotational Symmetry Breaking
EGP
[] =R2
dr iA
2
+
V(r) 2r2
4||2 + ||
4
2
Rotational Symmetry Breaking
The functionalEGP is invariant under rotations around the z-axis.
If is fixed and large, the GP minimizer is not an eigenfunction ofthe angular momentum (rotational symmetry breaking), due to theoccurrence of isolated vortices.
VorticesGP has a vortex at r0 (0 = x0 + iy0) with winding number d, if
GP(r0) = 0, (locally)GP
|GP
|
0
|
0
|
d.
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 7 / 35
Vortices and Rotational Symmetry Breaking
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Vortices and Rotational Symmetry Breaking
Vortices in a Rotating BE Condensate [Dalibard et al 05]
Formation of quantized vortices in a rotating Rb BE condensate.
Vortices in a fast rotating Rb BE condensate in a quartic+quadratic trap.
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 8 / 35
Vortices and Rotational Symmetry Breaking
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Vortices and Rotational Symmetry Breaking
Why Are Vortices Energetically Favorable?
EGP
[] =B1
dr||2 L + 2 ||4
Energy Compensation
For small rotational velocities the condensate is at rest in the inertial
frame (superfluidity) = |GP
| = const.At higher angular velocities vortices start to occur: For small , avortex of degree d at the origin has the form f(r)exp{id}, with fapprox. constant outside B (= nonlinear term).
Kinetic energyB1\Bdr ||2 d2
B1\B dr
1
r2 Cd2| log |.
Angular momentum B1\B
dr L d.
If
C
|log
|, a vortex can be energetically favorable.
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 9 / 35
Vortices and Rotational Symmetry Breaking
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Vortices and Rotational Symmetry Breaking
Why/When More Vortices?
EGP
[] =B1
dr||
2
L +
2
||4
Energy Optimization
So far we have not justified the breaking of the rotational symmetry,
since a vortex at the origin is an eigenfunction of L!GP can contain more than one vortex (= nonlinearity). fixes the total winding number d (angular momentum) of GP
and, if is sufficiently large, d > 1.
Suppose d = 2: The kinetic energy is d2
= 2 vortices of windingnumber 1 have a smaller kinetic energy (1 + 1 = 2) than 1 vortex ofwinding number 2 (22 = 4), but almost the same angular momentum.
If 1 the vortex cores are small ( ) and the interaction energycan be neglected =
many vortices can be energetically favorable.
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 10 / 35
Vortices and Rotational Symmetry Breaking
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y y g
Rotational Symmetry Breaking
Theorem (Symmetry Breaking [MC,Rindler-Daller,Yngvason 07])As 0, no minimizer ofEGP[] is an eigenfunction of the angularmomentum, if
6| log | + 3 < C
for any constant C R+.
Symmetry breaking is due to occurrence of isolated vortices outside ofthe origin.
The GP minimizer in no longer unique ( degeneracy).The estimate of the symmetry breaking threshold is not optimal (it isexpected to be 2| log | [Aftalion,Du 01]).The rotational symmetry is expected to be broken also for 1.
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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y
Nucleation of Vortices (Harmonic Traps) [Ignat,Millot 06]
EGP [] = R2
dr||2 L + 2V(x, y)||2 + 2 ||4
Harmonic trapping potential (rescaled): V(x, y) = (x2 + 2y2).0 <
1 measures the asymmetry of the potential.
The coefficient 2 of V is chosen so that the first critical velocity isO(| log |) (it is equivalent to rescale all lengths):
1 =(2+1)
2| log |.
The vortex free profile is the (unique L2 normalized) minimizer of
EGP [] =R2
dr||2 + 2V||2 + 2 ||4
.
is real and positive. If = 1, it is also radial.
As
0, 2
TF(x, y) = 12 [
V(x, y)]+ in L
(
D).
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 12 / 35
The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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y
Theorem (Absence of Vortices below 1 [Ignat,Millot 06])
For any > 0, if 1 log | log |, then
GP 0 TF
(x, y) in Lloc(R
2
\ D),EGP = EGP eiS + o(1), where S(x, y) = 212+1 xy,Up to a subsequence (and a global phase factor , || = 1)
GP
0
TF(x, y)eiS,
in H1loc(D) = no vortices, i.e., for any R0 < and sufficientlysmall, GP does not vanish inside the region where x2 + 2y2 < R20 .
Theorem (Occurence of Vortices above 1 [Ignat,Millot 06])
For any > 0, if 1 + log | log | and is sufficiently small, thenGP has at least one vortex atr such thatdist(r, D) C.If in addition 1 + O(log | log |), then the vortex remains closeto the origin, i.e.,
|r
|= o(1).
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Critical Velocities [Ignat,Millot 06]
d
=(1+2)
2(|
log |
+ (d
1)log|
log |) , d
N
Theorem (Number and Distribution of Vortices [Ignat,Millot 06])
For any 0 < 1, if d + log | log | d+1 log | log |, thenFor any R0 < and sufficiently small, GP has exactly d vorticesof winding number 1 atri,, i = 1, . . . , d inside x
2 + 2y2 < R20 ,
Vortices remain close to the origin and close one another, i.e.,
|ri| C1/2, |ri rj| C1/2.Setting i = ri, the configuration (1, . . . , d) minimizes therenormalized energy
W
1, . . . , d
=
i
=j
logi j
+
1 + 2
di=1
x2i +
2y2i
.
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Distribution of Vortices [Gueron,Shafrir 00]
The distribution of vortices is determined by the minimization of W:
If = 1 and d 6, regular polygons and stars (d 1 side regularpolygon plus the origin) centered at the origin are (local) minimizingconfigurations for the renormalized energy W.
If d 11 neither regular polygons nor stars are local minimizers ofW. As d increases the minimizers approach a triangular lattice.
Larger Angular Velocities [Baldo,Jerrard,Orlandi,Soner 08]
If d for any d but = O(| log |), the number of vortices isnot uniformly bounded in .
The vortex distribution minimizes a (rescaled) free boundary problem.
Landau Regime [Aftalion et al 06]
When
1, the occupation of Landau levels become relevant...
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Rotating BE Condensates in Anharmonic Traps[MC,Rindler-Daller,Yngvason 07]
EGP [] =B1dr
iA
2 2r2 ||2
4+
||42
Motivations
There is no upper bound on the angular velocity , i.e., thecondensate is confined for any = one can explore regimes of veryfast rotation.
The unitary disc is the strongest anharmonic trap one can think of
since it can be formally obtained as the limit s of ahomogeneous potential V(r) = rs.Any homoheneous potential V(r) = crs, s > 2 can be mapped to theabove model by means of a suitable rescaling of all lenghts[MC,Rindler-Daller,Yngvason 07].
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 16 / 35
The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Extraction of the TF Density
EGP [] = B1dr iA2
2r2
|
|2
4 + |
|4
2
If 2, the kinetic energy gives a smaller order correction.In anharmonic traps the second part of
EGP depends on .
TF Energy Functional
ETF[] =B1
dr
2
2
2r2
4
,
with ground state energy ETF = inf1=1 ETF [] and (L1 normalized)minimizer
TF(r) = 12 TF +
22r2
4 +.
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Asymptotics of the TF Functional
ETF[] =
B1
dr 2
2
2r2
4
Slow Rotation ( 1)ETF = 12 + O(2) and TF(r) = 1 + O(22).
Rapid Rotation ( 1)ETF = O(2) and TF(r) [C1 + C2r2]+.If > 4/(
), TF(r) = 0 for any r Rin
1 4
(hole).
Ultrarapid Rotation ( 1)ETF = 2/4 + O() and TF(r) = 228
r2 R2in
+
.
Rin = 1
O(11) =
TF approaches (1
r).
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 18 / 35
The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Experimental Observations [Engels et al 03]
Condensate Density
Giant Vortex (Hole) for-mation in a rotating 87RbBE condensate (induced bya laser beam).
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Slow Rotation ( 1)
Theorem (GP Asymptotics [MC,Rindler-Daller,Yngvason 07])If 0 as 0,
EGP =1
2+ O(2),
|GP|2 1
L1(B1)
= O ().
1 is the GS density of the GP functional without rotation (withenergy 12) = the rotation has no leading order effect on theGS asymptotics.
If | log |, the GP minimizer is unique and strictly positive. Inthis case GP 1 as 0 in H1(B1) (no vortex).If | log |, vortices start to occur in GP = rotational symmetrybreaking.
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Occurrence and Distribution of Vortices ( 1)1
2
|log
|The GP minimizer is a unique and a strictly positive radial function.2 2| log | + O(log | log |)
Uniformly bounded (in ) number of vortices at ri, i = 1, . . . , n.n is fixed by the remainder in the angular velocity asymptotics(coefficient of log
|log
|).
Vortices are very close to the origin: |ri| | log |1/2 and|ri rj| | log |1/2. The vortex core is a ball of radius .Vortices arrange in regular polygonscentered at the origin to minimizethe interaction energy.
3
C
|log
|, C > 2
The number of vortices is no longer uniformly bounded.Vortices are confined to a subset ofB1 (free boundary problem).
4 2| log | 1The number of vortices is /2.Vortices with winding number 1 are uniformlydistributed over
B1.
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 21 / 35
The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Vortex Energy Contribution ( 1)
Theorem (Improved Energy Asymptotics [MC,Yngvason 08])
For any | log | 1,
EGP = ETF +| log(2)|
2(1 + o(1)).
Since 1, ETF = 12(1 + o(1)).Vortices of winding number 1 are uniformly distributed over B1, theirnumber is /2 and their core is .Each vortex gives a kinetic contribution of order
|log(2)
|:
2
1/2
dr r1 | log(2)|.No proof of the existence of isolated vortices but only of a uniformdistribution of vorticity satisfying the above properties.
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Rapid Rotation ( 1): Numerical Simulations[Kasamatsu et al 02]
Condensate Density
Hole
The GP minimizer is exponen-tially small in a disc centeredat the origin (hole) and vortices
cover the whole trap.
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 23 / 35
The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Rapid Rotation ( 1)Theorem (GP Asymptotics [MC,Yngvason 08])
For any 1,
EGP = ETF +| log |
2(1 + o(1)),
|GP|2 TFL1(B1) = O | log |.
GP contains a number /2 of vortices with winding number 1uniformly distributed over a regular lattice with spacing
.
The vortex core is a ball of radius .Each vortex gives an kinetic contribution of order | log |:
2
dr r1 | log |.
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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Vortex Energy Contribution ( 1)
GP
TF(r)ei , where the phase minimizes the kinetic
energy and contains the vortices:
ei = i
| i| , =
arctan
y yix xi
.
Kinetic energy A
2
2
= Ar
2
2
, where we have
used the rotation =
log | i|, A Ar.
Electrostatic Analogy
Vortex at ri Unitary Point Charge + 1 at ri(vortex core of size ) (smeared over a ball of size )Vector Potential A Uniform Charge Density
2Vortex Kinetic Energy Energy of the Electric Field
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
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An Electrostatic Problem ( 1) [MC,Yngvason 08]Finding the optimal distribution of vortices is equivalent to minimize
the electrostatic energy of a charge distribution given by positivepoint charges and a negative uniform background.
As long as TF varies on a scale of order 1, the optimal distribution isuniform: Vortices on a regular lattice with fundamental cell Q andspacing
|Q|covering
B1 (triangular, square or hexagonal
lattice).
The cell volume is chosen so that the cell is neutral, i.e., |Q| = 2/.The dipole associated with any cell Qi vanishes because of the cellsymmetry = the electric field Ei generated by Qi decays very fast(at least r
3
) outside Qi = the leading order contribution is
given by the self-energy inside Qi (minimized by the triangularlattice).
Self-energy inside Qi:
Qi
\Bi
dr A
2= | log | + O(1).
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 26 / 35
The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
1
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Slow and Rapid Rotation ( 1): Vorticity
Theorem (Uniform Distribution of Vorticity [MC,Yngvason 08])
Let > 0 be sufficiently small and | log | 1. Then there exists afinite family of disjoint balls
Bi supp(TF) such that1 the radius of any ball is at most of order 1/
,
2 the sum of all the radii is at most of order
,
3
GP C| log(2)|1 on Bi, for some C > 0,and, denoting byri, the center of each ballBi and by di, the windingnumber of |GP|1GP on Bi,
2
di, (rri,) w
0TF(r) dr.
If one could prove that GP has only isolated vortices, then theTheorem would yield their number and uniform distribution.
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
( 1)
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Ultrarapid Rotation ( 1)
The rotational energy dominates: ETF =
2
41 + O
(11)and TF tends to a distribution supported on B1.
Theorem (Energy and Density Asymptotics [MC,Yngvason 08])
For any 1
1
2| log |,
EGP = ETF +| log |
2(1 + o(1)).
The density
|GP
|2 is exponentially small in almost everywhere, except
for a thin layer of width 11 around the boundary B1.
If 2 the occupation of Landau levels becomes relevant (Landauregime). Why the upper bound 1
2
|log
|?
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
( 1) f G
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Ultrarapid Rotation ( 1): Emergence of the GiantVortex
As long as 12| log | there are vortices in the support of
TF
even though it is very thin.
If 1
2
|log
|
one can concentrate the whole vorticity in the center
and lower the energy (giant vortex).
Heuristic Comparison
For 1 the number of cells inside supp(TF) is 1 = themutual interaction is
# of pairs
2.
Vortices (one inside each cell) neutralize the mutual interaction buthave an energy cost | log |.The vortex energy is of the same order of the mutual energy if 2
|log
|1 =
a transition takes place at that threshold.
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The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
N i l Si l i [K l 02]
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Numerical Simulations [Kasamatsu et al 02]
Condensate Density
Giant Vortex
The GP minimizer is concen-trated in a thin annulus nearB1 (giant vortex) and expo-nentially small everywhere elsebut the essential support of
GP contains no vortices.
M. Correggi (CIRM) Vortices in Rotating BE Condensates Verona 15/09/2009 30 / 35
The Thomas-Fermi Limit of the Gross-Pitaevskii Theory
Ul id R i ( 1) Th Gi V
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Ultrarapid Rotation ( 1): The Giant Vortex
Rigorous Comparison [MC,Rougerie,Yngvason 09]
The upper bound EGP ETF + | log |2
(1 + o(1)) holds for any
| log | 2.A trial function of the form giant(r)
TF(r)eiN, with winding
number N /2 yieldsEGP[giant] = ETF + O(2) + O(22| log |)
but 2 | log | in this regime = giant lowers the energy.If > c =
O(2
|log
|1), vortices are expelled from the essential
support of GP.Inside the hole the GP minimizer is exponentially small in andcontains a very large number of vortices, i.e., GP giant onlyinside its essential support
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