冷却原子気体,並びに固体物質における l off 超流動研究の現状
DESCRIPTION
07.5.10. 東大. 冷却原子気体,並びに固体物質における L OFF 超流動研究の現状. 岡山大学自然科学研究科 町田一成. 共同研究者 市岡優典,水島健,高橋雅裕. Outline. 1) General introduction to cold atom gases: BEC, BCS, and crossover 2) Resonace Fermionic superfluid with mismatched FS’s - PowerPoint PPT PresentationTRANSCRIPT
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冷却原子気体,並びに固体物質における LOFF超流動研究の現状
岡山大学自然科学研究科町田一成
07.5.10. 東大
共同研究者 市岡優典,水島健,高橋雅裕
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Outline1) General introduction to cold atom gases: BEC, BCS,
and crossover2) Resonace Fermionic superfluid with mismatched FS’s Possible realization of Fulde-Ferrell-Larkin-Ovchinniko
v state (FFLO)3) Microscopic calculation; Bogoliubov-de Gennes (Bd
G) equation 4) Topological structure of vortex in FFLO; physics of shift5) Condensed matter systems; superconductivity in Ce
CoIn5-- a heavy Fermion material; Quasi-classical Eilenberger formalism6) Conclusions
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Magnetic confinement
“Laser cooling”
Trapped atomic gasesTrapped atomic gasesNeutral atoms: Li, Na, K, Rb, Cs, Cr, Yb, H, He*
⇒ Hyperfine spin F (e.g., 6Li atoms, F = 9/2, 7/2)
Trapping potentialTrapping potential
⇒ 3-dimensional harmonic trap
Typically, Axial symmetry
Inter-atomic interactionInter-atomic interaction
a: s-wave scattering length (e.g., a = 2.75nm in 23Na)
⇒ By using Feshbach resonance, a → ±∞
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trapping
cooling
imaging
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What are statistics of Alkali atoms?
Alkali Atoms
Why Alkali’s?Strong transitions in optical/near IR:
Easily manipulated with lasers
Nuclear physics:Odd # neutrons + Odd # protons= Unstable
Alkali’s tend to be Bosons: odd p,e even n
Atom Isotope Abundance Half Life
HLiK
2640
0.01%8%0.01%
StableStable109 years
Only Fermionic Isotopes:
Composite Bosons: Made of even number of fermionsComposite Fermions: odd number of fermions
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Condensation in real &
momentum space!
in situ image
Observing statisticsObserving statistics
Hulet et al., Science (2001)
Fermi degeneracy
BosonsBosons FermionsFermions
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New Probes• in situ & TOF image (Density)• RF Spectroscopy (Tunneling current, density of states)• Noise Correlations
Settings• Low Dimension• Rotation• Optical lattices• Ring trap• Chips
Controls• Interactions• Population
States• Vortices (multiply-quantized & coreless vortices)• Soliton• Dipolar BEC• SF-Insulator Transition• Tonks-Girardeau gas & BKT phase• BCS-BEC crossover• Imbalanced Fermionic Superfluid
BEC: Li, Na, Rb, Yb, K, Cs, Cr, He, HFermionic SF: Li, K
Recent ProgressRecent Progress
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Controlling InteractionControlling Interaction
“Feshbach resonance” between two lowest hf states“Feshbach resonance” between two lowest hf states
Scattering is dominated by bound state closest to threshold
Bound state (spin singlet)
En
ergy
Magnetic field: B B0
Spin triplet channel
Bound state energy is shifted relative to continuum
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-1
0
1
834.15650-20
-10
0
10
20x10
3
Magnetic Field: B [G]
Atoms formstable molecules
bound state
Scatterin
g length
a
s-wave scattering length vs Magnetic field: 6Lis-wave scattering length vs Magnetic field: 6Li Zwierlein et al., Nature (2005)
Strong interactionsUniversality? Fraction of molecules?
Strong interactionsUniversality? Fraction of molecules?
← Molecular BEC BCS →
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Dance AnalogyE
(Figures: Markus Greiner)
Tightly bound pairs
Every boy is dancing with every girl: distance between pairs greater than distance between people
Slow Dance
Fast Dance
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Ultra cold atoms by laser cooling
Atomic Bose-Einsteincondensate (sodium)
Molecular Bose-Einsteincondensate (lithium 6Li2)
Pairs of fermionicatoms (lithium-6)
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UniversalityOnly length-scale near resonance is density:
No microscopic parameters enter equation of state
Hypothesis: is Universal parameter -- independent of system
Binding energy: 2 MeV << proton mass (GeV)
Implications: Heavy Ion collisions, Neutron stars
Nuclear matter is near resonance!!
Tune quark masses: drive QCD to resonance
Implications: Lattice QCD calculationsBraaten and Hammer, Phys. Rev. Lett. 91, 102002 (2003)
pion mass (140 MeV)
Bertsch: Challenge problem in many-body physics (1998): ground state of resonant gas
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Calculations
Fixed Node Greens Function Monte CarloJ. Carlson, S.-Y Chang, V. R. Pandharipande, and K. E. SchmidtPhys. Rev, Lett. 91, 050401 (2003)
Linked Cluster ExpansionG. A. Baker, Phys. Rev. C 60, 054311 (1999)
Lowest Order Constrained Variational MethodH. Heiselberg, J. Phys. B: At. Mol. Opt. Phys. 37, 1 (2004)
No systematic expansion
Ladder (Galitskii) approximationH. Heiselberg, Phys. Rev. A 63, 043606 (2003)
Mean field theoryEngelbrecht, Randeria, and Sa de Melo, Phys. Rev. B 55, 15153 (1997)
Resumation using an effective field theorySteele, nucl-th/0010066
Experiments:Duke: -0.26(7) ENS: -0.3 JILA: -0.4 Innsbruck: -0.68(1)
Fixed Node Diffusion Monte CarloG. E. Astrakharchik, J. Boroonat, J. Casulleras, and S. Giorgini,Phys. Rev. Lett. 93, 200404 (2004)
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Superfluidity near resonanceE
B
Molecules:
Bosons -- condense, form superfluid
Atoms:Fermions with attractive interactions -- pair (cf BCS) form superfluid
Theory: continuously deform one into other; BCS-BEC crossover
Superfluidity: Needs bosons which condense
Superfluidity: Needs bosons which condense
Leggett, J. Phys. (Paris) C7, 19 (1980)P. Nozieres and S. Schmitt-Rink, J. Low Temp Phys. 59, 195 (1985)
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Superfluidity near resonanceE
B
B
BCSBEC
All properties smooth across resonance
Pairs shrink
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Fermionic Superfluidity withImbalanced Spin Populations
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ExperimentsExperiments • Zwierlein et al., Science (2006); Nature (2006)• Zwierlein et al., Science (2006); Nature (2006)
TOF images (after expansion)TOF images (after expansion)
100nK100nK50nK50nK
350nK350nK 260nK260nK 190nK190nK
70nK70nK
70nK70nK
SF or Normal ?
Bimodal structure in minority component ⇒ the dense in the central area marks the onset of the condensation!
50-50% BCS ?Dashed lines: normal fermions(TF approximation)
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In situ & reconstructed 3D imagesIn situ & reconstructed 3D images
Columnar densitiesColumnar densities
Absorption images
Empty core! (P < Pc ~ 0.8) ⇒ locally 50-50% BCS pairing
Reconstructed 3D profiles from the integrated 2D distributions
Reconstructed 3D profiles from the integrated 2D distributions
⇒ Only assuming the axial symmetry
Cross section
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Observation of quantum phase transitionObservation of quantum phase transition
Critical population imbalance Pc
Critical difference in “Fermi energies”Critical difference in “Fermi energies”
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Summary of MIT experimentsSummary of MIT experiments
1. Density profile (integrated & cross-section profiles)Bimodal structure ⇒ Observation of fermionic superfluidity with mismatched spin populationPhase separation like profile
2. Direct observation of quantum phase transitionCritical population imbalance ~ pairing gap
Question
Is the density in the superfluid always 50-50%?
⇒ SF-N phase separation or other exotic pairing?
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On resonance a diverges
Only remaining energy scales are EF1 and EF2
Condition for breakdown universal constant ·
will relate EF1 to EF2 and thus pick out a
universal number mismatch for breakdown in a harmonic trap:
= 70(3) %
Universal Physics
The critical imbalance is a measure of the unitary interaction strength!
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A condensate emerges from the Fermi sea
Critical Imbalancec= 71(3)%
Increase atom number of smaller cloud:
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FFLO and -shift physics
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magnetization
(z)
Fulde & Ferrell, PR 135, A550 (1964) Larkin & Ovchinnikov, JETP 20, 762 (1965)
(k↑,-k+q↓): spatially inhomogeneous pairing field
Cooper pairing (k,-k)Cooper pairing (k,-k+q)
Cooper pairing has a non-vanishing center-of-mass momentum q
• Superfluid phase in unequal mixture of two species with mismatched FS’s
Introduction to FFLO
zexp(iqz) FF state
zsin(qz) LO state
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Physics ofshift
Order parameter changes signWhen connecting two ground states
Doubly degenerate ground state
Midgap state
K. Machida and H. Nakanishi, PRB30,122 (1984).
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THEORETICAL FRAMEWORK: Mean-field theory
• Self-consistent condition: Pairing field & particle density
Bogoliubov-de Genns (BdG) equation
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FFLO State in Uniform SystemFFLO State in Uniform System
-phase shift → “ mid-gap states” with zero-energy-phase shift → “ mid-gap states” with zero-energy
Pairing field |(r,z)|
①
① Local Density of States (LDOS): N(z, E)
up-spin down-spin
Zeeman splitting2 = 1.0
Local magnetization: ↑ - ↓
-phase shift
FFLOnodal plane
+(r)
-(r)
-shift
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P= 0.34P= 0.34
BCS
SF
FFLO
N
Ground state at T=0Ground state at T=0
Density (r)
Pairing field (r)
g=-1.5 ⇒ (0)/EF(0) =0.35 and Pc = 0.62
Shell structure!Shell structure!
FFLOnodal plane
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Pairing fieldPairing field
Spatial Profiles of Ground State in finite PSpatial Profiles of Ground State in finite P
Local “magnetization”Local “magnetization”
• Ground state in nonzero P at T = 0 ⇒ Spatially modulated “FFLO-like” pairing state• With increasing P, the area of suppressed polarization shrinks toward the trap center. i.e., the stable region of “BCS” pairing shrinks, BUT, the FFLO oscillation emerges outside region, which allows the coexistence with excess atoms ⇒ NOT simple BCS-Normal Phase separated state!
0(0)/EF(0) = 0.32
Condensation radiiCondensation radii
Locally equal population ⇒ BCS pairingLocally equal population ⇒ BCS pairing
FFLO modulated pairing⇒ SF is still robust!
FFLO modulated pairing⇒ SF is still robust!
Pc
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Depletion throughcondensate!
Difference Profiles on ResonanceDifference Profiles on Resonance
~ Pc = 0.7
Zwierlein et al., Science (2006); Nature (2006)Zwierlein et al., Science (2006); Nature (2006)
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Phase diagram at T = 0Phase diagram at T = 0
BdG in trapped systemBdG in trapped system
“FFLO” ⇒ Pairing state which changes sign“BCS” ⇒ Pairing state having a definite sign
On resonance?
Zwierlein et al., Science ‘06Zwierlein et al., Science ‘06
Critical population imbalance & pair potential⇒ Linear relation in WC limit
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Critical population imbalance at T = 0
Pc = 0.57
Critical temperature at P = 0
c0/ ~ 5.8
(0)/EF(0) =0.32
Generic phase diagram e.g., CDW, SDW, and stripe phase etc. ⇒ Transition from C (BCS) to IC (FFLO) phases
Generic phase diagram e.g., CDW, SDW, and stripe phase etc. ⇒ Transition from C (BCS) to IC (FFLO) phases
Phase diagramPhase diagram
Lifshitz (Leung) pointTL ~ 0.6Tc0
Lifshitz (Leung) pointTL ~ 0.6Tc0
Tc curve for BCS-Normal phase transition obtained from the gap equation
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Superfluidity of a two-component Fermi gas with asymmetric spin densitiesbased on the microscopic theory approaching from the WC towards SC limits.
1. Superfluid state of unequal mixture at T = 0
• The stable region of “BCS” pairing shrinks toward the trap center, with increasing P, while the “FFLO” pairing emerges in the outer region.• The strong suppression of the local “magnetization” ⇒ direct evidence of “superfluidity”.
2. Stable superfluid state in finite T’s and phase diagram
• “FFLO” pairing is favored in large P and low T’s• The T-dependence of Pc(T) is observable in the experiment!
Especially, enhancement of Pc in low T region• Generic phase diagram e.g., (i) Double-phase transition (BCS ⇒ FFLO ⇒ Normal), (ii) Two second-order phase transition lines merge in the L point with TL ~ 0.6 Tc0.
ConclusionsConclusions
50-50% BCS core + FFLO pairing +
surrounded by fully polarized normal cloud
50-50% BCS core + FFLO pairing +
surrounded by fully polarized normal cloud
For the details, Machida, Mizushima, Ichioka, PRL (2006)
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Side imaging
Top imaging
Vortex lattices in rotating Fermionic superflidVortex lattices in rotating Fermionic superflid
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Quantized vortices in imbalanced superfluidQuantized vortices in imbalanced superfluidDirect observation of “superfluidity” in unequal mixture
⇒ Quantized vortices induced by external rotation
P = 1 0.74 0.58 0.48 0.32 0.16 0.07 0
Zwierlein et al., Science 311, 492 (2006)
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Vortex core structure in population imbalance
-----why vortex is visible in balance case ?-----why vortex is invisible in imbalance case ?
BCS ----- OP ----- (r)
BEC ------ OP----- n (r)
vortex visibility or invisibility
M. Takahashi, et al, PRL (2006)
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Caroli-de Gennes-Matricon state & Quantum depletionCaroli-de Gennes-Matricon state & Quantum depletion
Hayashi et al., PRL 80, 2921 (1998); JPSJ 67, 3368 (1998) Hayashi et al., PRL 80, 2921 (1998); JPSJ 67, 3368 (1998)
Continuous 2 phase change around the singularity
Quasiparticles passing through vortex experience the -phase shift
↓Appearance of core-bound state
“Caroli-de Gennes-Matricon (CdGM) state”
-phase shift
Fermi level: EF
Local density of states(LDOS)
Vortexcenter
Lowest CdGM state (A) with finite amplitude at core ⇒ Positive shiftLowest CdGM state (A) with finite amplitude at core ⇒ Positive shift
P = 0
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“Quantum depletion”
Caroli-de Gennes-Matricon state & Quantum depletionCaroli-de Gennes-Matricon state & Quantum depletion
Hayashi et al., PRL 80, 2921 (1998); JPSJ 67, 3368 (1998) Hayashi et al., PRL 80, 2921 (1998); JPSJ 67, 3368 (1998)
Lowest CdGM state ⇒ Discretization & Positive shift⇒ Unoccupied at low T
Total density with vortex at T = 0 (solid line)
vortex core
Vortex center
cf Majorana zero mode when chiral p wave px+ipy
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Reduced quantum depletion inside core in imbalance caseReduced quantum depletion inside core in imbalance case
Vortex core structure at P = 0.3 and 0/EF = 0.32
(1) vortex with imbalance (2) vortex with balance (3) vortex free
Local “polarization”: m(r) = n↑(r) -n↓(r)⇒ Peak at core
DensitiesDensitiesPairing fieldPairing field
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Reduced quantum depletion inside core in imbalance caseReduced quantum depletion inside core in imbalance case
Total densityTotal density
“Core filling factor”
SC
WC
F = n(0)/nmax
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It is difficult to see the quantum depletion in imbalanced caseBut minority component core is still visible in density profile experiment.
Total@balanced case
Majority@P=0.3 Minority@P=0.3
Total@P=0.3
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It is difficult to see the quantum depletion in imbalanced caseBut minority component core is still visible in density profile experiment.
Total@balanced case
Majority@P=0.3 Minority@P=0.3
Total@P=0.3
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SummarySummary
Vortex core structure under population imbalance
Ref: Takahashi, Mizushima, Ichioka, Machida, PRL(2006)
• Core is filled in by majority component ⇒ difficult to see But minority component core is visible in density profile experiment This can be checked by in situ imaging (Zweirlein et al. 2006)• Local polarization shows a peak at vortex core• Splitting of CdGM states due to the resonance with mid-gap states at FFLO node
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Topological structure of a vortex in FFLO superfluidTopological structure of a vortex in FFLO superfluidTM, Ichioka, Machida, PRL 95, 117003 (2005)Ichioka, Adachi, TM, Machida, preprint (2006)
What happens in quasiparticle structure if there exists FFLO nodal plane crossing vortex line ?
What happens in quasiparticle structure if there exists FFLO nodal plane crossing vortex line ?
• FFLO modulation vector // Vortex line
+(r)
FFLOnodal plane
Vortex line
-(r)
“-phase shift”
Splitting of CdGM statesdue to resonance with mid-gap (surface) states
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• FFLO state (2D localization): 2-dimensional (planar) defects⇒ 2D localization of excess atoms at nodal plane
• Vortex state with BCS-pairing: 1-dimensional (line) defects⇒ 1D accumulation of excess atoms at vortex line
• Vortex state with FFLO modulated pairing→ ?
Paramagnetic moment ⇄ Electronic state (local density of states)Paramagnetic moment ⇄ Electronic state (local density of states)
Topological structure of the pair potential in the FFLO state
2 phase winding
Vortex line
phase shiftFFLO nodal plane
Topological structure of a vortex in FFLO superfluidTopological structure of a vortex in FFLO superfluidMizushima, Ichioka, Machida, PRL 95, 117003 (2005)
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FFLO States With a Vortex Line
Missing of local magnetization at the intersection point ?Missing of local magnetization at the intersection point ?
Local Polarization: m = ↑ - ↓Pairing field: 0/EF = 0.1, = 0.5
Quasiparticles crossing the intersection point cannot experience phase shift
-shift
-shift
-shift
Neglecting the background potential: V = 0
VortexFFLO
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Topology of FFLO vortex
0
0
+
-Two dimensional nodal plane: magnetization accumulation due to shift---planar defect m-sheet
vortex line: magnetization accumulation due to shift---line defect m-rod
Intersection point of node and vortex:+ shift non-singularNo bound stateLocalized magnetization absent
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NUMERICAL RESULTS: FFLO states
-phase shift → “ mid-gap states” with zero-energy-phase shift → “ mid-gap states” with zero-energy
Pairing field |(r,z)|
①
① Local Density of States (LDOS): N(z, E)
up-spin down-spin
Zeeman splitting2 = 1.0
Local magnetization: ↑ - ↓
0 = max[(r, z)] = 1.5
↑↓
≠ 0 ≠ 0
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NUMERICAL RESULTS: Vortex states
①
Particle-hole asymmetry at the coreParticle-hole asymmetry at the core
① LDOS: N(r, E) ② LDOS: N(r=0, z=0, E)
②
0
/2
/2
Pairing field |(r,z)|: = 0
q = 1/2
Vortex without FFLO
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Topology of FFLO vortex
Two dimensional nodal plane: magnetization accumulation due to shift---planar defect
vortex line: magnetization accumulation due to shift---line defect
Intersection point of node and vortex:+ shift non-singularNo bound stateLocalized magnetization absent
• Numerical results: 0/ = 0.1, = 0.5
Pairing field |(r,z)|Local magnetization: ↑ - ↓
0
0
+
-
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①
③
②
Pairing field |(r,z)|
② LDOS: N↑(r, E) ① LDOS: N↑(z, E)
FFLO modulation
ordinary vortex
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③ LDOS: N(r=0, z=0, E)
Disappearance of magnetization ← splitting of bound state
Doppler shift
LDOS at the intersection point
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Pauli paramagnetic effect on vortex lattice
Zeeman effect -----> up spin and down spin population imbalance
-------> Fulde-Ferrell-Larkin-Ovchinikov (FFLO)
CeCoIn5
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Phase diagram in H vs T
CeCoIn5
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準古典近似に基づく vortex 描像
Integrate out high frequency partsfrom Gorkov green functions
valid for long wave length description
reliable for quantitative predictions
cf BdG useful, but hard to handle
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Quasiclassical Eilenberger theory
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Free energy with paramagnetic effect
Normal state susceptibility
Average flux density
Internal field distribution
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Self-consistent equation
Pairing potential
Vector potential
Total internal fieldParamagnetic contribution
Diamagnetic (super-current) contribution
Paramagnetic magnetization
Normal state magnetization
B(r)/B0
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YNi2B2C
Nishida, et al, JPSJ 73, 3247 (04)
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CeCoIn5 ---A heavy Fermion supercondutorA. Bianchi et al, PRL91, 187004 (2003).
FFLO
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NMR experiement on CeCoIn5 Kakuyanagi et al, PRL 94, 047602 (2005)
FFLO
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Distribution of magnetization in FFLO vortex
Calculation by BdGCalculation by quasi-classical Eilenberger eq.
M
Normal component
Nodal sheet
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Conclusion Possible observation of FFLO state in resonance Fermion superfluids in cold atoms w
ith unequal populations for two species. Also possibly in CeCoIn5 in high field.
Topological structure of vortex in FFLO Missing of the magnetization at the intersection between nodal plane (sheet-like magnetization) and vortex line (rod-like magnetization)
-phase shift physics:doubly degenerate ground state yields a similar phenomena such as in incommensurate spin density wave states (Cr),
spin Peierls systems (CuGeO3, etc)
Refs. T. Mizushima, K.M. and M. Ichioka, PRL 94, 060404 (2005). T. Mizushima, K.M. and M. Ichioka, PRL 95, 117003 (2005).
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FFLO vortex and spin polarization
vortex line
vortex line
nodal plane
nodal plane
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FFLO in cigar shape geometry modulated spin polarization
bservable by species selective density profile experiment
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Midgap state
Local density of states in FFLO
Midgap state
+
+