Mutual inï¬‚uence of vortices and quasiparticles in high

Mutual influence ofvortices and quasiparticles

in high-temperature superconductorsPredrag Nikolic and Subir Sachdev

[email protected]

Harvard University

Mutual influence of vortices and quasiparticles in high-temperature superconductors – p. 1/30

OverviewQuantum vortices in high-temperature superconductors

introduction

experimental signatures

Vortex dynamics

introduction; microscopic model

mass renormalization and friction

quasiparticle mediated interactions

Quasiparticles near a vortex

electronic LDOS

comparison with experiments


Phenomenology of SuperconductivityBCS theory in metals

attractive interactions between electrons

Cooper pairs =⇒ charged superfluid

BCS: BEC:

Excitations:

quasiparticles plasmons vortices


Phenomena and MotivationApplicability of BCS theory:

conventional superconductors . . . microscopic description

high-temperature superconductors . . . phenomenology

High-Tc puzzles

transport in the “normal” phase

competing orders

vortex core structure

unified picture: quantum vortices

Quasiparticles are the key to vortex dynamics


Crucial Experiment: Nernst EffectVortices move in thermal gradient

Lorentz force gives rise to perpendicular voltage

Vortices exist in the “normal” phase, even at T = 0

Nernst effect

Yayu Wang, Lu Li,N. P. Ong ; Phys. Rev. B

73, 024510 (2006) 0.00 0.05 0.10 0.15 0.20 0.25 0.300

20

40

60

80

100

120

140

La2-x

SrxCuO4

T (

K)

Sr content x

20

500

100

50

10

180

160

onsetT T *

Tc


Quantum Vortices on a LatticeTheory of competing orders near a 2D superfluid-Mott transition

bosons on a lattice, fractional hole doping

duality: vortices on dual lattice, external flux = doping

Hofstadter: degenerate vortex flavors = density waves

T. Hanaguri, et al.Nature 430, 1001 (2004)

L. Balents, L. Bartosch, A. Burkov,S. Sachdev, K. Sengupta

A. Melikyan, Z. Tešanovic


Quantum Vortices and Fermions

Quantum d-wave vortices?

small vortex coresX

light vortices?

nearly frictionless

vortex dynamics?

BCS: vortices are classical

Vortex quantum fluctuations:

resonant scattering of

quasiparticles

sub-gap peaks in LDOSB.W.Hoogenboom, et al.

Phys.Rev.Lett. 87, 267001 (2001)


Quasiparticles and Vortex Dynamics

Conventional BCS superconductors (s-wave)

bound core states (Caroli, de Gennes, Matricon)

large core traps many quasiparticles

=⇒ semi-classical vortex dynamics

High-Tc superconductors (d-wave)

no bound core states (Wang, McDonald; Franz, Tešanovic)

small cores

but, gapless nodal quasiparticles. . .

=⇒ quantum vortex dynamics?


Vortex dynamics inclean d-wave superconductors

(contribution of nodal quasiparticles)


Vortex Dynamics

mv.uv= hρsz × (uv − vs)

︸︷︷︸

Magnus force

−D(uv − vn)︸︷︷︸

quasiparticle friction

−D′z × (uv − vn) −duv︸︷︷︸

impurity friction

+Fext

Fext . . . all external forces, vortex-vortex interactions, etc.

Magnus Force (Galilean invariance):FM = hρsuv


Vortex Mass

Bare “hydrodynamic” mass (any superfluid)

neutral: mv =Evs2 ∝ log

(Rξ

)

charged: mv is finite due to screening

��

��

��

��

��

��

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Fermionic superfluids

quasiparticles play very important role

in s-wave superconductors

vortex mass ≈ total mass of core states

vortex friction ≈ total friction of core states

d-wave: scattering of extended states


Microscopic ModelMain focus:

role of nodal quasiparticles in vortex dynamics

quasiparticle dynamics in presence of a vortex

Ingredients:

a single vortex

nodal quasiparticles: gapless Dirac fermions


Model: Bogoliubov-de Gennes

H = Hv(rv) +∑

nodes

∫

d2rΨ†(r)HBdG(r, rv)Ψ(r)

Linearized Bogoliubov-de Gennes Hamiltonian +

Franz-Tešanovic transformation:

HBdG =

vf(px + ax) v∆(py + ay)

v∆(py + ay) −vf(px + ax)

+ mvfvx

1 0

0 1

Berry phase effects Doppler shift

a(r) =~

2z × r

r· · · (ξ → 0)

v(r) =π~

m

∫

d2k

(2π)2

ik × z

k2

(

1− 11+ λ2k2

)

eikrλ

^z

φξ


Quasiparticle ContributionApproximations:

ignore quasiparticle interactions, disorder. . .

2nd order expansion in vortex displacement from the “origin”

“linear response” to external oscillating force

Main idea:∫

drv(τ)DΨ†DΨe−∫

dτd2rΨ†(∂∂τ+HBdG(rv)

)

Ψ=

∫

drv(τ)e−S v[rv(τ)]

integrate out massive Dirac quasiparticles,

then set their mass to zero

S v =

∫

dω2π

[

F‖(ω)|rv(ω)|2 + F⊥(ω)iz(

r∗v(ω) × rv(ω))]


Summary of ResultsVortex dynamics due to nodal quasiparticles:

F‖(ω) = −η|ω| + A1ω2 ln(|ω|) + mvω

2

2+ A2|ω|3 ; F⊥(ω) = 0

Analytical results with Doppler shift ignored:

η = π6

(

1v2

f+ 1

v2∆

)

T 2 Ohmic dissipation atT , 0

A1 = − ln(2)4

(

1v2

f+ 1

v2∆

)

T ln(T ) anomalous term atT , 0

mv ≈ 0.05(

1v2

f+ 1

v2∆

)

Λ vortex mass atT = 0

A2 . . . a universal function ofvfv∆

sub-Ohmic dissipation

Doppler shift introduces no qualitative changes.


Comparison With SemiclassicsDoppler: T = 0 vortex mass due to quasiparticles:

2 4 6

1

2

3

4

5

αDs( )

αDlog( )

Vortex mass for αD =vFv∆

:

mv =s(αD)

8

(

αD +1αD

)

me

Semi-classical: infra-red divergent vortex mass

infra-red cut-off: inter-vortex separation, mv ∝ 1√B

characteristic length-scale is absent (gapless Dirac qp.)

=⇒ spatial variations of potential are “large”

Kopnin, Volovik


Implications for the “Normal” Statesmall vortex motion damping

small vortex mass (of the order of electron mass)

Consequences:

large vortex quantum fluctuations =⇒ DW order

mostly responding to Magnus force =⇒ flux-flow resistivity

Nernst effect

Yayu Wang, Lu Li,N. P. Ong ; Phys. Rev. B

73, 024510 (2006) 0.00 0.05 0.10 0.15 0.20 0.25 0.300

20

40

60

80

100

120

140

La2-x

SrxCuO4

T (

K)

Sr content x

20

500

100

50

10

180

160

onsetT T *

Tc


Interactions Between Vortices

α

x

y

nodal

Doppler shift due to one vortex presents a

“chemical potential” for Dirac quasiparticles

near the other vortex

Effects:

dissipation to quasiparticles in vortex-vortex scattering

vortex lattice orientation pinned to the substrate

magnetic field dependent dynamics

relevant for flux-flow regime?


Influence ofvortex quantum fluctuations

on nodal quasiparticles


ModelVortex localization:

by neighboring vortices in a vortex lattice

by a pinning impurity

Model: vortex in a harmonic trap: Hv =p2

v

2mv+

12

mvω2vr

2v

Vortex is coupled to Dirac quasiparticles:

H = Hv(rv) +∑

nodes

∫

d2rΨ†(r)HBdG(r, rv)Ψ(r)

Vortex position rv and momentum pv are operators.

Vortex mass mv, and trap frequency ωv are “known” parameters.


Perturbation Theory

Vortex zero-point motion: H0 = ωvb†µbµ +∫

d2rΨ†V0Ψ

Resonant scattering: H1 =∫

d2rΨ†(

Vµb†µ + h.c.)

Ψ + · · ·

quasiparticle propagator

vortex propagator ν

simple scattering ν

2nd order scattering

µ

ν ν

µ

Small parameter:

α =

mvv2f

~ωv

12


Quasiparticle LDOSρ(ǫ, r) ∼ spectral weight in the quasiparticle Green’s function

ρ(ǫ, r) =ωv

~v2f

∞∑

n=0

α2nFn

(

ǫ

~ωv,ǫr~vf

;α

)

Effect of the vortex zero-point quantum motion:

0.5 1 1.5 2 2.5

0.1

0.2

0.3

0.4

=0.005=0.01=0.05=0.1=0.5=1=5

α 2α 2

α 2

α 2α 2

α 2

α 2

ρ 0 ω v/

ω vε /

ω vm vε /

ω vmv

ρ 0

finite LDOS at the origin

no zero-energy peak


Effects of Resonant ScatteringOne-loop correction to LDOS: ρ1(ǫ, r)

main peak

secondary features

discontinuity at ǫ = ωv


Full Quasiparticle LDOSEnergy scans at different radii:

0.5 0.51 11.5 1.52 2

0.10.1

0.2

0.2

0.3

0.3

0.4

0.4

0.5

0.5

0.62α =0.3

ε/ωv

ωv

ρ

2α =1

ωv

ρ

ε/ωv

sub-gap peak due to resonant scattering?

no bound states in d-wave vortex cores...


Further TestsScaling of the sub-gap peak:

0.1 0.2 0.3 0.4

0.2

0.4

0.6

0.8

α2

cε /ωv

0 10 20 30 40 50 600

2

4

6

8

10

12

14

16 1 T 2 T 4 T 6 T YBCO

Eco

re [

meV

]

∆p [meV]

ǫc ∝ αωv ∝√ωv∆0 ǫc ∝ ∆0

measure LDOS discontinuities =⇒ trapping potential

measure vortex size (STM) =⇒ zero-point amplitude

together: determine both ωv spectrum, mv


ConclusionsVortices are quantum particles in clean d-wave superconductors

Nernst effect, CDW, LDOS near vortex cores

microscopic theory of quasiparticle contribution to

vortex dynamics

finite and small vortex mass

dissipation: super-Ohmic at T = 0, Ohmic at T > 0

influence of vortex quantum fluctuations on electronic spectra

no zero-energy peak

sub-gap resonant peaks in LDOS


The Big Picture

Sv =

∫

dτ

[

m0v

2

(

drv(τ)dτ

)2

+idrv(τ)

dτ· ~A0(rv(τ))

]

+i∫

d2rdτAµ(r, τ)Jvµ(r, τ)

SA =∫

d2kdω

8π3

(

18π2ρs

[

k2|Aτ(k, ω)|2 + ω2|Ai(k, ω)|2]

+e∗2k4π

[

δi j −kik j

k2

]

Ai(−k,−ω)A j(k, ω)

)

SΨ = i∫

d2rdτ

{

αµ(r, τ)Jvµ(r, τ) +iπǫµνλaµ∂ναλ

− iΨγµ(∂µ − iaµ)Ψ +ivF

4πρsΨγ0Ψ

(

∂yAτ − ∂τAy

)}


Without Doppler Shift. . .Rescale coordinates and use gauge invariance to make the

Hamiltonian isotropic:

H =

px + ax py + ay

py + ay px + ax

The spectrum is gapless: ǫq,l,k = qk

q = ±1 . . . charge (particle or hole state)l ∈ Z . . . angular momentumk > 0 . . . radial wavevector

Wavefunctions are:

ψq,l,k(r, φ) =1√

4π

√ǫ · J−l+ 1

2(kr)ei(l−1)φ

−iq√ǫ · J−l− 1

2(kr)eilφ

, l < 0

√ǫ · Jl− 1

2(kr)ei(l−1)φ

iq√ǫ · Jl+ 1

2(kr)eilφ

, l > 0


Effects of the CoreSpecial attention is needed for zero angular momentum

there are two square-integrable solutions for ψ

both cannot be included =⇒ over-complete set of states

both cannot be excluded =⇒ incomplete set of states

must introduce a parameter: θ

ψq,0,k(r, φ) =1√

4π

sinθ

√ǫ · J− 1

2(kr)e−iφ

iq√ǫ · J 1

2(kr)

+ cosθ

√ǫ · J 1

2(kr)e−iφ

−iq√ǫ · J− 1

2(kr)

θ captures all details of the vortex core!

if all of flux is inside a finite disc, θ = 0

work with θ = 0: no qualitative changes for θ , 0


Transition Matrix ElementsNo Doppler shift, θ = 0, node p ≈ ~kfx:

U1,2 =

∫

d2rψ†1(r)∇ψ2(r) =18

eiπ4 (q2−q1)

(

σx

vF+ i

y

v∆

)

U1,2

Non-zero only for σ = l2 − l1 = ±1.

U1,2 =

4√

k1k2δ(ǫ2 − ǫ1) −Cσ

(k1k2

)σl− 12 ǫ1+ǫ2√

k1k2Θ (σ(k2 − k1)) , l > σ+1

2

−4√

k1k2δ(ǫ1 − ǫ2) −Cσ

(k1k2

)σl− 12 ǫ1+ǫ2√

k1k2Θ (σ(k1 − k2)) , l < σ+1

2

2σq1q2π

ǫ1+ǫ2ǫ1−ǫ2 +

12πCσ

(k1k2

)σl− 12 ǫ1+ǫ2√

k1k2log

(k1−k2k1+k2

)2, l = σ+1

2

Cσ =

q2 , σ = 1

q1 , σ = −1


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Mutual inï¬‚uence of vortices and quasiparticles in high

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