Role of amplitude fluctuations in the BKT transition Outline - - PowerPoint PPT Presentation

Role of amplitude fluctuations in the BKT transition

Outline

• Motivations
• XY model
• Villain model
• O(2) model
• Traditional FRG picture
• Amplitude and phase

representation

• Universality of 2d Bose

gas

• XY critical temperatures
• Conclusions
• Future perspectives

Motivations

• Topological phase transitions are widely

present in 2d

• Realization of 2d BEC-BCS crossover with

BKT physics (Heidelberg)

• Inconsistency of traditional FRG picture
• Vortex core energy effects in superconductors
• Long range Ising model and Kondo problem

Phase only action: XY model

βHXY = −K X

hiji

[cos (θi − θj) − 1]

Low temperature expansion Small displacements around

βHsw = K 2 Z (rθ)2 d2x.

Non-periodic

θi = θj

Critical phase at all temperatures

Gij = hcos(θi θj)i.

Spin-spin correlation: Power law behavior at all temperatures:

M(x) = e− 1

K G(0) → ML ∝

⇣ a L ⌘

1 2πK

G(x) = e

1 K [G(x)−G(0)] ∝

⇣aπ x ⌘

1 2πK

Single vortex configuration

F = U − TS = (πJ − 2T) log(L/a)

Energy Entropy We expect a proliferation of vortex configuration if

T > πJ 2

Villain model

Pure phase action with quadratic periodic term

S[θ] = X

n.n.

V (θ − θ0) eV (x) = X

m

eV0(x−2πm) V0(x) = −Kv 2 x2

Vortex unbinding Conjecture XY model can be described effectively by villain model with

Kv = f(K)

Spin waves vs vortexes

• Spin waves
• Vortexes

r ⇥ jk = 0

r · j⊥ = 0

rθ = j = j? + jk

I j⊥dl = 2π X

i

qi Z j? · jkdr = 0

βH = βHsw + βHv

Coulomb gas Hamiltonian

Continuous field theory: O(2) model

ϕ = √ρeiθ

Madelung representation

S[ϕ, ϕ∗] = Z d2x ⇢1 2∂µϕ∂µϕ∗ − µ|ϕ|2 + U 2 |ϕ|4

• Frozen amplitude

fluctuations

ρ = ρ0 + δρ δρ ⌧ ρ0

S[ρ, θ] = Z d2x ⇢ 1 8ρ∂µρ∂µρ + ρ 2∂µθ∂µθ − µρ + U 2 ρ2

• S[θ] =

Z d2x nρ0 2 ∂µθ∂µθ

Same universality of the O(2)

HXY = −J X

hiji

(sx,isx,j + sy,isy,j) ,

Transformation Hubbard-Stratonovich

S[ϕ] = Skin[ϕ] + Spot[ϕ]

Skin[ϕ] = 1 2 X

q

ϕqε(q)ϕ−q.

ε(q) = 2(Jd + µ) d − ε0(q) Jε0(q) + µ.

Spot[ϕ] = Z ddx " −U 2 r β J (Jd + µ)ϕ ! + Jd + µ J |ϕ|2 #

Dual Mapping

S[θ] = X

hiji

V (θi − θj)

Sum over first neighbors of a periodic interaction energy Transformation to variables on the dual lattice

Z dx 2π eV (x)−iqx = e

˜ V (q)

S[˜ θ] = X

hiji

˜ V (˜ θi − ˜ θi) + 2π i X

j

qj ˜ θj

Sine-Gordon Model

H = −J X

hiji

cos (θi − θj)

XY Model

H = − X

i6=j

qiqj log

• rj − ri

a

• Coulomb Gas

S = Z ddx {∂µϕ∗∂µϕ + U(ϕ∗ϕ)}

U(1) field theory sine-Gordon S = Z ddx {∂µϕ∂µϕ + u(1 − cos(βϕ))}

Unit charge approximation: sine-Gordon model

S[θ] = Z d2x ⇢1 2∂µ˜ θ∂µ˜ θ + u cos(β˜ θ)

• β = 2π

√ K

∂tKk = −πg2

kK2 k,

∂tgk = π ✓ 2 π − Kk ◆ gk

0.0 0.2 0.4 0.6 0.8 1.0

K

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

u

Functional RG

Exact flow equation for the effective action

scale k ∼ L−1 ∼ N − 1

d :

t = log ✓ k k0 ◆ ultraviolet scale

∂tΓk[ ˜ ϕ] = 1 2Tr ✓ ∂tRk Γ(2) + Rk ◆

k0 ∼ a−1 >> 1 :

Γk0[ ˜ ϕ] = S[ ˜ ϕ] = ⇒ Γk[ ˜ ϕ] = ⇒ Γ[ ˜ ϕ]

k0 > k > 0

k ≡ 0

Momentum shells

0.0 2.5 5.0 7.5 10.0

z

0.0 0.5 1.0 1.5 2.0

Rk

Exponential Power Law Optimized

0.0 2.5 5.0 7.5 10.0

z

0.00 0.25 0.50 0.75 1.00

Gk

Physical Exponential Power Law Optimized

Gk = ⇣ Γ(2) + Rk ⌘−1

Fourier space

Rk ≡ 0 = ⇒ Gk0(q) =

• q2 + m2−1

P.T.

fluctuations

m = 0 IR Regulator ∆ Sk = Z ddq (2π)d ϕ(q)Rk(q)ϕ(−q) Rk(q) 1 if q > k Rk(q) ⌧ 1 if q < k

0.0 0.3 0.6 1.0

˜ κk

5 10 15

˜ λk

FRG Picture

U(ρ) = λk 2 (ρ − κk)2

Smooth Crossover

0.0 0.2 0.4 0.6 0.8

κk

−0.1 0.0 0.1 0.2

η

Vanishing flow

Migdal Approximation

FRG Picture: Madelung representation

Γk[ρ, θ] = Z ddx ⇢ 1 8ρ∂µρ∂µρ + ρ 2∂µθ∂µθ + Uk(ρ)

• 0.0

0.5 1.0 1.5 2.0 2.5 3.0

˜ κk

0.0 0.5 1.0 1.5 2.0 2.5 3.0

˜ λk

Two phases? Irrelevance of the coupling?

NO

Z d2x ⇢ 1 8ρ∂µρ∂µρ + ρ 2∂µθ∂µθ + µρ

• Gaussian should not flow!

Subtraction of Gaussian contribution

0.0 0.5 1.0 1.5 2.0 2.5 3.0

˜ κk

0.0 0.5 1.0 1.5 2.0 2.5 3.0

˜ λk

Minimum Depletion Always relevant interaction Symmetric phase

2d Bose-Gas

Hbg = Z ⇢ ψ†(x) ✓ r2 2m µ ◆ ψ(x) + U 2 ψ†(x)ψ†(x)ψ(x)ψ(x)

• d2x

Quasi-condensation Non degenerate gas

µ T ! nk / e−βεk

Degenerate gas

µ ⌧ T ! nk / T εk + |µ|

Low energy action

S[ϕ, ϕ⇤] = Z d2x ⇢ 1 2m∂µϕ∂µϕ⇤ − µ0|ϕ|2 + U 2 |ϕ|4

Universality at “weak” coupling

• Small quantum renormalization of U can be neglected.
• Critical chemical potential
• Universal variable
• All models share same behavior for
• Superfluid density

µc = mTU π ln ξµ mU X ⌧ 1/mU X = µ − µc mTU ρs = 2mT π f(X)

• Run the amplitude flow with
• Extract the expectation 


value for the field

• Initiate the SG flow
• Extract the renormalized 


superfluid stiffness

FRG routine

K = κr K∗ = ρs κΛ = µ/U U 0(ρ)

• κr

= 0

Looking for

0.0 0.5 1.0 1.5 2.0 2.5 3.0

µ/U

0.0 0.5 1.0 1.5 2.0 2.5 3.0

ρs

6.0e-01 4.2e-01 3.0e-01 1.5e-01 7.5e-02 3.8e-02 1.9e-02

high U

µc

0.0 0.5 1.0

mU

0.75 1.00 1.25 1.50

µc mTU

Logarithmic correction

Universality recovered

0.0 0.5 1.0 1.5 2.0

X

0.0 0.5 1.0 1.5 2.0 2.5

ρs

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

X

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

f(X)

Collapsed curves Average Curve MC Data The universality in the X variable is confirmed. Incorrect large X behavior! Higher derivative of the phase?

Estimation of universal quantities

• Small X behavior:


• Large X behavior:

f(X) = 1 + √ 2κ0X κ0 = 0.61 ± 0.01 θ0 = log ⇣

ξ ξµ

⌘ π = 1.068 ± 0.01 κ0

(FRG) = 0.67 ± 0.07

θ0(FRG) = 1.033 ± 0.032 f(X) ≈ (π/2)θ(X) − 1/4

XY Model

Mean field initial condition: Approximate dispersion:

ε(q) = 2(Jd + µ) d ε0(q) Jε0(q) + µ ' q2 UΛ(ρ) = " − log πI0 2 r β J (Jd + µ)ϕ !! + Jd + µ J |ϕ|2 #

dependence recovered

µ

Optimal choice:

µ = 0

Recovers low temperature expansion

Spin stiffness

0.5 1 π/2 TBKT

T

0.00 0.25 0.50 0.75 1.00

Js(T)

In conclusion the method shows:

• Amplitude fluctuations irrelevant
• Non universal corrections
• Exact BKT features
• “Weak” Universality in 2d Bose gas
• Good estimation for XY critical temperature

Future perspectives

• Non perturbative SG treatment
• Inclusion of lattice dispersion relation
• Interplay between amplitude and phase excitations
• Application to the BEC-BCS crossover

Thank You