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

Role of amplitude fluctuations in the BKT transition

Outline • Motivations • Universality of 2d Bose gas • XY model • XY critical temperatures • Villain model • Conclusions • O(2) model • Future perspectives • Traditional FRG picture • Amplitude and phase representation

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 X β H XY = − K [cos ( θ i − θ j ) − 1] h ij i Low temperature Small displacements expansion around θ i = θ j β H sw = K Z ( r θ ) 2 d 2 x. 2 Non-periodic

Critical phase at all temperatures Spin-spin correlation: G ij = h cos( θ i � θ j ) i . Power law behavior at all temperatures: ⇣ a 1 ⌘ M ( x ) = e − 1 2 π K K G (0) → M L ∝ L 1 ⇣ a π ⌘ 1 2 π K K [ G ( x ) − G (0)] ∝ G ( x ) = e x

Single vortex configuration F = U − TS = ( π J − 2 T ) log( L/a ) Entropy Energy We expect a proliferation of vortex configuration if T > π J 2

Villain model Pure phase action with quadratic periodic term X V ( θ − θ 0 ) S [ θ ] = n.n. V 0 ( x ) = − K v X e V ( x ) = e V 0 ( x − 2 π m ) 2 x 2 m Vortex unbinding Conjecture XY model can be described effectively by villain model with K v = f ( K )

Spin waves vs vortexes r θ = j = j ? + j k • Spin waves • Vortexes r · j ⊥ = 0 r ⇥ j k = 0 I Z X j ⊥ dl = 2 π q i j ? · j k d r = 0 i β H = β H sw + β H v Coulomb gas Hamiltonian

Continuous field theory: O(2) model ⇢ 1 � Z 2 ∂ µ ϕ∂ µ ϕ ∗ − µ | ϕ | 2 + U d 2 x 2 | ϕ | 4 S [ ϕ , ϕ ∗ ] = ϕ = √ ρ e i θ Madelung representation ⇢ 1 � Z 8 ρ∂ µ ρ∂ µ ρ + ρ 2 ∂ µ θ∂ µ θ − µ ρ + U d 2 x 2 ρ 2 S [ ρ , θ ] = Frozen amplitude ρ = ρ 0 + δρ δρ ⌧ ρ 0 fluctuations Z n ρ 0 o d 2 x S [ θ ] = 2 ∂ µ θ∂ µ θ

Same universality of the O(2) X H XY = − J ( s x,i s x,j + s y,i s y,j ) , h ij i Hubbard-Stratonovich Transformation S [ ϕ ] = S kin [ ϕ ] + S pot [ ϕ ] S kin [ ϕ ] = 1 " ! # r + Jd + µ Z β X ϕ q ε ( q ) ϕ − q . d d x | ϕ | 2 S pot [ ϕ ] = 2 J ( Jd + µ ) ϕ − U 2 J q ε ( q ) = 2( Jd + µ ) d − ε 0 ( q ) J ε 0 ( q ) + µ.

Dual Mapping Sum over first neighbors of a periodic interaction energy X S [ θ ] = V ( θ i − θ j ) h ij i Transformation Z dx to variables on the dual ˜ 2 π e V ( x ) − iqx = e V ( q ) lattice S [˜ V (˜ θ i − ˜ q j ˜ ˜ X X θ ] = θ i ) + 2 π i θ j h ij i j

Sine-Gordon Model XY Model Coulomb Gas � � r j − r i X � � X H = − q i q j log H = − J cos ( θ i − θ j ) � � a � � i 6 = j h ij i sine-Gordon Z d d x { ∂ µ ϕ∂ µ ϕ + u (1 − cos( βϕ )) } S = U(1) field theory Z d d x { ∂ µ ϕ ∗ ∂ µ ϕ + U ( ϕ ∗ ϕ ) } S =

Unit charge approximation: sine-Gordon model ⇢ 1 � Z 2 ∂ µ ˜ θ∂ µ ˜ θ + u cos( β ˜ d 2 x S [ θ ] = θ ) √ β = 2 π K 0 . 7 0 . 6 0 . 5 ∂ t K k = − π g 2 k K 2 k , 0 . 4 u ✓ 2 ◆ 0 . 3 ∂ t g k = π π − K k g k 0 . 2 0 . 1 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 K

Functional RG Exact flow equation for the effective action ✓ ◆ ϕ ] = 1 ∂ t R k ∂ t Γ k [ ˜ 2Tr Γ (2) + R k k 0 ∼ a − 1 >> 1 : k ∼ L − 1 ∼ N − 1 scale ultraviolet scale d : Γ k 0 [ ˜ ϕ ] = S [ ˜ ϕ ] = ⇒ Γ k [ ˜ ϕ ] = ⇒ Γ [ ˜ ϕ ] k ≡ 0 k 0 > k > 0 ✓ k ◆ t = log k 0

Momentum shells 2 . 0 Exponential ⌘ − 1 Power Law ⇣ 1 . 5 Γ (2) + R k G k = Optimized R k 1 . 0 Fourier space 0 . 5 q 2 + m 2 � − 1 � R k ≡ 0 = ⇒ G k 0 ( q ) = 0 . 0 0 . 0 2 . 5 5 . 0 7 . 5 10 . 0 z P.T. m = 0 fluctuations 1 . 00 IR Regulator Physical Exponential d d q Z 0 . 75 ∆ S k = (2 π ) d ϕ ( q ) R k ( q ) ϕ ( − q ) Power Law G k Optimized 0 . 50 R k ( q ) � 1 if q > k 0 . 25 R k ( q ) ⌧ 1 if q < k 0 . 00 0 . 0 2 . 5 5 . 0 7 . 5 10 . 0 z

FRG Picture U ( ρ ) = λ k 2 ( ρ − κ k ) 2 15 0 . 2 10 0 . 1 λ k η ˜ 5 0 . 0 − 0 . 1 0 . 0 0 . 3 0 . 6 1 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 κ k ˜ κ k Migdal Smooth Vanishing Approximation Crossover flow

FRG Picture: Madelung representation ⇢ 1 � Z 8 ρ∂ µ ρ∂ µ ρ + ρ d d x Γ k [ ρ , θ ] = 2 ∂ µ θ∂ µ θ + U k ( ρ ) Two phases? 3 . 0 Irrelevance of the coupling? 2 . 5 NO 2 . 0 ⇢ 1 � λ k Z 8 ρ∂ µ ρ∂ µ ρ + ρ 1 . 5 ˜ d 2 x 2 ∂ µ θ∂ µ θ + µ ρ 1 . 0 Gaussian should not flow! 0 . 5 0 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 κ k ˜

Subtraction of Gaussian contribution 3 . 0 Always Minimum relevant Depletion 2 . 5 interaction 2 . 0 λ k 1 . 5 ˜ 1 . 0 Symmetric phase 0 . 5 0 . 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 κ k ˜

2d Bose-Gas � r 2 Z ⇢ ✓ ◆ � ψ ( x ) + U ψ † ( x ) 2 ψ † ( x ) ψ † ( x ) ψ ( x ) ψ ( x ) d 2 x H bg = 2 m � µ Quasi-condensation Non degenerate gas µ � T ! n k / e − βε k T Degenerate gas µ ⌧ T ! n k / ε k + | µ | Low energy action ⇢ 1 � Z 2 m ∂ µ ϕ∂ µ ϕ ⇤ − µ 0 | ϕ | 2 + U d 2 x 2 | ϕ | 4 S [ ϕ , ϕ ⇤ ] =

Universality at “weak” coupling • Small quantum renormalization of U can be neglected. ln ξ µ µ c = mTU • Critical chemical potential π mU • Universal variable X = µ − µ c mTU • All models share same behavior for X ⌧ 1 /mU • Superfluid density ρ s = 2 mT f ( X ) π

FRG routine • Run the amplitude flow with κ Λ = µ/U • Extract the expectation 
 � U 0 ( ρ ) = 0 � � value for the field κ r • Initiate the SG flow K = κ r • Extract the renormalized 
 K ∗ = ρ s superfluid stiffness

Looking for µ c 3 . 0 1 . 50 Logarithmic high U 6.0e-01 correction 4.2e-01 2 . 5 3.0e-01 1.5e-01 2 . 0 1 . 25 7.5e-02 3.8e-02 mTU ρ s 1 . 5 µ c 1.9e-02 1 . 0 1 . 00 0 . 5 0 . 0 0 . 75 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 0 . 0 0 . 5 1 . 0 µ/U mU

Universality recovered 4 . 0 2 . 5 3 . 5 2 . 0 MC Data 3 . 0 2 . 5 1 . 5 f ( X ) 2 . 0 ρ s 1 . 0 Collapsed curves 1 . 5 Average Curve 1 . 0 0 . 5 0 . 5 0 . 0 0 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 X X The universality in the X variable is confirmed. Incorrect large X behavior! Higher derivative of the phase?


 
 
 Estimation of universal quantities • Small X behavior: 
 √ f ( X ) = 1 + 2 κ 0 X κ 0 κ 0 = 0 . 61 ± 0 . 01 (FRG) = 0 . 67 ± 0 . 07 • Large X behavior: f ( X ) ≈ ( π / 2) θ ( X ) − 1 / 4 ⇣ ⌘ ξ log ξ µ θ 0 = = 1 . 068 ± 0 . 01 θ 0(FRG) = 1 . 033 ± 0 . 032 π

XY Model Mean field initial condition: " !! # r + Jd + µ β | ϕ | 2 U Λ ( ρ ) = − log 2 J ( Jd + µ ) ϕ π I 0 J ε ( q ) = 2( Jd + µ ) d � ε 0 ( q ) Approximate dispersion: J ε 0 ( q ) + µ ' q 2 dependence recovered µ Optimal choice: µ = 0 Recovers low temperature expansion

Spin stiffness 1 . 00 0 . 75 J s ( T ) 0 . 50 0 . 25 0 . 00 T BKT 0 0 . 5 1 π / 2 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

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