Sarma phase in relativistic and non-relativistic systems

Transcription

1 phase in relativistic and non-relativistic systems Tina Katharina Herbst In Collaboration with I. Boettcher, J. Braun, J. M. Pawlowski, D. Roscher, N. Strodthoff, L. von Smekal and C. Wetterich arxiv: and arxiv: ERG 214 Lefkada, Greece September 22-26, 214

2 The Phase homogeneous superfluid phase with gapless fermionic excitations [ (1963)] > 2 fermion species with spin imbalance Μ Μ Dispersion relation E (±) p = δµ = µ 1 µ 2 2 ε 2 p + 2 ± δµ [lowest branches] [εp... microscopic dispersion relation;... pairing gap] [Boettcher, TKH, Pawlowski, Strodthoff, von Smekal, Wetterich (214)]

3 The Phase homogeneous superfluid phase with gapless fermionic excitations [ (1963)] > 2 fermion species with spin imbalance Μ δµ = µ 1 µ 2 2 Μ Εpmin Εpmax Dispersion relation E (±) p = ε 2 p + 2 ± δµ [lowest branches] [εp... microscopic dispersion relation;... pairing gap] : > and lowest branch below zero [Boettcher, TKH, Pawlowski, Strodthoff, von Smekal, Wetterich (214)]

4 The Phase homogeneous superfluid phase with gapless fermionic excitations [ (1963)] > 2 fermion species with spin imbalance Μ Μ Εpmin Εpmax Dispersion relation E (±) p = δµ = µ 1 µ 2 2 ε 2 p + 2 ± δµ [lowest branches] [εp... microscopic dispersion relation;... pairing gap] occupation numbers 1.5 SF N fully polarized NF N : > and lowest branch below zero non-monotonous behavior of occupation numbers T > : Fermi surfaces smeared out no sharp distinction crossover [Boettcher, TKH, Pawlowski, Strodthoff, von Smekal, Wetterich (214)]

5 A Simple Criterion for the Phase zero-crossing of the lower branch if δµ > min ε 2 p + 2 p Μ assume: min pε p = = δµ c > c [NB: not valid on BEC-side, where µ < ] Μ Εpmin Εpmax

6 A Simple Criterion for the Phase zero-crossing of the lower branch if δµ > min ε 2 p + 2 p Μ assume: min pε p = = δµ c > c [NB: not valid on BEC-side, where µ < ] Μ Εpmin Εpmax 1 scenario I =δµ Second Order Transition: Condition always fulfilled / δµ c,i [Boettcher, TKH, Pawlowski, Strodthoff, von Smekal, Wetterich (214)]

7 A Simple Criterion for the Phase zero-crossing of the lower branch if δµ > min ε 2 p + 2 p Μ assume: min pε p = = δµ c > c [NB: not valid on BEC-side, where µ < ] Μ Εpmin Εpmax / 1 scenario II =δµ Second Order Transition: Condition always fulfilled First Order Transition: c vs δµ c decides c < δµ c: phase δµ c,ii [Boettcher, TKH, Pawlowski, Strodthoff, von Smekal, Wetterich (214)]

8 A Simple Criterion for the Phase zero-crossing of the lower branch if δµ > min ε 2 p + 2 p Μ assume: min pε p = = δµ c > c [NB: not valid on BEC-side, where µ < ] Μ Εpmin Εpmax / 1 scenario III =δµ Second Order Transition: Condition always fulfilled First Order Transition: c vs δµ c decides c < δµ c: phase c > δµ c: no phase δµ c,iii [Boettcher, TKH, Pawlowski, Strodthoff, von Smekal, Wetterich (214)]

9 A Simple Criterion for the Phase zero-crossing of the lower branch if δµ > min ε 2 p + 2 p Μ assume: min pε p = = δµ c > c [NB: not valid on BEC-side, where µ < ] Μ Εpmin Εpmax / 1 scenario I scenario II scenario III =δµ Second Order Transition: Condition always fulfilled First Order Transition: c vs δµ c decides c < δµ c: phase c > δµ c: no phase δµ c,iii δµ c,ii δµ c,i [Boettcher, TKH, Pawlowski, Strodthoff, von Smekal, Wetterich (214)]

10 Talk Outline 1 Motivation: Phase in a Relativistic System 2 A Potential Non-Relativistic Analog: The Unitary Fermi Gas 3 Phase in the BCS-BEC Crossover 4 Conclusions & Outlook

11 A Relativistic System: Quark-Meson Model at Finite Isospin Chemical Potential [Remember Monday s talks? e.g. W. Weise, B.-J. Schaefer, N. Strodthoff] [F. Rennecke, A. Juricic, N. Khan, J. Luecker,...] [

12 Quark-Meson Model with Isospin Chemical Potential [Kamikado, Strodthoff, von Smekal, Wambach (213) ] 2 flavors of quarks, ψ = (u, d) T, coupled to mesons, σ, π = (π, π +, π ) µ q = µ B /3: imbalance between quarks and antiquarks µ I : imbalance between up and down quarks

13 Quark-Meson Model with Isospin Chemical Potential [Kamikado, Strodthoff, von Smekal, Wambach (213) ] 2 flavors of quarks, ψ = (u, d) T, coupled to mesons, σ, π = (π, π +, π ) µ q = µ B /3: imbalance between quarks and antiquarks µ I : imbalance between up and down quarks BCS-BEC crossover analogy µ I > m π/2: pions condense in a Bose condensate µ I m π: Cooper-pairing of quarks and antiquarks

14 Quark-Meson Model with Isospin Chemical Potential [Kamikado, Strodthoff, von Smekal, Wambach (213) ] 2 flavors of quarks, ψ = (u, d) T, coupled to mesons, σ, π = (π, π +, π ) µ q = µ B /3: imbalance between quarks and antiquarks µ I : imbalance between up and down quarks BCS-BEC crossover analogy 2 effects: chiral symmetry breaking & pion condensation [Kamikado, Strodthoff, von Smekal, Wambach (213)] T[MeV] µ[mev] pion cond pion cond TCP order sigma SB region µ I [m π ]

15 Quark-Meson Model with Isospin Chemical Potential [Kamikado, Strodthoff, von Smekal, Wambach (213) ] 2 flavors of quarks, ψ = (u, d) T, coupled to mesons, σ, π = (π, π +, π ) µ q = µ B /3: imbalance between quarks and antiquarks µ I : imbalance between up and down quarks BCS-BEC crossover analogy 2 effects: chiral symmetry breaking & pion condensation T [MeV] QMiso - MFA CP µ q [MeV] fix µ I = m π > m π/2 (pion condensation possible) vary µ q order parameter: 2 π +π mean field approximation: no bosonic fluctuations

16 Including Fluctuations: Functional Renormalization Group bosonic fluctuations included (in LPA) tu k (χ, ) solve on 2-dimensional grid chiral SB ( χ) & pion condensation ( ) resolve first and second order transitions [Kamikado, Strodthoff, von Smekal, Wambach (213)] T [MeV] QMiso - MFA CP µ q [MeV] T [MeV] CP 5 QMiso - FRG µ q [MeV]

17 Including Fluctuations: Functional Renormalization Group bosonic fluctuations included (in LPA) tu k (χ, ) solve on 2-dimensional grid chiral SB ( χ) & pion condensation ( ) resolve first and second order transitions [Kamikado, Strodthoff, von Smekal, Wambach (213)] T [MeV] QMiso - MFA CP µ q [MeV] T [MeV] QMiso - FRG CP fluctuations strongly modify phase structure two transition branches at low T (first and second order) phase down to T = µ q [MeV]

18 A Potential Non-Relativistic Analog: The Imbalanced Unitary Fermi Gas [cf. talks by I. Boettcher, D. Roscher,...] T [MeV] QMiso - MFA CP very similar! T/µ UFG - MFA CP µ q [MeV] δµ/µ

19 Unitary Fermi Gas ultracold two-component fermions close to a broad Feshbach resonance 2 fermion species, ψ = (ψ 1, ψ 2 ) T bosonization in particle-particle channel: φ ψ 1 ψ 2 [diatomic molecule; Cooper pair] unitary regime: s-wave scattering length diverges, a 1 = ; strongly coupled

20 Unitary Fermi Gas ultracold two-component fermions close to a broad Feshbach resonance 2 fermion species, ψ = (ψ 1, ψ 2 ) T bosonization in particle-particle channel: φ ψ 1 ψ 2 [diatomic molecule; Cooper pair] unitary regime: s-wave scattering length diverges, a 1 = ; strongly coupled T/µ UFG - MFA δµ/µ CP µ > : condensation order parameter: 2 φφ mean field approximation: no bosonic fluctuations phase structure very similar to relativistic model

21 Including Fluctuations: Functional Renormalization Group bosonic order-parameter fluctuations included tu k ( ), tg 2 = η φ g 2 solve flow equation on a grid [g... Feshbach coupling] [Boettcher, Braun, TKH, Pawlowski, Roscher, Wetterich (214)] talk by Dietrich Roscher T/µ UFG - MFA CP δµ/µ.4 T/µ.3.2 UFG - FRG.1 CP δµ/µ

22 Including Fluctuations: Functional Renormalization Group bosonic order-parameter fluctuations included tu k ( ), tg 2 = η φ g 2 solve flow equation on a grid [g... Feshbach coupling] [Boettcher, Braun, TKH, Pawlowski, Roscher, Wetterich (214)] talk by Dietrich Roscher T/µ UFG - MFA CP δµ/µ T/µ UFG - FRG.1 CP δµ/µ Fluctuations: T c down critical imbalance δµ c(t = ) grows agreement with experiment & Monte Carlo [Ku et al. (212), Navon et al. (213)] [Goulko and Wingate (21)]

23 Including Fluctuations: Functional Renormalization Group bosonic order-parameter fluctuations included tu k ( ), tg 2 = η φ g 2 [g... Feshbach coupling] T [MeV] CP solve flow equation on a grid [Boettcher, Braun, TKH, Pawlowski, Roscher, Wetterich (214)] talk by Dietrich Roscher 5 QMiso - FRG µ q [MeV] T/µ UFG - FRG.1 CP δµ/µ Differences to the Relativistic System: NO splitting of transition line NO phase at T = phase SHRINKS

24 Phase in the BCS-BEC Crossover [Gubbels, Stoof (212) ]

25 Phase away from Unitarity Estimate possible: relativistic system slightly on BCS-side study full imbalanced BCS-BEC crossover (a 1 ) at T =

26 Phase away from Unitarity Estimate possible: relativistic system slightly on BCS-side study full imbalanced BCS-BEC crossover (a 1 ) at T = unitarity δµ BCS QCP onset BEC MFA: phase at T = occurs but on BEC-side of the crossover a -1 [cf. Sheehy, Radzihovsky (26), Parish, Marchetti, Lamacraft, Simons (27)]

27 Phase away from Unitarity Estimate possible: relativistic system slightly on BCS-side study full imbalanced BCS-BEC crossover (a 1 ) at T = δµ unitarity BCS QCP onset BEC a -1 [cf. Sheehy, Radzihovsky (26), Parish, Marchetti, Lamacraft, Simons (27)] MFA: phase at T = occurs but on BEC-side of the crossover Quantum Critical Point if transition of second order, there is always a phase!

28 Phase away from Unitarity Estimate possible: relativistic system slightly on BCS-side study full imbalanced BCS-BEC crossover (a 1 ) at T = δµ unitarity BCS QCP onset BEC a -1 MFA: phase at T = occurs but on BEC-side of the crossover Quantum Critical Point if transition of second order, there is always a phase! Impact of fluctuations?

29 Phase away from Unitarity Estimate possible: relativistic system slightly on BCS-side study full imbalanced BCS-BEC crossover (a 1 ) at T = unitarity δµ BCS QCP onset BEC FRG: transition line barely changed onset moves right a -1 [MFA: open symbols; FRG: full symbols]

30 Phase away from Unitarity Estimate possible: relativistic system slightly on BCS-side study full imbalanced BCS-BEC crossover (a 1 ) at T = unitarity δµ BCS QCP onset BEC FRG: transition line barely changed onset moves right QCP moves right a -1 [MFA: open symbols; FRG: full symbols]

31 Phase away from Unitarity Estimate possible: relativistic system slightly on BCS-side study full imbalanced BCS-BEC crossover (a 1 ) at T = unitarity δµ BCS QCP onset BEC FRG: transition line barely changed onset moves right QCP moves right NO on BCS-side! a -1 [MFA: open symbols; FRG: full symbols]

32 A Second Look at the Two Systems rel. non-rel. rel. system: additional SU(2) L SU(2) R chiral symmetry d.o.f. do not match! σ, π, π +, π φ, φ

33 A Second Look at the Two Systems rel. non-rel. σ, π, π +, π φ, φ rel. system: additional SU(2) L SU(2) R chiral symmetry d.o.f. do not match! MF phase structure agrees discrepancies in fermionic sector subleading large difference beyond MFA bosons and their fluctuations crucial!

34 A Second Look at the Two Systems rel. non-rel. σ, π, π +, π φ, φ rel. system: additional SU(2) L SU(2) R chiral symmetry d.o.f. do not match! MF phase structure agrees discrepancies in fermionic sector subleading large difference beyond MFA bosons and their fluctuations crucial! A Proposition non-relativistic system with four fermion species and interactions ] [ Ĥ λ (ψ 1ψ 2) ψ 1ψ 2 + (ψ 3ψ 2) ψ 3ψ 2 + (ψ 1ψ 4) ψ 1ψ 4 + (ψ 3ψ 4) ψ 3ψ 4 same SU(2) SU(2) symmetry as the rel. system phase structure likely similar phase at T = possible

35 Take-home Messages rel. and non-rel. systems for BCS-BEC crossover similar on MF-level but very different beyond bosonic d.o.f. and their fluctuations essential rel. system: non-trivial phase structure at low T (!) non-rel. system: good agreement with experiment and QMC for UFG phase at low T only on BEC-side proposition for a non-rel. system that might be more similar to the rel. one Where to go from here full imbalanced BCS-BEC crossover & QCP mass imbalance [Braun, Roscher] lower dimensions [Boettcher]... Stay Tuned & Thanks!

36 Backup: Quark-Meson Model L QMiso = ψ [Kamikado, Strodthoff, von Smekal, Wambach (213) ] (/ + g(σ + iγ 5 π τ) γ µ q γ τ 3µ I ) ψ ( νσ) ( νπ)2 + U(χ, ρ) cσ ( ν + 2µ I δ ν)π +( ν 2µ I δ ν)π, 2 flavors quark (µ q) and isospin (µ I ) chemical potentials SU(2) L SU(2) R U(1) V symmetry chiral symmetry breaking: χ ψψ pion condensation: 2 = g 2 ρ = g 2 π +π

37 Backup: FRG for the BCS-BEC Crossover L UFG = σ=1,2 ψ σ + φ ( Z φ τ A φ 2 2 species of fermions, σ = 1, 2 [Gurarie, Radzihovski (27); Zwerger (212); Diehl, Wetterich (27)] ) ( ) ( τ 2 µ σ ψ σ + g φ ψ 1ψ 2 + h.c. 2M σ 4M ) φ + ν Λ φ φ. bosonization: φ ψ 1 ψ 2 (particle-particle channel) spin imbalance by different chemical potentials µ 1, µ 2 ν Λ a 1 fine tuned to fix scattering length condensation: 2 = g 2 ρ = g 2 φ φ Renormalization of the Propagators P ψσ,k (iq, q) = iq + q 2 µ σ, ( ) P φ,k (iq, q) = A φ,k iq + q2. 2

38 Backup: Renormalization and the Condition Fluctuations modify chemical potentials and can thus influence the criterion, = δµ. polaron/frg studies of balanced UFG: fluctuations increase µ estimate: µ σ,eff µ σ +.6 µ σ for imbalance: δµ eff = (µ 1,eff µ 2,eff )/2.4 δµ criterion even less likely fulfilled [µ σ... chem. pot. of other species] here: unren. criterion not fulfilled at T = ren. criterion not fulfilled either