Universal quantum transport in ultracold Fermi gases How slowly can spins diffuse? Tilman Enss (U Heidelberg) Rudolf Haussmann (U Konstanz) Wilhelm Zwerger (TU München) Aarhus, 26 June 24
Transport in quantum fluids Schäfer & Teaney 29 can mass flow without friction? F = A @v x @y shear viscosity to entropy ratio: expt minimum values /s.4...8 ~/k B kinetic theory: s `mfp ` ~ & O() ~ k B k B s = ~ extrapolated from weak coupling gauge-gravity duality: perfect fluidity Kovtun, Son & Starinets 25 4 k B conjectured as universal lower bound transport near quantum critical point (QCP): incoherent relaxation
Strongly interacting Fermi gas Giorgini, Pitaevskii & Stringari 28 Bloch, Dalibard & Zwerger 28 dilute gas of and fermions, r ` Z H = dx X ~ 2 r 2 µ 2m =",# contact interaction: universal + g " # # " strong s-wave scattering, a ` (Feshbach resonance); scale invariance superfluid of fermion pairs below.2 T c /T F =.67(3) Ku et al. Science 22 Condensation T c Normal Fermi liquid Superfluid Unitarity 2 BCS /(k F a s ) BEC Attraction Randeria, Zwerger & Zwierlein 22
Quantum critical point resonant fixed point is Quantum Critical Point (QCP) at T=, μ=, /a= Nikolic & Sachdev 27 at unitarity /a= Enss PRA 22 abrupt change of ground state at QCP density n is order parameter: vacuum for T=, μ< gapless excitations above QCP: affect measurements in quantum critical regime
2 4 x (µm) 2 ρ (µm) 4 Universal properties at unitarity /a= scale invariance: properties depend only on μ/t ( angle ) Zhang+ Science 22 Figure 3 Constructing the EOS from in situ imaging. The atom cloud shown contains E F = 37 nk at the centre. a, Absorption image of the atomic cloud after quadrant ave averaging produces a low-noise density profile, n versus V. Thermometry is performed EOS (solid blue line), starting with the virial expansion for µ<.25 (green dashed the density profile yields T = 3 nk, and µ =.63. d, Given µ and T, the density profi a 3. b e.g. equation of state n = 3 T f n(µ/t ) measured by Zwierlein group (22), computed using Bold Diagrammatic MC quantum critical regime above QCP: T n /3 (T c. T. T F ) quantum and thermal fluctuations equally important, interplay challenging temperature only available scale for incoherent relaxation: Sachdev 999 n( µ,t) / n ( µ,t) 2.5 2..5. van Houcke+ 22 µ /k B T Figure 4 Equation of state of the unitary Fermi gas in the normal phase. Density n ( density n and the pressure P of a non-interacting Fermi gas, versus the ratio of chem work), red filled circles: experiment (this work). The BDMC error bars are estimated u deviation systematic plus statistical errors, with the additional uncertainty from the Fe red solid lines. Black dashed line and red triangles: Theory and experiment (this work) error. Green solid line: third order virial expansion. Open squares: first order bold diag superfluid transition point from Determinental Diagrammatic Monte Carlo 3. Filled di pressure EOS (ref. 23). ~ = O() k B T ) s = 2 5 T.7 ~ (µ = ) k B NATURE PHYSICS ADVANCE ONLINE PUBLICATION www.nature.com/naturephysics 2 large-n exp n Enss 22
Luttinger-Ward approach repeated particle-particle scattering dominant in dilute gas: self-consistent T-matrix Haussmann 993, 994; Haussmann et al. 27 self-consistent fermion propagator (3 momenta / 3 Matsubara frequencies) spectral function A(k,ε) at Tc works above and below Tc; directly in continuum limit Tc=.6() and ξ=.36() agree with experiment Haussmann et al. 29 conserving: exactly fulfills scale invariance and Tan relations Enss PRA 22
Transport in linear response shear viscosity from stress correlations (cf. hydrodynamics), (!) =! Re Z ˆ xy = X p, dt e i!t Z p x p y @v m c p c x p @y with stress tensor (cf. Newton ) correlation function (Kubo formula): Enss, Haussmann & Zwerger, Annals Physics 2 d 3 x D ˆ xy (x,t), ˆ xy (, ) E η(ω) = (resummed to infinite order) transport via fermions and bosonic molecules: very efficient description, satisfies conservation laws, scale invariance and Tan relations Enss PRA 22 assumes no quasiparticles: beyond Boltzmann kinetic theory, works near Tc; includes pseudogap and vertex corrections
High-energy tails in stress correlation (shear viscosity) exact viscosity sum rule (nonperturbative check): 2 Z η(ω) [ h n] d! [ (!) tail] = P hydrodynamics... C 4 ma P +(! ) 2 C 5 p m!. ω [E F / h] T= T= 5 T= 2 T= T=.5 T=.6 Enss, Haussmann & Zwerger 2 Taylor & Randeria 2; Enss, Haussmann & Zwerger 2; Enss 23
η/s [ h/k B ] superfluid phonon contrib. Luttinger-Ward theory kinetic theory phonon classical limit contribution T 8 agrees with large-n transport calculation Enss PRA 22 T 3/2 lowest friction of any nonrelativistic quantum fluid T c. T/T F cf. theory: Bruun, Massignan, Schäfer, Smith,... experiment: Cao+ Science 2 Shear viscosity/entropy of the unitary Fermi gas Enss, Haussmann & Zwerger 2
Spin transport with ultracold gases experiment: spin-polarized clouds in harmonic trap bounce! picture: J. Thomas 2 strongly interacting gas [movie courtesy Martin Zwierlein]: A.T. Sommer, M.J.H. Ku, G. Roati, M.W. Zwierlein, Nature 472, 2 (2)
Spin diffusion scattering conserves total + momentum: mass current preserved but changes relative - momentum: spin current decays c Position (mm).3.2....2.3 Sommer et al. 2 spin diffusion 4 8 2 6 4 8 Time (ms)
Spin conductivity and spin susceptibility use Einstein relation D s = s s spin conductivity: spin susceptibility: σ s me F / hn superfluid Luttinger-Ward theory classical gas kinetic theory w/o medium χ n,s / χ. superfluid χ n Sommer et al. (2) χ s Sommer et al. (2) χ n Luttinger-Ward χ s Luttinger-Ward free Fermi gas T c. T/T F T c. T/T F Enss & Haussmann PRL 22 medium effects important: large-n transport calculation Enss PRA 22 in two dimensions Enss, Küppersbusch, Fritz PRA 22 related work on spin transp.: Bruun 2; Duine et al. 2; Mink et al. 22, 23
Spin diffusivity obtain diffusivity from Einstein relation, D s = s s superfluid Sommer et al. (2) Luttinger-Ward theory kinetic theory (experiment rescaled from trap to infinite homogeneous box) md s / h minimum D s '.3 ~ m. T c T/T F Enss & Haussmann PRL 22 Quantum Monte Carlo simulation for finite range interaction: Wlazlowski et al. PRL 23 D s &.8 ~ m
Longitudinal vs transverse spin diffusion ' : J I ' ' ' ' ' ' ' ' I ' spin polarized x @ x+dx M M - dm r b / x x+dx J : i i I i i i i : i i i I lo T(mK) Mullin & Jeon 992 M(x) M(x+dx)
Spin-echo experiment (Thywissen group, Toronto) M? (t) = i exp[ t 3 /24 3 ]. A (ms) 2.5.5 M.8.6.4.2 2 3 4 Hold time t (ms) B 5 5 2 25 3 B-field gradient B (G/cm) Bardon+ Science 334, 722 (24) 8 D s ( /m) 6 4 2 2D: Koschorreck et al. Nature Physics 23.2.4.6.8 Temperature (T/T F ) i
Spin diffusion in kinetic theory ( ) local magnetization vector and gradient M(r,t) =M(r,t) ê(r,t) Boltzmann equation for spin distribution function Dσ p Dt + i σ p t transverse i v pi M r i ê σ v pi ê r i (n p+ n p )+Ω σ p = M r i = M r i ê + M ê r i longitudinal n pσ t σ ε p ( ) σp t spin rotation coll Landau 956, Silin 957; Leggett & Rice 968-7; Lhuillier & Laloë 982; Meyerovich 985; Jeon & Mullin 988, 992 many-body T-matrix in collision integral and spin rotation Enss 23 derived as leading order in large-n expansion Enss 22
Spin-rotation effect diffusion equation: Leggett 97; Jeon & Mullin 988; Enss 23 M + (r,t)=m x + im y : @M + @t ' i (r)m + + D? ( + iµm z )r 2 M + Leggett-Rice spin-rotation effect: complex diffusion constant, spin waves; spin current precesses around effective molecular field: spin-rotation parameter µ = Ω mf τ Enss PRA 23
Transverse spin diffusivity (3D) r b ' : J I ' ' ' ' ' ' ' ' I ' B D s ( /m) 8 6 4 2 md / h (3D) J : i i I i i i i : i i i I lo T(mK) long/trans M= trans M=. trans M=.3 trans M=.6 trans M=.9.. T/T F md / h (3D) at M=.999 5 2.5.2.2.4.6.8 Temperature (T/T F ) i Bardon+ Science 334, 722 (24) no medium with medium medium & spin rotation... T/T F using vacuum (two-body) scattering cross section: similar to helium case medium (finite density) scattering and spin-rotation effect: diffusivity much smaller! Enss PRA 23
. Transverse spin diffusivity (2D) md / h (2D) at ln(k F a)= long/trans M= trans M=. trans M=.3 trans M=.6 trans M=.9.. T/T F md / h (2D) at M=.999 5 2. ln(k F a 2D ) Koschorreck et al. 23.5 T=. T=., med.2 T=.5 T=.5, med. -2 - ln(k F a) 2 vacuum scattering Enss PRA 23 dependence on interaction and importance of medium effects cf. η: Enss, Küppersbusch & Fritz PRA 22 trap: Chiacchiera, Davesne, Enss & Urban, PRA 23
Conclusion and outlook lowest friction at strong interaction: almost perfect fluidity near QCP Luttinger-Ward: Enss, Haussmann & Zwerger, Ann. Phys. 326, 77 (2) large-n: Enss, PRA 86, 366 (22) quantitative understanding of spin diffusion: Luttinger-Ward transport calculation (tail, near Tc) slowest longitudinal spin diffusivity Enss & Haussmann, PRL 9, 9533 (22) transverse spin diffusion: D can be much lower than D in degenerate, polarized gas; Leggett-Rice spin-rotation effect Enss, PRA 88, 3363 (23) on c D s &.3 ~/m outlook: spin-rotation effect (Thywissen group) 2D Fermi gas: EoS and pseudogap Bauer, Parish & Enss, PRL 2, 3532 (24) md s / h md / h (2D) at M=.999 5 2 superfluid T c Sommer et al. (2) Luttinger-Ward theory kinetic theory. T/T F.5 T=. T=., med.2 T=.5 T=.5, med. -2 - ln(k F a) 2 @
BKT-BCS crossover in 2D Fermi gas Bauer, Parish & Enss, PRL 2, 3532 (24) Pressure EoS (vs Turlapov data): P/P..9.8.7.6.5 T =.2 T F T =.5 T F.4 T =.T F.3 Data from Ref. [] QMC T=.2 2 3 ln(k F a 2D ) 4 5 Phase diagram (Tc, T*): T/TF.4.2..8.6.4 T from Luttinger-Ward T c from Luttinger-Ward mean-field BCS.2 BCS+GMB from Ref. [2] BKT from Ref. [2]. 4 2 ln(k F a 2D ) 2 4 Tan contact density (vs Köhl data): C/k 4 F 2 Luttinger-Ward weak coupling limit Data from Ref. [3] T =QMC.5 5 x =/ ln(k F a 2D ) Spectral function/pseudogap: ρ(ω)/ρ..8.6.4.2. ω/εf 4 2 2 log (A(k, ω)) -.5 -. -.5 2 k/k F T/T F =.45 T/T F =.2 T/T F =. T/T F =.7 5 4 3 2 2 ω/ε F