Superfluid Hydrodynamics, Thermal Partition Function and Lifshitz - PowerPoint PPT Presentation

Superfluid Hydrodynamics, Thermal Partition Function and Lifshitz Scaling Crete Center for Theoretical Physics Shira Chapman, Carlos Hoyos, Yaron Oz arXiv: 1310.2247, 1402.2981 Tel Aviv University February 20, 2014

Outline ◮ Superfluids and Superconductors. ◮ Relativistic Superfluid Dynamics. ◮ Chiral Terms in Superfluids. ◮ Kubo Formulas From Equilibrium Partition Function. ◮ Lifshitz Scaling Symmetry. ◮ Quantum Critical Points. ◮ Experimental Implications.

Superfluidity and Superconductivity ◮ Pyotr Leonidovich Kapitsa - 1938 1978 - Nobel prize for discovery of superfluidity in 4 He ◮ Liquifies at T ≈ 4 . 2 ◦ K ◮ At T ≈ 2 . 17 ◦ K - Second order phase transition.

◮ Phase diagram of 4 He ◮ Remains liquid at absolute zero ◮ Condensate of atoms in ground state - Collective mode

◮ The λ point ◮ Heat conductivity increase by a factor of 10 6 ◮ Large part of atoms in ground state - Condensate - collective mode

The Two Fluid Picture ◮ Viscosity vs. no viscosity ◮ Effective picture - two fluid flows � v n , � v s ◮ Due to Landau

◮ Superfluid component carries no entropy

Superconductivity ◮ Heike Kamerlingh Onnes - 1911 1913 - Nobel prize for liquifying helium ◮ Zero resistance to DC current. ◮ Meissner effect (1933). ◮ Condensation of Cooper pairs. ◮ Two fluid picture - paired electrons form superfluid.

Relativistic Superfluids ◮ Spontaneous symmetry breaking ◮ S-wave superfluid - condensate - complex scalar operator ◮ Phase of scalar φ - Goldstone mode - participates in the hydrodynamics. ◮ Fluid variables: T - temperature, µ - chemical potential, u µ - normal fluid 4-velocity, ξ µ = − ∂ µ φ - Goldstone phase gradient, u µ s = − ξ µ /ξ - Superfluid velocity. ◮ Superconductors - broken gauge symmetry - ξ µ ≡ − ∂ µ φ + A µ .

Relativistic Superfluids ◮ New thermodynamic parameters ξ µ . ◮ Thermodynamic relations: ε n + P = sT + q n µ dP = sdT + q n d µ + f 2 d ξ 2 ◮ Stress tensor and current: T µν = ε n u µ u ν + P ( η µν + u µ u ν ) + ε s u µ s u ν s + π µν J µ = q n u µ + q s u µ s + j µ diss ◮ Josephson relation: u µ ξ µ = µ + µ diss .

◮ Hydrodynamic equations - conservation law: ∂ µ T µν = F νµ J µ ∂ µ J µ = CE µ B µ ∂ µ ξ ν − ∂ ν ξ µ = F µν ◮ Electric field: E µ = F µν u ν ◮ Magnetic field: B µ = 1 2 ǫ µνρσ u ν F ρσ ◮ C - triangular anomaly of three currents. ◮ Goal: constrain expressions for π µν , j µ diss , µ diss .

Chiral Effects in Superfluid ◮ Local second law - Constrain current and conductivities J µ = q n u µ + q s u µ s + σ E µ + B µ ( C µ + 2 Tg 1 ) + ω µ ( C µ 2 + 4 g 1 µ T − 2 g 2 T 2 ) + . . . π µν = ησ µν + ζ∂ µ u µ + . . . g 1 = g 1 ( T , µ, ξ 2 ); g 2 = g 2 ( T , µ, ξ 2 ). ◮ Vorticity: ω µ = ǫ µνρσ u ν ∂ ρ u σ ◮ Shear tensor: σ µν = P ρ ν ∂ ( ρ u σ ) − 1 µ P σ 3 P µν ∂ ρ u ρ . ◮ ◮ Comparison to normal fluid

◮ C - triangular anomaly of three currents. ◮ In the normal fluid g 2 integration constant - related to mixed chiral-gravitational anomaly. [Yarom: 1207.5824] � C β � ∂ µ J µ ∼ ǫ µνρσ 32 π 2 R α βµν R β 8 F µν F ρσ + . αρσ ◮ Numerical evidence that at T → 0 one restore the normal fluid values [Amado: 1401.5795] ◮ Chiral effects - 3 He , Neutron stars.

Equilibrium Partition Function ◮ Alternative method to derive hydrodynamic current ◮ Minwalla et al. - 1203.3544, Yarom et al. - 1203.3556 ◮ Consider equilibrated fluid on a curved manifold with non-trivial gauge fields x ) dx i � 2 + g ij ( � ds 2 = − e 2 σ ( � x ) � x ) dx i dx j , dt + a i ( � x ) dx 0 + A i ( � x ) dx i , A = A 0 ( � ◮ KK invariant gauge field: A 0 ≡ A 0 + µ 0 , A i ≡ A i − A 0 a i , ◮ Local temperature T ( � x ) = T 0 e − σ ◮ Local chemical potential µ ( � x ) = A 0 e − σ

Equilibrium Partition Function ◮ Build the most general equilibrium partition function [effective action] S = S 0 + S 1 , � d 3 x 1 T P ( T , µ, ˆ ζ 2 ) , S 0 = � d 3 x ˆ S 1 = ζ · ( g 1 ∂ × A + Tg 2 ∂ × a ) � µ 3 T ∂ × A + µ 2 � � d 3 x A · + C 6 T ∂ × a + . . . ◮ ˆ ζ ≡ − ∂ i φ + A i , transverse [spatial] part of goldstone field ◮ Differentiate with respect to the gauge field to obtain the current ◮ Advantage - algebraic rather then differential ◮ Disadvantage - only captures equilibrium properties

Linear Response Theory ◮ Relates transport coefficients to retarded correlation function of stress tensors and currents in terms of Kubo formulas ◮ Allow for a microscopic calculation e.g. Feynmann diagrams ◮ Deriving Kubo formulas - normally requires to solve the the equations of motion for a particular source of perturbation ◮ Alternative shorter algebraic method - from variations of the equilibrium partition function ◮ Reproduces known Kubo formulas for various fluid cases ◮ New Kubo formulas for superfluids

Results ◮ Kubo formulas - � µ i k � ζ − C � g 1 ( T , µ, ζ 2 ) = − lim � 4 Tk n ǫ ijn � J i ( k n ) J j ( − k n ) � ω =0 , 2 T k n → 0 ij � µ �� i − C � 2 � g 2 ( T , µ, ζ 2 ) = lim � � J i T 0 j � − µ � J i J j � � 2 T 2 k n ǫ ijn � � ω =0 2 T k n → 0 ij k � ζ ◮ Spatial superfluid velocity - new thermal parameter - ζ i . ◮ Similar role to chemical potential in the spatial direction. ◮ Substitution rules in propagators - q µ → ( i ω n + µ,� q + � ζ ). ◮ Holographic calculation also possible.

Lifshitz Superfluids - Quantum Critical Points ◮ Anisotropic Weyl - Lifshitz scaling symmetry: t → Ω z t x i → Ω x i z - dynamical critical exponent ◮ Must be accompanied by broken boost invariance ◮ Phase transition at zero temperature ◮ Driven by quantum fluctuations ◮ Quantum tuning parameter [ B , doping, pressure] ◮ First and Second order transition ◮ Infinite correlation length - scale invariance ◮ Hydrodynamic regime - l c ≫ L ≫ l T

Quantum Criticality ◮ Influence of quantum critical point felt way above T = 0. Strange Metal T SC g QCP ◮ Example: anti-ferromagnetic → heavy fermion metal transition. ◮ Strange metal behavior ρ ∼ T ( ∼ T 2 in normal metals) ◮ Characteristic of high T c superconductors in the non-superconducting regime.

Hydrodynamics with broken boost invariance ◮ Under lorentz transformations δ L = T µν ω µν ω µν antisymmetric parameter of Lorentz transformation ◮ Asymmetric stress tensor in time direction. ◮ Assumption - fluid can be described using former variables. ◮ No need of external time vector [phonons] ◮ Fluid velocity in the local rest frame points in the time direction ◮ Antisymmetric part of stress tensor: T [ µν ] = u [ µ V ν ] A

Constitutive relations ◮ Stress-tensor: T µν = ( ε n + p ) u µ u ν + p η µν + ε s u µ s + π ( µν ) + π [ µν ] s u ν . A ◮ Choice of frame - removing a redundancy by shift of thermal variables: u µ → u µ + δ u µ ; T → T + δ T ; µ → µ + δµ . ◮ Clark Putterman frame - no current corrections, j µ diss = 0 π µν u µ u ν = 0

◮ Decompose: π ( µν ) = ( Q µ u ν + Q ν u µ ) + Π P µν + Π µν , t where Π µν Π µν Q µ u µ = 0 , t u ν = 0 , t P µν = 0 . Q µ represent the heat flow.

Entropy Increase ◮ Entropy current: s = su µ − u ν T π µν + f T µ diss ζ µ . J µ ◮ Entropy production rate s = − [Π( ∂ µ u µ ) + Π µν t σ µν ] � a µ + P µν ∂ ν T � − Q µ ∂ µ J µ T T T � f ζ ν � � � − V A µ a µ − P µν ∂ ν T + µ diss P µν ∂ µ , 2 T T T ◮ Has to be positive sum of quadratic forms. ◮ Constraint dissipative corrections: Π µν t , Π , Q µ , V A µ , µ diss ◮ New vector - acceleration a µ ≡ u ν ∂ ν u µ . Two projections - in the direction and in the transverse direction to the superfluid velocity

◮ Number of transport terms in a superfluid T − preserving T − breaking non − Lifshitz 14 7 Lifshitz 22 13 ◮ More detailed results in the NR limit ◮ Only included parity preserving effects

The non-relativistic limit ◮ Fluid variables: ◮ ρ n , ρ s - mass densities, ◮ � v n ,� v s - velocities, ◮ � ω = � v s − � v n - counterflow w = δ ij − w i w j ◮ projector P ij . w 2 ◮ expansion in powers of c: ◮ Expand thermal parameters: ◮ u µ = (1 , � v n c ) ◮ ξ µ = − c (1 , � v s c ) ◮ µ rel = c + 1 c ( µ + ω 2 / 2) v 2 ◮ ǫ n = ρ n c 2 + U n − ρ n n 2 ◮ Expand constitutive relations 1 π µν = � c n π µν ( n ) n

◮ equations of motion: ◮ mass conservation: ∂ t ( ρ n + ρ s ) + ∂ i ( ρ n v i n + ρ s v i s ) = 0 ◮ Navier-Stokes: s ) + ∂ i p + ν i = 0 ∂ t ( ρ n v i n + ρ s v i s ) + ∂ k ( ρ n v i n v k n + ρ s v i s v k ◮ Energy conservation: ∂ t E + ∂ i [ Q i + Q ′ i ] + ν e = 0