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

Superfluid Hydrodynamics, Thermal Partition Function and Lifshitz Scaling

Oxford - Theoretical Physics Shira Chapman, Carlos Hoyos, Yaron Oz arXiv: 1310.2247, 1402.2981

Tel Aviv University March 04, 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 4He

◮ Liquifies at T ≈ 4.2◦K ◮ At T ≈ 2.17◦K - Second order phase transition.

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

◮ The λ point ◮ Heat conductivity increase by a factor of 106 ◮ Large part of atoms in ground state - Condensate - collective

mode

The Two Fluid Picture

◮ Viscosity vs. no viscosity ◮ Effective picture - two fluid flows

vn, vs

◮ 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, uµ - normal fluid 4-velocity, ξµ = −∂µφ - Goldstone phase gradient, uµ

s = −ξµ/ξ - Superfluid velocity. ◮ Superconductors - broken gauge symmetry - ξµ ≡ −∂µφ + Aµ.

Relativistic Superfluids

◮ New thermodynamic parameters ξµ. ◮ Thermodynamic relations:

εn + P = sT + qnµ dP = sdT + qndµ + f 2dξ2

◮ Stress tensor and current:

T µν = εnuµuν + P(ηµν + uµuν) + εsuµ

s uν s + πµν

Jµ = qnuµ + qsuµ

s + jµ diss ◮ Josephson relation:

uµξµ = µ + µdiss.

◮ Hydrodynamic equations - conservation law:

∂µT µν = FνµJµ ∂µJµ = CEµBµ ∂µξν − ∂νξµ = Fµν

◮ Electric field: E µ = Fµνuν ◮ Magnetic field: Bµ = 1 2ǫµνρσuνFρσ ◮ C - triangular anomaly of three currents. ◮ Goal: constrain expressions for πµν, jµ diss, µdiss.

Chiral Effects in Superfluid

◮ Local second law - Constrain current and conductivities

jµ

diss = σ

• E µ − TPµν∂ν

µ T

• + Bµ(Cµ + 2Tg1) + ωµ(Cµ2 + 4g1µT − 2g2T 2) + . . .

πµν = ησµν + ζ(∂αuα)Pµν + . . .

◮

g1 = g1(T, µ, ξ2); g2 = g2(T, µ, ξ2).

◮

Vorticity: ωµ = ǫµνρσuν∂ρuσ

◮

Shear tensor: σµν = Pρ

µPσ ν ∂(ρuσ) − 1 3Pµν∂ρuρ. ◮ Comparison to normal fluid

◮ C - triangular anomaly of three currents. ◮ In the normal fluid g2 integration constant - related to mixed

chiral-gravitational anomaly. [Yarom: 1207.5824] ∂µJµ ∼ ǫµνρσ C 8 FµνFρσ + β 32π2 Rα

βµνRβ αρσ

• .

◮ Numerical evidence that at T → 0 one restore the normal fluid

values [Amado: 1401.5795]

◮ Chiral effects - 3He, 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 ds2 = −e2σ(

x)

dt + ai( x)dxi2 + gij( x)dxidxj, A = A0( x)dx0 + Ai( x)dxi,

◮ KK invariant gauge field:

A0 ≡ A0 + µ0 , Ai ≡ Ai − A0ai ,

◮ Local temperature T(

x) = T0e−σ

◮ Local chemical potential µ(

x) = A0e−σ

Equilibrium Partition Function

◮ Build the most general equilibrium partition function [effective

action] S = S0 + S1 , S0 =

• d3x 1

T P(T, µ, ˆ ζ2) , S1 =

• d3x ˆ

ζ · (g1 ∂ × A + Tg2 ∂ × a) + C

• d3x A ·

µ 3T ∂ × A + µ2 6T ∂ × a

• + . . .

◮ ˆ

ζi ≡ −∂iφ + Ai, 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 -

g1(T, µ, ζ2) = − lim

kn→0

• ij

i 4Tkn ǫijnJi(kn)Jj(−kn) ω=0

kζ − C

2 µ T

• ,

g2(T, µ, ζ2) = lim

kn→0

• ij

i 2T 2kn ǫijn

• JiT 0j − µJiJj
• ω=0

kζ

− C 2 µ T 2

◮ Spatial superfluid velocity - new thermal parameter - ζi. ◮ Similar role to chemical potential in the spatial direction. ◮ Substitution rules in propagators - qµ → (iωn + µ,

q + ζ).

◮ Holographic calculation also possible.

Lifshitz Superfluids - Quantum Critical Points

◮ Anisotropic Weyl - Lifshitz scaling symmetry:

t → Ωzt xi → Ωxi 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 - lc ≫ L ≫ lT

Quantum Criticality

◮ Influence of quantum critical point felt way above T = 0.

Strange Metal SC QCP T g

◮ Example: anti-ferromagnetic → heavy fermion metal transition. ◮ Strange metal behavior ρ ∼ T (∼ T 2 in normal metals) ◮ Characteristic of high Tc superconductors in the

non-superconducting regime.

Hydrodynamics with broken boost invariance

◮ Under lorentz transformations

δL = Tµνωµν ωµν antisymmetric parameter of Lorentz transformation

◮ Asymmetric stress tensor in time direction ∼ T [0i]. ◮ 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[µV ν]

A

Constitutive relations

◮ Stress-tensor:

T µν = (εn + p)uµuν + pηµν + εsuµ

s uν s + π(µν) + π[µν] A

.

◮ Choice of frame - removing a redundancy by shift of thermal

variables: uµ → uµ + δuµ; T → T + δT; µ → µ + δµ .

◮ Clark Putterman frame - no current corrections,

jµ

diss = 0

πµνuµuν = 0

◮ Decompose:

π(µν) = (Qµuν + Qνuµ) + ΠPµν + Πµν

t

, where Qµuµ = 0, Πµν

t uν = 0,

Πµν

t Pµν = 0 .

Qµ represent the heat flow.

Entropy Increase

◮ Entropy current:

Jµ

s = suµ − uν

T πµν + f T µdissζµ .

◮ Entropy production rate

∂µJµ

s = − [Π(∂µuµ) + Πµν t σµν]

T − Qµ T

• aµ + Pµν ∂νT

T

• + µdissPµν∂µ

f ζν T

• − VA µ

2T

• aµ − Pµν ∂νT

T

• ,

◮ Has to be positive sum of quadratic forms. ◮ Constraint dissipative corrections: Πµν t , Π, Qµ, VA µ, µdiss ◮ New vector - acceleration aµ ≡ uν∂νuµ. 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:

◮ ρn, ρs - mass densities, ◮

vn, vs - velocities,

◮

ω = vs − vn - counterflow

◮ projector Pij

w = δij − w iw j w 2

.

◮ expansion in powers of c:

◮ Expand thermal parameters: ◮ uµ = (1, vn c ) ◮ ξµ = −c(1, vs c ) ◮ µrel = c + 1 c (µ + ω2/2) ◮ ǫn = ρnc2 + Un − ρn v2

n

2 ◮ Expand constitutive relations

πµν =

• n

1 cn πµν

(n)

◮ equations of motion:

◮ mass conservation:

∂t(ρn + ρs) + ∂i(ρnv i

n + ρsv i s) = 0

◮ Navier-Stokes:

∂t(ρnv i

n + ρsv i s) + ∂k(ρnv i nv k n + ρsv i sv k s ) + ∂ip + νi = 0

◮ Energy conservation:

∂tE + ∂i[Qi + Q′i] + νe = 0

◮ Dissipative corrections:

Q′i ∼ πi0

(1) =

= ωi(Q1ωj∂jT + Q2ωjDtvn j) + Q3Pij

ω∂jT + Q4Pij ωDtvn j + . . . νi = ∂tπ0i

(−1) + ∂kπ(ki) T (0) + . . .

πij

T (0) = −ησij − ζ∂kvkδij − t1∂k(ρsωk)

− wiwj(t2ωi∂iT + t3ωiDtvi) + . . . π0i

(−1) = −ωi(A1ωj∂jT + A2ωjDtvj) − A3Pij ω∂jT − A4Pij ωDtvj + . . .

◮ scaling of transport coefficients

∼ T

∆ z F

• µ

T

2(z−1) z

, w 2 T

2(z−1) z

• .

[Q1] = d − z; [Q2] = d − 2(z − 1); [Q3] = z + d − 2; [Q4] = d [A1] = d + 2 − 3z; [A2] = d − 4(z − 1); [A3] = d − z; [A4] = d − 2(z − 1)

Experimental Implications

◮ Part of heat flow proportional to acceleration ◮ Can be related to chemical potential:

Dtvi

n ≃∂i

• µ + w2

2

• + . . .

◮ Anisotropy between the direction of the counterflow and the

transverse direction.

◮ Hard to disentangle from effect of shear viscosity ◮ In superconductors alternating current create phase between

normal and super components

◮ Perhaps some unusual frequency dependence would reveal the

effect in superconductors

Outlook

◮ Evaluate g1 and g2 using the Kubo formulas

◮ Weakly coupled field theory ◮ Holographic model ◮ Perhaps explain temperature dependence of Amado [1401.5795]:

◮ Lifshitz Superfluid

◮ Suggest a measurement in superconductors. ◮ Include parity violating effects. ◮ Construct holographic model