Madrid, January 17, 2014 1

Andreas Schmitt

Institut für Theoretische PhysikTechnische Universität Wien

1040 Vienna, Austria

Sound modes and the two-stream instability inrelativistic superfluids

M.G. Alford, S.K. Mallavarapu, A. Schmitt, S. Stetina, PRD 87, 065001 (2013)

M.G. Alford, S.K. Mallavarapu, A. Schmitt, S. Stetina, arXiv:1310.5953 [hep-ph]

A. Schmitt, arXiv:1312.5993 [hep-ph]

• two-fluid picture of a superfluid• role reversal in first and second sound• two-stream instability

Madrid, January 17, 2014 2

• Superfluid hydrodynamics: relevance for compact stars

• r-mode instability• pulsar glitches• precession• asteroseismology• superfluid turbulence (?)

Cas A, Chandra X-Ray Observatory

• Superfluidity in dense matter

Nuclear matter Quark matter

neutrons (Tc . 10 keV) color-flavor locked phase (Tc ∼ 10 MeV)

hyperons color-spin locked phase (Tc ∼ 10 keV)

Madrid, January 17, 2014 3

• Two-fluid picture of a superfluid (liquid helium)London, Tisza (1938); Landau (1941)

relativistic: Khalatnikov, Lebedev (1982); Carter (1989)

• “superfluid component”:condensate, carries no entropy

• “normal component”: excitations(Goldstone mode), carries entropy

εp

p

phonon

roton

Hydrodynamic eqs. ⇒ two sound modes

1st sound 2nd sound

in-phase oscillation out-of-phase oscillation

(primarily) density wave (primarily) entropy wave

Madrid, January 17, 2014 4

• First and second sound in non-relativistic systems

liquid helium

K.R. Atkins et al. (1953)

ultracold fermionic gas (exp.)

L.A. Sidorenkov et al., Nature 498, 78 (2013)

weakly interacting Bose gas

H.Hu, et al., New Journ.Phys. 12, 043040 (2010)

unitary Fermi gas

E. Taylor et al., PRA 80, 053601 (2009)

Madrid, January 17, 2014 5

• Goals

How does the two-fluid picturearise from a microscopic field theory?

M.G. Alford, S.K. Mallavarapu, A. Schmitt, S. Stetina, PRD 87, 065001 (2013)

Compute sound modes in a relativistic superfluid(and in the presence of a superflow)

M.G. Alford, S.K. Mallavarapu, A. Schmitt, S. Stetina, arXiv:1310.5953 [hep-ph]

A. Schmitt, arXiv:1312.5993 [hep-ph]

Madrid, January 17, 2014 6

• Lagrangian and superfluid velocity

• starting point:complex scalar field

L = (∂ϕ)2 −m2|ϕ|2 − λ|ϕ|4

• Bose condensate 〈ϕ〉 = ρ eiψ spontaneously breaks U(1)

• zero temperature: single-fluid systemField theory Hydrodynamics

current jµ(∂ψ)2

λ∂µψ nvµ

stress-energy tensor T µν −gµνL + (∂ψ)2

λ∂µψ∂νψ (� + P )vµvν − gµνP

• superfluid velocity vµ = ∂µψ

µµ = |∂ψ|

Madrid, January 17, 2014 7

• Relativistic two-fluid formalism (page 1/2)• write stress-energy tensor as

Tµν = −gµνΨ + jµ∂νψ + sµΘν

• “generalized pressure” Ψ:– Ψ = P⊥ in superfluid and normal-fluid rest frames,– Ψ depends on momenta ∂µψ, Θµ

Ψ = Ψ[(∂ψ)2,Θ2, ∂ψ · Θ]

• “generalized energy density” Λ ≡ −Ψ + j · ∂ψ + s · Θ– Λ is Legendre transform of Ψ,

– Λ depends on currents jµ, sµ

Λ = Λ[j2, s2, j · s]

Madrid, January 17, 2014 8

• Relativistic two-fluid formalism (page 2/2)

jµ =∂Ψ

∂(∂µψ)= B ∂µψ +AΘµ

sµ =∂Ψ

∂Θµ= A ∂µψ + C Θµ

B = 2 ∂Ψ∂(∂ψ)2

, C = 2 ∂Ψ∂Θ2

A = ∂Ψ∂(∂ψ · Θ)

“entrainment coefficient”

• compute A, B, C from microscopic physics

B

C

A

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

1.5

2.0

2.5

TTc

A,B

,C@Μ

2 Λ

D

all temperatures

2PIEntrainment coefficient A

OHT5LOHT3L

0.000 0.001 0.002 0.003 0.004

0

1

2

3

4

TTc

106

´A

@Μ2

ΛD

(very) small temperatures

Madrid, January 17, 2014 9

• Microscopic calculation for arbitrary T (page 1/2)

• effective action density in the 2PI formalism (CJT)

Γ[ρ, S] = −U(ρ)− 12

Tr lnS−1 − 12

Tr[S−10 (ρ)S − 1]− V2[ρ, S]

• V2[ρ, S]: two-loop two-particle irreducible (2PI) diagrams

• use Hartree approximation

• impose Goldstone theorem by hand

• solve self-consistency equations for condensate ρ and M , δM

Madrid, January 17, 2014 10

• Microscopic calculation for arbitrary T (page 2/2)

•microscopic calculation done in normal-fluid rest frame

• identify effective action density with generalized pressure

Γ[µ, T,∇ψ] = Ψ

• restrict to weak coupling → no dependence onrenormalization scale

• consider uniform superflow v

• neglect dissipation → thermodynamics with (µ, T,v)

• compute entrainment coefficient, sound velocities etc.

Madrid, January 17, 2014 11

• Results I: critical velocity

• instability at v = vc• negative energies in Goldstone

dispersion �k(v) < 0

v =0.5

3> 0.29

T=0T=0.5 Tc HvLT=T

c HvL

-1.0 -0.5 0.0 0.5

0.0

0.1

0.2

0.3

0.4

kÈÈΜ

Ε kΜ

• generalization to Landau’s original argument �k − k · v < 0

non-superfluid

uniform superfluid

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

1.0

TTc

v

• dashed line: withoutbackreaction of condensate

• shaded region:dissipation, turbulence?

• similar phase diagram for holographic superfluid I. Amado, D. Arean,A. Jimenez-Alba, K. Landsteiner, L. Melgar and I. S. Landea, arXiv:1307.8100 [hep-th]

Madrid, January 17, 2014 12

• Results II: sound speeds and mixing angleultra-relativistic (towards) non-relativistic

u1

u2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

soun

dsp

eed

u

u1

u2u1

u2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.02 0.04 0.06

0.46

0.48

0.50

0.52

0.54

pure T wave

pure Μ wave

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTc

mix

ing

angl

eΑ

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTc

α = arctan δTδµ role reversal in first and second sound!

Madrid, January 17, 2014 13

• Sound speeds and mixing anglewith superflow

coupling

superflow

ultra-relativistic (towards) non-relativistic

anti-parallelparallel

u1

u2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

u

u1

u2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTcHvL

Α

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTcHvL

Madrid, January 17, 2014 13

• Sound speeds and mixing anglewith superflow

coupling

superflow

ultra-relativistic (towards) non-relativistic

u1

u2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

u

u1

u2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTcHvL

Α

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTcHvL

Madrid, January 17, 2014 13

• Sound speeds and mixing anglewith superflow

coupling

superflow

ultra-relativistic (towards) non-relativistic

u1

u2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

u

u1

u2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTcHvL

Α

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTcHvL

Madrid, January 17, 2014 13

• Sound speeds and mixing anglewith superflow

coupling

superflow

ultra-relativistic (towards) non-relativistic

u1

u2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

u

u1

u2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTcHvL

Α

Α1

Α2

0.0 0.2 0.4 0.6 0.8 1.0-

Π

2

-Π

4

0

Π

4

Π

2

TTcHvL

Madrid, January 17, 2014 14

• Results III: two-stream instability

• compute sound speed close toLandau’s critical velocity

non-superfluid

uniform superfluid

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

1.0

TTc

v

T = 0.4TcΘ = Π

0.988 0.989 0.990 0.991 0.992 0.993 0.994 0.995 0.9960.0

0.2

0.4

0.6

0.8

1.0

vvcHTL

ReH

uL

T = 0.4TcΘ = Π

0.988 0.989 0.990 0.991 0.992 0.993 0.994 0.995 0.996-0.10

-0.05

0.00

0.05

0.10

vvcHTL

ImHu

L

• complex sound speeds → one mode damped, one mode explodesplasma physics: O. Buneman, Phys.Rev. 115, 503 (1959); D.T. Farley, PRL 10, 279 (1963)

general two-fluid system: L. Samuelsson, C. S. Lopez-Monsalvo, N. Andersson, G. L. Comer,

Gen. Rel. Grav. 42, 413 (2010)

relevance for superfluids: N. Andersson, G. L. Comer, R. Prix, MNRAS 354, 101 (2004)

Madrid, January 17, 2014 15

• All directions

v = 0

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0v = 0.5vcHTL

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0v = 0.967vcHTL

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

v = 0.994vcHTL

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0v = 0.995vcHTL

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0v = 0.996vcHTL

-1.0 -0.5 0.5 1.0

-1.0

-0.5

0.5

1.0

(superflow pointing to the right)

Madrid, January 17, 2014 16

• Instability window in phase diagram

uniformsuperfluid

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

TTc

v

0.575 0.585

0.462

0.468

u > 1

ImHuL ¹ 0uniform superfluid

0.0 0.2 0.4 0.6 0.8

0.975

0.980

0.985

0.990

0.995

1.000

TTc

vv c

HTL

• tiny window for weak coupling λ = 0.05(varying λ shows that the window grows with λ)

• region with u > 1: problem in the formalism?(Hartree? enforced Goldstone theorem?)

• very small T : qualitatively different angular structure of instability

Madrid, January 17, 2014 17

• Summary

• a superfluid is a two-fluid system, and this can be derived frommicroscopic physics

• the two sound modes in a (weakly coupled, relativistic) superfluidcan reverse their roles (in terms of density and entropy waves)

• at large relative velocities of the two fluids, there is a dynamicalinstability (“two-stream instability”)

Madrid, January 17, 2014 18

• Outlook• start from fermionic theory

D. Müller, A. Schmitt, work in progress

• behavior beyond critical velocity• sound modes (role reversal):

– predictions for 4He or ultracold gases?

– apply to compact starsneutron superfluid & ion lattice: N. Chamel, D. Page and S. Reddy, PRC 87, 035803 (2013)

• two-stream instability:– instability more prominent at strong coupling?

holographic approach: C.P.Herzog and A.Yarom, PRD 80, 106002 (2009); I.Amado,

D.Arean, A.Jimenez-Alba, K.Landsteiner, L.Melgar, I.S.Landea, arXiv:1307.8100 [hep-th]

– time evolution of instabilityI. Hawke, G. L. Comer and N. Andersson, Class. Quant. Grav. 30, 145007 (2013)

– relevance for compact stars, e.g., pulsar glitchesN. Andersson, G. L. Comer, R. Prix, MNRAS 354, 101 (2004)