Formulation ofrelativistic spin hydrodynamics
based on the entropy-current analysis
Hidetoshi TAYA (Fudan University)
based on arXiv: 1901.06615 [hep-th]
in collaboration withK. Hattori (YITP), M. Hongo (Keio), X.-G. Huang (Fudan), M. Matsuo (UCAS)@The 5th Workshop on Chirality, Vorticity and Magnetic Field in Heavy Ion Collisions
2019/04/11

Ultra-relativistic heavy ion collisions

Aim: study quark-gluon plasma (QGP)
Found: QGP behaves like a perfect liquidand hydrodynamics works so well

Huge ω and Bω, B

Question: QGP under huge ω and/or B ?

Naïve expectation: QGP is polarized Magnetic field B effectZeeman splitting (Landau quantization)
𝐸𝐸 → 𝐸𝐸 − 𝑠𝑠 ⋅ 𝒒𝒒𝐵𝐵

➡ charge dependent spin polarization
 Rotation ω effectBarnett effect
𝐸𝐸 → 𝐸𝐸 − 𝑠𝑠 ⋅ 𝜔𝜔

➡ charge in dependent spin polarization

Experimental fact

How about theory?
Hydrodynamics for spin polarized QGP?
Far from completeHydrodynamics for spin polarized QGP
 “Hydro simulations” exist, but…
- usual hydro (i.e., hydro w/o spin) is solved- thermal vorticity 𝜔𝜔� 𝜇𝜇𝜇𝜇 ≡ 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 /𝑇𝑇 − 𝜕𝜕 𝜈𝜈 𝑢𝑢𝜇𝜇 /𝑇𝑇 is converted
into spin via Cooper-Frye formula (???)

 Formulation of relativistic hydrodynamics with spinis still developingCurrent status of formulation of spin hydro Non-relativistic casee.g. Eringen (1998); Lukaszewicz (1999)

Already established (e.g. micropolar fluid)- applied to pheno. and is successful

e.g. spintronics:
Takahashi et al. (2015)- spin must be dissipative because of mutual conversion
between spin and orbital angular momentum

 Relativistic caseSome trials exist, but- only for “ideal” fluid (no dissipative corrections)- some claim spin should be conservedPurpose of this talk Formulate relativistic spin hydrodynamics with
1st order dissipative corrections for the first time Clarify spin should be dissipative
Outline1. Introduction
2. Formulation based on an entropy-current analysis
3. Linear mode analysis
4. Summary

Outline

1. Introduction
2. Formulation based on an entropy-current analysis
3. Linear mode analysis
4. Summary

Introduction to hydrodynamics w/o spin (1/3)
Hydrodynamics is a low energy effective theory thatdescribes spacetime evolution of IR modes (hydro modes)
textbook by Landau, Lifshitz

 Phenomenological formulation (EFT construction)
Step 1: Write down the conservation law: 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇4 eqs

Step 2: Express 𝑻𝑻𝝁𝝁𝝁𝝁 i.t.o hydro variables (constitutive relation)
1 + (4-1) = 4 DoGs
- define hydro variables: {𝛽𝛽, 𝑢𝑢𝜇𝜇 } (𝑢𝑢2 = −1) chemical potential for 𝑃𝑃𝜇𝜇
- write down all the possible tensor structures of 𝑇𝑇𝜇𝜇𝜇𝜇

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑓𝑓1 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑓𝑓2 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈
+𝑓𝑓3 𝛽𝛽 𝜖𝜖 𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜎𝜎 + 𝑓𝑓4 𝛽𝛽 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑓𝑓5 𝛽𝛽 𝜕𝜕 𝜈𝜈 𝑢𝑢𝜇𝜇+𝑓𝑓6 𝛽𝛽 𝑔𝑔𝜇𝜇𝜇𝜇 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓7 𝛽𝛽 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 𝜕𝜕𝜌𝜌 𝑢𝑢𝜌𝜌 + 𝑓𝑓8 𝛽𝛽 𝑢𝑢𝜇𝜇 𝜕𝜕𝜇𝜇 𝑢𝑢𝜈𝜈 + ⋯ + 𝑂𝑂(𝜕𝜕 2 )- simplify the tensor structures by (assumptions in hydro)(1) symmetry(2) power counting ➡ gradient expansion(3) other physical requirements ➡ thermodynamics (see next slide) Hydrodynamic eq. = conservation law + constitutive relation

Introduction to hydrodynamics w/o spin (2/3)
 Constraints by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 i.t.o derivatives𝜇𝜇𝜇𝜇𝜇𝜇𝜈𝜈

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑇𝑇(0) + 𝑇𝑇(1) + 𝑂𝑂(𝜕𝜕 2 )In static equilibrium 𝑇𝑇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇→ 𝑇𝑇(0) = (𝑒𝑒, 𝑝𝑝, 𝑝𝑝, 𝑝𝑝), so that𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )1st law of thermodynamics says
𝑑𝑑𝑑𝑑 = 𝛽𝛽𝛽𝛽𝛽𝛽, 𝑠𝑠 = 𝛽𝛽(𝑒𝑒 + 𝑝𝑝)

By using EoM 0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇 , div. of entropy current 𝑆𝑆𝜇𝜇 = 𝑠𝑠𝑢𝑢𝜇𝜇 + 𝑂𝑂(𝜕𝜕) can be evaluated as𝜇𝜇𝜇𝜇

𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 = −𝑇𝑇 1 𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 + 𝑂𝑂(𝜕𝜕 3 )2st law of thermodynamics requires 𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 ≥ 0, which is guaranteed if RHS isexpressed as a semi-positive bilinear as𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇−𝑇𝑇 1 𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 = ∑𝑋𝑋 ∈ 𝑇𝑇 𝜆𝜆𝑖𝑖 𝑋𝑋𝑖𝑖 𝑋𝑋𝑖𝑖 𝜈𝜈𝜈𝜈 ≥ 0 with 𝜆𝜆𝑖𝑖 ≥ 0𝑖𝑖1𝜇𝜇ex) heat current: 2ℎ(𝜇𝜇 𝑢𝑢𝜈𝜈) ≡ ℎ𝜇𝜇 𝑢𝑢𝜈𝜈 + ℎ𝜈𝜈 𝑢𝑢𝜇𝜇 ∈ 𝑇𝑇 𝜇𝜇𝜇𝜇
1 (𝑢𝑢𝜇𝜇 ℎ = 0)𝜇𝜇𝜇𝜇

⇒ 𝑇𝑇 1 𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 = −𝛽𝛽ℎ𝜇𝜇 𝛽𝛽𝜕𝜕⊥𝜇𝜇 𝛽𝛽 −1 + 𝑢𝑢𝜈𝜈 𝜕𝜕𝜈𝜈 𝑢𝑢𝜇𝜇 ≥ 0⇒ ℎ𝜇𝜇 = −𝜅𝜅 𝛽𝛽𝜕𝜕⊥𝜇𝜇 𝛽𝛽−1 + 𝑢𝑢𝜈𝜈 𝜕𝜕𝜈𝜈 𝑢𝑢𝜇𝜇 with 𝜅𝜅 ≥ 0

Introduction to hydrodynamics w/o spin (3/3)
 Constitutive relation up to 1st order w/o spin
𝜇𝜇𝜇𝜇𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )(𝜇𝜇𝜇𝜇𝜇𝜇<𝜇𝜇𝑇𝑇(1) = −2𝜅𝜅 𝐷𝐷𝑢𝑢(𝜇𝜇 + 𝛽𝛽𝜕𝜕⊥ 𝛽𝛽−1 𝑢𝑢𝜈𝜈) − 2𝜂𝜂𝜕𝜕⊥ 𝑢𝑢𝜈𝜈> − 𝜁𝜁 𝜕𝜕𝜇𝜇 𝑢𝑢𝜇𝜇 Δ𝜇𝜇𝜇𝜇heat current

shear viscous effectbulk viscous effect Hydrodynamic equation w/o spin
Hydrodynamic eq. = conservation law + constitutive relation

Euler eq.
Navier-Stokes eq.
⋮𝜇𝜇𝜇𝜇

0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑇𝑇(0)⋮⋮0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑇𝑇(0) + 𝑇𝑇(1)Formulation of hydrodynamics with spin (1/4)
 Strategy is the same
 Phenomenological formulation
Step 1: Write down the conservation law
Step 2: Construct a constitutive relation- define hydro variables- write down all the possible tensor structures- simplify the tensor structures by e.g. thermodynamics

Formulation of hydrodynamics with spin (2/4)
Step 1: Write down the conservation laws
(1) energy conservation

0 = 𝜕𝜕𝜇𝜇 𝑇𝑇𝜇𝜇𝜇𝜇(canonical)(2) total angular momentum conservation

0 = 𝜕𝜕𝜇𝜇 𝑀𝑀𝜇𝜇,𝛼𝛼𝛼𝛼𝜓𝜓(𝑥𝑥) → 𝑆𝑆 Λ 𝜓𝜓(Λ−1 𝑥𝑥)= 𝜕𝜕𝜇𝜇 (𝐿𝐿𝜇𝜇,𝛼𝛼𝛼𝛼 +Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 )= 𝜕𝜕𝜇𝜇 (𝑥𝑥 𝛼𝛼 𝑇𝑇𝜇𝜇𝜇𝜇 − 𝑥𝑥 𝛽𝛽 𝑇𝑇𝜇𝜇𝜇𝜇 + Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 )∴ 𝜕𝜕𝜇𝜇 Σ𝜇𝜇,𝛼𝛼𝛼𝛼 = 𝑇𝑇 𝛼𝛼𝛼𝛼 − 𝑇𝑇𝛽𝛽𝛽𝛽𝜇𝜇𝜇𝜇

 Spin is not conserved if (canonical) 𝑇𝑇𝜇𝜇𝜇𝜇 has anti-symmetric part 𝑇𝑇(a) There’s no a priori reason (canonical) 𝑇𝑇𝜇𝜇𝜇𝜇 must be symmetricConsequence(1) Spin must not be a hydro mode in a strict sense𝜇𝜇𝜇𝜇(2) Nevertheless, it behaves like a hydro mode if 𝑇𝑇(a) ≪ 1➡ inclusion of dissipative nature is importantFormulation of hydrodynamics with spin (3/4)
Step 2: Construct a constitutive relation for 𝑇𝑇𝜇𝜇𝜇𝜇 , Σ 𝜇𝜇,𝛼𝛼𝛼𝛼(1) define hydro variables

4 + 6 = 10 DoGs = # of EoMs

Introduce spin chemical potential 𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇with 𝜔𝜔 𝜇𝜇𝜇𝜇 = −𝜔𝜔 𝜈𝜈𝜈𝜈𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇 are independent w/ each other at this stage (𝜔𝜔 𝜇𝜇𝜇𝜇 ≠thermal vorticity)(2) simplify the tensor structure by thermodynamics
Expand 𝑇𝑇𝜇𝜇𝜇𝜇 , Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 , i.t.o derivatives𝜇𝜇𝜈𝜈

𝑇𝑇𝜇𝜇𝜇𝜇 = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝 𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑇𝑇 1 + 𝑂𝑂 𝜕𝜕 2 ,where I defined spin density 𝝈𝝈𝜶𝜶𝜶𝜶

Σ 𝜇𝜇,𝛼𝛼𝛼𝛼 = 𝑢𝑢𝜇𝜇 𝜎𝜎 𝛼𝛼𝛼𝛼 + 𝑂𝑂(𝜕𝜕1 )Generalize 1st law of thermodynamics with spin as
𝑑𝑑𝑑𝑑 = 𝛽𝛽(𝑑𝑑𝑑𝑑 − 𝜔𝜔𝜇𝜇𝜇𝜇 𝑑𝑑𝑑𝑑𝜇𝜇𝜇𝜇 ), 𝑠𝑠 = 𝛽𝛽(𝑒𝑒 + 𝑝𝑝 − 𝜔𝜔𝜇𝜇𝜇𝜇 𝜎𝜎 𝜇𝜇𝜇𝜇 )With EoMs, div. of entropy current 𝑆𝑆𝜇𝜇 = 𝑠𝑠𝑢𝑢𝜇𝜇 + 𝑂𝑂(𝜕𝜕) can be evaluated as𝜇𝜇𝜇𝜇
𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 = −𝑇𝑇 1s

𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 + 𝜕𝜕𝜈𝜈 𝛽𝛽𝛽𝛽𝜇𝜇𝜇𝜇𝜇𝜇− 𝑇𝑇 1a2𝜕𝜕𝜇𝜇 𝛽𝛽𝛽𝛽𝜈𝜈 − 𝜕𝜕𝜈𝜈 𝛽𝛽𝛽𝛽𝜇𝜇− 2𝛽𝛽𝜔𝜔𝜇𝜇𝜇𝜇 + 𝑂𝑂(𝜕𝜕 3 )2𝜇𝜇𝜇𝜇

 2nd law of thermodynamics 𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 ≥ 0 gives strong constraint on 𝑇𝑇 1
 In global equilibrium 𝜕𝜕𝜇𝜇 𝑆𝑆𝜇𝜇 = 0, so that 𝜔𝜔 = thermal vorticity.

Formulation of hydrodynamics with spin (4/4)
 Constitutive relation for 𝑇𝑇𝜇𝜇𝜇𝜇𝑇𝑇(0) = 𝑒𝑒𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 + 𝑝𝑝(𝑔𝑔𝜇𝜇𝜇𝜇 + 𝑢𝑢𝜇𝜇 𝑢𝑢𝜈𝜈 )heat current(𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇up to 1st order with spin

shear viscous effect bulk viscous effect<𝜇𝜇𝑇𝑇(1) = −2𝜅𝜅 𝐷𝐷𝑢𝑢(𝜇𝜇 + 𝛽𝛽𝜕𝜕⊥ 𝛽𝛽−1 𝑢𝑢𝜈𝜈) − 2𝜂𝜂𝜕𝜕⊥ 𝑢𝑢𝜈𝜈> − 𝜁𝜁 𝜕𝜕𝜇𝜇 𝑢𝑢𝜇𝜇 Δ𝜇𝜇𝜇𝜇[𝜇𝜇[𝜇𝜇𝜇𝜇−2𝜆𝜆 −𝐷𝐷𝑢𝑢[𝜇𝜇 + 𝛽𝛽𝜕𝜕⊥ 𝛽𝛽−1 + 4𝑢𝑢𝜌𝜌 𝜔𝜔 𝜌𝜌[𝜇𝜇 𝑢𝑢𝜈𝜈] − 2𝛾𝛾 𝜕𝜕⊥ 𝑢𝑢𝜈𝜈] − 2Δ𝜌𝜌 Δ𝜆𝜆𝜈𝜈 𝜔𝜔 𝜌𝜌𝜌𝜌“boost heat current”

NEW !

“rotational (spinning) viscous effect”
e.g. Eringen (1998); Lukaszewicz (1999)

 Relativistic generalization of a non-relativistic micropolar fluid
 “boost heat current” is a relativistic effect Hydrodynamics equation up to 1st order with spin𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇

0 = 𝜕𝜕𝜇𝜇 (𝑇𝑇 0 + 𝑇𝑇 1 + 𝑂𝑂(𝜕𝜕 2 ))𝛼𝛼𝛼𝛼𝛽𝛽𝛽𝛽𝜕𝜕𝜇𝜇 (𝑢𝑢𝜇𝜇 𝜎𝜎 𝛼𝛼𝛼𝛼 ) = 𝑇𝑇(1) − 𝑇𝑇 1 + 𝑂𝑂(𝜕𝜕 2 )Outline

1. Introduction
2. Formulation based on an entropy-current analysis
3. Linear mode analysis
4. SummaryLinear mode analysis (1/2)
Setup: small perturbations on top of static equilibrium𝑖𝑖𝜕𝜕𝑡𝑡𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝜇𝜇= 𝑀𝑀 𝛿𝛿𝛿𝛿𝜇𝜇 + 𝑂𝑂(𝛿𝛿 2 )𝛿𝛿𝛿𝛿𝛿𝛿𝜔𝜔 𝜇𝜇𝜇𝜇𝛿𝛿𝜔𝜔 𝜇𝜇𝜇𝜇hydro evolution

𝛽𝛽 = 𝛽𝛽0

𝛽𝛽 = 𝛽𝛽0 + 𝛿𝛿𝛿𝛿𝜔𝜔 𝜇𝜇𝜇𝜇 = 0𝜔𝜔 𝜇𝜇𝜇𝜇 = 0 + 𝛿𝛿𝜔𝜔 𝜇𝜇𝜇𝜇𝑢𝑢𝜇𝜇 = 1, 𝟎𝟎𝑢𝑢𝜇𝜇 = 1, 𝟎𝟎 + 𝛿𝛿𝑢𝑢𝜇𝜇

propagate if Im 𝜔𝜔 = 0
dissipate if Im 𝜔𝜔 < 0unstable if Im 𝜔𝜔 > 0

Linear mode analysis (2/2)
 Hydro w/o spin {𝛽𝛽, 𝑢𝑢𝜇𝜇 }4 modes2 sound modes 𝜔𝜔 = ±𝑐𝑐s 𝑘𝑘 + 𝑂𝑂(𝑘𝑘 2 )𝜂𝜂𝑘𝑘 2
2 shear modes 𝜔𝜔 = −𝑖𝑖+ 𝑂𝑂(𝑘𝑘 4 )𝑒𝑒 + 𝑝𝑝where 𝑐𝑐s2 ≡ 𝜕𝜕𝜕𝜕/𝜕𝜕𝜕𝜕 Hydro with spin {𝛽𝛽, 𝑢𝑢𝜇𝜇 , 𝜔𝜔 𝜇𝜇𝜇𝜇 }4 modes2 sound modes 𝜔𝜔 = ±𝑐𝑐s 𝑘𝑘 + 𝑂𝑂(𝑘𝑘 2 )𝜂𝜂𝑘𝑘 22 shear modes 𝜔𝜔 = −𝑖𝑖+ 𝑂𝑂(𝑘𝑘 4 )𝑒𝑒 + 𝑝𝑝+ 6 dissipative modes

3 “boost” modes 𝜔𝜔 = −2𝑖𝑖𝜏𝜏b−1 + 𝑂𝑂(𝑘𝑘 2 )3 “spin” modes 𝜔𝜔 = −2𝑖𝑖𝜏𝜏s−1 + 𝑂𝑂(𝑘𝑘 2 )𝜕𝜕𝜎𝜎𝑖𝑖𝑖𝑖 ⁄𝜕𝜕𝜔𝜔𝑖𝑖𝑖𝑖
𝜕𝜕𝜎𝜎𝑖𝑖0 ⁄𝜕𝜕𝜔𝜔𝑖𝑖0where 𝜏𝜏s ≡, 𝜏𝜏b ≡4𝛾𝛾
4𝜆𝜆

 We explicitly confirmed that spin is dissipative The time-scale of the dissipation is controlled by the new viscous constants 𝛾𝛾, 𝜆𝜆Outline

1. Introduction
2. Formulation based on an entropy-current analysis
3. Linear mode analysis
4. Summary

Summary Spin polarization in QGP is one of the hottest topics inHIC. But, its theory, in particular hydrodynamic
framework, has not been developed well
 Relativistic spin hydrodynamics with 1st order dissipative
corrections is formulated based on the phenomenologicalentropy-current analysis
 Spin must be dissipative because of the mutual conversion
between the orbital angular momentum and spin Linear mode analysis of the spin hydrodynamic equation
also suggests that spin must be dissipative
Outlook: extension to 2nd order, Kubo formula, MHD,
application to cond-mat, numerical simulations

BACK UPLinearized hydro eq.
whereandDispersion relations

Further simplification by EoM
The 1st order constitutive relation reads

By using LO hydro eq.,
we can further simplify h,q as