Phenomenological formulation of relativistic spin hydrodynamics - - PowerPoint PPT Presentation

Phenomenological formulation of relativistic spin hydrodynamics

Hidetoshi TAYA (Fudan University)

based on arXiv: 1901.06615 [hep-th] in collaboration with

• K. Hattori (YITP), M. Hongo (RIKEN), X.-G. Huang (Fudan), M. Matsuo (UCAS)

@YITP 2019/03/27

Ultra-relativistic heavy ion collisions

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

Another interesting physics: Largest ω and B

Question: What happens under huge ω and/or B ? Specifically, any changes to QGP properties?

ω, , B

Naïve expectation: QGP is polarized

Zeeman splitting (Landau quantization) Bernett effect 𝐹 → 𝐹 − 𝑡 ⋅ 𝒓𝐶 𝐹 → 𝐹 − 𝑡 ⋅ 𝜕 ➡ charge dependent spin polarization ➡ charge in in dependent spin polarization  Magnetic field B effect  Rotation ω effect

Experimental fact

STAR (2018) See also talk by T.Niida

How about theory? Hydrodynamics for spin polarized QGP? ➡ Far from complete

See also talk by X.-G. Huang

 “Hydro simulations” exist, but…  Formulation of relativistic spin hydrodynamics is still developing

usual hydro (i.e., hydro w/o /o spin) is solved

(1) Compute velocity gradient at freezeout and define thermal vorticity ෥ 𝜕𝜈𝜉 ≡ 𝜖𝜈 𝑣𝜉/𝑈 − 𝜖𝜉 𝑣𝜈/𝑈 (2) Use Cooper-Frye formula with spin 𝑔 𝐹 → 𝑔(𝐹 − 𝑡 ⋅ 𝜕), where 𝜕 is spin chemical potential (≠ ෥ 𝜕 in general) (3) Assume 𝜕 = ෥ 𝜕 (true only for global equilibrium) (4) Get spin-dependent hadron yield

Hydrodynamics for spin polarized QGP

Becattini, Florkowski, Speranza (2018) Becattini, Chandra, Del Zanna, Grossi (2013)

Current status of formulation of spin hydro

 Non-relativistic case  Relativistic case

Already established (e.g. micropolar fluid)

• applied to pheno. and is successful

Some trials exist, but

e.g. Eringen (1998); Lukaszewicz (1999) e.g. spintronics: Takahashi et al. (2015)

• sp

spin in must t be dis issipati tive because of mutual conversion between spin and orbital angular momentum

• only for “ideal” fluid (no dissipative corrections)
• some claim sp

spin sh should be co conse served

Purpose of this talk

 Formulate relativistic spin hydrodynamics with 1st order dissipative corrections for the first time  Clarify spin should be dissipative 1. Introduction 2. Formulation based on an entropy-current analysis 3. Linear mode analysis

• 4. Summary

Outline

Outline

1. Introduction

• 2. Formulation based on an entropy-current analysis

3. Linear mode analysis

• 4. Summary

Step 2: Expr Express 𝑼𝝂𝝃 i.t. i.t.o hy hydro varia riables (constitutive relation)

• define hydro variables:
• write down all the possible tensor structures of 𝑈𝜈𝜉
• simplify the tensor structures by (assumptions in hydro)

(1) symmetry (2) power counting ➡ gradient expansion (3)

• ther physical requirements ➡ thermodynamics (see next slide)

 Phenomenological formulation (EFT construction)

0 = 𝜖𝜈𝑈𝜈𝜉

𝑈𝜈𝜉 = 𝑔

1 𝛾 𝑕𝜈𝜉 + 𝑔 2 𝛾 𝑣𝜈𝑣𝜉

{𝛾, 𝑣𝜈}

4 eqs 1 + (4-1) = 4 DoGs

+𝑔

3 𝛾 𝜗𝜈𝜉𝜍𝜏𝜖𝜍𝑣𝜏 + 𝑔 4 𝛾 𝜖𝜈𝑣𝜉 + 𝑔 5 𝛾 𝜖𝜉𝑣𝜈

Introduction to hydrodynamics w/o spin (1/3)

+𝑔

6 𝛾 𝑕𝜈𝜉𝜖𝜍𝑣𝜍 + 𝑔 7 𝛾 𝑣𝜈𝑣𝜉𝜖𝜍𝑣𝜍 + 𝑔 8 𝛾 𝑣𝜈𝜖𝜈𝑣𝜉 + ⋯ + 𝑃(𝜖2)

Step 1: Writ rite do down the the conservation law:

Hydrodynamics is a low energy effective theory that describes spacetime evolution of IR modes (hydro modes)

(𝑣2 = −1)

chemical potential for 𝑄𝜈

 Hydrodynamic eq. = conservation law + constitutive relation

textbook by Landau, Lifshitz

 Constraints by thermodynamics

Introduction to hydrodynamics w/o spin (2/3)

𝑈𝜈𝜉 = 𝑈

(0) 𝜈𝜉 + 𝑈 (1) 𝜈𝜉 + 𝑃(𝜖2)

1st law of thermodynamics says 2st law of thermodynamics requires 𝜖𝜈𝑇𝜈 ≥ 0, which is guaranteed if RHS is expressed as a semi-positive bilinear as Expand 𝑈𝜈𝜉 i.t.o derivatives 𝜖𝜈𝑇𝜈 = −𝑈 1

𝜈𝜉𝜖𝜈 𝛾𝑣𝜉 + 𝑃(𝜖3)

In static equilibrium 𝑈𝜈𝜉 → 𝑈

(0) 𝜈𝜉 = (𝑓, 𝑞, 𝑞, 𝑞), so that

𝑈

(0) 𝜈𝜉 = 𝑓𝑣𝜈𝑣𝜉 + 𝑞(𝑕𝜈𝜉 + 𝑣𝜈𝑣𝜉)

−𝑈 1

𝜈𝜉𝜖𝜈 𝛾𝑣𝜉 = σ𝑌𝑗 ∈ 𝑈 1 𝜇𝑗 𝑌𝑗 𝜈𝜉𝑌𝑗 𝜉𝜈 ≥ 0 with 𝜇𝑗 ≥ 0

2ℎ(𝜈𝑣𝜉) ≡ ℎ𝜈𝑣𝜉 + ℎ𝜉𝑣𝜈 ∈ 𝑈 1

𝜈𝜉

⇒ 𝑈 1

𝜈𝜉𝜖𝜈 𝛾𝑣𝜉 = −𝛾ℎ𝜈 𝛾𝜖⊥𝜈𝛾−1 + 𝑣𝜉𝜖𝜉𝑣𝜈 ≥ 0

⇒ ℎ𝜈 = −𝜆 𝛾𝜖⊥𝜈𝛾−1 + 𝑣𝜉𝜖𝜉𝑣𝜈 with 𝜆 ≥ 0 ex) heat current: (𝑣𝜈ℎ𝜈 = 0) 𝑒𝑡 = 𝛾𝑒𝑓, 𝑡 = 𝛾(𝑓 + 𝑞) By using EoM 0 = 𝜖𝜈𝑈𝜈𝜉, div. of entropy current 𝑇𝜈 = 𝑡𝑣𝜈 + 𝑃(𝜖) can be evaluated as

 Hydrodynamic equation w/o spin

Introduction to hydrodynamics w/o spin (3/3)

𝑈

(0) 𝜈𝜉 = 𝑓𝑣𝜈𝑣𝜉 + 𝑞(𝑕𝜈𝜉 + 𝑣𝜈𝑣𝜉)

0 = 𝜖𝜈𝑈𝜈𝜉

Hydrodynamic eq. = conservation law + constitutive relation

𝑈𝜈𝜉 = 𝑈

(0) 𝜈𝜉

𝑈𝜈𝜉 = 𝑈

(0) 𝜈𝜉 + 𝑈 (1) 𝜈𝜉

Euler eq. Navier-Stokes eq. 0 = 𝜖𝜈𝑈𝜈𝜉

⋮ ⋮ ⋮

 Constitutive relation up to 1st order w/o spin

hea eat t curr rren ent sh shear ear visc scous eff effect bulk visc scous eff effect

𝑈

(1) 𝜈𝜉 = −2𝜆 𝐸𝑣(𝜈 + 𝛾𝜖⊥ (𝜈𝛾−1 𝑣𝜉) − 2𝜃𝜖⊥ <𝜈𝑣𝜉> − 𝜂 𝜖𝜈𝑣𝜈 Δ𝜈𝜉

Formulation of hydrodynamics with spin (1/4)

Step 2: Co Construct a co constitutiv ive rela latio ion

• define hydro variables
• write down all the possible tensor structures
• simplify the tensor structures

Phenomenological formulation

Step 1: Writ rite do down the the co cons nservatio ion la law

 Strategy is the same

(1) symmetry (2) gradient expansion (3) thermodynamics

Formulation of hydrodynamics with spin (2/4)

Step 1: Write down the conservation law

0 = 𝜖𝜈𝑈𝜈𝜉 0 = 𝜖𝜈𝑁𝜈,𝛽𝛾 = 𝜖𝜈 (𝑀𝜈,𝛽𝛾+Σ𝜈,𝛽𝛾) = 𝜖𝜈 (𝑦𝛽𝑈𝜈𝛾 − 𝑦𝛾𝑈𝜈𝛽 + Σ𝜈,𝛽𝛾)

𝜔(𝑦) → 𝑇 Λ 𝜔(Λ−1𝑦)

𝜖𝜈Σ𝜈,𝛽𝛾 = 𝑈𝛽𝛾 − 𝑈𝛾𝛽

(1) Spin must not be a hydro mode in a strict sense  Spin is not conserved if (canonical) 𝑈𝜈𝜉 has anti-symmetric part 𝑈

(a) 𝜈𝜉

 There’s no a priori reason (canonical) 𝑈𝜈𝜉 must be symmetric

(1) (1) ene nergy co cons nservation (2) (2) total ang ngula lar r mom

• mentum co

cons nservatio ion

∴

Consequence (2) Nevertheless, it behaves like a hydro mode if 𝑈

(a) 𝜈𝜉 ≪ 1

➡ inclusion of dissipative nature is important

(canonical)

Formulation of hydrodynamics with spin (3/4)

(1) define hydro variables

(2) simplify the tensor structure by thermodynamics 𝛾, 𝑣𝜈, 𝜕𝜈𝜉 Introduce spin chemical potential

4 + 6 = 10 DoGs = # of EoMs

 𝛾, 𝑣𝜈, 𝜕𝜈𝜉 are independent w/ each other at this stage (𝜕𝜈𝜉 ≠thermal vorticity) 𝑈𝜈𝜉 = 𝑓𝑣𝜈𝑣𝜉 + 𝑞 𝑕𝜈𝜉 + 𝑣𝜈𝑣𝜉 + 𝑈 1

𝜈𝜉 + 𝑃 𝜖2 ,

Generalizing 1st law of thermodynamics with spin as Expand 𝑈𝜈𝜉, Σ𝜈,𝛽𝛾, i.t.o derivatives 𝜖𝜈𝑇𝜈 = −𝑈 1s

𝜈𝜉 𝜖𝜈 𝛾𝑣𝜉 + 𝜖𝜉 𝛾𝑣𝜈

2 − 𝑈 1a

𝜈𝜉

𝜖𝜈 𝛾𝑣𝜉 − 𝜖𝜉 𝛾𝑣𝜈 2 − 2𝛾𝜕𝜈𝜉 + 𝑃(𝜖3) 𝑒𝑡 = 𝛾(𝑒𝑓 − 𝜕𝜈𝜉𝑒𝜏𝜈𝜉), 𝑡 = 𝛾(𝑓 + 𝑞 − 𝜕𝜈𝜉𝜏𝜈𝜉) With EoMs, div. of entropy current 𝑇𝜈 = 𝑡𝑣𝜈 + 𝑃(𝜖) can be evaluated as

Step 2: Construct a constitutive relation for 𝑈𝜈𝜉, Σ𝜈,𝛽𝛾

Σ𝜈,𝛽𝛾 = 𝑣𝜈𝜏𝛽𝛾 + 𝑃(𝜖1) where I defined sp spin in den density 𝝉𝜷𝜸  In global equilibrium 𝜖𝜈𝑇𝜈 = 0, so that 𝜕 = thermal vorticity.  2nd law of thermodynamics 𝜖𝜈𝑇𝜈 ≥ 0 gives strong constraint on 𝑈 1

𝜈𝜉

with 𝜕𝜈𝜉 = −𝜕𝜉𝜈

𝑈

(0) 𝜈𝜉 = 𝑓𝑣𝜈𝑣𝜉 + 𝑞(𝑕𝜈𝜉 + 𝑣𝜈𝑣𝜉)

 Constitutive relation for 𝑈𝜈𝜉 up to 1st order with spin

𝑈

(1) 𝜈𝜉 = −2𝜆 𝐸𝑣(𝜈 + 𝛾𝜖⊥ (𝜈𝛾−1 𝑣𝜉) − 2𝜃𝜖⊥ <𝜈𝑣𝜉> − 𝜂 𝜖𝜈𝑣𝜈 Δ𝜈𝜉

−2𝜇 −𝐸𝑣[𝜈 + 𝛾𝜖⊥

[𝜈𝛾−1 + 4𝑣𝜍𝜕𝜍[𝜈 𝑣𝜉] − 2𝛿 𝜖⊥ [𝜈𝑣𝜉] − 2Δ𝜍 𝜈Δ𝜇 𝜉𝜕𝜍𝜇

 Hydrodynamics equation up to 1st order with spin

hea eat t curr rren ent sh shea ear visc scous eff effect bulk visc scous eff effect “boost heat current” “rotational (spinning) viscous effect”

NEW !

 Relativistic generalization of a non-relativistic micropolar fluid  “boost heat current” is a relativistic effect

e.g. Eringen (1998); Lukaszewicz (1999)

𝜖𝜈(𝑣𝜈𝜏𝛽𝛾) = 𝑈

(1) 𝛽𝛾 − 𝑈 1 𝛾𝛽 + 𝑃(𝜖2)

0 = 𝜖𝜈(𝑈 0

𝜈𝜉 + 𝑈 1 𝜈𝜉 + 𝑃(𝜖2))

Formulation of hydrodynamics with spin (4/4)

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 equilibrium

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

hydro evolution 𝑗𝜖𝑢 𝜀𝛾 𝜀𝑣𝜈 𝜀𝜕𝜈𝜉 = 𝑁 𝜀𝛾 𝜀𝑣𝜈 𝜀𝜕𝜈𝜉 + 𝑃(𝜀2) 𝛾 = 𝛾0 𝑣𝜈 = 1, 𝟏 𝜕𝜈𝜉 = 0 𝛾 = 𝛾0 + 𝜀𝛾 𝑣𝜈 = 1, 𝟏 + 𝜀𝑣𝜈 𝜕𝜈𝜉 = 0 + 𝜀𝜕𝜈𝜉

Linear mode analysis (2/2)

 Hydro w/o spin 4 modes {𝛾, 𝑣𝜈}

2 sound modes 2 shear modes

 Hydro with spin + 6 dissipative modes {𝛾, 𝑣𝜈, 𝜕𝜈𝜉}

𝜕 = ±𝑑s𝑙 + 𝑃(𝑙2)

4 modes

2 sound modes 2 shear modes where 𝑑s

2 ≡ 𝜖𝑞/𝜖𝑓

3 “boost” modes 3 “spin” modes where 𝜐s ≡

Τ 𝜖𝜏𝑗𝑘 𝜖𝜕𝑗𝑘 4𝛿

, 𝜐b ≡

Τ 𝜖𝜏𝑗0 𝜖𝜕𝑗0 4𝜇

 We explicitly confirmed that spin is dissipative  The time-scale of the dissipation is controlled by the new viscous constants 𝛿, 𝜇 𝜕 = −2𝑗𝜐s

−1 + 𝑃(𝑙2)

𝜕 = −2𝑗𝜐b

−1 + 𝑃(𝑙2)

𝜕 = ±𝑑s𝑙 + 𝑃(𝑙2) 𝜕 = −𝑗 𝜃𝑙2 𝑓 + 𝑞 + 𝑃(𝑙4) 𝜕 = −𝑗 𝜃𝑙2 𝑓 + 𝑞 + 𝑃(𝑙4)

Outline

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

• 4. Summary

Summary

 Relativistic spin hydrodynamics with 1st order dissipative corrections is formulated based on the phenomenological entropy-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  Spin polarization of QGP is one of the hottest topics in

• HIC. But, its theory, in particular hydrodynamic

framework, has not been developed well Outlook: extension to 2nd order, Kubo formula, application to cond-mat, numerical simulations, and start something new with you!