Operator Size and Symmetry Ying Zhao IAS <latexit - - PowerPoint PPT Presentation

Operator Size and Symmetry

Ying Zhao IAS

arXiv:1904.12820

• H. Lin, J. Maldacena, Y. Zhao
• L. Susskind arXiv:1904.12819

Number of simple operators affected S∞(ψ1) = 1

S∞(ψ1(u)) = X

k

X

i1<...<ik

k| √ 2ci1...ik|2

ψ1(u) = X

k

X

i1<...<ik

ci1...ik ⇣ 2

k 2 ψi1...ψik

⌘

Operator size

X.-L. Qi, A. Streicher arXiv:1810.11958

• D. Roberts, D. Stanford, A. Streicher

arXiv:1812.02633

S∞(O) = 1 2

N

X

i=1

2− N

2 tr([O, ψi]†[O, ψi])

Operator size at infinite temperature Size operator

ˆ S = X

j

iψj

l ψj r + N

2

S∞(O) = hI|O†

r ˆ

SOr|Ii

Falling toward black holes

• Y. Sekino, L. Susskind arXiv:0808.2096
• L. Susskind arXiv:1802.01198

Scrambling Local perturbation spreads

δSe

2π β t∗ = S,

t∗ = β 2π log S δS

Scrambling time Size growth

e

2π β t

δS

eiHτ

e−iHτ

W

τ = 1

τ = 2

τ = 3

τ

τ = 2π β t

Operator size at finite temperature

X.-L. Qi, A. Streicher arXiv:1810.11958

Sβ(O) = δ−1

β

h S∞(Oρ

1 2 ) − S∞(ρ 1 2 )

i

Sβ(O) = δ−1

β

" hTFD|O†

r ˆ

SOr|TFDi hTFD|O†

rOr|TFDi

hTFD| ˆ S|TFDi #

δβ = 2G(β 2 )

r = r+ r ∼ 2r+ − r− r ∼ 2r+ Newtonian ∆ρ ∼ r+ ∆ρ ∼ r+ log[β/r+]

= ( 2J2t2 t < β Throat regime

β2J2 2π2 e

2π β t

t > β Rindler regime

size = P ˜ β

arXiv:1804.04156

• A. Brown, H. Gharibyan, A. Streicher
• L. Susskind, L. Thorlacius, Y. Zhao

Sβ(ψ(u)) = 1 + 2 βJ π sinh(πt β ) 2

[ ˆ P, ˆ B] = −i ˆ E

Symmetry generators in JT gravity

[ ˆ P, ˆ E] = −i ˆ B

[ ˆ B, ˆ E] = i ˆ P

Xr

Xl

Xr

Xr

Xl

Xl

ˆ B

ˆ P

ˆ E

ˆ B = β 2π (Hr − Hl)

ˆ P = −i[ ˆ B, ˆ E] = −iµ β2 4π2 X

j

⇣ ψj

l ˙

ψj

r − ˙

ψj

l ψj r

⌘

ˆ E = β 2π 2 4Hr + Hl + iµ X

j

ψj

l ψj r hHr + Hl + iµ

X

j

ψj

l ψj riTFD

3 5

µ J = αs ∆ ✓ 2π βJ ◆2 1 G( β

2 )

Sβ(Ur) = ∆ 2αs βJ 2π hTFD|U†

r( ˆ

E ˆ B)Ur|TFDi

Approximate symmetry generators in SYK

X.-L. Qi, arXiv:1804.00491

• J. Maldacena,

Sβ(U) = δ−1

β

h hTFD|U†

r ˆ

SUr|TFDi hTFD| ˆ S|TFDi i

δβ = 2G(β 2 )

ˆ S = X

j

iψj

l ψj r + N

2

Example 1

ur

ϕb (ϕ )

|O

b

(a)

ϕt ϕt ϕ b

O(ϕ )

t |G ( ,0)|O (ϕ ) b

π

Α

(b) (c) O(ϕ )

t |

⇣ ˆ B, ˆ P, ˆ E ⌘ = ∆ 1 tan ϕt−ϕb

2

, −i∆sin ϕt+ϕb

2

sin ϕt−ϕb

2

, ∆cos ϕt+ϕb

2

sin ϕt−ϕb

2

!

Example 1

ur

⇣ ˆ B, ˆ P, ˆ E ⌘ ≈ ∆2βJ 2π ✓ 1, sinh(2π β ur), cosh(2π β ur) ◆

Sβ(ψ(ur)) = ∆ 2αs βJ 2π ( ˆ E − ˆ B) = 2∆2 αs ✓βJ 2π ◆2 sinh2 ✓π β ur ◆

ϕb = 2π β ✓ − 1 2J + iur ◆

ϕt = 2π β ✓ 1 2J + iur ◆

ψ

ψ

Throat

Rindler

ρ

Example 2

ρm

ρm = − log ⇣ tan(ϕt 2 ) ⌘

⇣ ˆ B, ˆ P, ˆ E ⌘ = (∆ sinh ρm, 0, ∆ cosh ρm)

Sβ(ρm) = ∆2 2αs βJ 2π e−ρm

= ⇒ ˜ βJ

Size growth from SL(2) algebra

d dρSβ( ˜ ψ(ρ)) = ∆2 2αs βJ 2π 2hTFD| ˜ ψ(ρ)[i ˆ P, ˆ E ˆ B] ˜ ψ(ρ)|TFDi

= −Sβ( ˜ ψ(ρ))

[ ˆ P, ˆ E − ˆ B] = i( ˆ E − ˆ B)

Example 1 revisited

Throat

Rindler

ρ

Example 2 revisited

˜ ψ(ρ) = e−i ˆ

P (ρ−ρb)ψei ˆ P (ρ−ρb)

∂Sβ(U(ur)) ∂ur = i ∆ 2αs βJ 2π hUr(ur)† 2π β ˆ B, ˆ E ˆ B

• Ur(ur)i = ∆

2αs J hUr(ur)† ˆ PUr(ur)i

Saturation of symmetry generators

(a) (b) (c)

5 10 15 0.2 0.4 0.6 0.8 1.0

P/ ✓S − S0 π ◆

5 10 15 0.2 0.4 0.6 0.8 1.0

E/ ✓S − S0 π ◆

Thank you.