Eigenstate Thermalization in Sachdev-Ye-Kitaev model & Schwarzian theory
by Pranjal Nayak
[basesd on 1901.xxxxx] with Julian Sonner and Manuel Vielma
University of Michigan, Ann Arbor January 23, 2019 Closed Quantum Systems
Quantum Mechanics is unitary!
(t) = (t, t ) (t ) |
How come we observe thermal physics?
How come we observe black hole formation? Plan of the talk
✓ Review of ETH
✓ Review of Sachdev-Ye-Kitaev model
✓ Numerical studies of ETH in SYK model
✓ ETH in the Schwarzian sector of the SYK model
✓ ETH in the Conformal sector of the SYK model
✓ A few thoughts on the bulk duals
✓ Summary and conclusion
!3 Eigenstate Thermalization Hypothesis (ETH) Quantum Thermalization
5 Quantum Thermalization
Classical thermalization: ergodicity & ⇐ chaos
5 Quantum Thermalization
Classical thermalization: ergodicity & ⇐ chaos
v.s.
Quantum thermalization: Eigenstate Thermalisation Hypothesis
S(E)/2 m n = (E) + e f(E,!)R h |O| i Omc mn mn
5 Quantum Thermalization
Classical thermalization: ergodicity & ⇐ chaos
v.s.
Quantum thermalization: Eigenstate Thermalisation Hypothesis
S(E)/2 m n = (E) + e f(E,!)R h |O| i Omc mn mn micro-canonical ensemble avg. energy of ensemble
5 Quantum Thermalization
Classical thermalization: ergodicity & ⇐ chaos
v.s.
Quantum thermalization: Eigenstate Thermalisation Hypothesis
S(E)/2 m n = (E) + e f(E,!)R h |O| i Omc mn mn micro-canonical entropy 1 in RMT random matrix
5 Quantum Thermalization
Classical thermalization: ergodicity & ⇐ chaos
v.s.
Quantum thermalization: Eigenstate Thermalisation Hypothesis
S(E)/2 m n = (E) + e f(E,!)R h |O| i Omc mn mn micro-canonical entropy 1 in RMT random matrix
A generic excited state will then thermalise by dephasing it(E E ) S = c⇤c e i j O E¯ + e h |O| i i j Oij !
!7 SYK model: a new Guinea Pig
1 J J4,5,7,8 10 1,2,4,6 7 4 9 6 3 J7,9,10,11 2 8 11 5 J2,3,5,6
[Sachdev-Ye ’93; Kitaev ’15]
!7 SYK model: a new Guinea Pig
1 J J4,5,7,8 10 1,2,4,6 7 4 9 6 3 J7,9,10,11 2 8 11 5 J2,3,5,6
[Sachdev-Ye ’93; ‣ a model of N Majorana fermions Kitaev ’15] ‣ with all-to-all couplings ‣ and quenched random couplings
!7 SYK model: a new Guinea Pig
1 J J4,5,7,8 10 1,2,4,6 7 4 9 6 3 J7,9,10,11 2 8 11 5 J2,3,5,6
[Sachdev-Ye ’93; ‣ a model of N Majorana fermions Kitaev ’15] ‣ with all-to-all couplings ‣ and quenched random couplings
H = − J ψ ψ ψ ψ ∑ i1i2i3i4 i1 i2 i3 i4 1≤i1 where, J is chosen from a Gaussian ensemble: 2 2 3! J ⟨Jijkl⟩ = 0 ⟨J ⟩ = ijkl N3 !7 Solvable Limit of SYK [Sachdev ’15; Parcollet, Georges ’00; Kitaev ’15; Polchinski, Rosenhaus ’16; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16] 8 Solvable Limit of SYK [Sachdev ’15; Parcollet, Georges ’00; Kitaev ’15; Polchinski, Rosenhaus ’16; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16] ‣ at large-N, it is often useful to use bi-local effective action 2 Scol J q 1 − = log Pf [∂ − Σ] + dτdτ′| G(τ, τ′) | − dτdτ′Σ (τ′, τ)G(τ, τ′) N τ 2q ∫ 2 ∫ 8 Solvable Limit of SYK [Sachdev ’15; Parcollet, Georges ’00; Kitaev ’15; Polchinski, Rosenhaus ’16; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16] ‣ at large-N, it is often useful to use bi-local effective action 2 Scol J q 1 − = log Pf [∂ − Σ] + dτdτ′| G(τ, τ′) | − dτdτ′Σ (τ′, τ)G(τ, τ′) N τ 2q ∫ 2 ∫ 1 is the Lagrange multiplier that imposes Σ(τ, τ′) G(τ, τ′) = ∑ ψi(τ) ψi(τ′) N i 8 Solvable Limit of SYK [Sachdev ’15; Parcollet, Georges ’00; Kitaev ’15; Polchinski, Rosenhaus ’16; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16] ‣ at large-N, it is often useful to use bi-local effective action 2 Scol J q 1 − = log Pf [∂ − Σ] + dτdτ′| G(τ, τ′) | − dτdτ′Σ (τ′, τ)G(τ, τ′) N τ 2q ∫ 2 ∫ 1 is the Lagrange multiplier that imposes Σ(τ, τ′) G(τ, τ′) = ∑ ψi(τ) ψi(τ′) N i ‣ In the large-N limit, the saddle point equations are given by, ∂τ G(τ, τ′) − [Σ * G](τ, τ′) = δ(τ − τ′) 2 q−1 J G (τ, τ′) = Σ(τ, τ′) 8 Solvable Limit of SYK [Sachdev ’15; Parcollet, Georges ’00; Kitaev ’15; Polchinski, Rosenhaus ’16; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16] ‣ at large-N, it is often useful to use bi-local effective action 2 Scol J q 1 − = log Pf [∂ − Σ] + dτdτ′| G(τ, τ′) | − dτdτ′Σ (τ′, τ)G(τ, τ′) N τ 2q ∫ 2 ∫ 1 is the Lagrange multiplier that imposes Σ(τ, τ′) G(τ, τ′) = ∑ ψi(τ) ψi(τ′) N i ‣ In the large-N limit, the saddle point equations are given by, ∂τ G(τ, τ′) − [Σ * G](τ, τ′) = δ(τ − τ′) 2 q−1 J G (τ, τ′) = Σ(τ, τ′) captures Melonic Physics! 8 Emergent Symmetry of SYK [Kitaev ’15; Polchinski, Rosenhaus ’16; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16] 9 Emergent Symmetry of SYK [Kitaev ’15; Polchinski, Rosenhaus ’16; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16] ‣ In the deep IR limit, | τ | J ≫ 1 , the saddle point equations can be replaced by ∂τ G(τ, τ′) − [Σ * G](τ, τ′) = δ(τ − τ′) 2 q−1 J G (τ, τ′) = Σ(τ, τ′) 9 Emergent Symmetry of SYK [Kitaev ’15; Polchinski, Rosenhaus ’16; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16] ‣ In the deep IR limit, | τ | J ≫ 1 , the saddle point equations can be replaced by ∂τ G(τ, τ′) − [Σ * G](τ, τ′) = δ(τ − τ′) 2 q−1 J G (τ, τ′) = Σ(τ, τ′) ‣ The action and the saddle point equations are symmetric under reparametrizations: 2/q Conformal Symmetry G(τ, τ′) → (f′( τ) f′( τ′) ) G (f(τ), f(τ′) ) of 1D 2(1−1/q) Σ(τ, τ′) → (f′( τ) f′( τ′) ) Σ (f(τ), f(τ′) ) 9 Emergent Symmetry of SYK [Kitaev ’15; Polchinski, Rosenhaus ’16; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16] ‣ In the deep IR limit, | τ | J ≫ 1 , the saddle point equations can be replaced by ∂τ G(τ, τ′) − [Σ * G](τ, τ′) = δ(τ − τ′) 2 q−1 J G (τ, τ′) = Σ(τ, τ′) ‣ The action and the saddle point equations are symmetric under reparametrizations: 2/q Conformal Symmetry G(τ, τ′) → (f′( τ) f′( τ′) ) G (f(τ), f(τ′) ) of 1D 2(1−1/q) Σ(τ, τ′) → (f′( τ) f′( τ′) ) Σ (f(τ), f(τ′) ) ‣ This symmetry is explicitly broken by considering the 1/J corrections. 9 Schwarzian in SYK [Kitaev ’15; Maldacena, Stanford ’16] 10 Schwarzian in SYK [Kitaev ’15; Maldacena, Stanford ’16] ‣ At low-energy break conformal symmetry. Leading soft- mode physics G0[f] f G0 G0'[f] f G ' 0 Schwarzian action 10 Schwarzian in SYK [Kitaev ’15; Maldacena, Stanford ’16] ‣ At low-energy break conformal symmetry. Leading soft- mode physics G0[f] f G0 G0'[f] f G ' 0 Schwarzian action ‣ Effective action on the ‘reparametrization modes’ f(τ) 1 exp − dτ {f(τ), τ} ∫ ��(2,ℝ) [ g2 ∫ ] 2 2 βJ f′′ ′ ( τ) 3 f′′ ( τ) g ∼ where, {f(τ), τ} = − N f′( τ) 2 ( f′( τ) ) 10 Spectrum [Kitaev ’15; Polchinski, Rosenhaus ’15; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16; Gross, Rosenhaus ’17] 11 Spectrum [Kitaev ’15; Polchinski, Rosenhaus ’15; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16; Gross, Rosenhaus ’17] discrete tower of states 2n+1 �n ∼ ψi ∂ ψi hn = 2n + 1 + ϵn 11 Spectrum [Kitaev ’15; Polchinski, Rosenhaus ’15; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16; Gross, Rosenhaus ’17] continuum: Schwarzian discrete tower of states 1 S = − dτ {f(τ), τ} � ∼ ψ ∂2n+1ψ g2 ∫ n i i Diff 1 f(τ) ∈ (S ) hn = 2n + 1 + ϵn 11 Spectrum [Kitaev ’15; Polchinski, Rosenhaus ’15; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16; Gross, Rosenhaus ’17] continuum: Schwarzian discrete tower of states 1 S = − dτ {f(τ), τ} � ∼ ψ ∂2n+1ψ g2 ∫ n i i Diff 1 f(τ) ∈ (S ) hn = 2n + 1 + ϵn 2D BH 11 Spectrum [Kitaev ’15; Polchinski, Rosenhaus ’15; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16; Gross, Rosenhaus ’17] continuum: Schwarzian discrete tower of states 1 S = − dτ {f(τ), τ} � ∼ ψ ∂2n+1ψ g2 ∫ n i i Diff 1 f(τ) ∈ (S ) hn = 2n + 1 + ϵn Limit of conformal six-pt functions OPE coeffs. 2D BH ETH 11 Spectrum [Kitaev ’15; Polchinski, Rosenhaus ’15; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16; Gross, Rosenhaus ’17] continuum: Schwarzian discrete tower of states 1 S = − dτ {f(τ), τ} � ∼ ψ ∂2n+1ψ g2 ∫ n i i Diff 1 f(τ) ∈ (S ) hn = 2n + 1 + ϵn Ward identities for Limit of conformal Schwarzian six-pt functions ~ monodromy method OPE coeffs. 2D BH weak ETH ETH 11 Spectrum [Kitaev ’15; Polchinski, Rosenhaus ’15; Jevicki, Suzuki, Yoon ’16; Maldacena, Stanford ’16; Gross, Rosenhaus ’17] continuum: Schwarzian discrete tower of states 1 S = − dτ {f(τ), τ} � ∼ ψ ∂2n+1ψ g2 ∫ n i i Diff 1 f(τ) ∈ (S ) hn = 2n + 1 + ϵn Ward identities for Limit of conformal Schwarzian six-pt functions ~ monodromy method OPE coeffs. 2D BH weak ETH ETH [PN, Julian Sonner & Manuel Vielma] 11 ETH in the SYK model: A numerical study Study by Exact Diagonalization 13 Study by Exact Diagonalization ‣ Solve SYK in exact diagonalization [Sonner, Vielma ’17] ETH is true in SYK! 13 Study by Exact Diagonalization ‣ Solve SYK in exact diagonalization [Sonner, Vielma ’17] ETH is true in SYK! S(E)/2 m n = (E) + e f(E,!)R h |O| i Omc mn mn 13 Study by Exact Diagonalization ‣ Solve SYK in exact diagonalization [Sonner, Vielma ’17] ETH is true in SYK! S(E)/2 m n = (E) + e f(E,!)R h |O| i Omc mn mn number operator, , at some site nk̂ k 13 Study by Exact Diagonalization ‣ Solve SYK in exact diagonalization [Sonner, Vielma ’17] ETH is true in SYK! S(E)/2 m n = (E) + e f(E,!)R h |O| i Omc mn mn number operator, , at some site nk̂ k 13 RMT or Not? ‣ Need to look at the off-diagonal matrix elements S(E)/2 m n = mc(E) mn + e f(E,!)Rmn h |O| i O !14 RMT or Not? ‣ Need to look at the off-diagonal matrix elements S(E)/2 m n = mc(E) mn + e f(E,!)Rmn h |O| i O Crossover from constant (RMT) to non-constant behaviour! !14 RMT or Not? ‣ Need to look at the off-diagonal matrix elements S(E)/2 m n = mc(E) mn + e f(E,!)Rmn h |O| i O Crossover from constant (RMT) to non-constant behaviour! Thouless energy !14 More evidence of ETH [Sonner, Vielma ’17] Compare OTOC in eigenstates to thermal result Become essentially indistinguishable as system size increases 2⇡ Conjecture: ETH = L (E) 9 !15 ? Can we understand the numerically observed thermalization in a finer detail? Analytically? ? Can we understand the numerically observed thermalization in a finer detail? Analytically? ? Can we understand how and why the results deviate from RMT behaviour? ? Can we understand the numerically observed thermalization in a finer detail? Analytically? ? Can we understand how and why the results deviate from RMT behaviour? ? Can we do a dual bulk computation to understand black hole formation in 2 dimensions? ETH in the Schwarzian Limit Correlation Functions !18 Correlation Functions • Correlation functions of the fermions as well as of the excited operators in SYK spectrum get contribution from: !18 Correlation Functions • Correlation functions of the fermions as well as of the excited operators in SYK spectrum get contribution from: τ1 τ3 ✦ Exchange of Schwarzian modes h1 h2 � � � � = τ2 τ4 ⟨ 1 2 3 4⟩ h1 h2 f(τ) 1 f′( τ1)f′( τ2) f′(τ3)f′(τ4) exp − dτ {f(τ), τ} ∫ [ 2 ∫ ] 2 2 ��(2,ℝ) g ( | f(τ1) − f(τ2)| ) ( | f(τ3) − f(τ4)| ) !18 Correlation Functions • Correlation functions of the fermions as well as of the excited operators in SYK spectrum get contribution from: τ1 τ3 ✦ Exchange of Schwarzian modes h1 h2 � � � � = τ2 τ4 ⟨ 1 2 3 4⟩ h1 h2 f(τ) 1 f′( τ1)f′( τ2) f′(τ3)f′(τ4) exp − dτ {f(τ), τ} ∫ [ 2 ∫ ] 2 2 ��(2,ℝ) g ( | f(τ1) − f(τ2)| ) ( | f(τ3) − f(τ4)| ) ✦ Exchange of excited modes (conformal limit) τ1 τ3 τ1 τ3 τ2 τ τ4 2 τ4 !18 Correlation Functions • Correlation functions of the fermions as well as of the excited operators in SYK spectrum get contribution from: τ1 τ3 ✦ Exchange of Schwarzian modes h1 h2 � � � � = τ2 τ4 ⟨ 1 2 3 4⟩ h1 h2 f(τ) 1 f′( τ1)f′( τ2) f′(τ3)f′(τ4) exp − dτ {f(τ), τ} ∫ [ 2 ∫ ] 2 2 ��(2,ℝ) g ( | f(τ1) − f(τ2)| ) ( | f(τ3) − f(τ4)| ) ✦ Exchange of excited modes (conformal limit) τ1 τ3 τ1 τ3 τ2 τ τ4 2 τ4 !18 The Schwarzian theory !19 The Schwarzian theory • For large but finite values of SYK coupling J, the conformal symmetry is explicitly broken G0[f] f G0 f(τ) 1 G0'[f] exp − dτ f(τ), τ f 2 { } ∫ ��(2,ℝ) [ g ∫ ] G0' Schwarzian action βJ g2 ∼ N !19 The Schwarzian theory • For large but finite values of SYK coupling J, the conformal symmetry is explicitly broken G0[f] f G0 f(τ) 1 G0'[f] exp − dτ f(τ), τ f 2 { } ∫ ��(2,ℝ) [ g ∫ ] G0' Schwarzian action βJ g2 ∼ N • f ( τ ) are the pseudo-Goldstone modes The leading contribution to the physical observables is due to exchange of these modes f(τ) 1 ⋅ = exp − dτ {f(τ), τ} ⋅ ⟨ ⟩ ∫ ��(2,ℝ) [ g2 ∫ ] ( ) !19 Ward Identities [PN, Sonner, Vielma ’19] !20 Ward Identities [PN, Sonner, Vielma ’19] • Within Schwarzian theory, one can derive the following ‘Ward’ identity m n h δ(x − x )+δ(x − x ) + sgn(x − x ) − sgn(x − x ) ∂ ⟨� x , x ⟩ ∑ [ i( 2i−1 2i ) ( 2i−1 2i ) x2i] ∏ j ( 2j−1 2j) i=1 j=1 1 n = − ⟨Sch(x) �i(x2j−1, x2j)⟩ + κ3 g2 ∏ i=1 �j(x2j−1, x2j) ≡ �hj(x2j−1)�hj(x2j) !20 Ward Identities [PN, Sonner, Vielma ’19] • Within Schwarzian theory, one can derive the following ‘Ward’ identity m n h δ(x − x )+δ(x − x ) + sgn(x − x ) − sgn(x − x ) ∂ ⟨� x , x ⟩ ∑ [ i( 2i−1 2i ) ( 2i−1 2i ) x2i] ∏ j ( 2j−1 2j) i=1 j=1 1 n = − ⟨Sch(x) �i(x2j−1, x2j)⟩ + κ3 g2 ∏ i=1 �j(x2j−1, x2j) ≡ �hj(x2j−1)�hj(x2j) • We are interested in computing ⟨HHLL⟩ 2 where, ‘heavy’ operators are defined by hi ∼ 1/g !20 Ward Identities [PN, Sonner, Vielma ’19] • Within Schwarzian theory, one can derive the following ‘Ward’ identity m n h δ(x − x )+δ(x − x ) + sgn(x − x ) − sgn(x − x ) ∂ ⟨� x , x ⟩ ∑ [ i( 2i−1 2i ) ( 2i−1 2i ) x2i] ∏ j ( 2j−1 2j) i=1 j=1 1 n = − ⟨Sch(x) �i(x2j−1, x2j)⟩ + κ3 g2 ∏ i=1 �j(x2j−1, x2j) ≡ �hj(x2j−1)�hj(x2j) • We are interested in computing ⟨HHLL⟩ 2 where, ‘heavy’ operators are defined by hi ∼ 1/g • In the semi-classical limit, g 2 → 0 , 2 {f(τ), τ} = − g ∑ [κ1,iδ(τ − τ2i−1) + κ1,iδ(τ − τ2i) i −κ sgn(τ − τ ) − sgn(τ − τ ) + g2κ 2,i( 2i−1 2i )] 3 !20 Ward Identities [PN, Sonner, Vielma ’19] • Within Schwarzian theory, one can derive the following ‘Ward’ identity m n h δ(x − x )+δ(x − x ) + sgn(x − x ) − sgn(x − x ) ∂ ⟨� x , x ⟩ ∑ [ i( 2i−1 2i ) ( 2i−1 2i ) x2i] ∏ j ( 2j−1 2j) i=1 j=1 1 n = − ⟨Sch(x) �i(x2j−1, x2j)⟩ + κ3 g2 ∏ i=1 �j(x2j−1, x2j) ≡ �hj(x2j−1)�hj(x2j) • We are interested in computing ⟨HHLL⟩ 2 where, ‘heavy’ operators are defined by hi ∼ 1/g can be obtained • In the semi-classical limit, g 2 → 0 , from saddle point of path 2 integral {f(τ), τ} = − g ∑ [κ1,iδ(τ − τ2i−1) + κ1,iδ(τ − τ2i) i −κ sgn(τ − τ ) − sgn(τ − τ ) + g2κ 2,i( 2i−1 2i )] 3 !20 Ward Identities [PN, Sonner, Vielma ’19] • Within Schwarzian theory, one can derive the following ‘Ward’ identity m n h δ(x − x )+δ(x − x ) + sgn(x − x ) − sgn(x − x ) ∂ ⟨� x , x ⟩ ∑ [ i( 2i−1 2i ) ( 2i−1 2i ) x2i] ∏ j ( 2j−1 2j) i=1 j=1 1 n = − ⟨Sch(x) �i(x2j−1, x2j)⟩ + κ3 g2 ∏ i=1 �j(x2j−1, x2j) ≡ �hj(x2j−1)�hj(x2j) • We are interested in computing ⟨HHLL⟩ 2 where, ‘heavy’ operators are defined by hi ∼ 1/g can be obtained • In the semi-classical limit, g 2 → 0 , from saddle point of path 2 integral will be used {f(τ), τ} = − g ∑ [κ1,iδ(τ − τ2i−1) + κ1,iδ(τ − τ2i) to solve for i f(τ) −κ sgn(τ − τ ) − sgn(τ − τ ) + g2κ 2,i( 2i−1 2i )] 3 !20 Hill’s Eq. & Monodromy !21 Hill’s Eq. & Monodromy • The solution for f ( τ ) can be found by solving the Hill’s equation: 1 p′′ ( τ) + T(τ) p(τ) = 0 2 where, T(τ) = − g2 κ δ(τ − τ ) + κ δ(τ − τ ) ∑ [ 1,i 2i−1 1,i 2i i 2 −κ2,i(sgn(τ − τ2i−1) − sgn(τ − τ2i))] + g κ3 !21 Hill’s Eq. & Monodromy • The solution for f ( τ ) can be found by solving the Hill’s equation: 1 p′′ ( τ) + T(τ) p(τ) = 0 2 where, T(τ) = − g2 κ δ(τ − τ ) + κ δ(τ − τ ) ∑ [ 1,i 2i−1 1,i 2i i 2 −κ2,i(sgn(τ − τ2i−1) − sgn(τ − τ2i))] + g κ3 • We do the computation with a finite background temperature, T = 1/β impose trivial monodromy around the thermal circle !21 Hill’s Eq. & Monodromy • The solution for f ( τ ) can be found by solving the Hill’s equation: 1 p′′ ( τ) + T(τ) p(τ) = 0 2 where, T(τ) = − g2 κ δ(τ − τ ) + κ δ(τ − τ ) ∑ [ 1,i 2i−1 1,i 2i i 2 −κ2,i(sgn(τ − τ2i−1) − sgn(τ − τ2i))] + g κ3 • We do the computation with a finite background temperature, T = 1/β impose trivial monodromy around the thermal circle O !21 O where, T(τ) = − g2 κ δ(τ − τ ) + κ δ(τ − τ ) ∑ [ 1,i 2i−1 1,i 2i i 2 −κ2,i(sgn(τ − τ2i−1) − sgn(τ − τ2i))] + g κ3 • We do the computation with a finite background temperature, T = 1/β impose trivial monodromy around the thermal circle ⟨�H(τ1)�H(τ2)⟩ where, T(τ) = − g2 κ δ(τ − τ ) + κ δ(τ − τ ) ∑ [ 1,i 2i−1 1,i 2i i 2 −κ2,i(sgn(τ − τ2i−1) − sgn(τ − τ2i))] + g κ3 • We do the computation with a finite background temperature, T = 1/β impose trivial monodromy around the thermal circle ⟨�H(τ1)�H(τ2)⟩ !22 Effective Temperature [PN, Sonner, Vielma ’19] • We find: !22 Effective Temperature [PN, Sonner, Vielma ’19] • We find: ✦ 2 T e f f > T and Teff ∝ g hH !22 Effective Temperature [PN, Sonner, Vielma ’19] • We find: ✦ 2 T e f f > T and Teff ∝ g hH ✦ Teff ∝ T !22 Effective Temperature [PN, Sonner, Vielma ’19] Light operators see an Effective temperature • We find: due to the presence of heavy operators ✦ 2 T e f f > T and Teff ∝ g hH ✦ Teff ∝ T !22 Effective Temperature [PN, Sonner, Vielma ’19] Light operators see an Effective temperature • We find: due to the presence of heavy operators ✦ 2 T e f f > T and Teff ∝ g hH but not when the ✦ Teff ∝ T !!! background temperature goes to ZERO! Effective temperature goes to ZERO with Background temperature !22 Effective Temperature [PN, Sonner, Vielma ’19] Light operators see an Effective temperature • We find: due to the presence of heavy operators ✦ 2 T e f f > T and Teff ∝ g hH but not when the ✦ Teff ∝ T !!! background temperature goes to ZERO! Effective temperature goes to ZERO with Weak ETH! Background temperature !22 A New Order of Limits !23 A New Order of Limits • Recall, in the previous computation we first took, N → ∞ followed by βJ → ∞ βJ ⇒ g2 ∼ → 0 N !23 A New Order of Limits • Recall, in the previous computation we first took, N → ∞ followed by βJ → ∞ βJ ⇒ g2 ∼ → 0 N • Alternatively, we could have considered βJ N , β J → ∞ simultaneously such that g2 ∼ stays finite followed by g → 0 N !23 A New Order of Limits • Recall, in the previous computation we first took, N → ∞ followed by βJ → ∞ βJ ⇒ g2 ∼ → 0 N • Alternatively, we could have considered βJ N , β J → ∞ simultaneously such that g2 ∼ stays finite followed by g → 0 N 2hL ⟨� (τ ) � (τ )� (τ ) � (τ )⟩ π Teff H 1 L 3 L 4 H 2 = ⟨�H(τ1)�H(τ2)⟩ sinh (πTeff |τ34 |) τ1 < τ3, τ4 < τ2 [LMTV ’18; Altland, Bagrets, Kamanev ’16] !23 Schwarzian Liouville !24 Schwarzian Liouville • Schwarzian theory is closely related to 1-D Liouville theory, [Altland, Bagrets, Kamanev ’16] as well as to dimensional reduction of 2-D Liouville theory [Lam, Mertens, Turiaci, Verlinde ’18] !24 Schwarzian Liouville • Schwarzian theory is closely related to 1-D Liouville theory, [Altland, Bagrets, Kamanev ’16] as well as to dimensional reduction of 2-D Liouville theory [Lam, Mertens, Turiaci, Verlinde ’18] • There is another class of states in the Liouville theory with energy E, Eigenfunctions of Liouville QM ⇄ FZZ boundary states in 2D Liouville !24 Schwarzian Liouville • Schwarzian theory is closely related to 1-D Liouville theory, [Altland, Bagrets, Kamanev ’16] as well as to dimensional reduction of 2-D Liouville theory [Lam, Mertens, Turiaci, Verlinde ’18] • There is another class of states in the Liouville theory with energy E, Eigenfunctions of Liouville QM ⇄ FZZ boundary states in 2D Liouville These states thermalize with 2g2E T˜ = eff 2π [Lam, Mertens, Turiaci, Verlinde ’18] !24 ETH in the Conformal Limit Correlation Functions • Correlation functions of the fermions as well as of the excited operators in SYK spectrum get contribution from: τ1 τ3 ✦ Exchange of Schwarzian modes h1 h2 � � � � = τ2 τ4 ⟨ 1 2 3 4⟩ h1 h2 f(τ) 1 f′( τ1)f′( τ2) f′(τ3)f′(τ4) exp − dτ {f(τ), τ} ∫ [ 2 ∫ ] 2 2 ��(2,ℝ) g ( | f(τ1) − f(τ2)| ) ( | f(τ3) − f(τ4)| ) ✦ Exchange of excited modes (conformal limit) τ1 τ3 τ1 τ3 τ2 τ τ4 2 τ4 !26 Correlation Functions • Correlation functions of the fermions as well as of the excited operators in SYK spectrum get contribution from: τ1 τ3 ✦ Exchange of Schwarzian modes h1 h2 � � � � = τ2 τ4 ⟨ 1 2 3 4⟩ h1 h2 f(τ) 1 f′( τ1)f′( τ2) f′(τ3)f′(τ4) exp − dτ {f(τ), τ} ∫ [ 2 ∫ ] 2 2 ��(2,ℝ) g ( | f(τ1) − f(τ2)| ) ( | f(τ3) − f(τ4)| ) ✦ Exchange of excited modes (conformal limit) τ1 τ3 τ1 τ3 τ2 τ τ4 2 τ4 !26 A Study of 3-point fns. !27 A Study of 3-point fns. discrete tower of states τ1 2n+1 τ3 �n ∼ ψi ∂ ψi τ 2 hn = 2n + 1 + ϵn • In the conformal limit, we consider the 3-point correlation functions. [Jevicki, Suzuki, Yoon ’16; Gross, Rosenhaus ’17] !27 A Study of 3-point fns. discrete tower of states τ1 2n+1 τ3 �n ∼ ψi ∂ ψi τ 2 hn = 2n + 1 + ϵn • In the conformal limit, we consider the 3-point correlation functions. [Jevicki, Suzuki, Yoon ’16; Gross, Rosenhaus ’17] • We are interested in the limit, m, n → ∞ ≫ k ∼ �(1) !27 A Study of 3-point fns. discrete tower of states τ1 2n+1 τ3 �n ∼ ψi ∂ ψi τ 2 hn = 2n + 1 + ϵn • In the conformal limit, we consider the 3-point correlation functions. [Jevicki, Suzuki, Yoon ’16; Gross, Rosenhaus ’17] • We are interested in the limit, m, n → ∞ ≫ k ∼ �(1) Γ(4E − 1) ⟨� � � ⟩ = �(E, d) × 2−2(2E−1) 2πE m k n [ Γ(2E − d)Γ(2E + d) ] −E ln 2 E → ∞ fk(E) δm,n + e R(E, d) [PN, Sonner, Vielma ’19] m + n E = , d = n − m 2 !27 A Study of 3-point fns. discrete tower of states τ1 2n+1 τ3 �n ∼ ψi ∂ ψi τ 2 hn = 2n + 1 + ϵn • In the conformal limit, we consider the 3-point correlation functions. [Jevicki, Suzuki, Yoon ’16; Gross, Rosenhaus ’17] • We are interested in the limit, m, n → ∞ ≫ k ∼ �(1) Γ(4E − 1) ⟨� � � ⟩ = �(E, d) × 2−2(2E−1) 2πE m k n [ Γ(2E − d)Γ(2E + d) ] −E ln 2 E → ∞ fk(E) δm,n + e R(E, d) [PN, Sonner, Vielma ’19] m + n E = , d = n − m 2 conformal sector of SYK satisfies ETH !27 A Bulk Story? Dual of Schwarzian theory in 1st Limit? !29 Dual of Schwarzian theory in 1st Limit? [Vos ’18] !29 Dual of Schwarzian theory in 2nd Limit? !30 Dual of SYK theory in Conformal Limit? BH geometry? !31 Dual of SYK theory in Conformal Limit? BH geometry? Can the ETH in conformal sector be understood as a correction? !31 Summary • We showed that Eigenstate thermalization can be studied analytically in SYK model in 2 limits: ✦ Schwarzian limit: where the contribution of pseudo-Goldstone modes is considered ‣ First Thermodynamic limit: N → ∞ then βJ → ∞ Weak ETH! βJ ‣ Second Thermodynamic limit: N, βJ → ∞ simultaneously, followed by g2 ∼ → 0 N No ETH! ‣ ETH in boundary states of Liouville theory: ETH! ✦ Conformal limit: where only the operators that are primaries of Virasoro (and their descendants) are considered ETH! TO DO • Bulk dual of the each of the above limits • Are ‘heavy’ states chaotic? !32 Summary • We showed that Eigenstate thermalization can be studied analytically in SYK model in 2 limits: ✦ Schwarzian limit: where the contribution of pseudo-Goldstone modes is considered ‣ First Thermodynamic limit: N → ∞ then βJ → ∞ Weak ETH! βJ ‣ Second Thermodynamic limit: N, βJ → ∞ simultaneously, followed by g2 ∼ → 0 N No ETH! ‣ ETH in boundary states of Liouville theory: ETH! ✦ Conformal limit: where only the operators that are primaries of Virasoro (and their descendants) are considered ETH! TO DO • Bulk dual of the each of the above limits • Are ‘heavy’ states chaotic? THANK YOU! !32