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 ) |<latexit sha1_base64="0mRkoTRzgQR4aO7anRXxchLVKBU=">AAACGnicbVBNSwMxFMzWr1q/Vj16CRahBSnbIqgHoejFYwXXFrpLyaZpG5rNLslboaz9H178K148qHgTL/4b03YP2joQGGbm8fImiAXX4DjfVm5peWV1Lb9e2Njc2t6xd/fudJQoylwaiUi1AqKZ4JK5wEGwVqwYCQPBmsHwauI375nSPJK3MIqZH5K+5D1OCRipY9cevFjzEpQ9RWRfMHyBU48Sgd1xCY6h45SzgGFZpGMXnYozBV4k1YwUUYZGx/70uhFNQiaBCqJ1u+rE4KdEAaeCjQteollM6JD0WdtQSUKm/XR62xgfGaWLe5EyTwKeqr8nUhJqPQoDkwwJDPS8NxH/89oJ9M78lMs4ASbpbFEvERgiPCkKd7liFMTIEEIVN3/FdEAUoWDqLJgSqvMnLxK3VjmvODcnxfpl1kYeHaBDVEJVdIrq6Bo1kIsoekTP6BW9WU/Wi/VufcyiOSub2Ud/YH39AEbgn+c=</latexit> i U 0 | 0 i 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)δ + 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)δ + 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)δ + 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)δ + 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 ! <latexit sha1_base64="HyA2Sf+KyMx+ET7x4XsHPBM8SRE=">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</latexit> i,j X dephasing: on average thermal expectation value spectral chaos up to exponential in S of non-extensive operator 5 Sachdev-Ye-Kitaev (SYK) model: a Review SYK model: a new Guinea Pig !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<i2<i3<i4≤N 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.