M C Q S T Motivation Configuration space of N spin- 1 Q N = { 1 - PowerPoint PPT Presentation

Phase Transition in the Quantum Random Energy Model Simone Warzel Venice – Quantissima in the Serenissima III August 23, 2019 M C Q S T

Motivation Configuration space of N spin- 1 Q N = {− 1 , 1 } N 2 : Random Energy Model Derrida ’80 √ U ( σ ) = N g ( σ ) , σ ∈ Q N with g ( σ ) i.i.d. standard Gaussian random variables. • Simplest extreme case in family of mean-field spin-glass models, i.e. p -spin model N � 1 p � � E [ U ( σ ) U ( σ ′ )] = N � σ j σ ′ E [ U ( σ )] = 0 , � � j N � j = 1 Special cases: p = 2 Sherrington-Kirkpatrick ’75 p = ∞ REM √ • Asymptotically almost surely max | U | = N β c + O ( 1 ) β c := 2 ln 2 with

Motivation F j σ = ( σ 1 , . . . , − σ j , . . . , σ N ) Transversal magnetic field induces spin flips: N N C 2 ≡ ℓ 2 ( Q N ) � � ( T ψ ) ( σ ) = − ψ ( F j σ ) , ψ ∈ j = 1 j = 1 • Spectrum of T : − N , − N + 2 , . . . , N − 2 , N Quantum Random Energy Model H = Γ T + U with Γ ≥ 0 strength of the transversal field. • Simple model for studying quantum effects in unstructured energy landscape, e.g. in the context of: − mean-field quantum spin glasses Goldschmidt ’90, . . . − quantum annealing algorithms Jörg/Krzakala/Kurchan/Maggs ’08, . . . − many-body localized systems Laumann/Pal/Scardiccio ’14, . . . • Model for mutation of genotypes in random fitness landscape Schuster/Eigner ’77, . . . , Baake/Wagner ’01

Motivation Quantum Random Energy Model H = Γ T + U with Γ ≥ 0 strength of the transversal field. • Simple model for studying quantum effects in unstructured energy landscape, e.g. in the context of: − mean-field quantum spin glasses Goldschmidt ’90, . . . − quantum annealing algorithms Jörg/Krzakala/Kurchan/Maggs ’08, . . . − many-body localized systems Laumann/Pal/Scardiccio ’14, . . . • Model for mutation of genotypes in random fitness landscape Schuster/Eigner ’77, . . . , Baake/Wagner ’01 Predicted features: I. Spin-glass transition & free-energy Manai/W.’ 19 II. Quantum phase transition in ground-state and exponential run time of adiabatic search Adame/W.’ 16 III. Localization/delocalization transitions of eigenvectors W. ≥ ’16

Free energy Z ( β, Γ) := Tr e − β H Partition fuction at inverse temperature β ∈ [ 0 , ∞ ] : p N ( β, Γ) := N − 1 ln Z ( β, Γ) Pressure: • Freezing transition at β = β c for REM: Derrida ’80, . . .  β 2 β ≤ β c  2 N →∞ p N ( β, 0 ) = p REM ( β ) = lim β 2 2 + ( β − β c ) β c β > β c c  Entropy vanishes in low-temperature phase! • Self-averaging through gaussian fluctuation bounds: � � t − ct 2 � � | p N ( β, Γ) − E [ p N ( β, Γ)] | > √ ≤ C exp P N

Free energy Z ( β, Γ) := Tr e − β H Partition fuction at inverse temperature β ∈ [ 0 , ∞ ] : p N ( β, Γ) := N − 1 ln Z ( β, Γ) Pressure: • Freezing transition at β = β c for REM: Derrida ’80, . . .  β 2 β ≤ β c  2 N →∞ p N ( β, 0 ) = p REM ( β ) = lim β 2 2 + ( β − β c ) β c β > β c c  Entropy vanishes in low-temperature phase! • Self-averaging through gaussian fluctuation bounds: � � t − ct 2 � � | p N ( β, Γ) − E [ p N ( β, Γ)] | > √ ≤ C exp P N Proof: McDiarmid & Lipschitz estimate � ∂ p N � 2 1 � σ | e − β H | σ � 2 ≤ 1 � � = N Z 2 ∂ g ( σ ) N σ σ For p -spin generalization see: Crawford ’07

Phase diagram Replica method and static approximation in path-integral representation of E [ Z ( β, Γ) n ] Goldschmidt ’90, . . . , Obuchi/Nishimori/Sherrington ’07,. . . Theorem (Manai/W. ’19) Quantum Paramagnet N →∞ p N ( β, Γ) = max { p REM ( β ) , p PAR ( β Γ) } p PAR ( β Γ) = β 2 lim 2 + ln cosh ( β Γ) c

Proof ideas Lower bounds is based on Gibb’s variational principle , i.e. Tr Ue − β Γ T p N ( β, Γ) − p PAR ( β Γ) ≥ − β β � � 1 � = − 2 N √ g ( σ ) = O √ Tr e − β Γ T N 2 N N N σ Tr Te − β U p N ( β, Γ) − p REM ( β Γ) ≥ − β Tr e − β U = 0 . N N →∞ p N ( β, Γ) ≥ max { p REM ( β ) , p PAR ( β Γ) } . lim inf Hence:

Proof ideas Upper bound is based on absence of percolation of large deviation sites X ε := { σ ∈ Q N | U ( σ ) < − ε N } with ε > 0 arbitrary. Spin flips to/from X ε with Hamming distance d = 1: � � ( | σ �� σ ′ | + h . c . ) ∆ ε = − σ ∈ X ε σ ′ : d ( σ,σ )= 1 ¥ * • For every σ ∈ X ε there are at most K ε other large deviation ¥ ¥ * . sites in the ball B δ ε N centered at σ with radius δ ε N . ¥¥ ¥ � ∆ ε ≥ T Consequently: B δ N . � • Use Golden-Thomson for decomposition H = H ε + Γ∆ ε Z ( β, Γ) ≤ Tr e − β H ε e − β Γ∆ ε Z PAR ( β Γ) e β Γ ε N + Z REM ( β ) ≤ e − β Γ inf ∆ ε � � β Γ inf ∆ ε + max { p REM ( β ) , p PAR ( β Γ) + β Γ ǫ } . Hence: lim sup p N ( β, Γ) ≤ lim sup N N →∞ N →∞

Confinement to Hamming ball Lemma (cf. Friedman/Tillich ’05, . . . ) For any δ ∈ ( 0 , 1 / 2 ) the Dirichlet restriction to a ball in the Hamming cube is bounded: � � � � � ≤ 2 N δ ( 1 − δ ) + o ( N ) � T � � � B δ N B δ N = A + A † with � W.l.og. center ball at σ 0 = ( 1 , 1 , . . . , 1 ) and write − T Proof: � � 1 if � σ ′ | A | σ � = 0 else � � � A † A � and Estimate � T B δ N � ≤ 2 � A � = 2 � � � = N δ × N ( 1 − δ ) + o ( N 2 ) � A † A � ≤ max � � ′ σ | A † A | σ � � � σ σ ′

Summary & Outlook I. Spin glass perspective: • Proof of conjectured thermodynamic phase diagram � • Fluctuation properties of the partition function, stochastic stability, . . . ?? • Extension of qualitative properties to p -spin models ?? II. Quantum annealing & ground-state transition: � Farhi/Goldstone/Gutmann/Nagaj ’08, Adame/W. ’16. III. Localization/delocalization properties of eigenvectors: • Low-energy spectrum � • Multifractality • Delocalization of bulk states . . . ??

Localization/delocalization Laumann/Pal/Scardiccio ’14 Faoro/Feigelman/Ioffe ’18 Smelyanskiy/Kechedzhi/Boixo/Neven/Altshuler ’19 Main claim: Main claim: Eigenstates are delocalized vs localized on Hammingcube Multifractality of intermediate eigenstates Similar to Rosenzweig-Porter, cf von Soosten/W. 18

Low-energy spectrum of QREM Theorem ( Γ > β c ) For any ε > 0 there is N ε ∈ N , s.t. with asympt. full probability and for all N ≥ N ε , the eigenvalues E of H with E ≤ − ( β c − ε ) N are found in intervals centered at � � κ 2 Γ 2 n − N − , n ∈ { 0 , 1 , . . . } , 1 − 2 n N �� ln N � with radius O . N � N � There are exactly eigenvalues in each ball and the corresponding normalized eigenfunctions n ψ E are delocalized: xE ∞ ≤ 2 − N e Γ ( 2 ) N � ψ E � 2 where Γ( x ) := − x ln x − ( 1 − x ) ln( 1 − x ) and x E := E N Γ − min U . N

Low-energy spectrum of QREM Theorem ( Γ < β c ) For any ε > 0 there is N ε ∈ N , s.t. with asympt. full probability and for all N ≥ N ε , the eigenvalues √ � � E of H with E ≤ − max(Γ , β c / 2 ) − ε N are each exponentially localized in a (single) large-deviation site.

Thank You! Based on joint work with Ch. Manai.