Spectral Properties of the Quantum Random Energy Model Simone - PowerPoint PPT Presentation

Spectral Properties of the Quantum Random Energy Model Simone Warzel Zentrum Mathematik, TUM Cargese September 4, 2014

1. The Quantum Random Energy Model Q N := {− 1 , 1 } N Hamming cube: configuration space of N spins ( − ∆ ψ )( σ ) := N ψ ( σ ) − � N Laplacian on Q N : j = 1 ψ ( F j σ ) Spin flip: F j σ = ( σ 1 , . . . , − σ j , . . . , σ N ) − ∆ = N − � N j = 1 σ x Hence the Laplacian acts as a transversal magnetic field : j � N � Eigenvalues: 2 | A | , A ⊂ { 1 , . . . , N } Degeneracies: | A | � 1 Normalized Eigenvectors: f A ( σ ) = j ∈ A σ j √ 2 N Perturbation by a multiplication operator U : H = − ∆ + κ U U = U ( σ z 1 , . . . , σ z j ) ; κ ≥ 0; � U � ∞ ≈ O ( N ) Coupling constant √ In this talk: U ( σ ) = N g ( σ ) with { g ( σ ) } σ ∈ QN i.i.d. standard Gaussian r.v. REM

Some motivations and related questions 1. Adiabatic Quantum Optimization: Farhi/Goldstone/Gutmann/Snipser ’01, . . . Question: Find minimum in a complex energy landscape U ( σ ) e.g. REM, Exact Cover 3, . . . Idea: Evolve the ground state through adiabatic quantum evolution, i.e. i ∂ t ψ t = H ( t /τ ) ψ t generated by H ( s ) := ( 1 − s )( − ∆) + s U , s ∈ [ 0 , 1 ] τ ≈ c ∆ − 2 Required time : min 2. Mean field model for localization transition in disordered N particle systems Altshuler ’06 3. Evolutionary Genetics: Rugged fitness landscape for quasispecies . . . Schuster/Eigner ’77, Baake/Wagner ’01, . . .

Predicted properties for QREM Predicted low-energy spectrum: √ √ � 2 Γ) − 1 H = Γ ( − ∆ − N ) + U / 2, i.e. κ = ( Jörg/Krzakala/Kurchan/Maggs ’08 Presilla/Ostilli ’10, . . . 1 First order phase transition of the ground state at κ c = 2 ln 2 : √ κ < κ c : Extended ground state with non-random ground-state energy E 0 = − κ 2 + o ( 1 ) κ > κ c : Low lying eigenstates are concentrated on lowest values of U . In particular: E 0 = N + κ min U + O ( 1 ) κ = κ c : Energy gap ∆ min = E 1 − E 0 vanishes exponentially in N

Some heuristics √ Known properties of the REM : U ( σ ) = N g ( σ ) Location of the minimum: U 0 := min U = − κ − 1 N + O ( ln N ) c The extreme values U 0 ≤ U 1 ≤ . . . form a Poisson process about − κ − 1 N + O ( ln N ) of exponentially increasing intensity. c Perturbation theory: Fate of localized states: � δ σ , H δ σ � = N + κ U ( σ ) . √ � 1 N 2 − N / 2 ) . Fate of delocalized states: � f A , U f A � = σ U ( σ ) = O ( 2 N

2. Low-energy regime of the QREM in case κ < κ c Theorem (Case κ < κ c ) For any ε > 0 and except for events of exponentially small probability, the � � κ eigenvalues of H below 1 − κ c − ε N are within balls centred at κ 2 2 n − , n ∈ { 0 , 1 , . . . } , 1 − 2 n N � 2 + δ � N − 1 of radius O with δ > 0 arbitrary. � N � There are exactly eigenvalues in each ball and their eigenfunctions are n 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 κ N + κ c + ε .

Sketch of the proof – delocalization regime Step 1: Hypercontractivity of the Laplacian | ψ E ( σ ) | 2 ≤ � δ σ , P ( −∞ , E ] ( H ) δ σ � = inf t > 0 e tE � δ σ , e − tH δ σ � t > 0 e t ( E − κ U 0 ) � δ σ , e t ∆ δ σ � = 2 − N e Γ ( xE 2 ) N . = inf Step 2: Reduction of fluctuations Projection on centre of band and its complement: Q ε := 1 − P ε := 1 [ N ( 1 − ε ) , N ( 1 + ε )] ( − ∆) . � � N − 1 dim P ε ≤ 2 N e − ε 2 N / 2 2 + δ Note: – take ε = O .

Sketch of the proof – delocalization regime Lemma There exist constants C , c < ∞ such that for any ε > 0 and any λ > 0 : � � � � � dim P ε ≤ C e − c λ 2 � � P ε UP ε � − E [ � P ε UP ε � ] � > λ P 2 N � dim P ε = C N e − ε 2 N / 4 . E [ � P ε UP ε � ] ≤ C N 2 N 1 Concentration of measure using Talagrand inequality: Lipschitz continuity of F : R Q N → R , F ( U ) := � P ε UP ε � : F ( U ) − F ( U ′ ) ≤ � ψ, U ψ � − � ψ, U ′ ψ � � dim P ε ≤ � U − U ′ � 2 � ψ � ∞ ≤ � U − U ′ � 2 . 2 N � Tr ( P ε UP ε ) 2 N � ) 1 / 2 N . . . Moment method to estimate E [ � P ε UP ε � ] ≤ ( E 2

Sketch of the proof – delocalization regime Step 3: Schur complement formula � � − 1 P ε ( H − z ) − 1 P ε = P ε HP ε − z − κ 2 P ε UQ ε ( Q ε HQ ε − z ) − 1 Q ε UP ε . . . and using Step 2: � 2 + δ � N P ε UQ ε ( Q ε HQ ε − z ) − 1 Q ε UP ε ≈ N − 1 N − z P ε + O .

Low-energy regime of the QREM Main idea: Geometric decomposition � � κ For energies below E δ := 1 − κ c + δ N the localized eigenstates originate in large negative deviation sites : � � σ | κ U ( σ ) < − κ X δ := κ c N + δ N For δ > 0 small enough and except for events of exponentially small probability (e.e.p.): X δ consists of isolated points which are separated by a distance greater than 2 γ N with some γ > 0. On balls B γ,σ := { σ ′ � � dist ( σ, σ ′ ) < γ N } the potential is larger than − ǫ N aside from at σ .

Low-energy regime of the QREM Theorem E.e.p. and for δ > 0 sufficiently small, there is some γ > 0 such that all � � κ eigenvalues of H below E δ = 1 − κ c + δ N coincide up to an exponentially small error with those of � � H δ := H R ⊕ H B γ,σ . σ ∈ X δ where R := Q N \ � σ ∈ X δ B γ,σ . Low energy spectrum of H R looks like H in the delocalisation regime Low energy spectrum of H B γ,σ is explicit consisting of exactly one eigenstate below E δ . . .

Some spectral geometry on Hamming balls Known properties of Laplacian on B γ,σ : � E 0 ( − ∆ B γ,σ ) = N ( 1 − 2 γ ( 1 − γ )) + o ( N ) Adding a large negative potential κ U at σ and some more moderate background elsewhere, rank-one analysis yields: E 0 ( H B γ,σ ) = N + κ U ( σ ) − s γ ( N + κ U ( σ )) + O ( N − 1 / 2 ) where s γ is the self-energy of the Laplacian on a ball of radius γ N . for the corresponding normalised ground state: � � � � 2 ≤ e − L γ N � ψ 0 ( σ ′ ) for some L γ > 0. σ ′ ∈ ∂ B γ,σ | ψ 0 ( σ ) | 2 ≥ 1 − O ( N − 1 ) H B γ,σ has a spectral gap of O ( N ) above the ground state.

3. Comment on adiabatic quantum optimization √ 2 N and i ∂ t ψ t = H ( t /τ ) ψ t with ψ 0 ( σ ) = 1 / Study H ( s ) := ( 1 − s )( − ∆) + s κ U , s ∈ [ 0 , 1 ] . Farhi/Goldstone/Gutmann/Negaj ’06 1 Let σ 0 ∈ Q N be minimizing configuration for { U ( σ ) } and |� ψ τ , δ σ 0 �| 2 ≥ b . √ 2 N b − 2 2 N σ ( U ( σ ) − U ( σ 0 )) 2 ≈ O ( 2 N / 2 ) . Then τ ≥ �� 4 Adiabatic theorem of Jansen/Ruskai/Seiler ’07 as used in 2 Farhi/Goldstone/Gosset/Gutmann/Shor ’10 yields: Typically, the minimum ground-state gap of H ( s ) along the path s ∈ [ 0 , 1 ] is exponentially small in N .

4. Conclusion: Complete description of the low-energy spectrum of the QREM 1 . . . and generalisations to non-gaussian r.v.’s Ground-state phase transition at κ = κ c with an exponentially closing gap. Jörg/Krzakala/Kurchan/Maggs ’08 Open problem: Resonant delocalisation conjecture in QREM 2 with eigenfunctions possibly violating ergodicity are expected to occur closer to centre of band within renormalised gaps of Laplacian. Laumann/Pal/Scardicchio ’14

Appendix: Adiabatic theorem Jansen/Ruskai/Seiler ’07 Theorem Let H ( s ) , s ∈ [ 0 , 1 ] , be a twice differentiable family of self-adjoint operators with non-degenerate ground-state eigenvectors φ s and ground-state gaps γ ( s ) . Then the solution of i ∂ t ψ t = H ( t /τ ) ψ t , ψ 0 = φ 0 , satisfies: � � 1 − |� ψ τ , φ 1 �| 2 ≤ 1 1 1 γ ( 0 ) 2 � H ′ ( 0 ) � + γ ( 1 ) 2 � H ′ ( 1 ) � τ � � 1 7 1 γ ( s ) 3 � H ′ ( s ) � + γ ( s ) 2 � H ′′ ( s ) � ds + . 0