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

2:

QN = {−1, 1}N

Random Energy Model Derrida ’80 U(σ) =

√

N g(σ),

σ ∈ QN

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

E [U(σ)] = 0, E [U(σ)U(σ′)] = N

• 1

N

N

• j=1

σjσ′

j

• p

Special cases: p = 2 Sherrington-Kirkpatrick ’75 p = ∞ REM

• Asymptotically almost surely

max |U| = N βc + O(1)

with

βc := √

2 ln 2

Motivation

Transversal magnetic field induces spin flips: Fjσ = (σ1, . . . , −σj, . . . , σN)

(Tψ) (σ) = −

N

• j=1

ψ(Fjσ), ψ ∈

N

• j=1

C2 ≡ ℓ2(QN)

• 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

Partition fuction at inverse temperature β ∈ [0, ∞]: Z(β, Γ) := Tr e−βH Pressure: pN(β, Γ) := N−1 ln Z(β, Γ)

• Freezing transition at β = βc for REM:

Derrida ’80, . . .

lim

N→∞ pN(β, 0) = pREM(β) =

  

β2

2

β ≤ βc

β2

c

2 + (β − βc)βc

β > βc

Entropy vanishes in low-temperature phase!

• Self-averaging through gaussian fluctuation bounds:

P

• |pN(β, Γ) − E [pN(β, Γ)]| >

t

√

N

• ≤ C exp
• −ct2

Free energy

Partition fuction at inverse temperature β ∈ [0, ∞]: Z(β, Γ) := Tr e−βH Pressure: pN(β, Γ) := N−1 ln Z(β, Γ)

• Freezing transition at β = βc for REM:

Derrida ’80, . . .

lim

N→∞ pN(β, 0) = pREM(β) =

  

β2

2

β ≤ βc

β2

c

2 + (β − βc)βc

β > βc

Entropy vanishes in low-temperature phase!

• Self-averaging through gaussian fluctuation bounds:

P

• |pN(β, Γ) − E [pN(β, Γ)]| >

t

√

N

• ≤ C exp
• −ct2

Proof: McDiarmid & Lipschitz estimate

• σ

∂pN ∂g(σ) 2 =

1 N Z 2

• σ

σ|e−βH|σ2 ≤ 1

N 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)

lim

N→∞ pN(β, Γ) = max{pREM(β), pPAR(βΓ)}

Quantum Paramagnet pPAR(βΓ) = β2

c

2 + ln cosh (βΓ)

Proof ideas

Lower bounds is based on Gibb’s variational principle, i.e. pN(β, Γ) − pPAR(βΓ) ≥ − β N

Tr Ue−βΓT Tr e−βΓT = − β

2N√ N

• σ

g(σ) = O

• 1

√

2NN

• pN(β, Γ) − pREM(βΓ) ≥ − β

N

Tr Te−βU Tr e−βU = 0 .

Hence:

lim inf

N→∞ pN(β, Γ) ≥ max{pREM(β), pPAR(βΓ)}.

Proof ideas

Upper bound is based on absence of percolation of large deviation sites Xε := {σ ∈ QN | U(σ) < −εN} with ε > 0 arbitrary. Spin flips to/from Xε with Hamming distance d = 1:

∆ε = −

• σ∈Xε
• σ′:d(σ,σ)=1

(|σσ′| + h.c.)

• For every σ ∈ Xε there are at most Kε other large deviation

sites in the ball BδεN centered at σ with radius δεN. Consequently:

∆ε ≥ T

• BδN.
• Use Golden-Thomson for decomposition H = Hε + Γ∆ε

Z(β, Γ) ≤ Tr e−βHεe−βΓ∆ε

≤ e−βΓ inf ∆ε

Z PAR(βΓ) eβΓεN + Z REM(β)

• ¥

*

¥

¥

*

¥¥

¥

Hence:

lim sup

N→∞

pN(β, Γ) ≤ lim sup

N→∞

βΓ inf ∆ε

N

+ max{pREM(β), pPAR(βΓ) + βΓǫ}.

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:

• T
• BδN
• ≤ 2N
• δ(1 − δ) + o(N)

Proof: W.l.og. center ball at σ0 = (1, 1, . . . , 1) and write −T

• BδN = A + A† with

σ′|A|σ =

• 1 if

0 else Estimate T

• BδN ≤ 2A = 2
• A†A and

A†A ≤ max

σ

• σ′
• ′σ|A†A|σ
• = Nδ × N(1 − δ) + o(N2)

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: Eigenstates are delocalized vs localized on Hammingcube Main claim: Multifractality of intermediate eigenstates Similar to Rosenzweig-Porter, cf von Soosten/W. 18

Low-energy spectrum of QREM

Theorem (Γ > βc)

For any ε > 0 there is Nε ∈ N, s.t. with asympt. full probability and for all N ≥ Nε, the eigenvalues E of H with E ≤ − (βc − ε) N are found in intervals centered at

Γ

• 2n − N −

κ2

1 − 2n

N

• ,

n ∈ {0, 1, . . . } , with radius O

• ln N

N

• .

There are exactly

N

n

• eigenvalues in each ball and the corresponding normalized eigenfunctions

ψE are delocalized: ψE2

∞ ≤ 2−N eΓ(

xE 2 )N

where Γ(x) := −x ln x − (1 − x) ln(1 − x) and xE := E NΓ − min U N .

Low-energy spectrum of QREM

Theorem (Γ < βc)

For any ε > 0 there is Nε ∈ N, s.t. with asympt. full probability and for all N ≥ Nε, the eigenvalues E of H with E ≤ −

• max(Γ, βc/

√

2) − ε

• N are each exponentially localized in a (single)

large-deviation site.

Thank You!

Based on joint work with Ch. Manai.