P. Zanardi USC

Geometry of quantum criticality

Simons Centre For Geometry and Physics November 2010€

⟨Ψ | Φ⟩

€

⟨Ψ | Φ⟩

Qualitative: the difference between the means should be(much) bigger than the finite size variances….

Quantitative: Fisher Information metric2

11

2

)log()(

),( i

E

ii

E

i i

iiF pdp

p

qpdPPQPd ∑∑

==

=−

=+=

Problem: Quantumly each Q-preparation defines infinitelymany prob distributions, one for each observable: one has to maximize over all possible experiments!

⟩P|2||| ⟩⟨= Pipi

d(P,Q)= number of (asymptotically) distinguishable preparations between P and Q: ds=dX/Variance(X)d(P,Q)= number of (asymptotically) distinguishable preparations between P and Q: ds=dX/Variance(X)

Distinguishability of Exps Distance in Prob space

Surprise!

=),(max QPdFExps Projective Hilbert space distance!

)(cos||||||||

|||cos),( 11 F

PQ

PQQPDFS

−− ≡⋅

⟩⟨≡ )(cos

||||||||

|||cos),( 11 F

PQ

PQQPDFS

−− ≡⋅

⟩⟨≡

(Wootters 1981)

Statistical distance and geometrical one collapse:

Hilbert space geometry is (quantum) information geometry…..Question: What about non pure preparations ?!?Pρ

Answer: Bures metric & Uhlmann fidelity!(Braunstein Caves 1994)

2/110

2/1110

110

),(

)(cos),(

ρρρρρ

ρρ

trF

Fd

=

= −

2/110

2/1110

110

),(

)(cos),(

ρρρρρ

ρρ

trF

Fd

=

= −

Remark: In the classical case one has commuting objects, simplification ….

Q-fidelity!

(Quantum) Phase Transition: dramatic change of the (ground) state properties of a quantum system with respectA smooth change of some control parameter e.g., temperature, external field, coupling constant,…

Characterization of PTs:“Traditional”:Local Order parameter (OP), symmetry breaking (SB)(correlation functions/length) Landau-Ginzburg frameworkSmoothness of the (GS) Free-energy (I, II,.. order QPTs)Quantum Information views:Quantum Entanglement (concurrence, block entanglement,..)Geometrical Phases

Question: How to map out the phase diagram of a system with no a priori knowledge of its symmetries and ops?Question: How to map out the phase diagram of a system with no a priori knowledge of its symmetries and ops?

Differential Geometry of QPTs:QGT)()( HLH ∈λ

H∈⟩Ψ )(| λ

Hamiltonians

Ground states

M∈λControl parameters

⟩Ψ∂Ψ⟩⟨ΨΨ⟨∂−⟩Ψ∂Ψ⟨∂= νμνμμν |||Q ⟩Ψ∂Ψ⟩⟨ΨΨ⟨∂−⟩Ψ∂Ψ⟨∂= νμνμμν |||Q

νμν μμν λλ∑=⟩ΨΨ⟩⟨Ψ−Ψ⟨=ΨΨ ddQddddQ |)|1(,:),(

Hermitean metric over the projective space

⟩Ψ∂Ψ⟨=

∂−∂∝⟩Ψ∂Ψ⟨∂==

νμ

μννμνμμνμν

|:

|ImIm:

A

AAQF

Riemannian metric

2-form (=Berry curvature!)

∑ −≈−≈+

=

μν νμμν

μνμν

λλλλλ2

2/11),(

Re:

dseddgdF

Qg

Free Fermi Systems

∑ ∑ ++= +++

ij ijjijijijiZ chcBccAcH ).(

2

1∑ ∑ ++= +++

ij ijjijijijiZ chcBccAcH ).(

2

1

)(

,

RMBAZ

BBAA

L

TT

∈−≡−== L x L Matrix of coupling constants

00|

0|)exp(|

=⟩

⟩∝⟩Ψ ++∑

i

jij

ijiZ

c

cGcGaussian ground-state(even number of particles)

Problem: given Z find the antisymmetric matrix G

0|,,],[

)(}{}{

00

0

11

=⟩Ψ+Λ=Λ−=

+=⏐⏐⏐ →⏐⏐⏐ →⏐

∑

∑

>

+

+==

kk

kkkkkk

i

ikiikikBogoliubovL

ikFourierL

ii

EHH

chcgcc

ηηηηη

η

Diagonalization (Lieb-Schulz-Mattis 61)

Coherent states (Perelemov))(2 RO L

Ground-State Fidelity

|2

det||)(|)(|),( 212121

TTZZZZF

+=⟩ΨΨ⟨≡ |

2det||)(|)(|),( 21

2121

TTZZZZF

+=⟩ΨΨ⟨≡

⇒Θ±=⇒∈ =−− 2/

121

121

1 )}{exp()()( LL iTTSpROTT νν

∏∏==

Θ

Θ=+

=2/

1121 |)cos(||

2

1|),(

LL ieZZF

νν

μ

μ

∏∏==

Θ

Θ=+

=2/

1121 |)cos(||

2

1|),(

LL ieZZF

νν

μ

μ

∏=

−=⇒=

2/

1

212121 |

2cos|),(0],[

L

ZZFTTν

θθ

1

1

1

T

ThgG

The polar decomposition of contains the physical relevant data

TZ ΦΛ=

+Φ ⊂Λ 0)( RSp Quasi-particle spectrum

)(ROT L∈ Many-body GS structure

TZRORM LL →→ :)()( Continuous map for Z non-singular

)(),(

)()(,)(

21

)(

δλλλλλ λ

+==∈=

ZZZZRoKeT L

K

))(16

exp(),( 422

21 δλλ

δλo

KTrZZF +⎟

⎠

⎞⎜⎝

⎛∂∂

= ))(16

exp(),( 422

21 δλλ

δλo

KTrZZF +⎟

⎠

⎞⎜⎝

⎛∂∂

=

∑=

= ⎟⎠

⎞⎜⎝

⎛∂∂

−=⇒⊕=2/

1

22'

)(2/1 2)()()(

LyL KTriK

ν

νννν λ

θσλθλ

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

2/

1

2

2

''

2 ||

)Re()Im()Im()Re(L

v

v

z

zzzzS

ν

ννν∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

2/

1

2

2

''

2 ||

)Re()Im()Im()Re(L

v

v

z

zzzzS

ν

ννν

When the quasi-particle spec has a zero has a sharp increase and F a sharp decrease

0|| →vz

2S

)(1 λTT = )(2 δλλ +=TT 21

1 TT −If moving from to makes to develop an eigenvalue close –1 one has a sharp fidelity drop

∞−

∞ −=−≡ ||1||||||),( 21

12121 TTTTTTd has a sharp increase

1 2 42

Im( ) 1tan exp( ( ))

Re( ) 8

zF S o

zν

νν

θ δλ δλ−= ⇒ = − +

{ } 1( )

| |

L

v

v

Sp Z z

z

ν

ν

==

Λ =

THE XY Model (I)

)2

1

2

1(),( 11

1

zi

yi

yi

xi

xi

i

LH λσσσγσσγγλ +−

++

= ++=∑ )

2

1

2

1(),( 11

1

zi

yi

yi

xi

xi

i

LH λσσσγσσγγλ +−

++

= ++=∑

=anysotropy parameter, =external magnetic fieldγ λ

QCPs:

0=γ

1±=λ

XX line II-order QPT

Ferro/para-magnetic II order QPT

Jordan-Wigner mapping H Free-Fermion system: EXACTLY SOLVABLE!

L

k

L

kk

πγλπ 2sin)

2(cos 222 +−=Λ

Quasi-particle spectrum: zeroes in the TDL in all the QCPs Gaplessness of the many-body spectrum

)exp(),( 22

22 γλ δγδλδγγδλλ SSF +−≈++

),(2 γλγS ),(2 γλλS

XY Model (III): Overlap Functions

Metric scaling behaviour: the XY model

μνμνμν

νμν

μμν λλλλ

λλ ddKK

Trddgds ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂

∂

∂

∂∝= ∑∑2

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

=22 1

1,

1

1

||16 γλγdiag

Lg ⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

=22 1

1,

1

1

||16 γλγdiag

Lg

( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−++−=>

+=≤

1

||1

16

1)1|(|

||1||16

1)1|(|

22 γ

λλ

γγ

λ

hLR

LR

Ricci Scalar

∑ −⟩Ψ∂Ψ⟩⟨Ψ∂Ψ⟨

=n n

nn

EE

HHQ

20

00

))()((

)(||)()(||)(

λλ

λλλλ νμμν ∑ −

⟩Ψ∂Ψ⟩⟨Ψ∂Ψ⟨=

n n

nn

EE

HHQ

20

00

))()((

)(||)()(||)(

λλ

λλλλ νμμν

QGT: Spectral representations

( )

)(min:

:

1:||

1||||||||

0

2*

EE

HH

XHHHQQQ

nn −=Δ

∂Λ=Δ

=⟩⟨−⟩⟨Δ

=≤⟩ΛΛ⟨=≤

∑Λ

ΛΛΛ

μ μμ

μν

Local operator (trans inv)

Spectral gap

0)(lim >Δ∞→ LL ∞<−∞→ )(lim LXL d

L (Hastings 06)

∞<−∞→ ||lim μνQL d

LGapped system

For gapped systems the QGT entries scales at most extensively

Superextensive scaling implies gaplessness

∫∞

∞−= =−= τττω

ω μνωμνμν

μν dGGd

diQ )(|)(ˆ

0 ∫∞

∞−= =−= τττω

ω μνωμνμν

μν dGGd

diQ )(|)(ˆ

0

( )

ττ

νμνμμν

τ

ττθτ

HH XeeX

HHHHG

−=

⟩∂⟩⟨∂⟨−⟩∂∂⟨=

:)(

)()(:)(

Dynamical response functions

∫∞

∞−

−= ττω μνω

μν dGeG ti )()(ˆ

The information-geometrical object Q is expressed in terms of dynamical susceptibilities (You et al 07)

The integral representation of Q

Question:Does gaplessnes imply super-extensivity?

∫→∂ )(xxVdH dμμ

QGT Critical scaling

Continuum limit

Scaling transformations

μμ

ζ

μα

τατα

VV

xxΔ−→

→→ ;

Q

cdSing LQq Δ−→= ν

μνμν λλ ||/: Proximity of the critical point

QLLQq dSing Δ−→= /: μνμν At the critical points

dQ −−Δ+Δ=Δ ζνμ 2: dQ −−Δ+Δ=Δ ζνμ 2:

Scal dim of QGT: the smaller the faster the orthogonalization rateSuper-extensivity 2/)( dd −+≤Δ ζ

νλλξ −−= || c

Criticality it is not sufficient, one needs enough relevance….

Quantum Fidelity and Thermodynamics

)()(

)2/2/(),(

10

1010 ββ

ββββZZ

ZF

+=

Uhlmann fidelity between Gibbs states of the same Hamiltonianat different temperatures =partition function

))(exp()exp()( ββββ AHtrZ −=−=

)2/)2/2/()(2/)(2/)(exp(),( 1010110010 ββββββββββ ++−+= AAAF

)8

)(exp(),(

22

T

TcTTTTF Vδδ −≈+ )

8

)(exp(),(

22

T

TcTTTTF Vδδ −≈+

Singularities of the specific heat shows up at the fidelity level:Divergencies e.g., lambda points, results in a singular drop of fidelity

)(TcV

Temperature driven phase transitions can be detected by mixed state fidelity in both classical and quantum systems!

Temperature driven phase transitions can be detected by mixed state fidelity in both classical and quantum systems!

THANKS!

Joint work with

Nikola PaunkovicMarco Cozzini Paolo GiordaRadu IonicioiuL. Campos-Venuti

and now @USDamian AbastoToby JacobsonSilvano GarneroneStephan Haas

•Unitary Evolution ==>no non-trivial fixed points for t=∞ I.e., no strong (norm) convergence

Hey wait a sec: Equilibration of a finite closed quantumsystem?!? What are you talking about dude???

•Finite size ==>Point spectrum ==>A(t)=measurable quantity is a quasi-periodic function ==> no t=∞ limit (quasi-returns/revivals) ==> not even weak op convergence€

limt→∞U(t) | Ψ⟩=| Ψ⟩∞ ⇒ | Ψ⟩=| Ψ⟩∞

€

A(t) = D Ap exp(iωp t)⇒ ∀ε > 0∃T(ε,D) /p=1

∑ | A(T) − A(0) |≤ ε

Unitary equilibrations will have to be a different kind of convergence….Unitary equilibrations will have to be a different kind of convergence….

Unitary Equilibration??

€

L(t) =|⟨Ψ | exp(−iHt) | Ψ⟩ |2

€

L(t) =|⟨Ψ | exp(−iHt) | Ψ⟩ |2

Loschmidt Echo:Loschmidt Echo:

€

= pn pm exp[−it(En − Em )]n,m

∑

€

H = En | n⟩⟨n |n

∑

Spectral resolution Probability distribution(s)

Different Time-Scales & Characteristic quantities

•Relaxation Time (to get to a small value by dephasing and oscillate around it)

•Revivals Time (signal strikes back due to re-phasing)

Q1: how all these depend on H, , and system size?Q1: how all these depend on H, , and system size?

€

| Ψ⟩

Q2: how the global statistical features of L(t) depend on H, and system size N?Q2: how the global statistical features of L(t) depend on H, and system size N?

€

| Ψ⟩

€

pn :=|⟨Ψ | n⟩ |2

€

=Tr[| Ψ⟩⟨Ψ | e−iHt | Ψ⟩⟨Ψ | e iHt ] = ⟨ρΨ (t)⟩Ψ

Typical Time Pattern of L(t)

Transverse Ising (N=100)

€

P(y = L(t)) = limT →∞

1

Tδ(y − L(t)) =

0

T

∫ ⟨δ(L − L(t))⟩t

€

P(y = L(t)) = limT →∞

1

Tδ(y − L(t)) =

0

T

∫ ⟨δ(L − L(t))⟩t

For a given initial state L-echo is a RV over the time line [0,∞) with Prob Meas

€

A⊂[0,∞)

€

μ∞(A) := limT →∞

1

Tχ A (t)dt

0

T

∫ Characteristic function of

Probability distribution of L-echo

Goal: study P(y) to extract global information about theEquilibration process Goal: study P(y) to extract global information about theEquilibration process

€

χA

1 Moments of P(y)

€

κn := y nP(y)dy = limT →∞∫ 1

TLk (t)dt

0

T

∫

Each moment is a RV over the unit sphere (Haar measure) of initial states

Mean:

Long time average of L(t) is the purity of the time-averagedensity matrix (or 1 -Linear Entropy)

Question: How about the other moments e.g., variance andinitial state dependence? Are there “typical” values?Question: How about the other moments e.g., variance andinitial state dependence? Are there “typical” values?

€

κ1 = limT →∞

1

T⟨ρΨ,ρΨ,(t)⟩dt

0

T

∫ = ⟨ρΨ,Π nρΨ,Πm⟩m,n

∑ limT →∞

1

Te−i(En−Em )tdt

0

T

∫

= ⟨ρΨ,Π nρΨ,Π n⟩=n

∑ ⟨ρΨ,D1(ρΨ,)⟩= ⟨D1(ρΨ ),D1(ρΨ,)⟩= TrD1(ρΨ )2

€

limT →∞

1

Te−iHtρΨe

iHtdt0

T

∫ = Π nρΨ,Π n =:D1(n

∑ ρΨ )

€

limT →∞

1

Te−iHtρΨe

iHtdt0

T

∫ = Π nρΨ,Π n =:D1(n

∑ ρΨ )

Remark

€

D1Is a projection on the algebra of the fixed pointsOf the (Heisenberg) time-evolution generated by H

Remark dephased state min purity given the constraints I.e., constant of motion

€

Dn (X) = limT →∞

1

T(e−iHt )⊗n X

0

T

∫ (e iHt )⊗nDephasing CP-map of the n-copies Hamiltonian, S is a swap in

€

κn (ψ ) = Tr[Dn⊗2(S) |ψ⟩⟨ψ |⊗2n ]

€

κn (ψ ) = Tr[Dn⊗2(S) |ψ⟩⟨ψ |⊗2n ]

€

κn (ψ )ψ

=Tr[Dn

⊗2(S)P2n+ ]

Tr(P2n+ )

€

P2n+ = Projection on the totally symm ss of

€

Hilb⊗2n

€

(Hilb⊗n )⊗2

€

|κ n (ψ ) −κ n (φ) |≤||Dn⊗2(S) ||∞ |||ψ⟩⟨ψ |⊗2n − |φ⟩⟨φ |⊗2n ||1≤ 4n |||ψ −⟩ |φ⟩ ||

All L(t) moments are Lipschitz functions on the unit sphere of Hilb ==> Levi’s Lemma implies exp (in d) concentration around

Remark We assumed NO DEGENERACY, in general bounded above by

€

κn (ψ )ψ

€

κ1(ψ )ψ

=Tr[D1

⊗2(S)(1+ S)]

d(d +1)=

2

d +1

€

κ1(ψ )ψ

=Tr[D1

⊗2(S)(1+ S)]

d(d +1)=

2

d +1

d=dim(Hilb)==> exp (in N) small =(positivity)=> exp (in N) state-space concentration of around

€

κn (ψ )ψ

2 Moments of P(y)

€

κ1(ψ )ψ

€

κn (ψ )

Remark

€

κn (ψ )ψ

≈d>>n d−2n Tr[Dn

⊗2(S)σ ]σ ∈S2n

∑

€

κ1(ψ )ψ

=1+ d j (d j /d∑ )2

d +1

€

σ 2(L) := κ 2 −κ12 = pi

2p j2

i≠ j

∑ = κ12 − tr(ρ eq

4 ) ≤ κ12

€

σ 2(L) := κ 2 −κ12 = pi

2p j2

i≠ j

∑ = κ12 − tr(ρ eq

4 ) ≤ κ12

€

μ∞{t / | L(t) − ⟨L(t)⟩t |≥ Mσ } ≤ M−2

Chebyshev’s inequality

goes to zero with N system size ==> M can diverge While ==>Flucts of L(t) are exponentially rare in time …..! goes to zero with N system size ==> M can diverge While ==>Flucts of L(t) are exponentially rare in time …..!

€

κ1(ψ ) = ⟨L(t)⟩t€

σ(L)

€

∀ε,δ > 0,∃N |N ≥ N ⇒ μ∞{t / | Lψ NN (t) −κ1,N (ψ N )

ψ N |≥ ε} ≤ δ

∀ψ N ∈ SN ⊂HilbN & μH (SN ) ≥1− exp(−cN)

€

∀ε,δ > 0,∃N |N ≥ N ⇒ μ∞{t / | Lψ NN (t) −κ1,N (ψ N )

ψ N |≥ ε} ≤ δ

∀ψ N ∈ SN ⊂HilbN & μH (SN ) ≥1− exp(−cN)

In the overwhelming majority of time instants L(t) is exponentially close to the “equilibrium value” €

Mσ → 0

Remark: non-resonance assumed

This what we (morally) got :

€

c = c(ε,δ) > 0

Q: Can we do better? E.g., exp in d concentration?

A: yes we can!

€

t →α := (E1t,...,Ed t)∈ Td →| pne

iα n∑ |2

HP: energies rationally independent ==> motion on the d-torus ergodic==>Temporal averages=phase-space averages

€

| L(α ) − L(β ) |≤ 2 pn | e iα n − e iβ n |≤∑ pn |α n −β n |=: 4πD(α ,β )∑

L is Lipschitz on the d-torus with metric D ==> known measure concentration phenomenon!

€

μ∞{t / | LN (t) − ⟨LN (t)⟩t |≥ ε} = Pr{α ∈ T d / | L(α ) − ⟨L⟩α |≥ ε} ≤ exp(−cε 2

pn4∑)

€

μ∞{t / | LN (t) − ⟨LN (t)⟩t |≥ ε} = Pr{α ∈ T d / | L(α ) − ⟨L⟩α |≥ ε} ≤ exp(−cε 2

pn4∑)

€

c = (128π 2)−1

€

c = (128π 2)−1

Remark The rate of meas-conc is the inv of purity of the dephasedStates I.e., mean of L==> Typically order d=epx(N), as promised…

Far from typicality: Small Quenches

€

H0 | Ψ0⟩= E0 | Ψ0⟩

H = H0 +V

||V ||= o(ε)

€

H0 | Ψ0⟩= E0 | Ψ0⟩

H = H0 +V

||V ||= o(ε)

Ground State of an initial HamiltonianQuench-Ham= init-Ham + perturbation

€

pn =|⟨ΨQuenchn | Ψ0⟩ |2 Distribution on the eigenbasis of

€

p0 ≈1− χ F = o(1)

€

Slin =1−Tr[D1(ρΨ0)2] =1− ⟨L(t)⟩t =1− pn

2 ≈1− p02 ≈ 2χ F

n

∑

The linear entropy of the dephased state for a small quenchIs given by the fidelity susceptibility: a well-known object!The linear entropy of the dephased state for a small quenchIs given by the fidelity susceptibility: a well-known object!

=measures how initial state fails to be a quenched Hamiltonian Eigenstate. For H(quench) close to H(0) we expect it to be small….

€

Slin :=1−κ 1

€

H = H0 +V

€

pn≠0 ≈|⟨Ψ0 |V | Ψn⟩ |2

(En − E0)2= o(ε 2)GS Fidelity: leading term!

Remark:

€

σ 2(L) ≈ p0(1− p0) ≈ χ F

€

σ 2(L) ≈ p0(1− p0) ≈ χ F

€

χF =|⟨Ψ0 |V | Ψn⟩ |2

(E0 − En )2n≠0

∑ ≤1

Δ2(⟨V 2⟩− ⟨V⟩2) :=

1

Δ2X

€

χF =|⟨Ψ0 |V | Ψn⟩ |2

(E0 − En )2n≠0

∑ ≤1

Δ2(⟨V 2⟩− ⟨V⟩2) :=

1

Δ2X

€

V := V j∑Δ := minn (En − E0)

Local operator (trans inv)

Spectral gap

0)(lim >Δ∞→ LL ∞<−∞→ )(lim LXL d

L (Hastings 06)

€

limL→∞ L−dχ F < ∞Gapped system

For gapped systems 1-LE mean scales at most extensivelyFor gapped systems 1-LE mean scales at most extensively

Superextensive scaling implies gaplessnessSuperextensive scaling implies gaplessness

Small Quenches: The Role of Criticality

€

V → dd xV (x)∫

Small Quenches: FS Critical scaling

Continuum limit

Scaling transformations

€

x →αx;τ →α ζ τ

V →α −ΔV

€

χFSing /Ld →| λ − λ c |νΔQ Proximity of the critical point

€

χFSing /Ld → L−ΔQ At the critical points

€

ΔQ := 2Δ − 2ζ − d

€

ΔQ := 2Δ − 2ζ − d

Scal dim of FS: the smaller the faster the orthogonalization rateSuper-extensivity 2/)( dd −+≤Δ ζ

νλλξ −−= || c

Criticality it is not sufficient, one needs enough relevance….

THE XY Model

)2

1

2

1(),( 11

1

zi

yi

yi

xi

xi

i

LH λσσσγσσγγλ +−

++

= ++=∑ )

2

1

2

1(),( 11

1

zi

yi

yi

xi

xi

i

LH λσσσγσσγγλ +−

++

= ++=∑

=anysotropy parameter, =external magnetic fieldγ λ

QCPs:

0=γ

1±=λ

XX line III-order QPT

Ferro/para-magnetic II order QPT

Jordan-Wigner mapping H Free-Fermion system: EXACTLY SOLVABLE!

L

k

L

kk

πγλπ 2sin)

2(cos 222 +−=Λ

Quasi-particle spectrum: zeroes in the TDL in all the QCPs Gaplessness of the many-body spectrum

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L(t) = (1− sin2(2α k )sin2(Λk2 t))

k

∏

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L(t) = (1− sin2(2α k )sin2(Λk2 t))

k

∏

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α k =θk (λ1) −θk (λ 2)H.T Quan et al, Phys. Rev. Lett. 96, 140604 (2006)

Ising in transverse field:

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cosθν (λ ) =cos

2πk

L− λ

Λν

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γ=1

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γ=1

Large size limit (TDL)==> spec(H) quasi-continuous ==> Large t limits exist (R-L Lemma) =time averages

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L(t) = e−Ls( t )

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s(t) =1

2πln[(1− sin2(2α k )sin2(Λk

2 t))]dk∫ t→∞ ⏐ → ⏐ ⏐ s(∞) − Am | t |−3 / 2 cos(Emt + 3π /4) + (m↔ M)

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s(∞) = −1

πln[(1− | cos(α k ) | /2)]dk∫

Inverting limits I.e., 1st t-average, 2nd TDL

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⟨L(t)⟩t = e−Lg(λ1 ,λ2 )

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g = −1

2πln[(1− sin2(α k ) /2)]dk∫

•g and s are qualitatively the same but when we consider different phases

(m=band min, M=band max

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κ1 = Π k[1−α k /2],

N =100

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var(L) = Π k[1−α k +3

8α 2

k ] − Π k[1−α k +1

4α 2

k ]

N =100

First Two Moments

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α k := sin2(θk (h1) −θk (h1)

2)

P(L=y) Different Regimes

• Large ==>L for (moderately) large (quasi) exponential

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δh

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δh• Small and close to criticality a) Exponentialb) Quasi critical I.e., universal “Batman Hood”

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δh• Small and off critical a) Exponential b) Otherwise Gaussian

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L >>|δh |−2

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L >>|δh |−1

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L <<| h(i) −1 |∝ξ

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κn ≤ n!(κ1)n ⇒ eλ ⟨L ⟩ ≤ χ (λ ) := ⟨eλL⟩≤

1

1− λ ⟨L⟩

L=20,30,40,60,120

h(1)=0.2, h(2)=0.6

L=10,20,30,40

h(1)=0.9, h(2)=1.2

Approaching exp for large sizes

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P(y = L) ≈ θ(y)exp(−y /⟨L⟩)

⟨L⟩

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P(y = L) ≈ θ(y)exp(−y /⟨L⟩)

⟨L⟩

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κn ≤ n!(κ1)n ⇒ eλ ⟨L ⟩ ≤ χ (λ ) := ⟨eλL⟩≤

1

1− λ ⟨L⟩