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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⟩