localization and entanglement in disordered harmonic ...houssam/pdf/talks/hossam-scgp1… ·...

Localization and Entanglement inDisordered Harmonic Oscillator Systems

Houssam Abdul-RahmanUniversity of Arizona

Based on joint work with R. Sims (UA) and G. Stolz (UAB).

Entanglement and Dynamical Systems 2018Simons Center for Goemetry and Physics

December 12, 2018

Houssam Abdul-Rahman Disordered Oscillator Systems 1 / 29

Many-body Localization (MBL)

In single-body quantum systems, sufficiently strong disorder localizeswave functions in space. This is the essence of Anderson localization.

By now, Anderson localization is well understood, both physically andmathematically.

Question:What happens in a quantum system

when both disorder and interactions are present?

This is the many-body setting where the situation is fundamentallydifferent from the single particle case.

Considerable analytical and numerical challenges persist even for thesimplest many-body models.

There are extensive efforts in the physics literature to understand thephenomenon of MBL as well as the many-body transition.

e.g. arXiv:1705.09103, arXiv:1804.11065


Many-Body Localization indicators

The many-body dynamics:I zero-velocity Lieb-Robinson bounds.

I quasi-locality of observables.

States localization:I decay of correlations.

I small entanglement (area laws).


MBL Rigorous Results- Models

XX/XY chain.Hamza/Sims/Stolz ’12Pastur/Slavin ’14AR/Stolz ’15Sims/Warzel ’16

AR/Nachtergale/Sims/Stolz ’16, ’17

Tonks-Girardeau gas.Seiringer/Warzel ’16

Oscillator systems.Nachtergaele/Sims/Stolz ’12,’13AR/Sims/Stolz ’17AR ’18

AR/Sims/Stolz arXiv:1810.12769

XXZ spin chain in the Ising phase.Elgart/Klein/Stolz ’18a, ’18b

Beaud/Warzel ’17, ’18

Holstein model.Mavi/Schencker ’18


Outline

I A disordered oscillator model.

II The regime of localized excitations.I Zero velocity Lieb-Robinson bounds.

I Quasi-locality.

I Decay of correlations in eigenstates.

III EntanglementI An area law for non-Gaussian states above the ground state.

I Dynamical Entanglement (work in progress).


A Disordered Oscillator ModelThe Hamiltonian

HΛ =∑x∈Λ

(p2x + kxq

2x

)+

∑{x, y} ⊂ Λ|x− y| = 1

(qx − qy)2

Λ := [−L,L]ν ∩ Zν where L ≥ 1 and ν ≥ 1.

qx and px = −i ∂∂qx

are the position and momentum operators.

The Hilbert space HΛ =⊗x∈Λ

L2(R, dqx).

{kx}x are i.i.d. random variables with absolutely continuousdistribution given by a bounded density supported in [0, kmax].


The Effective One-Particle Hamiltonian

HΛ =∑x∈Λ

p2x + qThΛq, q := (qx)x∈Λ

hΛ is the Anderson model on `2(Λ), i.e., hΛ = h0,Λ + k, whereI h0,Λ is the negative discrete Laplacian over `2(Λ).I k := diag{kx, x ∈ Λ}.

Recall that:I spec(hΛ) is almost surely simple.

I spec(hΛ) ⊂[minx∈Λ

kx, 4ν + kmax

].

hΛ is almost surely positive, and ‖hΛ‖ ≤ 4ν + kmax.

h−1/2Λ does not have a deterministic upper bound.


HΛ as a free boson system

Since hΛ is positive with simple spectrum (almost surely), it can be

diagonalized with eigenvalues {γ2j }|Λ|j=1 and unique (up to a phase)

orthogonal eigenvectors {φj}|Λ|j=1, hΛ =

|Λ|∑j=1

γ2j |φj〉〈φj |.

For j = 1, . . . , |Λ|, define

bj :=1√2

(γ12j φ

Tj q + iγ

− 12

j φTj p), q := (qx)x∈Λ, p := (px)x∈Λ.

Note that each operators bj is fully determined by (γj , φj).

{bj}j satisfy the CCR: [bj , bk] = 0 and [bj , b∗k] = δj,k1l.

HΛ can be written as a free boson system

HΛ =

|Λ|∑j=1

γj(2b∗jbj + 1l)


Eigenvalues and eigenfunctions of HΛ

HΛ =

|Λ|∑j=1

γj(2b∗jbj + 1l), γj ’s are the eigenvalues of h

1/2Λ .

There exists a unique vacuum ψ0 of the b’s, i.e., bjψ0 = 0 for all j.

For every α = (α1, . . . , α|Λ|) ∈ N|Λ|0 , the eigen-pair (ψα, Eα) is givenas

ψα =

|Λ|∏j=1

1√αj !

(b∗j )αjψ0, Eα =

|Λ|∑j=1

γj(2αj + 1).

{bj}j are the modes (or quasi-particles).

α ∈ N|Λ| describes the occupations of modes.

The ground state energy is∑

j γj = Trh1/2Λ .

The gap above the ground state is 2 minj γj


Outline



I Quasi-locality.





hΛ is fully localized ⇒ HΛ is localized

hΛ being fully localized at all energies. i.e.,

E

(sup|u|≤1

|〈δx, h−1/2Λ u(hΛ)δy〉|

)≤ Ce−µ|x−y| (1)

implies the following localization results

Zero-velocity Lieb-Robinson bound.

Exponential clustering of the ground/thermal states.

Area laws for the entanglement of ground/thermal states.

Exponential clustering of eigenstates and after a quantum quench.Nachtergaele/Sims/Stolz ’12,’13

AR/Sims/Stolz ’17

The singular eigencorrelator localization (1) is

known for kx with sufficiently large disorder.


Low-energy localization of hΛ

Given any dimension ν ≥ 1, ∃λ0 > 0 and C <∞, µ > 0 (independent ofL) such that

E

(sup|u|≤1

|〈δx, h−1/2Λ u(hΛ)χ[0,λ0](hΛ)δy〉|

)≤ Ce−µ|x−y|

for all x, y ∈ Λ.Nachtergaele/Sims/Stolz ’12

What is the corresponding localization regime of HΛ?


The regime of localized excitations

For fixed λ0, let

Sλ0 := {j ∈ {1, . . . , |Λ|}; γ2j ∈ [0, λ0]}

I :={α = (α1, . . . , α|Λ|) ∈ N|Λ|0 ; suppα ⊆ Sλ0

}The regime of localized excitations:

PI := PI(HΛ) :=∑α∈I|ψα〉〈ψα|

ψα for α ∈ I is the eigenfunstion of HΛ that results from modesassociated with the bottom of the spectrum of hΛ.

PI is the spectral projection of HΛ associated with the energies

Eα = E0 +∑

j; γ2j∈[0,λ0]

2γjαj .

If we choose λ0 ≥ 4ν + kmax, then PI = 1l.


Weyl operators

To quantify localization for the oscillator system, it will be useful toidentify a convenient class of observables.

Let f : Λ→ C, the associated Weyl operator is defines as

W(f) :=⊗x∈Λ

exp (i(Re[fx]qx + Im[fx]px)) .

Note that supp(f) = supp(W(f)).

The Heisenberg dynamics: τt(W(f)) = eitHΛW(f)e−itHΛ .

W(f)I := PIW(f)PI = CfW(χ(hΛ)[0,λ0]f

)PI , where 0 < Cf ≤ 1.

Note that τt(W(f))I = τt(W(f)I).


Restricted Lieb-Robinson Bounds

Theorem (AR/Sims/Stolz arXiv:1810.12769 )

For any f, g : Λ→ C,

E(

supt‖[τt(W(f)I),W(g)I ]‖

)≤ C(1 + λ

1/20 )2

∑x,y∈Λ

|f(x)||g(x)|e−µ|x−y|

where C and µ are the constants in the eigencorrelator localization bound.

Note: in the case where f and g have disjoint supports,∑x,y∈Λ

|f(x)||g(x)|e−µ|x−y| ≤ Const. e−µ′d(supp(f),supp(g)).

Note: Similar restricted LR version was established in Elgart/Klein/Stolz ’18


Quasi-Locality

For X ⊂ Λ, let f : Λ→ C, such that supp f ⊆ X.

supp(W(f)) ⊆ X but suppτt(W(f)) = Λ.

Can τt(W(f)) be “approximated” by a local operator supported “near” X?

n

Λ

X

X(n)

Figure: Define X(n) := {x ∈ Λ; d(x,X) ≤ n}


Quasi-Locality

Define the surface area of X

∂X := {x ∈ X, there exists y ∈ Λ \X; |x− y| = 1}.


Let X ⊂ Λ and f : Λ→ C satisfy supp(f) ⊂ X. For any n ∈ N0 and

t > 0, there is an operator Wt,n ∈ B(HΛ) supported on X(n), such thatfor any κ > 0 we have

E

(sup

α∈I;‖α‖∞≤κsupt∈R

∣∣∣⟨ψα, (τt(W(f))− Wt,n)ψα

⟩∣∣∣) ≤ C|∂X|‖f‖2/3∞ e−µn/3.

Note: Here

C = 25/3C

(∑z∈Zν

e−µ|z|/6

)4

(1 + κ)1/3(1 + λ1/20 )2. (2)


Exponential clustering of the eigenstates

The (restricted) dynamical correlations of local Weyl operators is given by

CIα(f, g; t) := 〈τt(W(f))IW(g)I〉ψα − 〈W(f)I〉ψα〈W(g)I〉ψα

where for any observable A, 〈A〉ψα := 〈ψα, Aψα〉.


For κ ∈ N0, and any functions f, g : Λ→ C,

E

(sup

α; ‖α‖∞≤κsupt∈R|CIα(f, g, t)|

)≤ 8C(1 + λ

1/20 )2

( ∑x,y∈Λ

|f(x)g(y)|e−µ|x−y|) 1κ+1

where C and µ are the constants in the eigencorrelator localization.


Outline



I Quasi-locality.





Bipartite EntanglementThe Logarithmic Negativity

Fix a subregion Λ0 ⊂ Λ and decompose the Hilbert spaceHΛ = HΛ0 ⊗HΛ\Λ0

, where

HΛ0 =⊗x∈Λ0

L2(R), HΛ\Λ0=

⊗x∈Λ\Λ0

L2(R)

For any state ρ ∈ B(HΛ), the logarithmic negativity N (ρ) is definedas

N (ρ) = log ‖ρT1‖1 (= log ‖ρT2‖1)

where ρT1 is the partial transpose of ρ with respect to Λ0 (the firstcomponent).

Some properties:I If ρ is a separable state then N (ρ) = 0.I If ρ is a pure state then N (ρ) is an upper bound for the von Neumann

entanglement entropy.


Entanglement of the Eigenstates of HΛ

Known:

Area laws of the ground state and thermal states.Nachtergaele/Sims/Stolz (2013), Vidal/Werner (’02)

Exponential clustering results of arbitrary eigenstates.AR/Sims/Stolz (’17,’18).

Open problem:Finding/Studying/Understanding the entanglement for the

eigenstates of HΛ.


Entanglement of (Non-)Gaussian States

The ground state and the thermal states of free boson systems areGaussian states (quasi-free).

All eigenstates above the ground state are non-Gaussian. i.e.,

〈ψα,W(f)ψα〉 = e−14〈f ,Mf〉

|Λ|∏k=1

Lαk

(〈f ,Mχk(M)f〉

2

).

Here f =

(Re[f ]Im[f ]

), Lαk(·) is the Laguerre polynomial of degree αk,

M = h−1/2 ⊕ h1/2, and χk(M) := χ{γ−1k }

(h−1/2)⊕ χ{γk}(h1/2).

Note: M is the correlation matrix 〈RRT 〉ψα , where R =

[qp

].

There are (almost) NO rigorous results about the entanglement ofnon-Gaussian states.


The N-modes ensemble

For each N ∈ N0, let JN be the set all occupations α associated with atotal of N modes,

JN = {α = (α1, . . . , α|Λ|) ∈ N|Λ|0 ;∑j

αj = N}.

We define the N-modes ensemble stateρN :=1

|JN |∑α∈JN

|ψα〉〈ψα|.

ρN is the orthogonal projection onto the Fock space sectorspan{ψα;

∑j αj = N}.

Tr[HΛρN ] =

(1 +

2N

|Λ|

)∑k

γk −→|Λ|→∞ g. s. energy∑k

γk


The N-modes ensemble

For each N ∈ N0, let JN be the set all occupations α associated with atotal of N modes,

JN = {α = (α1, . . . , α|Λ|) ∈ N|Λ|0 ;∑j

αj = N}.

We define the N-modes ensemble stateρN :=1

|JN |∑α∈JN

|ψα〉〈ψα|.

For all N ∈ N, ρN is non-Gaussian, in fact

Tr[ρNW(f)] = e−14〈f ,Mf〉QN

(〈f ,Mf〉

2

).

where QN is a polynomial of degree N .

The exact logarithmic negativity can be found using correlationmatrices.


An Area Law

ρN :=1

|JN |∑α∈JN

|ψα〉〈ψα|, and we have E(|〈δx, h−1/2

Λ δy〉|)≤ C e−µ|x−y|.

Let ∂Λ0 := {x ∈ Λ0; ∃ y ∈ Λ \ Λ0 with |x− y| = 1}.

Theorem (AR ’18)

For any Λ0 ⊂ Λ, N ∈ N0, and the corresponding N-modes ensemble ρN ,there exists C <∞ such that

E (N (ρN )) ≤ C(2N + 1)|∂Λ0| (3)

where the constant C is independent of N , Λ0 and Λ.

Note: C = C(4dλ+ kmax)1/2

∑x∈Zd

e−µ|x|

2

.


Dynamical EntanglementA simple case

Let HΛ0 and HΛ\Λ0be the restrictions of HΛ to Λ0 and Λ \ Λ0,

respectively.

Let ρ1 and ρ2 be any thermal/ground states of HΛ0 and HΛ\Λ0,

respectively.

Note that (for example) if ρ1 and ρ2 are ground states then ρ1 ⊗ ρ2 isthe ground state of HΛ0 ⊗ 1l + 1l⊗HΛ\Λ0

We study ρt := e−itHΛ(ρ1 ⊗ ρ2

)eitHΛ .

ρt is and entangled state with respect to HΛ0 ⊗HΛ\Λ0.

Question: What can we say about the entanglement of ρt?


Dynamical EntanglementWhat about correlations of ρt?

Define the positions-momenta correlations

Corρt(Ax, By) := 〈AxBy〉ρt − 〈Ax〉ρt〈By〉ρt , A,B ∈ {p, q}, x, y ∈ Λ.

Then the following are upper bounds for

E(

supt|Cρt(Ax, By)|1/3

), for all A,B ∈ {q, p},

ρ0 = ρα1 ⊗ ρα2 (eigenstate-eigenstate):

≤ C1

(1 +N

)2/3e−η1|x−y|, where max

{‖α1‖∞, ‖α2‖∞

}≤ N.

ρ0 = ρβ ⊗ ρβ (thermal-thermal):

≤ C2 max{

1, β−1/3}e−η2|x−y|.

AR/Sims/Stolz ’17


Dynamical EntanglementInitial Results

In the gaped case with gap γ:

N(ρt)≤ Cγ |∂Λ0| where Cγ →∞ as γ → 0.

If ρβ11 and ρβ2

2 are thermal states (with inverse temperatures β1 andβ2) and

(ρβ1,β2)t = e−itH(ρβ1

1 ⊗ ρβ22

)eitH

then

E(N((ρβ1,β2)t

))≤ C(1 + max{β1, β2})

(max{2 + C2

h, 2 + 8t2})1/2 |∂Λ0|.

If ρ1 and ρ2 are ground states then

E(N (ρt)

1/2)≤ C

(max{2 + C2

h, 2 + 8t2})1/4 |∂Λ0|.


Thank you.


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