Entanglement and transport in two disordered quantum many-body “toy” systems Houssam Abdul-Rahman University of Arizona Based on a joint work with B. Nachtergaele, R. Sims, and G. Stolz. 35th Annual Western States Mathematical Physics Meeting, Caltech Feb. 2017 Houssam Abdul-Rahman XY and Harmonic Oscillators 1 / 22
Overview The Disordered XY Chain. ◮ Dynamical entanglement. ◮ The transport of energy and particles. The Disordered Harmonic Oscillators System. ◮ Dynamical correlations in the eigenstates. ◮ Correlations in initially product states. Houssam Abdul-Rahman XY and Harmonic Oscillators 2 / 22
The XY Chain An Anisotropic XY Chain in Random Transversal Magnetic Field n − 1 n � j +1 + (1 − γ j ) σ y j σ y � µ j [(1 + γ j ) σ x j σ x ν j σ z H = − j +1 ] − j j =1 j =1 Λ = [1 , n ] , Λ 0 a block of spins (subinterval of Λ ). � H x = ( C 2 ) ⊗ n , dim H = 2 n . The Hilbert space: H := x ∈ Λ µ j , γ j and ν j are i.i.d. Houssam Abdul-Rahman XY and Harmonic Oscillators 3 / 22
The XY Chain Jordan-Wigner Transform ↓ Jordan-Wigner ↓ H = C ∗ M C , C := ( c 1 , c ∗ 1 , c 2 , c ∗ 2 , . . . , c n , c ∗ n ) t . M is the block Jacobi matrix − ν 1 σ z µ 1 S ( γ 1 ) ... ... µ 1 S ( γ 1 ) t M := , ... ... µ n − 1 S ( γ n − 1 ) µ n − 1 S ( γ n − 1 ) t − ν n σ z � 1 � � 1 � γ 0 , σ z = S ( γ ) = . − γ − 1 0 − 1 Houssam Abdul-Rahman XY and Harmonic Oscillators 4 / 22
The XY Chain Assumptions Assumptions: The XY chain H has almost sure simple spectrum. M satisfies eigencorrelator localization, i.e � � ≤ C 0 (1 + | j − k | ) − β , for some β > 6 . sup � g ( M ) jk � E | g |≤ 1 Applications: µ j = µ , γ j = γ for all j ∈ N . ν j are i.i.d from an absolutely continuous, compactly supported distribution. Isotropic case ( γ = 0 ): M − → Anderson Model. Anisotropic case ( γ � = 0 ): ◮ Large disorder case. Elgart/Shamis/Sodin (2012). ◮ Uniform spectral gap for M around zero. Chapman /Stolz (2014). Houssam Abdul-Rahman XY and Harmonic Oscillators 5 / 22
Dynamical Entanglement The Entanglement Entropy and the Entanglement of Formation Λ 0 Fix Λ 0 ⊆ Λ , consider the decomposition: � � H = H Λ 0 ⊗ H Λ \ Λ 0 , where H Λ 0 = H x , H Λ \ Λ 0 = H x . (1) x ∈ Λ 0 x ∈ Λ \ Λ 0 Let ρ be a pure state in B ( H ) , then ρ 1 log ρ 1 � where ρ 1 = Tr H 2 ρ. � E ( ρ ) = − Tr , For any (mixed) state ρ ∈ B ( H ) , then � E f ( ρ ) = inf p k E ( | ψ k �� ψ k | ) . p k ,ψ k k Houssam Abdul-Rahman XY and Harmonic Oscillators 6 / 22
Dynamical Entanglement Motivation Question For 1 ≤ ℓ ≤ n , let H [1 ,ℓ ] and H [ ℓ +1 ,n ] be the restrictions of H to the corresponding interval. Let ρ (1) and ρ (2) be any eigenstates states of H [1 ,ℓ ] and H [ ℓ +1 ,n ] , respectively. We study ρ t := e − itH � ρ (1) ⊗ ρ (2) � e itH . ρ t is an entangled state with respect to H [1 ,ℓ ] ⊗ H [ ℓ +1 ,n ] . Question: What can we say about the entanglement of ρ t ? Houssam Abdul-Rahman XY and Harmonic Oscillators 7 / 22
Dynamical Entanglement Problem Setting Λ 0 Λ 3 Λ 1 Λ 2 Λ 4 In general Decompose Λ into disjoint intervals Λ 1 , Λ 2 , . . . , Λ m . H Λ k is the restriction of H to Λ k . ψ k is an eigenfunction of H Λ k , and ρ k = | ψ k �� ψ k | . Define ρ = � m k =1 ρ k , and its dynamics ρ t = e − itH ρe itH . Houssam Abdul-Rahman XY and Harmonic Oscillators 8 / 22
Dynamical Entanglement : Main Theorem Dynamics of products of eigenstates Λ 0 Λ 2 Λ 3 Λ 1 Λ 4 Theorem There exists C < ∞ such that � � E sup E ( ρ t ) ≤ C t, { ψ k } k =1 , 2 ,...,m for all n , m , any choice of the interval Λ 0 ⊂ Λ and all decompositions Λ 1 , . . . , Λ m of Λ = [1 , n ] . Houssam Abdul-Rahman XY and Harmonic Oscillators 9 / 22
Dynamical Entanglement : Corollaries 1 Let ρ β be the tensor product of local thermal states, then � � E sup E f (( ρ β ) t ) ≤ C . t,β 2 For α = ( α 1 , . . . , α n ) ∈ {↑ , ↓} n , the up-down configuration associated with α is given by: e α = e α 1 ⊗ e α 2 ⊗ . . . ⊗ e α n , � � α E ( e − itH | e α �� e α | e itH ) E sup < C . 3 Let ψ be an eigenfunction of the full XY chain H . � � E ( | ψ �� ψ | ) E sup < C . Pastur/Slavin (2014). AR/Stolz (2015). ψ 4 Let ρ β be a thermal state of the full XY chain H . � � E sup E f ( ρ β ) < C . β Houssam Abdul-Rahman XY and Harmonic Oscillators 10 / 22
An Isotropic XY Chain in Random Transversal Magnetic Field n − 1 n � [ σ x j σ x j +1 + σ y j σ y � ν j σ z H iso = − j +1 ] − j j =1 j =1 ↓ Jordan-Wigner ↓ �� � H iso = c ∗ Ac + l , where c := ( c 1 , c 2 , . . . , c n ) t . j ν j 1 − ν 1 µ � � ... ... µ ≤ Ce − η | j − k | . |� e j , g ( A ) e k �| A := , E sup ... ... µ | g |≤ 1 − ν n µ Houssam Abdul-Rahman XY and Harmonic Oscillators 11 / 22
Particle Number Operator � � N := | e ↑ �� e ↑ | j and N S := | e ↑ �� e ↑ | j . j ∈ Λ j ∈ S N e α = ke α , where k = |{ j : α j = ↑}| . Let ρ = | e α �� e α | then �N� ρ := Tr N ρ = k is the expected number of up-spins. [ H, N ] = 0 ⇒ The number of up-spins is conserved in time. ρ t = e − itH iso ρe itH iso is the time evolution of ρ . �N S � ρ t is the expected number of up-spins in S at time t . Houssam Abdul-Rahman XY and Harmonic Oscillators 12 / 22
Particle Number/Energy Transport Results S 2 S 1 S 2 Fix Fix S 1 = [ a, b ] ⊂ Λ and S 2 ⊂ Λ \ S 1 . n � η j � 0 � Initial state: ρ = , with η j = 0 for all j / ∈ S 2 . 1 − η j 0 j =1 � � 4 C (1 + e − η ) 2 e − ηdist ( S 1 ,S 2 ) �N S 1 � ρ t ≤ E sup t Similar results for disordered Tonks-Girardeau gas, Seiringer/Warzel (2016). � � 4 CD (1 + e − η ) 2 e − ηdist ( S 1 ,S 2 ) , E sup |� H S 1 � ρ t − � H S 1 � ρ | ≤ t where D = sup n � A n � . Houssam Abdul-Rahman XY and Harmonic Oscillators 13 / 22
The Harmonic Oscillators Houssam Abdul-Rahman XY and Harmonic Oscillators 14 / 22
The Harmonic Oscillators The Hamiltonian � 1 � x + k x � � 2 mp 2 2 q 2 λ ( q x − q y ) 2 H = + x x ∈ Λ { x, y } ∈ Λ | x − y | = 1 Λ := [ − L, L ] d ∩ Z d where L ≥ 1 and d ≥ 1 . q x and p x = − i ∂ ∂q x are the position and momentum operators. � L 2 ( R , dq x ) . The Hilbert space H = x ∈ Λ m, λ ∈ (0 , ∞ ) . { k x } x are i.i.d. random variables with absolutely continuous distribution given by a bounded density ρ supported in [0 , k max ] . Houssam Abdul-Rahman XY and Harmonic Oscillators 15 / 22
The Harmonic Oscillators On a Lattice The Effective one-particle Hamiltonian | Λ | � γ k (2 B ∗ H = k B k + 1 l) ← − Free boson system. k =1 The operators B k satisfy the CCR [ B j , B k ] = [ B ∗ j , B ∗ [ B j , B ∗ k ] = 0 , k ] = δ j,k 1 l for all j, k ∈ { 1 , . . . , | Λ |} . 1 2 where { γ k } k are the eigenvalues of h k x 2 + 2 dλ, if x = y , � δ x , hδ y � = − λ, if | x − y | = 1 , 0 , else. Houssam Abdul-Rahman XY and Harmonic Oscillators 16 / 22
The Harmonic Oscillators The Eigencorrelator Localization Assumption: There exist constants C < ∞ and η > 0 such that � � α < Ce − η | x − y | , for α ∈ { 0 , 1 , − 1 } , 2 g ( h ) δ y �| |� δ x , h E sup | g |≤ 1 for all x, y ∈ Λ . Satisfied for d = 1 . d > 1 in the large disorder case. Houssam Abdul-Rahman XY and Harmonic Oscillators 17 / 22
The Harmonic Oscillators Eigenstates | Λ | � γ k (2 B ∗ H = k B k + 1 l) . k =1 There is a unique vacuum Ω b (the ground state of H ). The eigen-pair of H associated with α = ( α 1 , . . . , α | Λ | ) ∈ N | Λ | is 0 ( ψ α , E α ) , | Λ | 1 � α j !( B ∗ � j ) α j Ω b , ψ α = E α = (2 α j + 1) γ j � j =1 j For any α , the corresponding eigenstate is ρ α = | ψ α �� ψ α | . Houssam Abdul-Rahman XY and Harmonic Oscillators 18 / 22
The Harmonic Oscillators Correlations at the eigenstates Let C α ( A, B, t ) := � τ t ( A ) B � ρ α − � A � ρ α � B � ρ α , where τ t ( A ) = e itH Ae − itH . In the following Theorem: A ∈ { q x , p x } , B ∈ { q y , p y } . Theorem For any x, y ∈ Λ and α ∈ ℓ ∞ ( N | Λ | 0 ) , there exist constants C < ∞ and η > 0 such that � � < C (1 + � α � ∞ ) 2 e − η | x − y | . sup | C α ( A, B, t ) | E t Houssam Abdul-Rahman XY and Harmonic Oscillators 19 / 22
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