Correlations in disordered quantum harmonic oscillator systems: The effects of excitations and quantum quenches
aa r X i v : . [ m a t h - ph ] F e b Contemporary Mathematics
Correlations in disordered quantum harmonic oscillatorsystems: The effects of excitations and quantum quenches
Houssam Abdul-Rahman, Robert Sims, and G¨unter Stolz
Abstract.
We prove spatial decay estimates on disorder-averaged position-momentum correlations in a gapless class of random oscillator models. First,we prove a decay estimate on dynamic correlations for general eigenstateswith a bound that depends on the magnitude of the maximally excited mode.Then, we consider the situation of a quantum quench. We prove that the fulltime-evolution of an initially chosen (uncorrelated) product state has disorder-averaged correlations which decay exponentially in space, uniformly in time.
1. Introduction
The mathematical investigation of disorder effects on quantum many-body sys-tems, including, in particular, the phenomenon of many-body localization (MBL),is still in the early stages of its development. It has recently received strong atten-tion in the physics and quantum information theory literature, see, e.g., [
15, 5, 4 ]for recent reviews with extensive lists of references. Most mathematical resultshave been for models whose study can be fully reduced to the investigation of aneffective one-particle Hamiltonian (i.e., without interaction) such as the Andersonmodel. Only few results go beyond such models. This includes [ ], which proposesa derivation of MBL for certain disordered quantum spin chains, based on an asyet unproven assumption on level statistics for these models. Also, [ ] establishesexponential decay of ground state correlations for the Aubry-Andr´e model (an XXZchain in quasi-periodic field, which maps via the Jordan-Wigner transform to inter-acting Fermions). Recently, fully rigorous proofs of MBL in the droplet spectrum ofthe XXZ chain in random field (a regime extending beyond the ground state) havebeen given in [ ] and [ ]. In particular, the latter establishes exponential clus-tering of all eigenstates throughout the droplet spectrum with respect to arbitrarylocal observables.Models which can be fully reduced to an effective one-particle Hamiltonian in-clude the XY spin chain in random transversal field, see the review [ ], and theTonks-Girardeau gas subject to a random potential [ ]. Here we present somenew results on localization properties for another such model, disordered harmonicoscillator systems, as previously studied in [
13, 14, 1 ]. In [ ] results on themany-body dynamics in the form of zero-velocity Lieb-Robinson bounds as well asexponential decay of dynamical correlations (exponential clustering) of the ground c (cid:13) state and of thermal states of such systems were shown. [ ] further investigatedground and thermal states by establishing an area law for their bipartite entangle-ment entropy. More recently in [ ], area laws are proven for uniform ensembles ofequal-excitation energy eigenstates where the surface area bound increases linearlyin the total number of excitations (modes).There is, of course, a long history of interesting results for deterministic os-cillator models. It is well-known, see for example [
7, 17 ] and references therein,that ground states of uniformly gapped oscillator models satisfy exponential decayof correlations. Moreover, area laws for both ground and thermal states of gappedoscillator models can be found e.g. in [ ], see also the well-referenced review [ ].By contrast, as in [
13, 14, 1 ], we will consider models where the gap above theground state energy vanishes in the thermodynamic limit. For the results we willprove, estimates like the above mentioned deterministic results will not suffice, andwe instead exploit disorder effects.Our first new result here, Theorem 2.1 below, shows that exponential cluster-ing in disordered oscillator systems also holds for the dynamic position-momentumcorrelations of excited states. The bound obtained will only depend on the max-imal local excitation number of these states, i.e., when expressed in terms of thecorresponding free Boson system, for states with positive particle number density.This is desirable to show that the model is in the many-body localized phase, asthe latter, if it exists for a given model, should cover an extended part of the energyspectrum of the system. We also point out that, as opposed to ground and thermalstates, the excited states of oscillator systems are not quasi-free, a property usedin most of the works on exactly solvable models mentioned above. While the cre-ation of excited states out of the ground state of an oscillator system is a simplealgebraic process, our result can still be seen as a simple example of the possibilityto go beyond quasi-free states in the study of disordered many-body systems.In our second result, Theorem 2.2, we study quenched position-momentumcorrelations of disordered oscillator systems. Quantum quenches and their effect onthe non-equilibrium dynamics of quantum many-body systems have been frequentlyconsidered in physics, see, e.g., [ ] for a survey with many related references, aswell as [ ] for a discussion of quantum quenches in the context of many-bodylocalization.In its simplest form, a quantum quench refers to the investigation of a quan-tum state which is initially prepared with respect to one Hamiltonian and thentime-evolved with respect to another. A common scenario is as follows. Consider asystem defined on a Hilbert space for which there is a natural bipartite decompo-sition into two subsystems, i.e. H = H ⊗ H . Denote by H = H ⊗
1l + 1l ⊗ H + I the Hamiltonian for the full system where I represents the interaction between thetwo subsystems. As an initial state take a product ϕ = ϕ ⊗ ϕ which is uncorre-lated with respect to the bipartite decomposition. The time-evolution ϕ t = e − iHt ϕ of this initial state ϕ under the full Hamiltonian dynamics may exhibit interest-ing behavior, for example, non-trivial correlations may develop in time due to theinteraction I .In the disordered oscillator systems considered here we will assume that thetwo states ϕ and ϕ have exponentially clustered correlations with respect tothe Hamiltonians H and H of the subsystem. We will show that the position-momentum correlations of the state will remain globally exponentially clustered, uniform in time and in the sizes of both subsystems. In fact, Theorem 2.2 willbe more general in allowing for the decomposition into an arbitrary number ofsubsystems.Applications of Theorem 2.2, which we discuss in Section 6, include the casewhere the initial product state consists of thermal states of the subsystems, sothat the result of [ ] on exponential clustering of these states applies, or whereone starts with products of eigenstates of the subsystems, so that our first result,Theorem 2.1, can be applied in the subsystems. In physical terms these applicationssay that if each of the subsystems is localized in the sense of exponential decay ofstatic correlations of eigenstates and thermal states within the subsystem, thenthis form of quantum quench yields no thermalization. In this context we includeTheorem 6.1, as a technical result, proven in Appendix A, which improves resultson thermal state correlations in [ ] by quantifying the temperature dependence.
2. Model and Results2.1. The Model.
For any dimension d ≥
1, we consider a coupled harmonicoscillator system, on a finite rectangular box Λ := ([ a , b ] × . . . × [ a d , b d ]) ∩ Z d ,given by the Hamiltonian(1) H Λ = X x ∈ Λ (cid:18) p x + k x q x (cid:19) + X { x, y } ⊂ Λ : | x − y | = 1 λ ( q x − q y ) . This Hamiltonian acts on the Hilbert space(2) H Λ = O x ∈ Λ L ( R ) = L ( R Λ )and q x and p x are, respectively, the position and momentum operators at the site x ∈ Λ. By standard results, these operators are self adjoint, on suitable domains,and satisfy the commutation relations(3) [ q x , q y ] = [ p x , p y ] = 0 , and [ q x , p y ] = iδ x,y
1l for all x, y ∈ Λ . Here δ x,y is the Kronecker delta function.For each x ∈ Λ, k x represents a variable spring constant. We introduce disorderby allowing the sequence { k x } to be chosen as a sequence of i.i.d. random variables.More precisely, we will assume that their common distribution µ is absolutely con-tinuous,(4) dµ ( k ) = ν ( k ) dk, with k ν k ∞ < ∞ and supp ν = [0 , k max ]for some constant k max < ∞ .The Hamiltonian H Λ describes a convenient family of harmonic oscillators thatinteract through nearest neighbor terms with a coupling strength of λ >
0. To beclear, the second sum in (1) is taken over all undirected edges {{ x, y } ⊂ Λ : | x − y | =1 } where | · | denotes the 1-norm. With methods similar to those of [ ], the resultswe prove below generalize to a larger class of disordered oscillator models; thecaveat being that the assumptions on the effective one-particle Hamiltonian, i.e.the analogue of (7) below, would need to be verified on a case-by-case basis. Forease of presentation, we restrict our attention to the model defined by (1) above.As is well known, the analysis of general oscillator systems reduces, via Bo-goliubov transformation, to the analysis of an effective one-particle Hamiltonian, H. ABDUL-RAHMAN, R. SIMS, AND G. STOLZ see, e.g., [ ] where this is reviewed for more general systems. In the specific caseof (1), the corresponding one-particle Hamiltonian is the finite volume Andersonmodel on ℓ (Λ), i.e.(5) h Λ = λh , Λ + 12 k, where h , Λ is the negative discrete Laplacian on Λ and k an i.i.d. random po-tential. As we work in finite volume, the spectrum of h Λ is discrete and, underour assumption of an absolutely continuous distribution for the k x , almost surelysimple (as is seen easily by an analyticity argument).By our assumptions, it is clear that h Λ is self-adjoint and moreover, using that0 ≤ h , Λ ≤ d ,(6) σ ( h Λ ) ⊆ (cid:20)
12 min x ∈ Λ k x , (cid:18) dλ + k max (cid:19)(cid:21) . This means that h Λ is invertible almost surely, but its inverse will not be uniformlybounded in the disorder. In fact, for large boxes Λ, the minimum of σ ( h Λ ) will beclose to zero with high probability, due to the presence of large clusters where all k x are close to zero.As a localization characteristic of h Λ , we will assume that its singular eigen-function correlators decay exponentially. More precisely, we assume that there existconstants C < ∞ , η > < s ≤
1, independent of Λ, such that(7) E sup | g |≤ |h δ x , h − Λ g ( h Λ ) δ y i| s ! ≤ Ce − η | x − y | , for all x, y ∈ Λ, where E ( · ) is the disorder average and { δ x } x ∈ Λ the canonical basisof ℓ (Λ). The supremum is taken over all functions g : R → C with | g ( x ) | ≤ g ( h Λ ) is defined by the functional calculus of symmetric matrices.The non-standard feature of (7) is the term h − / . By the discussion above,this term does not have an a-priori norm bound and can thus not be absorbedinto g ( h Λ ). This term also is the reason for the inclusion of the moment s , whichincreases the applicability of (7).In the absence of the factor h − / the bound(8) E sup | g |≤ |h δ x , g ( h Λ ) δ y i| ! ≤ Ce − η | x − y | is a standard result for two regimes, e.g. [ ]: (i) in dimension d = 1 (where Λ is aninterval) for any choice of the distribution density ν , and (ii) in the large disorderregime for d > k ν k ∞ is sufficiently small, or k x is rescaled by alarge parameter).The singular eigenfunction correlators in (7) were first introduced in [ ]. Asis discussed there in Appendix A, (7) holds for d ≥ s = 1(combing Propositions A.1(b) and A.3(b)), and for d = 1 and any ν with s = 1 / s = 1 / s replaced by 1,due to the fact that |h δ x , h − Λ g ( h Λ ) δ y i| does not satisfy a uniform a-priori bound.Applications such as this are the reason we allow for the flexibility of s in (7). As further discussed in Section 3, the diagonalization of h Λ transforms H Λ intoa model of free bosons(9) H Λ = | Λ | X k =1 γ k (2 B ∗ k B k + 1l) , where γ k are the eigenvalues of h Λ , and the operators { B k } satisfy canonical com-mutation relations (CCR) namely(10) [ B j , B k ] = [ B ∗ j , B ∗ k ] = 0 , [ B j , B ∗ k ] = δ j,k . In this case, there is a unique normalized vacuum state Ω ∈ H Λ corresponding tothese B -operators, i.e., Ω satisfies B k Ω = 0 for all k . An explicit orthonormal basis(ONB) of eigenvectors of H Λ is then given by(11) ψ α = | Λ | Y j =0 p α j ! ( B ∗ j ) α j Ωfor an excitation number configuration α = ( α , . . . , α | Λ | ) ∈ N | Λ | (here N := { , , , · · · } ). One easily checks that these excited states satisfy(12) H Λ ψ α = E α ψ α with E α = | Λ | X k =1 (2 α k + 1) γ k and therefore, the gap above the ground state energy of H Λ is 2 min k γ k . One goal of this work is toestimate dynamic correlations of position and momentum operators in arbitraryeigenstates. To make this more precise, let τ t ( A ) denote the Heisenberg evolutionof an operator A under H Λ , i.e.,(13) τ t ( A ) = e itH Λ Ae − itH Λ , and for any trace-class operator ρ on H Λ take(14) ρ t = e − itH Λ ρe itH Λ to be the Schr¨odinger evolution of ρ . In this case, if h A i ρ = Tr[ Aρ ] denotes the ρ -expectation of the observable A , then the Heisenberg and Schr¨odinger evolutionsare related by h τ t ( A ) i ρ = h A i ρ t .It is convenient to introduce a 2 | Λ | × | Λ | correlation matrix(15) Γ ρ ( t, t ′ ) := (cid:28) τ t (cid:18) qp (cid:19) ( q T , p T ) (cid:29) ρ t ′ − (cid:28) τ t (cid:18) qp (cid:19)(cid:29) ρ t ′ (cid:10) ( q T , p T ) (cid:11) ρ t ′ which collects mixed-time dynamic correlations of position and momentum oper-ators corresponding to ρ . Here (cid:18) qp (cid:19) and ( q T , p T ) are 2 | Λ | column and row vec-tors, the time-evolution and ρ -expectation of vectors and matrices are understoodcomponent-wise, and columns are multiplied with rows in the usual sense of matrixmultiplication to form matrices. Use of these mixed-time correlations Γ ρ ( t, t ′ ) isconvenient when formulating our main results below; in one we set t ′ = 0 and inthe other t = 0. H. ABDUL-RAHMAN, R. SIMS, AND G. STOLZ
Our first result concerns disorder-averaged correlations in eigenstates, i.e. weconsider ρ = ρ α = | ψ α ih ψ α | for some α ∈ N | Λ | . Since eigenstates are time-invariant,we set t ′ = 0 in (15) and denote by Γ α ( t ) := Γ ρ α ( t, h τ t ( q x ) i ρ α = h τ t ( p x ) i ρ α = 0for all x , t and α (in fact, we will also see this directly in the proof of Theorem 2.1below). Thus (15) simplifies to(17) Γ α ( t ) = (cid:28) τ t (cid:18) qp (cid:19) ( q T , p T ) (cid:29) ρ α . Lastly, for a 2 | Λ | × | Λ | block-matrix(18) M = (cid:18) A BC D (cid:19) , let M xy = (cid:18) A xy B xy C xy D xy (cid:19) be 2 × k M xy k . For definiteness we choose thelatter to be the Euclidean operator norm. Theorem . Assume that the effective Hamiltonian h Λ satisfies (7), withbounds uniform in Λ . Then (19) E (cid:18) sup t k (Γ α ( t )) xy k s (cid:19) ≤ CC ′ (1 + k α k ∞ ) s e − η | x − y | for all finite rectangular boxes Λ ⊂ Z d , x, y ∈ Λ and α ∈ N | Λ | . Here C , η and s areas in (7) and C ′ < ∞ depends on d , λ , s and k max , but is independent of Λ . We finally note that our proof of Theorem 2.1 in Section 4 below will show, thebound in (19) can be slightly improved to(20) ≤ ˜ C (cid:0) {k α k , k α k s ∞ } (cid:1) e − η | x − y | , which is better for excitation vectors with only a few large excitations α j (say, justone of them). Our second result concerns the position andmomentum correlations when a quantum quench is applied. In particular, we de-compose the rectangular box Λ into M disjoint rectangular sub-boxes(21) Λ = M ] ℓ =1 Λ ℓ . For ℓ = 1 , , . . . , M , consider the restrictions H Λ ℓ of the harmonic system H Λ toΛ ℓ . Let H , Λ denote the Hamiltonian of the non-interacting system on H Λ ,(22) H , Λ = M X ℓ =1 H Λ ℓ ⊗ Λ \ Λ ℓ . For each ℓ = 1 , . . . , M , let ρ ℓ be a state acting on the Hilbert space H Λ ℓ := L ( R Λ ℓ ).In particular, we will consider the cases where the ρ ℓ are either eigenstates orthermal states of H Λ ℓ . We are interested in the Schr¨odinger time evolution ρ t ,under the full Hamiltonian H Λ given in (1), of the state initially given by theproduct state(23) ρ := M O ℓ =1 ρ ℓ . This quantum quench is understood as a sudden change in the Hamiltonian H , Λ at t = 0, consisting in switching on the interactions between the subsystems H Λ ℓ .To describe the dynamic correlations in this case we set the first argumentequal to zero in (15) and define(24) ˜Γ ρ ( t ) := Γ ρ (0 , t ) . That the local systems H Λ ℓ are initially uncorrelated means that for x ∈ Λ j and y ∈ Λ ℓ with j = ℓ ,(25) (˜Γ ρ (0)) xy = 0 . The following result says that if each of the subsystems H Λ ℓ is localized in thesense of exponential decay of static correlations within the subsystem, then theirquantum quench, described above, yields no thermalization. More precisely, the so-called quenched dynamic correlations of the product state ρ remain exponentiallydecaying in the fully interacting system for all times (here we also use the local2 | Λ ℓ | × | Λ ℓ | correlation matrices Γ ρ ℓ with t = t ′ = 0 in (15)). Theorem . Assume that the effective Hamiltonian h Λ satisfies (7), withbounds uniform in Λ . Let ρ ℓ ∈ B ( H Λ ℓ ) , ℓ = 1 , . . . , M , be a family of states suchthat, for some C ′ < ∞ , and η ′ > , (26) E ( k (Γ ρ ℓ ) xy k s ) ≤ C ′ e − η ′ | x − y | for all ℓ and all x, y ∈ Λ ℓ , where < s ≤ is as in (7).Then, for η from (7), ˜ η := min { η, η ′ } and ρ = N ℓ ρ ℓ , there exists a constant C ′′ < ∞ such that (27) E (cid:18) sup t ∈ R k (˜Γ ρ ( t )) xy k s (cid:19) ≤ ( C ′ ) / C ′′ e − ˜ η | x − y | for all x, y ∈ Λ . Here C ′ is the constant from (26) and C ′′ depends on d , λ , s , k max and ˜ η , but is independent of Λ and the number of subregions M . This will be proven in Section 5.By results in [ ], special cases where condition (26) is known to hold includethe ground state and thermal states of the subsystems H Λ ℓ . Theorem 2.1 aboveextends this to excited states. In each of these cases, (26) actually follows from (7).Theorem 2.2 allows for the additional freedom to choose different temperaturesand different maximal excitation numbers in each of the subsystems, or even tochoose some of the factors in the initial product state as thermal states and othersas excited states. It is then of some interest to understand the dependence of theconstants in (27) on these additional parameters. We include a more thoroughdiscussion of this in Section 6 at the end of this paper.
3. Reduction to the Effective Hamiltonian
In this section, we briefly review the previously mentioned reduction of themany-body Hamiltonian H Λ to the effective one-particle h Λ as a means to introducesome relevant notation. Once this is done, we provide a simple lemma concerningmixed-time correlations, i.e. (15). H. ABDUL-RAHMAN, R. SIMS, AND G. STOLZ
Keeping with the vector notation established in Section 2.2, one readily seesthat the oscillator Hamiltonian H Λ in (1) can be re-written as(28) H Λ = ( q T , p T ) (cid:18) h Λ
00 1l (cid:19) (cid:18) qp (cid:19) with h Λ the effective one-particle Hamiltonian described in (5). The real non-negative matrix h Λ can be diagonalized in terms of a real orthogonal O : C | Λ | → ℓ (Λ) and its transpose O T = O − , i.e.(29) O T h Λ O = γ where γ = diag( γ k ) with 1 ≤ k ≤ | Λ | . Here the numbers γ k are the eigenvalues of h Λ counted according to multiplicity. By our assumptions on the spring constants,the eigenvalues of h Λ are almost surely positive, and we will denote by γ = diag( γ k )with γ k > ≤ k ≤ | Λ | . As discussed in Section 2.1, the γ k are almostsurely non-degenerate.As is well-known, see [ ] for more details in this specific setting, H Λ can bereduced to a system of free Bosons. In fact, consider the mapping(30) V − = 1 √ (cid:18) i − i (cid:19) (cid:18) γ / O T γ − / O T (cid:19) . Our assumptions guarantee this map is almost surely well-defined, invertible, andone readily checks that the product(31) (cid:18) BB ∗ (cid:19) := V − (cid:18) qp (cid:19) produces a collection of operators { B k } | Λ | k =1 on H Λ which, together with their ad-joints, satisfy the CCR, i.e. (10). Moreover, in terms of these B -operators(32) H Λ = | Λ | X k =1 γ k (2 B ∗ k B k + 1l) , a model of free Bosons.Due to the simple form of (32), the dynamics of these B -operators is(33) τ t (cid:18) BB ∗ (cid:19) = (cid:18) e − itγ e itγ (cid:19) (cid:18) BB ∗ (cid:19) from which the dynamics of position and momentum operators readily follows,(34) τ t (cid:18) qp (cid:19) = V (cid:18) e − itγ e itγ (cid:19) (cid:18) BB ∗ (cid:19) , where we have used (31). It will also be convenient to note that(35) V = 1 √ (cid:18) O γ − / O γ / (cid:19) (cid:18)
1l 1l − i i (cid:19) . As indicated in Section 2.2, much of our analysis reduces to the investigationof the mixed-time correlation function Γ ρ ( t, t ′ ) in (15) for a state ρ . Since (34)shows that the dynamics of position and momentum operators can be expressed interms of the B -operators, up to scalar-valued coefficients, one immediately has thefollowing. Lemma . Let ρ be a state on H Λ . Suppose that the matrix (36) Γ Bρ = (cid:28)(cid:18) BB ∗ (cid:19) ( B T , ( B ∗ ) T ) (cid:29) ρ − (cid:28)(cid:18) BB ∗ (cid:19)(cid:29) ρ (cid:10) ( B T , ( B ∗ ) T ) (cid:11) ρ is well-defined. Then, for all t, t ′ ∈ R , (37) Γ ρ ( t, t ′ ) = V (cid:18) e − i ( t + t ′ ) γ e i ( t + t ′ ) γ (cid:19) Γ Bρ (cid:18) e − it ′ γ e it ′ γ (cid:19) V T where V is as in (35).
4. Proof of Theorem 2.1
We begin with a calculation which evaluates the eigenstate correlation matrixΓ α ( t ) given by (17) in terms of the effective Hamiltonian h Λ , using Lemma 3.1. Lemma . We have the identity Γ α ( t ) = (cid:18) O α O T O α O T (cid:19) h − / cos(2 th / ) sin(2 th / ) − sin(2 th / ) h / cos(2 th / ) ! +(38) + 12 h − / e − ith / ie − ith / − ie − ith / h / e − ith / ! . Here, in a slight abuse of notation, we use α also to denote the diagonal matrixwith entries α k , 1 ≤ k ≤ | Λ | . Proof.
By orthogonality of the eigenvectors ψ α , it is clear that each of(39) (cid:28)(cid:18) BB ∗ (cid:19)(cid:29) ρ α , h BB T i ρ α , and h B ∗ ( B ∗ ) T i ρ α vanish identically. We note that this and (34) implies (16). Moreover, using alsothe commutation relations (10), we find that for all 1 ≤ j, k ≤ | Λ | ,(40) h B ∗ k B j i ρ α + δ j,k = h B j B ∗ k i ρ α = h B ∗ j ψ α , B ∗ k ψ α i = ( α j + 1) δ j,k . and therefore, for Γ Bρ α as in (36), we have that(41) Γ Bρ α = (cid:18) α + 1l) α (cid:19) An application of Lemma 3.1 yields(42) Γ α ( t ) = V (cid:18) α + 1l) e − iγt αe iγt (cid:19) V T . A short calculation shows that(43) 12 (cid:18)
1l 1l − i i (cid:19) (cid:18) α + 1l) e − iγt αe iγt (cid:19) (cid:18) − i i (cid:19) can be rewritten as(44) (cid:18) α α (cid:19) (cid:18) cos(2 γt ) sin(2 γt ) − sin(2 γt ) cos(2 γt ) (cid:19) + 12 (cid:18) e − iγt e − iγt (cid:19) (cid:18) i − i
1l 1l (cid:19) . Using the form of V and V − in (35) and (30) this givesΓ α ( t ) = (cid:18) O γ − / O γ / (cid:19) (cid:18) α cos(2 γt ) α sin(2 γt ) − α sin(2 γt ) α cos(2 γt ) (cid:19) (cid:18) γ − / O T γ / O T (cid:19) + 12 (cid:18) O γ − / O γ / (cid:19) (cid:18) e − iγt ie − iγt − ie − iγt e − iγt (cid:19) (cid:18) γ − / O T γ / O T (cid:19) . (45)This is the same as (38), due to(46) O γ − / α cos(2 γt ) γ − / O T = O α O T h − / cos(2 th − / )and similar consequences of the functional calculus. (cid:3) We can now present the proof of Theorem 2.1.
Proof. (of Theorem 2.1) We first consider the most singular case, and thencomment on how the remaining cases follow similarly.Lemma 4.1 demonstrates that(47) h τ t ( q x ) q y i α = h δ x , O α O T h − / cos(2 th / ) δ y i + 12 h δ x , h − / e − ith / δ y i . Using (7), it is clear that(48) E (cid:18) sup t ∈ R (cid:12)(cid:12)(cid:12) h δ x , h − / e − ith / δ y i (cid:12)(cid:12)(cid:12) s (cid:19) ≤ Ce − η | x − y | , and so we need only estimate the first term in (47) above. If the eigenvalues of h Λ are non-degenerate, which holds almost surely, we can write(49) O α O T = | Λ | X j =1 α j χ { γ j } ( h Λ ) = k α k ∞ X a =0 aχ J ( a ) ( h Λ ) . Here χ { γ j } ( h Λ ) is the projection onto the eigenvector of h Λ to γ j , and χ J ( a ) ( h Λ )is the spectral projection for h Λ onto J ( a ) := { γ j : α j = a } . Given this, oneimmediately sees that(50) (cid:12)(cid:12)(cid:12) h δ x , O α O T h − / cos(2 th / ) δ y i (cid:12)(cid:12)(cid:12) s ≤ k α k ∞ X a =0 a s (cid:12)(cid:12)(cid:12) h δ x , h − / χ J ( a ) ( h Λ ) cos(2 th / ) δ y i (cid:12)(cid:12)(cid:12) s and therefore an application of (7) again implies(51) E (cid:18) sup t ∈ R (cid:12)(cid:12)(cid:12) h δ x , O α O T h − / cos(2 th / ) δ y i (cid:12)(cid:12)(cid:12) s (cid:19) ≤ C k α k s ∞ ( k α k ∞ + 1) e − η | x − y | . This completes the argument for the most singular correlations.As is clear from Lemma 4.1, the other correlations in the 2 × α ( t )) xy produce similar terms. These terms require bounds on eigenfunction correlators lesssingular than (7), in the sense that the term h − / is replaced by 1l or h / . They canbe bounded by (7) due to the uniform spectral bound (6). For example, associating˜ g ( x ) := x / g ( x ) with each g such that | g | ≤
1, one gets(52) E sup | g |≤ |h δ x , g ( h Λ ) δ y i| s ! ≤ (4 dλ + k max s/ E sup | g |≤ |h δ x , h − / g ( h Λ ) δ y i| s ! , and similar for E (cid:16) sup | g |≤ |h δ x , h / g ( h Λ ) δ y i| s (cid:17) .Finally, the bound in terms of k α k in (20) follows by directly considering themiddle term in (49).This completes the proof of Theorem 2.1. (cid:3)
5. Proof of Theorem 2.2
By Lemma 3.1, the qp -correlations (24) corresponding to the time-evolution ofany initially chosen density matrix ρ can be evaluated as:˜Γ ρ ( t ) = (cid:28)(cid:18) qp (cid:19) ( q T , p T ) (cid:29) ρ t − (cid:28)(cid:18) qp (cid:19)(cid:29) ρ t (cid:10) ( q T , p T ) (cid:11) ρ t (53) = V (cid:18) e − itγ e itγ (cid:19) Γ Bρ (cid:18) e − itγ e itγ (cid:19) V T where V and Γ Bρ are as in (35) and (36), respectively. For our arguments here, weprefer to re-express this in terms of the time-zero qp -correlations, i.e., we write(54) ˜Γ ρ ( t ) = V t Γ ρ V Tt , where we have set(55) V t = V (cid:18) e − itγ e itγ (cid:19) V − = cos(2 th / ) h − / sin(2 th / ) − h / sin(2 th / ) cos(2 th / ) ! . The final equality is a direct calculation.By (54) one has, for any x, y ∈ Λ,(56) (˜Γ ρ ( t )) xy = X z,z ′ ∈ Λ ( V t ) xz (Γ ρ ) zz ′ ( V Tt ) z ′ y with 2 × ρ ( t )) xy , ( V t ) xz , (Γ ρ ) zz ′ and ( V Tt ) z ′ y defined according to (18).Similar to the arguments in Section 4, our basic assumption (7), guarantees theexistence of ˜ C < ∞ , depending on d , λ , s and k max , such that(57) E (cid:18) sup t ∈ R k ( V t ) xy k s (cid:19) ≤ ˜ Ce − η | x − y | for every rectangular box Λ and all x, y ∈ Λ. It is clear that the same bound alsoholds for V Tt .For the product state ρ = ⊗ Mℓ =1 ρ ℓ in (23), the qp -correlation matrix is the directsum of the correlation matrices of the factors ρ ℓ . More precisely, for x, y ∈ Λ,(58) (Γ ρ ) xy = (cid:26) (Γ ρ ℓ ) xy if x, y ∈ Λ ℓ for some ℓ ,0 otherwise.Thus, by condition (26)(59) E ( k (Γ ρ ) xy k s ) ≤ C ′ e − η ′ | x − y | for all x, y ∈ Λ.For all x, y ∈ Λ we have that E (cid:18) sup t ∈ R k (˜Γ ρ ( t )) xy k s/ (cid:19) ≤ X z,z ′ ∈ Λ E (cid:18) sup t ∈ R k ( V t ) xz k s (cid:19) / ×× E ( k (Γ ρ ) zz ′ k s ) / E (cid:18) sup t ∈ R k ( V Tt ) z ′ y k s (cid:19) / , (60)where we have used (56) and H¨older’s inequality. Thus (57) and (59) yield E (cid:18) sup t ∈ R k (˜Γ ρ ( t )) xy k s/ (cid:19) ≤ ˜ C / C ′ X z,z ′ ∈ Λ e − η | x − z | / e − η ′ | z − z ′ | / e − η | z ′ − y | /
32 H. ABDUL-RAHMAN, R. SIMS, AND G. STOLZ ≤ C ′′ e − ˜ η | x − y | . (61)Here one may take ˜ η = min { η, η ′ } and(62) C ′′ = ˜ C / ( C ′ ) / (cid:18) − e − ˜ η (cid:19) d .
6. Applications of Theorem 2.2 (i) As a first application of Theorem 2.2 we consider the case where the factorsin the product state (23) are thermal states of the subsystems. Assumption (26)in Theorem 2.2 is then a consequence of Theorem 6.1 in [ ] on the position-momentum correlations of thermal states of oscillator systems. We start by statingan improved version of this result, which makes the temperature dependence of thebound explicit, a fact of some interest by itself which was not addressed in [ ]. InAppendix A we sketch the modifications of the argument in [ ] needed to get thisimprovement.While more general systems are considered in [ ], we will continue to focus onthe model (1). Here we only require the general assumption (4) on the distributionof the k x , and, in particular, we do not require to be in a fully localized regime asneeded for (7). Theorem . For a rectangular box Λ ⊂ Z d and β ∈ (0 , ∞ ) , let H Λ be givenby (1), ρ β = e − βH Λ / Tr[ e − βH Λ ] its thermal states, and Γ ρ β = Γ ρ β (0 , their staticposition-momentum correlation matrices.There exist C < ∞ and µ > , dependent on d , λ and the distribution of therandom variables k x , but independent of Λ and β , such that (63) E (cid:16) k (Γ ρ β ) xy k (cid:17) ≤ C max (cid:26) , β (cid:27) e − µ | x − y | for all x, y ∈ Λ . In Section A below we will briefly discuss how the β dependence in (63) can beextracted from the bounds provided in [ ].Let us consider the quantum quench with respect to the decomposition Λ = U Mℓ =1 Λ ℓ , and assume that the local states are the thermal states of H Λ ℓ with inversetemperatures β ℓ , ℓ = 1 , . . . , M , i.e.,(64) ρ ℓ,β ℓ = e − β ℓ H Λ ℓ Tr[ e − β ℓ H Λ ℓ ] . Then condition (26) is satisfied by Theorem 6.1 when applied to each of the localHamiltonians. In this case, and for the remainder of this section, we will furtherassume that (7) holds with s = 1 /
2. As is discussed in Section 2.1, this will be thecase for the model we are considering when either d = 1 or d ≥ ρ β ,...,β M := M O ℓ =1 ρ ℓ,β ℓ . Theorem 2.2 implies that(66) E (cid:18) sup t k (˜Γ ρ β ,...,βM ( t )) xy k (cid:19) ≤ C ′ max n , β − / o e − ˜ η | x − y | for all x, y ∈ Λ. Here β = min ℓ β ℓ and ˜ η = min { η, µ } where η and µ are as in (7)and (63), respectively, and C ′ is independent of Λ, β and the number of subsystems M . (ii) Next we discuss the case where the initial state is a product of eigenstatesof the subsystems. Fix a nonnegative integer N < ∞ , and let α ℓ ∈ N | Λ ℓ | with k α ℓ k ∞ ≤ N for all ℓ = 1 , . . . , M . Consider any family ρ α ℓ = | ψ α ℓ ih ψ α ℓ | , ℓ =1 , . . . , M , of eigenstates of H Λ ℓ corresponding to the excitation vectors α ℓ , with ψ α ℓ given by (11) when used for the subsystem Λ ℓ . Theorem 2.1 implies thatcondition (26) is satisfied for all ℓ = 1 , . . . , M , in particular, there exist constants C ′ > η < ∞ such that(67) E (cid:16) k (Γ ρ αℓ ) xy k (cid:17) ≤ C ′ (1 + k α ℓ k ∞ ) e − η | x − y | , for all ℓ and all x, y ∈ Λ ℓ . Here η is as in (7), which we have again taken to holdwith s = 1 /
2, and C ′ is independent of Λ, N , and of M . With ρ α the correspondingproduct, i.e. ρ α = ⊗ Mℓ =1 ρ α ℓ , an application of Theorem 2.2 shows that(68) E (cid:18) sup t k (˜Γ ρ α ( t )) xy k (cid:19) ≤ ˜ C (1 + N ) e − η | x − y | for all x, y ∈ Λ. Here again ˜
C < ∞ is independent of Λ, the number M of decom-positions, and of the highest excitation N , and moreover, η is as above.(iii) One can combine the cases (i) and (ii) and consider a product state ρ as in (23) where each local state is either a thermal state or an eigenstate of theHamiltonian H Λ ℓ . The arguments in cases (i) and( ii) above provide a proof of thefollowing result, which summarizes all the cases considered so far. Corollary . Fix β > and N < ∞ . Let ρ = ⊗ Mℓ =1 ρ ℓ where each of thelocal states ρ ℓ is either a thermal state of H Λ ℓ with inverse temperature β ℓ ∈ [ β, ∞ ) ,or an eigenstate associated with an excitation vector α ℓ such that k α ℓ k ∞ ≤ N . If ˜ η = min { η, µ } , where η is as in (7) with s = 1 / and µ is as in (63), then thereexists C < ∞ such that (69) E (cid:18) sup t k (˜Γ ρ ( t )) xy k (cid:19) ≤ C max (cid:26) (1 + N ) , β (cid:27) e − ˜ η | x − y | for all x, y ∈ Λ . Here C is independent of Λ , N , M and of β . (iv) In the extreme case where each subsystem consists of only one site, i.e., M = | Λ | , the initial Hamiltonian H , Λ is a system of non-interacting harmonicoscillators over the d dimensional lattice Λ,(70) H , Λ = X x ∈ Λ H { x } ⊗ Λ \{ x } , where H { x } is the one dimensional harmonic oscillator(71) H { x } = p x + k x q x . The eigenstates of H { x } are known to be the Hermite functions(72) φ n x ( q x ) = 1 √ n x n x ! (cid:18) √ k x π (cid:19) H n x ( r k x q x ) e − √ kx q x , where n x ∈ N the excitation number at vertex x ∈ Λ, H n x ( · ) is the Hermitepolynomial of degree n x . In this special case, the following corollary improves onthe bound in (68) for the correlations of the dynamics of the product state(73) ρ = O x ∈ Λ ρ n x , where ρ n x = | φ n x ih φ n x | . Corollary . Let ρ be as in (73) and let N = max x n x . If η is as in (7)with s = 1 / , then there exists C < ∞ , independent of Λ and N , such that (74) E (cid:18) sup t k (˜Γ ρ ( t )) xy k (cid:19) ≤ C (1 + 2 N ) e − η | x − y | for all x, y ∈ Λ . Proof.
In the current case the local Hamiltonians h { x } reduce to the singlenumbers k x /
2. This means that in the case of an n x excitation at site x , thecorrelation matrix from (38) reduces to the 2 × n x (0) = √ k − x (1 + 2 n x ) i − i √ k x (1 + 2 n x ) . Since k x is a random variable with a bounded density ν and supported on thecompact set [0 , k max ], one gets(76) E (max { k x , k − x } ) ≤ k ν k ∞ max n ( k max ) , ( k max ) / o . Hence, there exists C ′ < ∞ such that(77) E ( k Γ n x (0) k ) ≤ C ′ (1 + 2 n x ) . Given (77), the full correlation matrix for the product state ρ = ⊗ x ρ n x satisfies(78) E ( k (Γ ρ ) xy k ) ≤ C ′ (1 + 2 N ) δ x,y . Arguing as in (60), see also (61), we conclude that(79) E (cid:18) sup t ∈ R k (˜Γ ρ ( t )) xy k (cid:19) ≤ ˜ C X z,z ′ ∈ Λ e − η | x − z | / E (cid:16) k (Γ ρ ) zz ′ k (cid:17) e − η | z ′ − y | / where ˜ C is as in (57) with s = 1 /
2. Since Holder and (78) imply that(80) E (cid:16) k (Γ ρ ) zz ′ k (cid:17) ≤ E ( k (Γ ρ ) zz ′ k ) ≤ p C ′ (1 + 2 N ) δ z,z ′ the claim in (74) now follows as in the end of the proof of Theorem 2.2. (cid:3) Appendix A. Proof of Theorem 6.1
In the following we use and refine several results from [ ] to prove Theorem 6.1.We start by noting that these results are only formulated for cubes in [ ], but thatthey extend to the rectangular boxes considered here.By Lemma 5.4 of [ ],(81) (Γ ρ β ) xy = 12 h δ x , h − / ϕ ( h Λ ) δ y i iδ xy − iδ xy h δ x , h / ϕ ( h Λ ) δ y i ! , where we have set ϕ ( t ) = coth( βt / ). Thus for (63) it suffices to show that(82) E (cid:16) |h δ x , h ± / ϕ ( h Λ ) δ y i| / (cid:17) ≤ C max (cid:26) , β (cid:27) e − µ | x − y | . Expanding 1l = P z | δ z ih δ z | and an application of H¨older’s inequality show that theleft hand side of (82) can be bounded by(83) X z (cid:16) E ( |h δ x , h ± / δ z i| ) (cid:17) / ( E ( |h δ z , ϕ ( h Λ ) δ y i| )) / . The two factors in the sum can both be bounded using Proposition A.3(c) of[ ] and the method of its proof, respectively. For the first factor we can citeProposition A.3(c) directly to conclude the existence of C < ∞ and µ > E ( |h δ x , h ± / δ z i| ) ≤ C e − µ | x − z | for all x and z .To understand the β -dependence of the second factor, we need to analyze theproof of Proposition A.3(c) of [ ]. It requires splitting low and high energies of h Λ . At low energies, we can use localization of h Λ : Our assumptions yield thatthere exists E > h Λ has localized s -fractionalmoments in [0 , E ] for all s ∈ (0 , ]. By Proposition A.3(b) of [ ] thisimplies the existence of C < ∞ and µ > E sup | g |≤ |h δ z , h − / g ( h Λ ) χ [0 ,E ] ( h Λ ) δ y i| ! ≤ C e − µ | z − y | . Using the elementary bound(86) | ϕ ( t ) | ≤ β √ E + 1 β t − for all t ∈ [0 , E ], (85) gives(87) E (cid:0) |h δ z , ϕ ( h Λ ) χ [0 ,E ] ( h Λ ) δ y i| (cid:1) ≤ C β √ E + 1 β e − µ | z − y | . We will further prove that there are C < ∞ and µ > E (cid:0) |h δ z , ϕ ( h Λ ) χ ( E , ∞ ) ( h Λ ) δ y i| (cid:1) ≤ C coth ( β p E ) e − µ | z − y | (88) ≤ C (cid:18) β √ E (cid:19) e − µ | z − y | for all z and y .Inserting all of (84), (87) and (88) into (83), ultimately gives the bound (82).We still owe the proof of the first claim in (88). This is done by an analysis ofthe proof of Proposition A.3(c) in [ ]. This proof, see (A.15) in [ ], uses that(89) |h δ z , ϕ ( h Λ ) χ ( E , ∞ ) ( h Λ ) δ y i| ≤ C ′ Z Γ |h δ z , ( h Λ − ζ ) − χ ( E ,M ] ( h Λ ) δ y i| | dζ | , where Γ is the rectangular contour with vertices E ± i and ( M + 1) ± i , M thea-priori upper bound for σ ( h Λ ) from (6), and C ′ = max {| ϕ ( ζ ) | : ζ ∈ Γ } / (2 π ). Using the elementary bound | coth( ζ ) | ≤ coth (Re ζ ) for ζ ∈ C \ { } one has(90) | ϕ ( ζ ) | ≤ coth ( β Re ζ ) = coth β r Re ζ + | ζ | ! . This means that(91) C ′ ≤ π coth β min ζ ∈ Γ r Re ζ + | ζ | ! = 12 π coth (cid:16) β p E (cid:17) . The argument in [ ] shows that the ( β -independent) integral in (89) is boundedby C ′′ e − µ | z − y | for some C ′′ < ∞ , µ >
0. Combined with (91) this yields (88).
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