Entanglement Dynamics of Disordered Quantum XY Chains

Abdul‐Rahman, Houssam; Nachtergaele, Bruno; Sims, Robert; Stolz, Guenter

doi:10.1007/s11005-016-0835-9

Cited by 29 publications

(56 citation statements)

References 41 publications

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“…Here d(S 1 , S 2 ) = min{|x − y| : x ∈ S 1 , y ∈ S 2 } is the distance of S 1 and S 2 . This bound follows as a special case of Theorem 1.1 in [1]. The argument is similar to the proof of Theorem 7.1 below, provided in Appendix A.…”

Section: Absence Of Particle and Energy Transportmentioning

confidence: 74%

“…proving (29). Similar arguments lead to the bound (30) on dynamical entanglement in [1]. In particular, this uses that the product states ρ = |ϕ A ⊗ ϕ B ϕ A ⊗ ϕ B | as well as their time evolution τ t (ρ) under H are still quasi-free, so that E (τ t (ρ)) can be calculated using (33).…”

Section: Area Laws For the Bipartite Entanglementmentioning

confidence: 83%

“…Bounds on the dynamical entanglement in disordered spin chains were previously discussed in [9,32,53,60]. A rigorous result for the XY chain, proven in [1], is presented as (30)…”

Section: Results Surveyed In This Workmentioning

confidence: 99%

“…Here we will discuss a broad range of results of this type. In particular, we will describe recent localization results in the disordered XY chain and closely related disordered free Fermion systems from [1,2,29,48,55]. In the last two subsections we will also include several previously unpublished results, where we will provide detailed proofs.…”

Section: The Many-body Localization Contextmentioning

confidence: 99%

“…In fact, this result extends to any finite number of quenches in the initial state, with bounds not depending on the number of quenches. The proofs of (29) and (30) in [2] and [1] rely heavily on the fact that the eigenstates ρ of H are quasi-free, as already used in the previous section, so that they are entirely determined by their two-point function with respect to the fermionic modes c j , c * j , summarized in the correlation matrix…”

Section: Area Laws For the Bipartite Entanglementmentioning

confidence: 99%

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Localization properties of the disordered XY spin chain

et al. 2017

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We review several aspects of Many-Body Localization-like properties exhibited by the disordered XY chains: localization properties of the energy eigenstates and thermal states, propagation bounds of Lieb-Robinson type, decay of correlation functions, absence of particle transport, bounds on the bipartite entanglement, and bounded entanglement growth under the dynamics. We also prove new results on the absence of energy transport and Fock space localization. All these properties are made accessible to mathematical analysis due to the exact mapping of the XY chain to a system of quasi-free fermions given by the Jordan-Wigner transformation. Motivated by these results we discuss conjectured properties of more general disordered quantum spin and other systems as possible directions for future mathematical research.

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Section: Absence Of Particle and Energy Transportmentioning

confidence: 74%

Section: Area Laws For the Bipartite Entanglementmentioning

confidence: 83%

“…Bounds on the dynamical entanglement in disordered spin chains were previously discussed in [9,32,53,60]. A rigorous result for the XY chain, proven in [1], is presented as (30)…”

Section: Results Surveyed In This Workmentioning

confidence: 99%

Section: The Many-body Localization Contextmentioning

confidence: 99%

Section: Area Laws For the Bipartite Entanglementmentioning

confidence: 99%

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Localization properties of the disordered XY spin chain

et al. 2017

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Out‐of‐time‐ordered correlators in many‐body localized systems

2016

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In many-body localized systems, propagation of information forms a light cone that grows logarithmically with time. However, local changes in energy or other conserved quantities typically spread only within a finite distance. Is it possible to detect the logarithmic light cone generated by a local perturbation from the response of a local operator at a later time? We numerically calculate various correlators in the random-field Heisenberg chain. While the equilibrium retarded correlator A(t = 0)B (t > 0) is not sensitive to the unbounded information propagation, the out-of-time-ordered corre- In the presence of disorder, localization can occur not only in single-particle systems [1], but also in interacting many-body systems [2][3][4][5][6][7][8][9][10][11][12]. The former is known as Anderson localization (AL), and the latter is called manybody localization (MBL). Neither AL nor MBL systems transfer energy, charge, or other local conserved quantities: Changes in energy or charge at position x = 0 from equilibrium can spread and lead to changes in the corresponding quantity only within a finite distance |x| < L 0 , where L 0 is the localization length.A characteristic feature that distinguishes MBL from AL lies in the dynamics of entanglement after a global quench. Initialized in a random product state at time t = 0, the half-chain entanglement entropy remains bounded in AL systems [13], but grows logarithmically with time in MBL systems [14][15][16][17][18][19][20][21]. In sharp contrast to the transport phenomena, the unbounded growth of entanglement in MBL systems suggests that information propagates throughout the system, although very slowly.The propagation of information can be formulated by adapting the Lieb-Robinson (LR) bound [22][23][24] to the present context. In particular, it is manifested as the noncommutativity of a local operator A at x = 0 and t = 0 with another local operator B at position x and evolved for some time t . In MBL systems, the operator norm of the commutator [A(0, 0), B (x, t )] is non-negligible inside a light cone whose radius is given by |x| ∼ log |t |, and decays exponentially with distance outside the light cone, i.e. [25],after averaging over disorder. Here, B (x, t ) = e i H t B (x, 0)e −i H t is the time-evolved operator; · is the operator norm (the largest singular value); C , v LR , ξ are positive constants.Is it possible to detect the logarithmic light cone (LLC) with equilibrium correlators of A(0, 0) and B (x, t )? Arguably the most straightforward approach is to measure the commutator in the LR bound (1) on equilibrium states, i.e., thermal states or eigenstates, using the Kubo formula in linear response theory: (2) where U = e −i Aτ with τ 1 is a local unitary perturbation if A is Hermitian. The first and second terms on the lefthand side of (2) are the expectation values of B in the presence and absence of the perturbation U , respectively. The difference is the effect of U observed by measuring B . Note that U and B can be, but do not have to be, chosen as the op...

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Low-Energy Fock-Space Localization for Attractive Hard-Core Particles in Disorder

Beaud

Warzel

2017

Ann. Henri Poincaré

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Abstract. We study a one-dimensional quantum system with an arbitrary number of hard-core particles on the lattice, which are subject to a deterministic attractive interaction as well as a random potential. Our choice of interaction is suggested by the spectral analysis of the XXZ quantum spin chain. The main result concerns a version of high-disorder Fock-space localization expressed here in the configuration space of hard-core particles. The proof relies on an energetically motivated Combes-Thomas estimate and an effective one-particle analysis. As an application, we show the exponential decay of the two-point function in the infinite system uniformly in the particle number.Mathematics Subject Classification (2010). 82B44.

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Entanglement Dynamics of Disordered Quantum XY Chains

Cited by 29 publications

References 41 publications

Localization properties of the disordered XY spin chain

Localization properties of the disordered XY spin chain

Out‐of‐time‐ordered correlators in many‐body localized systems

Low-Energy Fock-Space Localization for Attractive Hard-Core Particles in Disorder

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