Persistence of Locality in Systems with Power-Law Interactions

Gong, Zhe-Xuan; Foss‐Feig, Michael; Michalakis, Spyridon; Gorshkov, Alexey V.

doi:10.1103/physrevlett.113.030602

Cited by 129 publications

(203 citation statements)

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“…We then use the powerful formalism of quasiadiabatic continuation [44] to relate such a state to the ground state of a spectrally gapped long-range interacting Hamiltonian. This strategy is made possible by the recent proof of Kitaev's small incremental entangling (SIE) conjecture [43,45], and by significant recent improvements in Lieb-Robinson bounds [4] for long-range interacting systems [46,47].…”

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Entanglement Area Laws for Long-Range Interacting Systems

et al. 2017

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We prove that the entanglement entropy of any state evolved under an arbitrary 1=r α long-rangeinteracting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α > D þ 1. We also prove that for any α > 2D þ 2, the ground state of such a Hamiltonian satisfies the entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known to satisfy the area law. These results significantly generalize their existing counterparts for short-range interacting systems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presence of long-range interactions. DOI: 10.1103/PhysRevLett.119.050501 Quantum many-body systems often have approximately local interactions, and this locality has profound effects on the entanglement properties of both ground states and the states created dynamically after a quantum quench. For example, the entanglement entropy, defined as the entropy of the reduced state of a subregion, often scales as the boundary area of the subregion for ground states of shortrange interacting Hamiltonians [1]. This "area law" of entanglement entropy is in sharp contrast to the behavior of thermodynamic entropy, which typically scales as the volume of the system. While the study of area laws originates from black hole physics [2,3], area laws have received considerable attention recently in the fields of quantum information and condensed matter physics. In particular, area laws are known to be closely related to the velocity of information propagation in quantum lattices [4], quantum critical phenomena [5], bulk-boundary correspondence [6], efficient classical simulation of quantum systems [7], topological order [8], and many-body localization [9].However, the description of many-body systems in terms of local interactions is often only an approximation, and not always a good one; in numerous systems of current interest, ranging from frustrated magnets and spin glasses [10,11] to atomic, molecular, and optical systems [12][13][14][15][16][17], longrangeinteractions are ubiquitous and lead to qualitatively new physics, e.g., giving rise to novel quantum phases and dynamical behaviors [18][19][20][21][22][23][24][25], and enabling speedups in quantum information processing [26][27][28][29][30]. Particles in these systems generally experience interactions that decay algebraically (∼1=r α ) in the distance (r) between them. As might be expected, α controls the extent to which the system respects notions of locality developed for short-range interacting systems: For α sufficiently small, it is well established [19] that locality may be completely lost, and for α sufficiently large there is ample numerical and analytical evidence [31][32][33][34] that area laws may persist. However, there is currently no general and rigorous understanding of when area laws do or do not survive the presence of long-range interactions.The modern understanding of area laws draws heavily from several rigor...

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Entanglement Area Laws for Long-Range Interacting Systems

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“…In a very recent paper [9], a logarithmic light cone was obtained for long-range, i.e., power-law decaying, interactions. The…”

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“…In a very recent paper [9], a logarithmic light cone was obtained for long-range, i.e., power-law decaying, interactions. The anomalous LR bound we find yields a qualitatively completely different, anomalously slow many-body transport.…”

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New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

et al. 2014

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We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson (LR) bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light cone jxj ≤ vjtj, we obtain jxj ≤ vjtj α for some 0 < α < 1. We can characterize the allowed values of α exactly as those exceeding the upper transport exponent α þ u of a one-body Schrödinger operator. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. We also discuss anomalous LR bounds with power-law tails for a random dimer field. Introduction.-Relativistic systems are local in the sense that information propagates at most at the speed of light. In their seminal paper [1], Lieb and Robinson found that nonrelativistic quantum spin systems described by local Hamiltonians satisfy a similar "quasilocality" under the Heisenberg dynamics. Their Lieb-Robinson (LR) bound and its recent generalizations [2,3] implies the existence of a "light cone" jxj ≤ vjtj in space-time, outside of which quantum correlations (concretely: commutators of local observables) are exponentially small. In other words, the LR bound shows that, to a good approximation, quantum correlations propagate at most ballistically, with a systemdependent "Lieb-Robinson velocity" v.About ten years ago, the general interest in LR bounds resurged when Hastings and coworkers realized that they are the key tool for deriving exponential clustering, a higher-dimensional Lieb-Schultz-Mattis theorem, and the celebrated area law for the entanglement entropy in onedimensional systems with a spectral gap [2,4,5]. These results highlight the role of entanglement in constraining the structure of ground states in gapped systems and yield many applications to quantum information theory, e.g., in developing algorithms to simulate quantum systems on a classical computer [6,7].In this Letter, we announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. The LR bound is anomalous in the sense that the forward half of the ordinary light cone is changed to the region jxj ≤ vjtj α for some 0 < α < 1.Previous study has focused on the dependence of the Lieb-Robinson velocity v on the system details [3], with particular interest in the case v ¼ 0, since it may be interpreted as dynamical localization [8]. In a very recent paper [9], a logarithmic light cone was obtained for long-range, i.e., power-law decaying, interactions. The

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“…While spin-wave theories can be useful in treating long-range interactions 44,45 , they are unable to distinguish major differences in quantum phases between integer and half-integer spin chains. Exact numerical studies for long-range interacting spin models are restricted to small system sizes and usually inconclusive [46][47][48][49] , since the correlation length is generally divergent 32,50 . Approximate numerical techniques such as the density matrix renormalization group (DMRG) method have been adapted to treat long-range interactions 51 , but determining complete diagrams with large-system-size calculations remains challenging, and those that exist are primarily for spin-1/2 chains 20,29,52,53 .…”

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confidence: 99%

“…While long-range interacting classical models have been studied in considerable detail for some time [24][25][26][27][28] , there is a relative lack of in-depth studies of quantum phase transitions in long-range interacting systems, despite the emerging experimental prospects for studying both their equilibrium and nonequilibrium properties [15][16][17][18][29][30][31][32][33][34][35] . One reason is that many analytically solvable lattice models become intractable when interactions are no longer short-ranged, a well-known example being the spin-1/2 XXZ model.…”

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Kaleidoscope of quantum phases in a long-range interacting spin-1 chain

Gong

Maghrebi

et al. 2016

Phys. Rev. B

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Motivated by recent trapped-ion quantum simulation experiments, we carry out a comprehensive study of the phase diagram of a spin-1 chain with XXZ-type interactions that decay as $1/r^{\alpha}$, using a combination of finite and infinite-size DMRG calculations, spin-wave analysis, and field theory. In the absence of long-range interactions, varying the spin-coupling anisotropy leads to four distinct phases: a ferromagnetic Ising phase, a disordered XY phase, a topological Haldane phase, and an antiferromagnetic Ising phase. If long-range interactions are antiferromagnetic and thus frustrated, we find primarily a quantitative change of the phase boundaries. On the other hand, ferromagnetic (non-frustrated) long-range interactions qualitatively impact the entire phase diagram. Importantly, for $\alpha\lesssim3$, long-range interactions destroy the Haldane phase, break the conformal symmetry of the XY phase, give rise to a new phase that spontaneously breaks a $U(1)$ continuous symmetry, and introduce an exotic tricritical point with no direct parallel in short-range interacting spin chains. We show that the main signatures of all five phases found could be observed experimentally in the near future.Comment: 11 pages, 6 figure

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Persistence of Locality in Systems with Power-Law Interactions

Cited by 129 publications

References 42 publications

Entanglement Area Laws for Long-Range Interacting Systems

Entanglement Area Laws for Long-Range Interacting Systems

New Anomalous Lieb-Robinson Bounds in QuasiperiodicXYChains

Kaleidoscope of quantum phases in a long-range interacting spin-1 chain

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