Entanglement versus gap for one-dimensional spin systems

Gottesman, Daniel; Hastings, Matthew B.

doi:10.1088/1367-2630/12/2/025002

Cited by 64 publications

(76 citation statements)

References 23 publications

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“…When it is gapless, the scaling by which the gap vanishes as a function of the system's size has important consequences for its physics. For example, gapped systems have exponentially decaying correlation functions (22), and quantum critical systems are necessarily gapless (31). Moreover, systems that obey a CFT are gapless but the gap must vanish as 1=n (32).…”

Section: Significancementioning

confidence: 99%

“…Motivated by hardness results in quantum complexity theory, there are various interesting examples of 1D Hamiltonian constructions (20)(21)(22) that can have larger, even linear, scaling of entanglement entropy with the system's size. In condensed matter physics, nontranslationally invariant models have been proposed and argued to violate the area law maximally (i.e., linearly for a chain) (23); Huijse and Swingle gave a supersymmetric model with some degree of fine-tuning that violates the…”

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confidence: 99%

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Supercritical entanglement in local systems: Counterexample to the area law for quantum matter

Movassagh

Shor

2016

Proc. Natl. Acad. Sci. U.S.A.

108

178

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Quantum entanglement is the most surprising feature of quantum mechanics. Entanglement is simultaneously responsible for the difficulty of simulating quantum matter on a classical computer and the exponential speedups afforded by quantum computers. Ground states of quantum many-body systems typically satisfy an "area law": The amount of entanglement between a subsystem and the rest of the system is proportional to the area of the boundary. A system that obeys an area law has less entanglement and can be simulated more efficiently than a generic quantum state whose entanglement could be proportional to the total system's size. Moreover, an area law provides useful information about the low-energy physics of the system. It is widely believed that for physically reasonable quantum systems, the area law cannot be violated by more than a logarithmic factor in the system's size. We introduce a class of exactly solvable one-dimensional physical models which we can prove have exponentially more entanglement than suggested by the area law, and violate the area law by a square-root factor. This work suggests that simple quantum matter is richer and can provide much more quantum resources (i.e., entanglement) than expected. In addition to using recent advances in quantum information and condensed matter theory, we have drawn upon various branches of mathematics such as combinatorics of random walks, Brownian excursions, and fractional matching theory. We hope that the techniques developed herein may be useful for other problems in physics as well.quantum matter | local Hamiltonians entanglement entropy | area law | spin chains | Hamiltonian gap S tudy of quantum many-body systems (QMBSs) is the study of quantum properties of matter and quantum resources (e.g., entanglement) provided by matter for building revolutionary new technologies such as a quantum computer. One of the properties of the QMBS is the amount of entanglement among parts of the system (1, 2). Entanglement can be used as a resource for quantum technologies and information processing (2-5); however, at a fundamental level it provides information about the quantum state of matter, such as near-criticality (6, 7). Moreover, systems with high entanglement are usually hard to simulate on a classical computer (8). How much entanglement do natural QMBSs possess? What are the fundamental limits on simulation of physical systems?The area law says that entanglement entropy between two subsystems of a system is proportional to the area of the boundary between them. A generic state does not obey an area law (9); therefore, obeying an area law implies that a QMBS contains much less quantum correlation than generically expected. One can imagine that any given system has inherent constraints such as underlying symmetries and locality of interaction that restrict the states to reside on special submanifolds rendering their simulation efficient (10).Since the discovery that the Affleck-Kennedy-Lieb-Tasaki (AKLT) model (11) is exactly solvable, and that the density matrix re...

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Section: Significancementioning

confidence: 99%

mentioning

confidence: 99%

Supercritical entanglement in local systems: Counterexample to the area law for quantum matter

Movassagh

Shor

2016

Proc. Natl. Acad. Sci. U.S.A.

108

178

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“…The well known area law of the bipartite entanglement entropy restricts the Hilbert space accessible to a ground state of gapped systems [51,52], while the area law is violated by a leading logarithmic correction in critical systems, whose prefactor is determined by the number of chiral modes and precisely given by Widom conjecture [53]. In this respect, the application of the entanglement entropy in describing quantum criticality in many-body Hamiltonian merits a lot of studies [42,54].…”

Section: Introductionmentioning

confidence: 99%

Quantum entanglement in the one-dimensional spin-orbitalSU(2)⊗XXZmodel

2015

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We investigate the phase diagram and the spin-orbital entanglement of a one-dimensional SU(2)⊗XXZ model with SU(2) spin exchange and anisotropic XXZ orbital exchange interactions and negative exchange coupling. As a unique feature, the spin-orbital entanglement entropy in the entangled ground states increases here linearly with system size. In the case of Ising orbital interactions we identify an emergent phase with long-range spin-singlet dimer correlations triggered by a quadrupling of correlations in the orbital sector. The peculiar translational invariant spin-singlet dimer phase has finite von Neumann entanglement entropy and survives when orbital quantum fluctuations are included. It even persists in the isotropic SU(2)⊗SU(2) limit. Surprisingly, for finite transverse orbital coupling the long-range spin singlet correlations also coexist in the antiferromagnetic spin and alternating orbital phase making this phase also unconventional. Moreover we also find a complementary orbital singlet phase that exists in the isotropic case but does not extend to the Ising limit. The nature of entanglement appears essentially different from that found in the frequently discussed model with positive coupling. Furthermore we investigate the collective spin and orbital wave excitations of the disentangled ferromagnetic-spin/ferro-orbital ground state and explore the continuum of spin-orbital excitations. Interestingly one finds among the latter excitations two modes of exciton bound states. Their spin-orbital correlations differ from the remaining continuum states and exhibit logarithmic scaling of the von Neumann entropy with increasing system size. We demonstrate that spin-orbital excitons can be experimentally explored using resonant inelastic x-ray scattering, where the strongly entangled exciton states can be easily distinguished from the spin-orbital continuum.

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“…The TEBD algorithm can be seen as a descendant of the density matrix renormalization group [12] method and is based on a matrix product state (MPS) representation [13,14] of the wavefunctions. Algorithms of this type are efficient because they exploit the fact that the ground-state wave functions are only slightly entangled, especially away from criticality [15]. As the entanglement grows linearly as a function of time, the simulations of long time evolutions is numerically very difficult.…”

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confidence: 99%

Bound states andE8symmetry effects in perturbed quantum Ising chains

2011

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In a recent experiment on CoNb2O6, Coldea et al. [1] found for the first time experimental evidence of the exceptional Lie algebra E8. The emergence of this symmetry was theoretically predicted long ago for the transverse quantum Ising chain in the presence of a weak longitudinal field. We consider an accurate microscopic model of CoNb2O6 incorporating additional couplings and calculate numerically the dynamical structure function using a recently developed matrix-productstate method. The excitation spectra show bound states characteristic of the weakly broken E8 symmetry. We compare the observed bound state signatures in this model to those found in the transverse Ising chain in a longitudinal field and to experimental data. The one-dimensional (1D) quantum Ising model in transverse and longitudinal fields is one of the most studied theoretical models in condensed matter physics. It is a relatively simple model that contains very rich physics; for example, it contains a quantum critical point (QCP) at zero longitudinal field related to the 2D classical Ising model. A remarkable fact is that the integrability present at the critical point remains under addition of a longitudinal field as a mass-generating perturbation. Zamolodchikov conjectured in 1989 an S-matrix describing eight emergent particles whose mass ratios are connected to the roots of the Lie algebra E 8 [2,3]. Recently, Coldea et al. performed neutron scattering experiments on CoNb 2 O 6 (cobalt niobate), a material that to a good approximation can be described by a quantum Ising chain. At low temperatures and in the presence of a strong external transverse magnetic field which tunes the system to near criticality, the observed spectrum shows characteristic excitations of the E 8 symmetry [1].However, a serious problem in comparing theory and experiment is that the real material has additional couplings that strictly speaking invalidate the exact solution, and until recently it was impractical to extend the theory non-perturbatively to include these couplings. In this Letter, we study a theoretical model for CoNb 2 O 6 which includes in addition to the Ising interaction other interactions arising from the lattice structure and the weak coupling between the chains. Using this model, we calculate the dynamical spectral function and compare the results to the observed spectra. Close to the QCP, the model retains features expected from the quantum Ising model, in particular the characteristic particles of the E 8 symmetry.We begin by deriving the theoretical model used to describe the low-energy physics of CoNb 2 O 6 . The spin lattice structure consists of chains of easy axis spins, realizing a two level system, on the Co 2+ ions coupled by a ferromagnetic Ising interaction along the chain direction, see Fig. 1A. We thus start from the quantum Ising chain, described by the Hamiltonian

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Entanglement versus gap for one-dimensional spin systems

Cited by 64 publications

References 23 publications

Supercritical entanglement in local systems: Counterexample to the area law for quantum matter

Supercritical entanglement in local systems: Counterexample to the area law for quantum matter

Quantum entanglement in the one-dimensional spin-orbitalSU(2)⊗XXZmodel

Bound states andE8symmetry effects in perturbed quantum Ising chains

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