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arXiv:1906.11954 · Journal of Statistical Physics

Bounded Entanglement Entropy in the Quantum Ising Model

Geoffrey R. Grimmett, Tobias J. Osborne, Petra F. Scudo

Published Dec 2, 2019• 5 citations
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Abstract

Abstract A rigorous proof is presented of the boundedness of the entanglement entropy of a block of spins for the ground state of the one-dimensional quantum Ising model with sufficiently strong transverse field. This is proved by a refinement of the stochastic geometric arguments in the earlier work by Grimmett et al. (J Stat Phys 131:305–339, 2008). The proof utilises a transformation to a model of classical probability called the continuum random-cluster model. Our method of proof is fairly robust, and applies also to certain disordered systems.

DOI
10.1007/s10955-019-02432-y

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Abstract A rigorous proof is presented of the boundedness of the entanglement entropy of a block of spins for the ground state of the one-dimensional quantum Ising model with sufficiently strong transverse field. This is proved by a refinement of the stochastic geometric arguments in the earlier work by Grimmett et al. (J Stat Phys 131:305–339, 2008). The proof utilises a transformation to a model of classical probability called the continuum random-cluster model. Our method of proof is fairly robust, and applies also to certain disordered systems.
A rigorous proof is presented of the boundedness of the entanglement entropy of a block of spins for the ground state of the one-dimensional quantum Ising model with sufficiently strong transverse field. This is proved by a refinement of the stochastic geometric arguments in the earlier work by Grimmett et al. (J Stat Phys 131:305–339, 2008). The proof utilises a transformation to a model of classical probability called the continuum random-cluster model. Our method of proof is fairly robust, and applies also to certain disordered systems. © 2019, The Author(s).
We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse field. When the field is sufficiently strong, the entanglement grows at most logarithmically in the number of spins. The proof utilises a transformation to a model of classical probability called the continuum random-cluster model, and is based on a property of the latter model termed ratio weak-mixing. Our proof applies equally to a large class of disordered interactions. 1 The quantum Ising model The quantum Ising model in a transverse magnetic field is one of the most famous examples of exactly solvable one-dimensional quantum models. The solution was first given by Pfeuty in [26], based on earlier works by Lieb, Schultz, and Mattis [18] and by McCoy [21]. The diagonalisation of the Hamiltonian and the determination of the energy eigenstates is based on methods developed by Jordan and Wigner [16] in the theory of second quantisation of fermion fields, and by Bogoliubov [7] in the theory of superconductivity. This model exhibits a second-order phase transition in the ground state when the temperature of the system is zero. The existence of the phase transition and
We consider the ground state of the one-dimensional quantum Ising model with transverse field hx in one dimension depending on the site x∈Z in a finite volume. We assume hx = λξx, where (ξx),x∈Z is a family of strictly positive i.i.d. random variables with a positive density at 0 satisfying the condition. E(ξ0−δ)<∞ for some δ > 0 and λ > 0 is the disorder parameter. We prove that if λ is sufficiently large, the entanglement entropy of the ground state has a stochastic bound uniform with respect to the volume.
We consider the ground state of the one-dimensional quantum Ising model with transverse field hx in one dimension depending on the site x∈Z in a finite volume Λm ≔ {−m, −m + 1, …, m + L}. We make suitable assumptions on the regions where the field is small and prove that if the field is sufficiently large on the complementary set, then the entanglement of the interval Λ0≔0,..,L relative to its complement Λm\Λ0 is bounded uniformly in m and L. The result applies in particular to periodic transverse fields. The bound is established by means of a suitable cluster expansion.
Abstract For pure states of multi-dimensional quantum lattice systems, which in a convenient computational basis have amplitude and phase structure of sufficiently rapid decorrelation, we construct high fidelity approximations of relatively low complexity. These are used for a conditional proof of area-law bounds for the states’ entanglement entropy. The condition is also shown to imply exponential decay of the state’s mutual information between disjoint regions, and hence exponential clustering of local observables. The applicability of the general results is demonstrated on the quantum Ising model in transverse field. Combined with available model-specific information on spin-spin correlations, we establish an area-law type bound on the entanglement in the model’s subcritical ground states, valid in all dimensions and up to the model’s quantum phase transition.
The scaling behavior of the entanglement entropy in the two-dimensional random transverse field Ising model is numerically studied through the strong disordered renormalization group method. We find that the leading term of the entanglement entropy always linearly scales with the block size. However, besides this area law contribution, we find a subleading logarithmic correction at the quantum critical point. This correction is discussed from the point of view of an underlying percolation transition, both at finite and at zero temperature.