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arXiv:2310.15838 · Journal of Mathematical Physics

Uniform bound of the entanglement for the ground state of the one-dimensional quantum Ising model with non-homogeneous transverse field

Massimo Campanino

Published Nov 1, 2023• 0 citations
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Abstract

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.

DOI
10.1063/5.0156436

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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.
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.
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
Ground state of the one-dimensional transverse field Ising model is investigated under the hyperbolic deformation, where the energy scale of j-th bond is proportional to the function \cosh ( j λ) that contains a parameter λ. Although the Hamiltonian is position dependent, the ground state is nearly uniform and finitely correlated. We observe the energy cross over between the ordered and disordered state with respect to the transverse field. The model shows first order phase transition, and the discontinuities in the magnetization and entanglement entropy at the transition point detect the Ising universality.
We study ground-state pairwise entanglement of spins in mean field clusters of spin-one particles interacting via the Ising interaction in the presence of longitudinal and transverse crystal fields. A general result is found that for all kinds of the spin-one Ising models, the pairwise negativities are symmetric with respect to the zero values of the applied transverse or longitudinal magnetic fields. For models considered in this paper, we show numerically that the crystal fields can serve as an alternative tool to create "on-demand" ground-state entanglement for small clusters of spins.