Automorphic equivalence preserves the split property

Moon, Alvin

doi:10.1016/j.jfa.2019.05.021

Cited by 8 publications

(4 citation statements)

References 19 publications

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“…As already stated, Propositions 6.12, 6.14 and Remark 6.13 have shown stability properties of the Z 2 -index, though the assumptions are quite strong. A more systematic treatment similar to recent studies of Z 2 -indices of ground states of spin chains satisfying the split property with time-reversal or reflection symmetry [48,54,55] will hopefully give more optimised results.…”

Section: Discussionmentioning

confidence: 85%

“…The split property has its roots in algebraic quantum field theory [29] but was adapted to fermion and spin chains by Matsui [44,45]. More recently, the application of the split property to the analytic approach to SPT phases has been developed by Ogata et al [54,55,56] and Moon [48]. Given a subset Λ ⊂ Z with complement Λ c = Z \ Λ and a Θ-invariant state ω, one introduces the product state of the restrictions by…”

Section: The Split Propertymentioning

confidence: 99%

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On ℤ2-indices for ground states of fermionic chains

Bourne

Schulz-Baldes

2020

Rev. Math. Phys.

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For parity-conserving fermionic chains, we review how to associate Z 2 -indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the Z 2 -valued spectral flow provides a topological obstruction for two systems to have the same Z 2 -index. A rudimentary definition of a Z 2 -phase label for a class of parity-invariant and pure ground states of the one-dimensional infinite CAR algebra is also provided. Ground states with differing phase labels cannot be connected without a closing of the spectral gap of the infinite GNS Hamiltonian.

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Section: Discussionmentioning

confidence: 85%

Section: The Split Propertymentioning

confidence: 99%

On ℤ2-indices for ground states of fermionic chains

Bourne

Schulz-Baldes

2020

Rev. Math. Phys.

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“…In one dimension, a non-vanishing gap for the infinite system implies the split property [71], which in turn plays a crucial role in definition of a topological index for symmetry-protected topological phases [86][87][88]. More generally, the presence of a spectral gap features as an assumption in the theories classifying topological phases of matter [26,27,73,75,85,90] and the derivation of the quantum Hall effect and similar properties [5][6][7]50].…”

Section: Introduction 1stability Of the Ground-state Gapmentioning

confidence: 99%

Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

Nachtergaele

Sims

Young

2021

Ann. Henri Poincaré

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We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.

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“…This factorization property was first used in [Oga20], in proving the stability of the index of 1D SPT. Following this idea, in [Moo19] the stability of split property in 1D was shown. Its 2-dimensional version is essential here, but an extra complication is that in 2D or higher, the boundary between the regions we will consider is infinite.…”

Section: Introductionmentioning

confidence: 99%

The Split and Approximate Split Property in 2D Systems: Stability and Absence of Superselection Sectors

Naaijkens

Ogata

2022

Commun. Math. Phys.

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The split property of a pure state for a certain cut of a quantum spin system can be understood as the entanglement between the two subsystems being weak. From this point of view, we may say that if it is not possible to transform a state $$\omega $$ ω via sufficiently local automorphisms (in a sense that we will make precise) into a state satisfying the split property, then the state $$\omega $$ ω has a long-range entanglement. It is well known that in 1D, gapped ground states have the split property with respect to cutting the system into left and right half-chains. In 2D, however, the split property fails to hold for interesting models such as Kitaev’s toric code. Here we show that this failure is the reason that anyons can exist in that model. There is a folklore saying that the existence of anyons, like in the toric code model, implies long-range entanglement of the state. In this paper, we prove this folklore in an infinite dimensional setting. More precisely, we show that long-range entanglement, in a way that we will define precisely, is a necessary condition to have non-trivial superselection sectors. Anyons in particular give rise to such non-trivial sectors. States with the split property for cones, on the other hand, do not admit non-trivial sectors. A key technical ingredient of our proof is that under suitable assumptions on locality, the automorphisms generated by local interactions can be “approximately factorized.” That is, they can be written as the tensor product of automorphisms localized in a cone and its complement respectively, followed by an automorphism acting near the “boundary” of $$\Lambda $$ Λ , and conjugation with a unitary. This result may be of independent interest. This technique also allows us to prove that the approximate split property, a weaker version of the split property that is satisfied in e.g. the toric code, is stable under applying such automorphisms.

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Automorphic equivalence preserves the split property

Cited by 8 publications

References 19 publications

On ℤ2-indices for ground states of fermionic chains

On ℤ2-indices for ground states of fermionic chains

Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States

The Split and Approximate Split Property in 2D Systems: Stability and Absence of Superselection Sectors

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