Boundedness of Entanglement Entropy and Split Property of Quantum Spin Chains

Abstract

We show the boundedness of entanglement entropy for (bipartite) pure states of quantum spin chains implies split property of subsystems. As a corollary the infinite volume ground states for 1-dim spin chains with the spectral gap between the ground state energy and the rest of spectrum have the split property. We see gapless excitation exists for the spinless Fermion on Z if the ground state is non-trivial and translationally invariant and the U (1) gauge symmetry is unbroken. Here we do not assume uniqueness of ground states for all finite volume Hamiltonians.Keywords: quantum spin chain, spectral gap, split property, boundedness of entanglement entropy.

On ℤ2-indices for ground states of fermionic chains

“…A gapped ground state ω is equivalent (see e.g. [45]) to the condition that there is a γ > 0 such that −i ω a * δ(a) ≥ γ ω(a * a) − |ω(a)| 2 , a ∈ (A car Λ ) loc. .…”

“…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].…”

“…While much of the manuscript is review, we do provide a candidate for a Z 2 -index of pure, gapped and parity-invariant ground states on the one-dimensional infinite CAR algebra that can be used as a phase label. To the best of our knowledge, the construction is new, though it heavily relies on the split property of one-dimensional ground states [44,45] as well as the infinite Jordan-Wigner transform [30,Chapter 6.5]. The use of the split property as a tool to characterise ground state SPT phases was first noted by Ogata [54].…”

Automorphic equivalence preserves the split property

“…A significant mathematical obstruction to assuming weaker decay conditions is in proving that certain gapped ground states of interactions satisfy the split property. So far, general results on sufficient conditions for the split property to hold critically use characteristics of finite-range or exponentially decaying one-dimensional interactions, such as boundedness of the entanglement entropy or the validity of Haag duality for the spin chain interactions [5,7]. We comment on the relationship between split property for translation invariant ground states and Haag duality in Section 2.…”

“…Lastly, we state the split property from [5,7] which will be best suited for our analysis. For brevity, we will refer to states which satisfy Definition 1.1 as split states.…”