Abstract
We study the entanglement structure and the topological edge states of the ground state of the spin-1/2 XXZ model with bond alternation. We employ parity-density matrix renormalization group with periodic boundary conditions. The finite-size scaling of Rényi entropies S2 and S∞ are used to construct the phase diagram of the system. The phase diagram displays three possible phases: Haldane type (an example of symmetry protected topological ordered phases), Classical Dimer and Néel phases, the latter bounded by two continuous quantum phase transitions. The entanglement and non-locality in the ground state are studied and quantified by the entanglement convertibility. We found that, at small spatial scales, the ground state is not convertible within the topological Haldane dimer phase. The phenomenology we observe can be described in terms of correlations between edge states. We found that the entanglement spectrum also exhibits a distinctive response in the topological phase: the effective rank of the reduced density matrix displays a specifically large “susceptibility” in the topological phase. These findings support the idea that although the topological order in the ground state cannot be detected by local inspection, the ground state response at local scale can tell the topological phases apart from the non-topological phases.
Highlights
Is the von Neuman entropy of ρA = TrB( ψAB ψAB ) and ρA′ is the scenario is very different when only a single copy the reduced state of the subsystem A of the state is available for converting into another single copy of the target state through LOCC
We study the Differential Local Convertibility (DLC), which is defined as the convertibility between two ground states |ψ(g)〉and ψ (g + ), corresponding to two Hamiltonians described by parameters g and g +
We focus on studying the Differential Local Convertibility (DLC), and study the edge states in the HD phase and the transitions from HD to the Néel phase, from Classical Dimers (CD) to HD phase, as well as from CD to Néel phase and to HD phase
Summary
We remark that we have drawn it using the finite-size scaling of Rényi entropies and the second derivative of ground state energy. This is because the local conversion cannot increase the entanglement which diverges at the critical point (quantified by the von Neumann entropy, see Fig. 2(c) with α = 1). Note that Δ= 1 no longer separates the positive and negative DLC regions: all the regions in HD phase are locally inconvertible due to the recombination of edge states for small LA.
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