Abstract
In this letter, we propose an application of string order parameter (SOP), commonly used in quantum spin systems, to identify symmetry-protected topological phase (SPT) in fermionic systems in the example of the dimerized fermionic chain. As a generalized form of dimerized model, we consider a one-dimensional spin-1/2 XX model with alternating spin couplings. We employ Jordan–Wigner fermionization to map this model to the spinless Su–Schrieffer–Heeger fermionic model (SSH) with generalized hopping signs. We demonstrate a phase transition between a trivial insulating phase and the Haldane phase by the exact analytical evaluation of reconstructed SOPs which are represented as determinants of Toeplitz matrices with the given generating functions. To get more insight into the topological quantum phase transition (tQPT) and microscopic correlations, we study the pairwise concurrence as a local entanglement measure of the model. We show that the first derivative of the concurrence has a non-analytic behaviour in the vicinity of the tQPT, like in the second order trivial QPTs.
Highlights
A detailed study of the phases of matter and the trans ition between them have long been an actual problem of condensed matter physics
The current wellstudied model is considered due to it’s simplicity and it can demonstrate clearly the physical picture of string order parameter (SOP). This result, expands the list of exactly calculable order parameters of symmetry-protected topological phase (SPT) phases. Another question which we address in this letter is the behaviour of local two-site entanglement in the vicinity of topological quantum phase transitions
We demonstrated that the derived expressions of SOPs take their maximal value of unity in the fully dimerized SPT Haldane phase and vanish in the trivial insulating phase
Summary
A detailed study of the phases of matter and the trans ition between them have long been an actual problem of condensed matter physics. Kennedy and Tasaki [5] explained the origin of the gap in terms of the hidden Z2 × Z2 symmetry breaking using non-local unitary transformation This trans formation usually converts SOPs to a simple ferromagnetic order parameter shedding light on the symmetry structure which is protecting the Haldane phase. SOP was used to identify the Haldane phase in a 1D bosonic lattice, a topological Kondo insulator and a Kitaev Ladder [14,15,16], demonstrating their solid application for non-spin systems These works were performed numerically using DMRG [17, 18] and SOPs were not evaluated analytically. This result, expands the list of exactly calculable order parameters of SPT phases Another question which we address in this letter is the behaviour of local two-site entanglement in the vicinity of topological quantum phase transitions (tQPT). Conclusive notes and future outlooks are presented in the last section section 5
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