Abstract
We study the $t-V-V'$ model in one dimension at half-filling. It is known that for large enough $V$ fixed, as $V'$ is varied, the system goes from a charge-density wave into a Luttinger liquid, then a bond-order, and then a second charge density wave phase. We find that the Luttinger liquid state is further split into two, separating parts with distinct values of the many-body polarization Berry phase. Inside this phase, the variance of the polarization is infinite in the thermodynamic limit, meaning that even if the polarization differs, it would not be measurable. However, in the gapped phases on each side of the Luttinger liquid, the polarization takes a different measurable value, implying topologically distinction. The key difference is that the large-$V'$ phases are link-inversion symmetric, while the small-$V'$ one is site-inversion symmetric. We show that large-$V'$ phase can be related to an $S=1$ spin chain, and exhibits many features of the Haldane phase. The lowest lying states of the entanglement spectrum display different degeneracies in the two cases, and we also find string order in the large-$V'$ phase. We also study the system under open boundary conditions, and suggest that the number of defects is related to the topology.
Highlights
Topological condensed-matter systems constitute an active research area
Quantum phase transitions occur when the relevant topological invariant (Z or Z2) undergoes a finite change at a gap closure point. These systems obey the bulk-boundary correspondence principle, which predicts the existence of edge states in the topologically nontrivial phases
An early result is the Haldane conjecture [14,15,16], which is based on a field-theoretical mapping of the Heisenberg model to a continuum one, and states that S = 1 spin chains are topologically nontrivial and exhibit spin
Summary
Topological condensed-matter systems constitute an active research area. Topological band insulators are well understood [1,2,3]. Topological edge states can arise in three ways: poles or zeros in the Green’s function (single-particle effects) or spontaneous symmetry breaking at the edge (many-body effect). The latter is not necessarily picked up by a topological invariant defined based on the single-particle Green’s function. In particular we find that the CDW-2 exhibits parallels to a Haldane phase [14,15] We show this via a mapping of our original Hamiltonian to an S = 1 spin model, by calculating the entanglement spectrum, and by showing that hidden antiferromagnetic (HAFM) order as well as finite range string correlation, as defined by den Nijs and Rommelse [21], is present.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.