Abstract
The bulk polarization is a $\mathbb{Z}_2$ topological invariant characterizing non-interacting systems in one dimension with chiral or particle-hole symmetries. We show that the bulk polarization can always be determined from the single-particle entanglement spectrum, even in the absence of symmetries that quantize it. In the symmetric case, the known relation between the bulk polarization and the number of virtual topological edge modes is recovered. We use the bulk polarization to compute Chern numbers in 1D and 2D, which illuminates their known relation to the entanglement spectrum. Furthermore we discuss an alternative bulk polarization that can carry more information about the surface spectrum than the conventional one and can simplify the calculation of Chern numbers.
Highlights
Topological phases of matter have attracted a lot of attention during the last decades, not the least because a large variety of relevant systems have been realized experimentally
We show that the bulk polarization can always be determined from the single-particle entanglement spectrum, even in the absence of symmetries that quantize it
It was previously known that the bulk polarization [Zak phase, or geometric phase for U(1) flux insertion] was encoded in the entanglement spectrum (ES)
Summary
Topological phases of matter have attracted a lot of attention during the last decades, not the least because a large variety of relevant systems have been realized experimentally. Certain short-range entangled states cannot be continuously transformed between themselves unless certain symmetries are broken These are the SPT phases that we focus on in this paper. Closely related, tool is the entanglement spectrum (ES), originally introduced for fractional quantum Hall systems [12] It provides information about the edge spectrum and has proven useful for other topologically ordered phases such as fractional Chern insulators [13] and certain quantum spin liquids [14]. The modern theory of polarization [18,19,20,21] related the bulk polarization to a geometric phase, which is nothing but the Zak phase for translationally invariant systems Due to this, it has found its way into topological physics [22]. VII we conclude and discuss possible extensions of our work
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