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Long/short range order v.s. long/short range interactions v.s. long-short range entanglement

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The following terms have been widely used to characterized the phases of quantum matter or the gauge theory:

i. Long/short range order,

ii. long/short range interactions,

iii. long-short range entanglement

How are they different from each other?

Can one give examples for certain phases of matter that has long/short range order, long/short range interactions, long-short range entanglement for each choices of long/short? There are totally $2^3=8$ possible characterizations in terms of these terms.

asked Nov 13, 2016 in Theoretical Physics by RKKY (320 points) [ no revision ]

Topology from entanglement properties , see the Xie Chen document ( just after the introduction ) : "A state has only short-range entanglement iff it can be transformed into an unentangled state (ie a direct-product state) through a local unitary evolution." and "If a state cannot be transformed into an unentangled state through a local unitary evolution, then the state has long-range entanglement". Range order is a too vague notion unless it is the "topological range order". They are not directly related to the short/long interactions ( which is the classic reverse approach ).

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