Universal entanglement signatures of foliated fracton phases

Fracton models exhibit a variety of exotic properties and lie beyond the conventional framework of gapped topological order. In a previous work, we generalized the notion of gapped phase to one of foliated fracton phase by allowing the addition of layers of gapped two-dimensional resources in the adiabatic evolution between gapped three-dimensional models. Moreover, we showed that the X-cube model is a fixed point of one such phase. In this paper, according to this definition, we look for universal properties of such phases which remain invariant throughout the entire phase. We propose multi-partite entanglement quantities, generalizing the proposal of topological entanglement entropy designed for conventional topological phases. We present arguments for the universality of these quantities and show that they attain non-zero constant value in non-trivial foliated fracton phases.Comment: 17 pages, 7 figure

Foliated fracton order in the checkerboard model

In this work, we show that the checkerboard model exhibits the phenomenon of foliated fracton order. We introduce a renormalization group transformation for the model that utilizes toric code bilayers as an entanglement resource, and show how to extend the model to general three-dimensional manifolds. Furthermore, we use universal properties distilled from the structure of fractional excitations and ground-state entanglement to characterize the foliated fracton phase and find that it is the same as two copies of the X-cube model. Indeed, we demonstrate that the checkerboard model can be transformed into two copies of the X-cube model via an adiabatic deformation.Comment: 8 pages, 9 figure

Foliated fracton order in the Majorana checkerboard model

We establish the presence of foliated fracton order in the Majorana checkerboard model. In particular, we describe an entanglement renormalization group transformation which utilizes toric code layers as resources of entanglement, and furthermore discuss entanglement signatures and fractional excitations of the model. In fact, we give an exact local unitary equivalence between the Majorana checkerboard model and the semionic X-cube model augmented with decoupled fermionic modes. This mapping demonstrates that the model lies within the X-cube foliated fracton phase.Comment: 13 pages, 17 figure

ECONOMIC IMPORTANCE OF EXPORT-PRODUCING INDUSTRY IN MINNEAPOLIS-ST. PAUL METROPOLITAN REGION

Community/Rural/Urban Development, International Relations/Trade,

ECONOMIC IMPORTANCE OF EXPORT-PRODUCING INDUSTRY IN MINNESOTA

In this report, the role and importance of export-producing industries in Minnesota are discussed. Major emphasis is on agricultural-related industries and their economic importance to the State in value of gross output and contribution to gross state product.Community/Rural/Urban Development, International Relations/Trade,

ECONOMIC IMPORTANCE OF AGRICULTURE-RELATED INDUSTRY IN MINNESOTA

This report is the third in a series on interindustry and interregional relationships and their implications for the economy of Minnesota and its substate development regions. In this report, the role and importance of agriculture-related industries in Minnesota are discussed. A total of 19 agriculture and 35 food products manufacturing industries are identified as the principal agriculture-related industries in the state and the nation. Their interindustry and interregional (Minnesota and rest-of-nation) linkages are derived for the 1977 calendar year.Community/Rural/Urban Development,

EMPLOYMENT TRENDS AND PROJECTIONS FOR MINNESOTA AND ITS SUBSTATE DEVELOPMENT REGIONS

Labor and Human Capital,