TY - JOUR

T1 - Generalized string-net models

T2 - A thorough exposition

AU - Lin, Chien Hung

AU - Levin, Michael

AU - Burnell, Fiona J.

N1 - Funding Information:
We thank T. Lan, C. Wang, and Y. Wan for helpful discussions. F.J.B. and C.-H.L. acknowledge support from NSF-DMR Grant No. 1352271 and the Sloan Foundation (Grant No. FG-2015-65927). F.J.B. is grateful for the financial support of NSF DMR Grant No. 1928166, the Carnegie Corporation of New York, and the Institute for Advanced Study.
Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/5/26

Y1 - 2021/5/26

N2 - We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of two-dimensional topologically ordered phases of matter. The ground states of these models can be thought of as superpositions of different "string-net configurations,"where each string-net configuration is a trivalent graph with labeled edges, drawn in the xy plane. What makes this construction more general than the original string-net construction is that, unlike the original construction, tetrahedral reflection symmetry is not assumed, nor is it assumed that the ground-state wave function φ is "isotropic": i.e., in the generalized setup, two string-net configurations X1,X2 that can be continuously deformed into one another can have different ground-state amplitudes φ(X1)≠φ(X2). As a result, generalized string-net models can realize topological phases that are inaccessible to the original construction. In this paper, we provide a more detailed discussion of ground-state wave functions, Hamiltonians, and minimal self-consistency conditions for generalized string-net models than what exists in the previous literature. We also show how to construct string operators that create anyon excitations in these models, and we show how to compute the braiding statistics of these excitations. Finally, we derive necessary and sufficient conditions for generalized string-net models to have isotropic ground-state wave functions on the plane or the sphere, a property that may be useful in some applications.

AB - We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of two-dimensional topologically ordered phases of matter. The ground states of these models can be thought of as superpositions of different "string-net configurations,"where each string-net configuration is a trivalent graph with labeled edges, drawn in the xy plane. What makes this construction more general than the original string-net construction is that, unlike the original construction, tetrahedral reflection symmetry is not assumed, nor is it assumed that the ground-state wave function φ is "isotropic": i.e., in the generalized setup, two string-net configurations X1,X2 that can be continuously deformed into one another can have different ground-state amplitudes φ(X1)≠φ(X2). As a result, generalized string-net models can realize topological phases that are inaccessible to the original construction. In this paper, we provide a more detailed discussion of ground-state wave functions, Hamiltonians, and minimal self-consistency conditions for generalized string-net models than what exists in the previous literature. We also show how to construct string operators that create anyon excitations in these models, and we show how to compute the braiding statistics of these excitations. Finally, we derive necessary and sufficient conditions for generalized string-net models to have isotropic ground-state wave functions on the plane or the sphere, a property that may be useful in some applications.

UR - http://www.scopus.com/inward/record.url?scp=85107128853&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.103.195155

DO - 10.1103/PhysRevB.103.195155

M3 - Article

AN - SCOPUS:85107128853

VL - 103

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 19

M1 - 195155

ER -