TY - JOUR
T1 - Generalized string-net models
T2 - A thorough exposition
AU - Lin, Chien Hung
AU - Levin, Michael
AU - Burnell, Fiona J.
N1 - 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.
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U2 - 10.1103/PhysRevB.103.195155
DO - 10.1103/PhysRevB.103.195155
M3 - Article
AN - SCOPUS:85107128853
SN - 2469-9950
VL - 103
JO - Physical Review B
JF - Physical Review B
IS - 19
M1 - 195155
ER -