3,010 research outputs found
Modular Matrices as Topological Order Parameter by Gauge Symmetry Preserved Tensor Renormalization Approach
Topological order has been proposed to go beyond Landau symmetry breaking
theory for more than twenty years. But it is still a challenging problem to
generally detect it in a generic many-body state. In this paper, we will
introduce a systematic numerical method based on tensor network to calculate
modular matrices in 2D systems, which can fully identify topological order with
gapped edge. Moreover, it is shown numerically that modular matrices, including
S and T matrices, are robust characterization to describe phase transitions
between topologically ordered states and trivial states, which can work as
topological order parameters. This method only requires local information of
one ground state in the form of a tensor network, and directly provides the
universal data (S and T matrices), without any non-universal contributions.
Furthermore it is generalizable to higher dimensions. Unlike calculating
topological entanglement entropy by extrapolating, which numerical complexity
is exponentially high, this method extracts a much more complete set of
topological data (modular matrices) with much lower numerical cost.Comment: 5+3 pages; 4+2 figures; One more appendix is adde
Gapped spin liquid with -topological order for kagome Heisenberg model
We apply symmetric tensor network state (TNS) to study the nearest neighbor
spin-1/2 antiferromagnetic Heisenberg model on Kagome lattice. Our method keeps
track of the global and gauge symmetries in TNS update procedure and in tensor
renormalization group (TRG) calculation. We also introduce a very sensitive
probe for the gap of the ground state -- the modular matrices, which can also
determine the topological order if the ground state is gapped. We find that the
ground state of Heisenberg model on Kagome lattice is a gapped spin liquid with
the -topological order (or toric code type), which has a long
correlation length unit cell length. We justify that the TRG
method can handle very large systems with over thousands of spins. Such a long
explains the gapless behaviors observed in simulations on smaller systems
with less than 300 spins or shorter than 10 unit cell length. We also discuss
experimental implications of the topological excitations encoded in our
symmetric tensors.Comment: 10 pages, 7 figure
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