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Identifying topological order by entanglement entropy

Nature Physics volume 8, pages 902–905 (2012)Cite this article

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Abstract

Topological phases are unique states of matter that incorporate long-range quantum entanglement and host exotic excitations with fractional quantum statistics. Here we report a practical method to identify topological phases in arbitrary realistic models by accurately calculating the topological entanglement entropy using the density matrix renormalization group (DMRG). We argue that the DMRG algorithm systematically selects a minimally entangled state from the quasi-degenerate ground states in a topological phase. This tendency explains both the success of our method and the absence of ground-state degeneracy in previous DMRG studies of topological phases. We demonstrate the effectiveness of our procedure by obtaining the topological entanglement entropy for several microscopic models, with an accuracy of the order of 10−3, when the circumference of the cylinder is around ten times the correlation length. As an example, we definitively show that the ground state of the quantum S = 1/2 antiferromagnet on the kagome lattice is a topological spin liquid, and strongly constrain the conditions for identification of this phase of matter.

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Figure 1: Evidence that the DMRG favours MESs.
Figure 2: The von Neumann entropy S(Ly) for the toric-code model in magnetic fields.
Figure 3: The entanglement entropy S(Ly) of the kagome J1–J2 model in equation (2), with Ly = 4–12 at .

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Acknowledgements

We thank T. Grover and A. Vishwanath for a helpful explanation of their work, and S. White for helpful discussions. H.C.J. thanks H. Yao for collaboration on related projects. This work was supported by the NSF through grant DMR 0804564 (L.B.), the NSF MRSEC Program under DMR 1121053, the NBRPC (973 Program) 2011CBA00300 (2011CBA00302), and benefited from the facilities of the KITP, supported by NSF PHY05-51164.

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Authors and Affiliations

  1. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA

    Hong-Chen Jiang & Leon Balents

  2. Microsoft Station Q, University of California, Santa Barbara, California 93106, USA

    Zhenghan Wang

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  1. Hong-Chen Jiang
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H.C.J. developed the simulation codes and performed the numerical experiments. All authors were equally responsible for writing the manuscript.

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Correspondence to Leon Balents.

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Jiang, HC., Wang, Z. & Balents, L. Identifying topological order by entanglement entropy. Nature Phys 8, 902–905 (2012). https://doi.org/10.1038/nphys2465

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