TY - JOUR

T1 - Entanglement entropy from tensor network states for stabilizer codes

AU - He, Huan

AU - Zheng, Yunqin

AU - Bernevig, B. Andrei

AU - Regnault, Nicolas

N1 - Funding Information:
H. He and Y. Zheng are grateful for the support from the Physics Department of Princeton University. N.R. acknowledges M. Hermanns and O. Petrova for fruitful discussions. The authors are grateful to G. Sierra and B. Bradlyn for numerous and enlightening discussions about related topics of fracton models. B.A.B. wishes to thank Ecole Normale Superieure, UPMC Paris, and the Donostia International Physics Center for their generous sabbatical hosting during some of the stages of this work. B.A.B. acknowledges support for the analytic work from NSF EAGER Grant No. DMR-1643312, ONR-N00014-14-1-0330, and NSF-MRSEC DMR-1420541. The computational part of the Princeton work was performed with support from the Department of Energy Grant No. de-sc0016239, Simons Investigator Award, the Packard Foundation, and the Schmidt Fund for Innovative Research. N.R. was supported by Grants No. ANR-17-CE30-0013-01 and No. ANR-16-CE30-0025.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - In this paper, we present the construction of tensor network states (TNS) for some of the degenerate ground states of three-dimensional (3D) stabilizer codes. We then use the TNS formalism to obtain the entanglement spectrum and entropy of these ground states for some special cuts. In particular, we work out examples of the 3D toric code, the X-cube model, and the Haah code. The latter two models belong to the category of "fracton" models proposed recently, while the first one belongs to the conventional topological phases. We mention the cases for which the entanglement entropy and spectrum can be calculated exactly: For these, the constructed TNS is a singular value decomposition (SVD) of the ground states with respect to particular entanglement cuts. Apart from the area law, the entanglement entropies also have constant and linear corrections for the fracton models, while the entanglement entropies for the toric code models only have constant corrections. For the cuts we consider, the entanglement spectra of these three models are completely flat. We also conjecture that the negative linear correction to the area law is a signature of extensive ground-state degeneracy. Moreover, the transfer matrices of these TNSs can be constructed. We show that the transfer matrices are projectors whose eigenvalues are either 1 or 0. The number of nonzero eigenvalues is tightly related to the ground-state degeneracy.

AB - In this paper, we present the construction of tensor network states (TNS) for some of the degenerate ground states of three-dimensional (3D) stabilizer codes. We then use the TNS formalism to obtain the entanglement spectrum and entropy of these ground states for some special cuts. In particular, we work out examples of the 3D toric code, the X-cube model, and the Haah code. The latter two models belong to the category of "fracton" models proposed recently, while the first one belongs to the conventional topological phases. We mention the cases for which the entanglement entropy and spectrum can be calculated exactly: For these, the constructed TNS is a singular value decomposition (SVD) of the ground states with respect to particular entanglement cuts. Apart from the area law, the entanglement entropies also have constant and linear corrections for the fracton models, while the entanglement entropies for the toric code models only have constant corrections. For the cuts we consider, the entanglement spectra of these three models are completely flat. We also conjecture that the negative linear correction to the area law is a signature of extensive ground-state degeneracy. Moreover, the transfer matrices of these TNSs can be constructed. We show that the transfer matrices are projectors whose eigenvalues are either 1 or 0. The number of nonzero eigenvalues is tightly related to the ground-state degeneracy.

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U2 - 10.1103/PhysRevB.97.125102

DO - 10.1103/PhysRevB.97.125102

M3 - Article

AN - SCOPUS:85043995190

VL - 97

JO - Physical Review B

JF - Physical Review B

SN - 2469-9950

IS - 12

M1 - 125102

ER -