Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (United States)
We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors T{sub inv} plus the symmetry group G{sub sym} of the tensors (i.e., the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional (2D) statistical Ising model, 2D statistical loop-gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a (G{sub sym},T{sub inv}) characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry-protected topological phase. The (G{sub sym},T{sub inv}) characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.
- OSTI ID:
- 21287084
- Journal Information:
- Physical Review. B, Condensed Matter and Materials Physics, Vol. 80, Issue 15; Other Information: DOI: 10.1103/PhysRevB.80.155131; (c) 2009 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA); ISSN 1098-0121
- Country of Publication:
- United States
- Language:
- English
Similar Records
Exponential Thermal Tensor Network Approach for Quantum Lattice Models
Twisted injectivity in projected entangled pair states and the classification of quantum phases