2018/07/31 by Wen-Tao Xu, Guang-Ming Zhang
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Anyon #Lambda #Mathematical physics #Mathematics #Matrix multiplication #Matrix product state #Partition function (quantum field theory) #Physics #Physics of Superconductivity and Magnetism #Pure mathematics #Quantum #Quantum many-body systems #Quantum mechanics #Tensor product #Topological order #Topological quantum computer #Toric code #cond-mat.stat-mech #cond-mat.str-el #quant-ph
paper · pdf · doi:10.1103/physrevb.98.165115
published as Phys. Rev. B 98, 165115 (2018) · 17 pages, 16 figures, including three appendices, revised version
openalex created_date 2018/08/03 · arxiv created 2018/09/21 · openalex publication_date 2018/10/10 · arxiv updated 2018/10/17 · openalex updated_date 2026/08/05
Due to the absence of local order parameters, it is a challenging task to characterize the quantum topological phase transitions between topologically ordered phases in two dimensions. In this paper, we construct a topologically ordered tensor network wave function with one parameter \ensuremathλ, describing both the toric code state (\ensuremathλ=1) and double semion state (\ensuremathλ=\ensuremath-1). Via calculating the correlation length defined from the one-dimensional quantum transfer operator of the wave function norm, we can map out the complete phase diagram in terms of the parameter \ensuremathλ, and three different quantum critical points (QCPs) at \ensuremathλ=0,\ifmmode±\else\textpm\fi1.73 are identified. The first one separates the toric code phase and double semion phase, while latter two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry-breaking phase, respectively. When mapping the quantum tensor network wave function to the exactly solved statistical model, the norm of the wave function is identified as the partition function of the classical eight-vertex model, and both QCPs at \ensuremathλ=\ifmmode±\else\textpm\fi1.73 correspond to the eight-vertex model at the critical point \ensuremathλ=√(3), while the QCP at \ensuremathλ=0 corresponds to the critical six-vertex model. Actually such a quantum-classical mapping cannot yield the complete low-energy excitations at these three QCPs. We further demonstrate that the full eigenvalue spectra of the transfer operators without/with the flux insertions can give rise to the complete quantum criticalities, which are described by the (2+0)-dimensional free boson conformal field theories (CFTs) compactified on a circle with the radius R=√(6) at \ensuremathλ=\ifmmode±\else\textpm\fi√(3) and R=√(8/3) at \ensuremathλ=0. From the complete transfer operator spectra, the finite-size spectra of the CFTs for the critical eight-vertex model are obtained, and the topological sectors of anyonic excitations are yielded as well. Furthermore, for the QCP at \ensuremathλ=0, no anyon condensation occurs, but the emerged symmetries of the matrix product operators significantly enrich the topological sectors of the CFT spectra. Finally, we provide our understanding of the (2+0)-dimensional conformal quantum criticalities and their possible connection with the generic (2+1)-dimensional CFTs for quantum topological phase transitions.