Tensor network state approach to quantum topological phase transitions and their criticalities of ${\mathbb{Z}}_2$ topologically ordered states

Abstract

We construct a general wave function with the topological order by introducing the ${\mathbb{Z}}_2$ gauge degrees of freedom, characterizing both the toric code state and double semion state. 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 $\lambda$ and identify three different quantum critical points (QCPs) at $\lambda =0$, $\pm 1.73$. The first one separates the toric code phase and double semion phase, while later two describe the topological phase transitions from the toric code phase or double semion phase to the symmetry breaking phase, respectively. When mapping to the exactly solved statistical models, the norm of the tensor network wave function is transformed into the partition function of the eight-vertex model. Actually such a quantum-classical mapping can not reveal the rich structures of low-energy excitations at these three QCPs. So we further demonstrate that the full eigenvalue spectra of the transfer operators with/without the flux insertions can describe the complete quantum criticalities, which are characterized by the two-dimensional compactified free boson conformal field theories (CFTs) with the compactified radii $R=\sqrt{6}$ for the QCPs at $\lambda =\pm \sqrt{3}$ and $R=\sqrt{8/3}$ for the QCP at $\lambda =0$. For the QCP at $\lambda =0$, there are no anyon condensation, and the emerged matrix product operator symmetries result in a rich structure of the low-energy excitations, distinct from those of both toric code and double semion phases. Finally, we discuss the possible relation between our conformal quantum criticalities and the general (2+1) spatial-time dimensional CFTs for quantum topological phase transitions.

17 pages, 16 figures, including three appendices, revised version

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Cited by in corpus (16)

• Gapless Coulomb state emerging from a self-dual topological tensor-network state
• Evidence for deconfined $U(1)$ gauge theory at the transition between toric code and double semion
• From trivial to topological paramagnets: The case of ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_2^3$ symmetries in two dimensions
• Entanglement order parameters and critical behavior for topological phase transitions and beyond
• Building models of topological quantum criticality from pivot Hamiltonians
• Complete characterization of non-Abelian topological phase transitions and detection of anyon splitting with projected entangled pair states
• Characterization of topological phase transitions from a non-Abelian topological state and its Galois conjugate through condensation and confinement order parameters
• Non-Hermitian effects of the intrinsic signs in topologically ordered wavefunctions
• Quantum phase transition between symmetry enriched topological phases in tensor-network states
• Simulating 2D topological quantum phase transitions on a digital quantum computer
• Anomalous domain wall condensation in a modified Ising chain
• Continuous topological phase transition between ${\mathbb{Z}}_2$ topologically ordered phases
• Bridging Rokhsar-Kivelson Type and Generic Quantum Phase Transitions via Thermofield Double States
• Transition between 2D Symmetry Protected Topological Phases on a Klein Bottle
• Twisted quantum doubles are sign problem-free
• Skeleton of isometric Tensor Network States for Abelian String-Net Models