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arXiv:1307.2916

Quantum phase transition from the topological viewpoint

Izumi Tanaka

Published Jul 7, 2013• 0 citations
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Abstract

This study targets quantum phases which are characterized by topological properties and no associated with the symmetry breaking. We concern ourselves primarily with the transitions among these quantum phases. This type of quantum phase transition was investigated by $G$-cobordism in unified framework. This framework provides a useful method to investigate a new quantum phase.

DOI
10.48550/arxiv.1307.2916

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This study targets quantum phases which are characterized by topological properties and no associated with the symmetry breaking. We concern ourselves primarily with the transitions among these quantum phases. This type of quantum phase transition was investigated by GGG-cobordism in unified framework. This framework provides a useful method to investigate a new quantum phase.
I would claim that we do not have a suitably general definition of what a topological phase is, or more importantly, any robust understanding of how to enter one even in the world of mathematical models. The latter is, of course, the more important issue and the main subject of this note. But a good definition can sharpen our thinking and a poor definition can misdirect us. I will not attempt a final answer here but merely comment on the strengths and weaknesses of possible definitions and argue for some flexibility. In particular, I describe a rather simple class of ``quantum gravity'' models which are neither lattice nor field theoretic but appear to contain strong candidates for topological phases.
Symmetry-protected topological (SPT) phases exhibit nontrivial order if symmetry is respected but are adiabatically connected to the trivial product phase if symmetry is not respected. However, unlike the symmetry-breaking phase, there is no local order parameter for SPT phases. Here we employ a tensor-network method to compute the topological invariants characterized by the simulated modular SSS and TTT matrices to study transitions in a few families of two-dimensional (2D) wave functions which are ZN{\mathbb{Z}}_{N}ZN​ (N=2and3)(N=2\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}3)(N=2and3) symmetric. We find that in addition to the topologically ordered phases, the modular matrices can be used to identify nontrivial SPT phases and detect transitions between different SPT phases as well as between symmetric and symmetry-breaking phases. Therefore modular matrices can be used to characterize various types of gapped phases in a unifying way.
We study the symmetry-protected topological (SPT) orders for bosonic systems from an information-theoretic viewpoint. We show that with a proper choice of the onsite basis, the degenerate ground-state space of SPT orders (on a manifold with boundary) is a quantum error-correcting code with macroscopic classical distance, hence is stable against any local bit-flip errors. We show that this error-correcting property of the SPT orders has a natural connection to that of the symmetry-breaking orders, whose degenerate ground-state space is a classical error-correcting code with a macroscopic distance, providing a new angle for the hidden symmetry-breaking properties in SPT orders. We further propose new types of topological entanglement entropy that probe the SPT orders hidden in their symmetric ground states, which also signal the topological phase transitions protected by symmetry. Combined with the original definition of topological entanglement entropy that probes the “intrinsic topological orders”, and the recent proposed one that probes the symmetry-breaking orders, the set of different types of topological entanglement entropy may hence distinguish topological orders, SPT orders, and symmetry-breaking orders, which may be mixed up in a single system.