Distinguishing different topologically ordered phases and characterizing phase transitions between them is a difficult task due to the absence of local order parameters. In this paper, we use a combination of analytical and numerical approaches to distinguish two such phases and characterize a phase transition between them. The ``toric code'' and ``double semion'' models are simple lattice models exhibiting ${\mathbb{Z}}_{2}$ topological order. Although both models express the same topological ground state degeneracies and entanglement entropies, they are distinct phases of matter because their emergent quasiparticles obey different statistics. For a 1D model, we tune a phase transition between these two phases and obtain an exact solution to the entire phase diagram, finding a second-order Ising$\ifmmode\times\else\texttimes\fi{}$Ising transition. We then use exact diagonalization to study the 2D case and find indications of a first-order transition. We show that the quasiparticle statistics provides a robust indicator of the distinct topological orders throughout the whole phase diagram.Distinguishing different topologically ordered phases and characterizing phase transitions between them is a difficult task due to the absence of local order parameters. In this paper, we use a combination of analytical and numerical approaches to distinguish two such phases and characterize a phase transition between them. The ``toric code'' and ``double semion'' models are simple lattice models exhibiting ${\mathbb{Z}}_{2}$ topological order. Although both models express the same topological ground state degeneracies and entanglement entropies, they are distinct phases of matter because their emergent quasiparticles obey different statistics. For a 1D model, we tune a phase transition between these two phases and obtain an exact solution to the entire phase diagram, finding a second-order Ising$\ifmmode\times\else\texttimes\fi{}$Ising transition. We then use exact diagonalization to study the 2D case and find indications of a first-order transition. We show that the quasiparticle statistics provides a robust indicator of the distinct topological orders throughout the whole phase diagram.

arXiv:1403.0768Physical Review BJul 14, 201457 citations

Entanglement, fidelity, and topological entropy in a quantum phase transition to topological order

Alioscia Hamma, Wenqiong Zhang, Stephan Haas +1 more

We present a numerical study of a quantum phase transition from a spin-polarized to a topologically ordered phase in a system of spin- $1∕2$ particles on a torus. We demonstrate that this non-symmetry-breaking topological quantum phase transition (TOQPT) is of second order. The transition is analyzed via the ground state energy and fidelity, block entanglement, Wilson loops, and the recently proposed topological entropy. Only the topological entropy distinguishes the TOQPT from a standard QPT, and remarkably, does so already for small system sizes. Thus the topological entropy serves as a proper order parameter. We demonstrate that our conclusions are robust under the addition of random perturbations, not only in the topological phase, but also in the spin-polarized phase and even at the critical point.We present a numerical study of a quantum phase transition from a spin-polarized to a topologically ordered phase in a system of spin-$1∕2$ particles on a torus. We demonstrate that this non-symmetry-breaking topological quantum phase transition (TOQPT) is of second order. The transition is analyzed via the ground state energy and fidelity, block entanglement, Wilson loops, and the recently proposed topological entropy. Only the topological entropy distinguishes the TOQPT from a standard QPT, and remarkably, does so already for small system sizes. Thus the topological entropy serves as a proper order parameter. We demonstrate that our conclusions are robust under the addition of random perturbations, not only in the topological phase, but also in the spin-polarized phase and even at the critical point.

arXiv:0705.0026Physical Review BApr 9, 2008112 citations

Numerical studies of topological quantum phase transitions in low dimensional systems

Yasuhiro Hatsugai

May 31, 20040 citations

Topological Phases in Condensed Matter Physics

Saurabh Basu

Springer eBooksJan 1, 20232 citations

Local integrals of motion detection of localization-protected topological order

Thorsten B. Wahl, Florian Venn, Benjámin Béri

Many-body-localized (MBL) phases can be topologically distinct, but distinguishing these phases using order parameters can be challenging. Here we show how topologically distinct local integrals of motion, variationally parametrized by quantum circuits, can be used to numerically demonstrate the topological inequivalence of MBL phases. We illustrate our approach on a fermionic chain where both topologically distinct MBL phases and benchmark comparisons to order parameters are possible. We also use our approach, augmented by the excited state density-matrix renormalization group algorithm, to extract high-energy topological doublets. We describe applying our methods to higher dimensions to identify MBL topological order and topological multiplets hidden by the dense many-body spectrum.Many-body-localized (MBL) phases can be topologically distinct, but distinguishing these phases using order parameters can be challenging. Here we show how topologically distinct local integrals of motion, variationally parametrized by quantum circuits, can be used to numerically demonstrate the topological inequivalence of MBL phases. We illustrate our approach on a fermionic chain where both topologically distinct MBL phases and benchmark comparisons to order parameters are possible. We also use our approach, augmented by the excited state density-matrix renormalization group algorithm, to extract high-energy topological doublets. We describe applying our methods to higher dimensions to identify MBL topological order and topological multiplets hidden by the dense many-body spectrum.

arXiv:2111.11543Physical review. B./Physical review. BApr 29, 202220 citations

{\mathbb{Z}}_{2}

topological order. Although both models express the same topological ground state degeneracies and entanglement entropies, they are distinct phases of matter because their emergent quasiparticles obey different statistics. For a 1D model, we tune a phase transition between these two phases and obtain an exact solution to the entire phase diagram, finding a second-order Ising$\ifmmode\times\else\texttimes\fi{}$Ising transition. We then use exact diagonalization to study the 2D case and find indications of a first-order transition. We show that the quasiparticle statistics provides a robust indicator of the distinct topological orders throughout the whole phase diagram.

We present a numerical study of a quantum phase transition from a spin-polarized to a topologically ordered phase in a system of spin-

1∕2

particles on a torus. We demonstrate that this non-symmetry-breaking topological quantum phase transition (TOQPT) is of second order. The transition is analyzed via the ground state energy and fidelity, block entanglement, Wilson loops, and the recently proposed topological entropy. Only the topological entropy distinguishes the TOQPT from a standard QPT, and remarkably, does so already for small system sizes. Thus the topological entropy serves as a proper order parameter. We demonstrate that our conclusions are robust under the addition of random perturbations, not only in the topological phase, but also in the spin-polarized phase and even at the critical point.