We show that a topological superconductor made of four chains of superconducting spinless fermions characterized by four Majorana edge states can adiabatically be deformed into a trivial band insulator. To unwind this time-reversal invariant topological superconductor, interactions to spinful fermions are switched on along an adiabatic path. Thereby, we couple modes which belong to two different representations of the time-reversal symmetry operator $\mathcal{T}$ with ${\mathcal{T}}^{2}=1$ and ${\mathcal{T}}^{2}=\ensuremath{-}1$, respectively. This observation can be understood by investigating how the relevant symmetries act on the entanglement spectrum giving rise to four instead of eight different topological phases with Majorana edge modes. We also show that a simple level crossing of doubly and singly degenerate states occurs in the entanglement spectrum on deforming the quantum state.We show that a topological superconductor made of four chains of superconducting spinless fermions characterized by four Majorana edge states can adiabatically be deformed into a trivial band insulator. To unwind this time-reversal invariant topological superconductor, interactions to spinful fermions are switched on along an adiabatic path. Thereby, we couple modes which belong to two different representations of the time-reversal symmetry operator $\mathcal{T}$ with ${\mathcal{T}}^{2}=1$ and ${\mathcal{T}}^{2}=\ensuremath{-}1$, respectively. This observation can be understood by investigating how the relevant symmetries act on the entanglement spectrum giving rise to four instead of eight different topological phases with Majorana edge modes. We also show that a simple level crossing of doubly and singly degenerate states occurs in the entanglement spectrum on deforming the quantum state.

arXiv:1203.5541Physical Review BSep 14, 20124 citations

Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases

Peng Ye, Zheng‐Cheng Gu

Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state (or atomic insulator) as long as certain global symmetry $G$ is unbroken. At low energies, most of the two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of the multicomponent Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ${\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...$). In addition to the usual multicomponent $BF$ topological term at level-1, we find that there are new topological terms with quantized coefficients (e.g., ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}$ and ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}$) in TQFT actions, where ${a}^{1},{a}^{2},...$ are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as $G$ is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.Symmetry-protected topological phases (SPT) are short-range entangled gapped states protected by global symmetry. Nontrivial SPT phases cannot be adiabatically connected to the trivial disordered state (or atomic insulator) as long as certain global symmetry $G$ is unbroken. At low energies, most of the two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of the multicomponent Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ${\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...$). In addition to the usual multicomponent $BF$ topological term at level-1, we find that there are new topological terms with quantized coefficients (e.g., ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}$ and ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}$) in TQFT actions, where ${a}^{1},{a}^{2},...$ are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as $G$ is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.

arXiv:1508.05689Physical review. B./Physical review. BMay 31, 201654 citations

Topological Phases in the Quantum Theory.

Tom D. Imbo

Jan 1, 198839 citations

Dynamical classification of topological quantum phases

Lin Zhang, Long Zhang, Sen Niu +1 more

arXiv:1802.10061中国科学通报：英文版Sep 29, 2018150 citations

Topological Phase Transition and Topological Quantum State Transfer in Periodically Modulated Circuit‐QED Lattice

Ji Cao, Wen‐Xue Cui, Yi Xue-Xi +1 more

Abstract The topological properties of a circuit‐QED lattice are studied and it is found that the system undergoes a topological phase transition accompanied by the changes of Chern numbers from 2 to −1. For different lattice sizes, the energy eigenvalue spectra and the probability distributions of edge states are shown. Moreover, it is revealed that, depending on the strictly adiabatic evolution of the time‐dependent Hamiltonian, several different topological quantum state transfers between the first two sites and the last two sites can be achieved by employing the topologically protected edge states as the transfer channels. Further, it is demonstrated that, by using resonator‐based input–output process, the photonic topological edge state can be detected by measuring the average photon number in the steady state. The scheme opens up a new prospect for investigating the potential application of topological matter in quantum information processing.Abstract The topological properties of a circuit‐QED lattice are studied and it is found that the system undergoes a topological phase transition accompanied by the changes of Chern numbers from 2 to −1. For different lattice sizes, the energy eigenvalue spectra and the probability distributions of edge states are shown. Moreover, it is revealed that, depending on the strictly adiabatic evolution of the time‐dependent Hamiltonian, several different topological quantum state transfers between the first two sites and the last two sites can be achieved by employing the topologically protected edge states as the transfer channels. Further, it is demonstrated that, by using resonator‐based input–output process, the photonic topological edge state can be detected by measuring the average photon number in the steady state. The scheme opens up a new prospect for investigating the potential application of topological matter in quantum information processing.

Annalen der PhysikJul 5, 202110 citations

\mathcal{T}

with

{\mathcal{T}}^{2}=1

and ${\mathcal{T}}^{2}=\ensuremath{-}1$, respectively. This observation can be understood by investigating how the relevant symmetries act on the entanglement spectrum giving rise to four instead of eight different topological phases with Majorana edge modes. We also show that a simple level crossing of doubly and singly degenerate states occurs in the entanglement spectrum on deforming the quantum state.

G

is unbroken. At low energies, most of the two-dimensional SPTs with Abelian symmetry can be described by topological quantum field theory (TQFT) of the multicomponent Chern-Simons type. However, in contrast to the fractional quantum Hall effect where TQFT can give rise to interesting bulk anyons, TQFT for SPTs only supports trivial bulk excitations. The essential question in TQFT descriptions for SPTs is to understand how the global symmetry is implemented in the partition function. In this paper, we systematically study TQFT of three-dimensional SPTs with unitary Abelian symmetry (e.g., ${\mathbb{Z}}_{{N}_{1}}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{{N}_{2}}\ifmmode\times\else\texttimes\fi{}...$). In addition to the usual multicomponent

BF

topological term at level-1, we find that there are new topological terms with quantized coefficients (e.g., ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}d{a}^{2}$ and ${a}^{1}\ensuremath{\wedge}{a}^{2}\ensuremath{\wedge}{a}^{3}\ensuremath{\wedge}{a}^{4}$) in TQFT actions, where

{a}^{1},{a}^{2},...

are 1-form U(1) gauge fields. These additional topological terms cannot be adiabatically turned off as long as

G

is unbroken. By investigating symmetry transformations for the TQFT partition function, we end up with the classification of SPTs that is consistent with the well-known group cohomology approach. We also discuss how to gauge the global symmetry and possible TQFT descriptions of Dijkgraaf-Witten gauge theory.