Topology, edge states, and zero-energy states of ultracold atoms in one-dimensional optical superlattices with alternating on-site potentials or hopping coefficients

Yan He, Kevin Wright, Saïd Kouachi +1 more

arXiv:1712.06538Physical Review AFeb 12, 201823 citations

Springer eBooks

Symmetry and Topology of the Configuration Space and Quantization

H. D. Doebner, J. Tolar

Published Jan 1, 1986• 8 citations

Publisher

DOI: 10.1007/978-1-4757-1472-2_10

Related papers

Configuration space topology and quantum internal symmetries

E. C. G. Sudarshan, Tom D. Imbo, T. R. Govindarajan

Physics Letters BNov 1, 198839 citations

An interplay of topology and quantized geometric phase for two different symmetry-class Hamiltonians

Rahul S, Ranjith Kumar R, Y R Kartik +2 more

Abstract Study of symmetry, topology and geometric phase can reveal many new and interesting results on the topological states of matter. Here we present a completely new and interesting result of symmetry, topology and quantization of geometric phase along with the physical explanation for two different symmetry classes. We present a detailed study of the auxiliary space for two different symmetry classes of Hamiltonians. We show explicitly that the origin of the auxiliary space inside the curve is only a necessary condition but it is not a sufficient condition for the topological state. One of the most interesting results is that same symmetry-class Hamiltonians show different behavior in topology and quantized geometric phase.Abstract Study of symmetry, topology and geometric phase can reveal many new and interesting results on the topological states of matter. Here we present a completely new and interesting result of symmetry, topology and quantization of geometric phase along with the physical explanation for two different symmetry classes. We present a detailed study of the auxiliary space for two different symmetry classes of Hamiltonians. We show explicitly that the origin of the auxiliary space inside the curve is only a necessary condition but it is not a sufficient condition for the topological state. One of the most interesting results is that same symmetry-class Hamiltonians show different behavior in topology and quantized geometric phase.

arXiv:1809.09821Physica ScriptaApr 29, 20196 citations

Quantized topological terms in weak-coupling gauge theories with a global symmetry and their connection to symmetry-enriched topological phases

Ling-Yan Hung, Xiao-Gang Wen

We study the quantized topological terms in a weak-coupling gauge theory with gauge group ${G}_{g}$ and a global symmetry ${G}_{s}$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,{\ensuremath{\nu}}_{d})$, where $G$ is an extension of ${G}_{s}$ by ${G}_{g}$ and ${\ensuremath{\nu}}_{d}$ an element in group cohomology ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ . When $d=3$ and/or when ${G}_{g}$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e., gapped long-range-entangled phases with symmetry). Thus, those SET phases are classified by ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$ , where $G/{G}_{g}={G}_{s}$ . We also apply our theory to a simple case ${G}_{s}={G}_{g}={Z}_{2}$ , which leads to 12 different SET phases in $2+1$ dimensions [(2+1)D], where quasiparticles have different patterns of fractional ${G}_{s}={Z}_{2}$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry ${G}_{s}$ , which may lead to different fractionalizations of ${G}_{s}$ quantum numbers and different fractional statistics [if in ( $2+1$ )D].We study the quantized topological terms in a weak-coupling gauge theory with gauge group ${G}_{g}$ and a global symmetry ${G}_{s}$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,{\ensuremath{\nu}}_{d})$, where $G$ is an extension of ${G}_{s}$ by ${G}_{g}$ and ${\ensuremath{\nu}}_{d}$ an element in group cohomology ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$. When $d=3$ and/or when ${G}_{g}$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e., gapped long-range-entangled phases with symmetry). Thus, those SET phases are classified by ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$, where $G/{G}_{g}={G}_{s}$. We also apply our theory to a simple case ${G}_{s}={G}_{g}={Z}_{2}$, which leads to 12 different SET phases in $2+1$ dimensions [(2+1)D], where quasiparticles have different patterns of fractional ${G}_{s}={Z}_{2}$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry ${G}_{s}$, which may lead to different fractionalizations of ${G}_{s}$ quantum numbers and different fractional statistics [if in ($2+1$)D].

arXiv:1212.1827Physical Review BApr 5, 201394 citations

Configuration Space and Second Quantization

CRC Press eBooksMay 21, 20048 citations

Symmetry and quantization

V. Aldaya, Manuel Calixto Molina, Julio Guerrero García

Jan 1, 19990 citations

Topology, edge states, and zero-energy states of ultracold atoms in one-dimensional optical superlattices with alternating on-site potentials or hopping coefficients

Yan He, Kevin Wright, Saïd Kouachi +1 more

One-dimensional superlattices with periodic spatial modulations of onsite potentials or tunneling coefficients can exhibit a variety of properties associated with topology or symmetry. Recent developments of ring-shaped optical lattices allow a systematic study of those properties in superlattices with or without boundaries. While superlattices with additional modulating parameters are shown to have quantized topological invariants in the augmented parameter space, we also found localized or zero-energy states associated with symmetries of the Hamiltonians. Probing those states in ultracold atoms is possible by utilizing recently proposed methods analyzing particle depletion or the local density of states. Moreover, we summarize feasible realizations of configurable optical superlattices using currently available techniques.One-dimensional superlattices with periodic spatial modulations of onsite potentials or tunneling coefficients can exhibit a variety of properties associated with topology or symmetry. Recent developments of ring-shaped optical lattices allow a systematic study of those properties in superlattices with or without boundaries. While superlattices with additional modulating parameters are shown to have quantized topological invariants in the augmented parameter space, we also found localized or zero-energy states associated with symmetries of the Hamiltonians. Probing those states in ultracold atoms is possible by utilizing recently proposed methods analyzing particle depletion or the local density of states. Moreover, we summarize feasible realizations of configurable optical superlattices using currently available techniques.

arXiv:1712.06538Physical Review AFeb 12, 201823 citations

We study the quantized topological terms in a weak-coupling gauge theory with gauge group

{G}_{g}

and a global symmetry

{G}_{s}

d

space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,{\ensuremath{\nu}}_{d})$, where

G

is an extension of

{G}_{s}

{G}_{g}

and ${\ensuremath{\nu}}_{d}$ an element in group cohomology

{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})

. When

d=3

and/or when

{G}_{g}

is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e., gapped long-range-entangled phases with symmetry). Thus, those SET phases are classified by

{\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})

, where

G/{G}_{g}={G}_{s}

. We also apply our theory to a simple case

{G}_{s}={G}_{g}={Z}_{2}

, which leads to 12 different SET phases in

2+1

dimensions [(2+1)D], where quasiparticles have different patterns of fractional

{G}_{s}={Z}_{2}

quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry

{G}_{s}

, which may lead to different fractionalizations of

{G}_{s}

quantum numbers and different fractional statistics [if in (

2+1

)D].