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arXiv:1411.0260 · Europhysics Letters (EPL)

Hidden-symmetry–protected topological phases on a one-dimensional lattice

Linhu Li, Shu Chen

Published Feb 1, 2015• 25 citations
arXivHTMLPDF

Abstract

We demonstrate the existence of a topologically nontrivial phase in a one-dimensional fermionic lattice system subjected to synthetic gauge fields, which is beyond the standard Altland-Zirnbauer classification of topological insulators. The topological phase can be characterized by the presence of degenerate zero-mode edge states or a quantized Berry phase of the occupied Bloch band. By analyzing symmetries of the system, we identify that the topological phase and zero-mode edge states are protected by two hidden symmetries. An extended model with hidden symmetry breaking is also studied in order to reveal the effect of hidden symmetries on the symmetry-protected topological phase.

DOI
10.1209/0295-5075/109/40006

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We demonstrate the existence of a topologically nontrivial phase in a one-dimensional fermionic lattice system subjected to synthetic gauge fields, which is beyond the standard Altland-Zirnbauer classification of topological insulators. The topological phase can be characterized by the presence of degenerate zero-mode edge states or a quantized Berry phase of the occupied Bloch band. By analyzing symmetries of the system, we identify that the topological phase and zero-mode edge states are protected by two hidden symmetries. An extended model with hidden symmetry breaking is also studied in order to reveal the effect of hidden symmetries on the symmetry-protected topological phase.
We show that a one-dimensional lattice of spinless fermions can undergo a topological phase transition only due to the effect of interaction. By allowing a dimerized interaction among the particles we show that the system exhibits a symmetry protected topological phase which does not exist in the absence of interaction and the system is a gapless state. The nontrivial topological character appears due to the onset of two degenerate bond-order phases as a function of dimerized interaction which are found to be topologically distinct from each other. As a result a topological phase transition occurs between these bond-order phases through a gap closing point. However, in the limit of strong interaction, the topologically distinct bond-order phases are separated from each other through a gapped charge density wave phase possessing a local antiferromagnetic order. We characterize this emergent topological nature by the edge states, Berry phase, and a nonlocal string order parameter and provide possible experimental signatures in terms of Thouless charge pumping and density-density correlation.
We identify the existence of various symmetry-protected topological states in one-dimensional superlattices with periodically modulated hopping amplitudes or on-site potentials, which can be characterized by the quantized Berry phase $\ensuremath{\pi}$ or the emergence of a pair of degenerate boundary states. It is shown that there may exist three types of topological phases, which are protected by inversion symmetry, chiral symmetry, and both of them, respectively, depending on the modulations, the odd, or the even modulation period. The connection between the hopping and potential modulations is also discussed. Furthermore, we demonstrate that the topological phase protected by the inversion symmetry can be realized in the interacting boson systems trapped in the same superlattices. The results can very possibly be studied experimentally in superlattice systems engineered with state-of-the-art technologies.
Symmetry plays a fundamental role in understanding complex quantum matter, particularly in classifying topological quantum phases, which have attracted great interests in the recent decade. An outstanding example is the time-reversal invariant topological insulator, a symmetry-protected topological (SPT) phase in the symplectic class of the Altland-Zirnbauer classification. We report the observation for ultracold atoms of a noninteracting SPT band in a one-dimensional optical lattice and study quench dynamics between topologically distinct regimes. The observed SPT band can be protected by a magnetic group and a nonlocal chiral symmetry, with the band topology being measured via Bloch states at symmetric momenta. The topology also resides in far-from-equilibrium spin dynamics, which are predicted and observed in experiment to exhibit qualitatively distinct behaviors in quenching to trivial and nontrivial regimes, revealing two fundamental types of spin-relaxation dynamics related to bulk topology. This work opens the way to expanding the scope of SPT physics with ultracold atoms and studying nonequilibrium quantum dynamics in these exotic systems.
The low energy behavior of a huge variety of one-dimensional interacting spinful fermionic systems exhibits spin-charge separation, described in the continuum limit by two sine-Gordon models decoupled in the charge and spin channels. Interaction is known to induce, besides the gapless Luttinger liquid phase, eight possible gapped phases, among which are the Mott, Haldane, charge-/spin-density, and bond-ordered wave insulators, and the Luther Emery liquid. Here we prove that some of these physically distinct phases have nontrivial topological properties, notably the presence of degenerate protected edge modes with fractionalized charge/spin. Moreover, we show that the eight gapped phases are in one-to-one correspondence with the symmetry-protected topological (SPT) phases classified by group cohomology theory in the presence of particle-hole symmetry P. The latter result is also exploited to characterize SPT phases by measurable nonlocal order parameters which follow the system evolution to the quantum phase transition. The implications on the appearance of exotic orders in the class of microscopic Hubbard Hamiltonians, possibly without P symmetry at higher energies, are discussed.
We propose a realizable experiment scheme to construct a one-dimensional synthetic magnetic flux lattice with spin-tensor-momentum coupled spin-1 atoms and explore its exotic topological states. Different from the Altland-Zirnbauer classification, we show that our system hosts a symmetry-protected phase protected by a magnetic group symmetry (M\mathcal{M}M) and characterized by a Z2{Z}_{2}Z2​ topological invariant. In single-particle spectra, we show that the topological nontrivial phase supports two kinds of edge states, which include two (four) zero-energy edge modes in the absence (presence) of two-photon detuning. We further study the bulk-edge correspondence in a non-Hermitian model by taking into account the particle dissipation. It is shown that the non-Hermitian system preserves the bulk-edge correspondence under the M\mathcal{M}M symmetry but exhibits the non-Hermitian skin effect with breaking the M\mathcal{M}M symmetry at nonzero magnetic flux. This work provides insights in understanding the exotic topological quantum states of high-spin systems and facilitating their experimental explorations.
We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single-particle excitations in the bulk. We consider specific microscopic models and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically nontrivial phase.