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Acta Physica Sinica

Topological phase in one-dimensional momentum space lattice of ultracold atoms without chiral symmetry

Chao Zeng, Yi-Yi Mao, Jizhou Wu, Tao Yuan, Han-Ning Dai, Yu-Ao Chen

Published Nov 17, 2023• 0 citations
Publisher

Abstract

Symmetry plays a crucial role in understanding topological phases in materials. In one-dimensional systems, such as the Su-Schrieffer-Heeger (SSH) model, chiral symmetry is thought to ensure the quantization of the Zak phase and the nontrivial topological phase. However, our work demonstrates that the one-dimensional lattice system with broken chiral symmetry can still possess quantized Zak phase and nontrivial topological phase. Specifically, we use a Bose-Einstein condensate of 87Rb atoms in a momentum space lattice of ultracold atoms to effectively simulate a one-dimensional Zigzag model of 26 sites, which intrinsically breaks the chiral symmetry by additional next-nearest-neighbor coupling. To ensure the existence of the nontrivial topological phase, where the Zak phase can be measured from the time-averaged displacement during the system’s evolution, we need to preserve the inversion symmetry by modulating laser power so that all next-nearest-neighbor coupling strengths are equal. Furthermore, by changing the ratio of nearest-neighbor coupling strengths, we observe a topological phase transition from a nontrivial topological phase to a trivial topological phase at the point where the ratio equals 1. Our work demonstrates that the ultracold atom system provides a controllable platform for studying the symmetrical phase and topological phase, with the potential to explore nonlinear topological phenomena by increasing the interactions among atoms. In addition, our system can be used to investigate other interesting topological phenomena with more complex models, such as critical phenomena at the phase transitions and flat band structures in the extended SSH model with long-range coupling. By controlling the coupling strengths, we can also explore the influence of different symmetries on the topological properties of extended SSH models in the future. Moreover, our platform makes it possible to studythe models with more energy bands, such as the Aharonov-Bohm caging model with a three-level structure, which shows peculiar flat-band properties. This work provides opportunities for various studies in the fields of symmetry, topology, and the interaction of controllable quantum systems.

DOI
10.7498/aps.73.20231566

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Symmetry plays a crucial role in understanding topological phases in materials. In one-dimensional systems, such as the Su-Schrieffer-Heeger (SSH) model, chiral symmetry is thought to ensure the quantization of the Zak phase and the nontrivial topological phase. However, our work demonstrates that the one-dimensional lattice system with broken chiral symmetry can still possess quantized Zak phase and nontrivial topological phase. Specifically, we use a Bose-Einstein condensate of 87Rb atoms in a momentum space lattice of ultracold atoms to effectively simulate a one-dimensional Zigzag model of 26 sites, which intrinsically breaks the chiral symmetry by additional next-nearest-neighbor coupling. To ensure the existence of the nontrivial topological phase, where the Zak phase can be measured from the time-averaged displacement during the system’s evolution, we need to preserve the inversion symmetry by modulating laser power so that all next-nearest-neighbor coupling strengths are equal. Furthermore, by changing the ratio of nearest-neighbor coupling strengths, we observe a topological phase transition from a nontrivial topological phase to a trivial topological phase at the point where the ratio equals 1. Our work demonstrates that the ultracold atom system provides a controllable platform for studying the symmetrical phase and topological phase, with the potential to explore nonlinear topological phenomena by increasing the interactions among atoms. In addition, our system can be used to investigate other interesting topological phenomena with more complex models, such as critical phenomena at the phase transitions and flat band structures in the extended SSH model with long-range coupling. By controlling the coupling strengths, we can also explore the influence of different symmetries on the topological properties of extended SSH models in the future. Moreover, our platform makes it possible to studythe models with more energy bands, such as the Aharonov-Bohm caging model with a three-level structure, which shows peculiar flat-band properties. This work provides opportunities for various studies in the fields of symmetry, topology, and the interaction of controllable quantum systems.
"As the progress of science pushes the frontiers of studying nature to both the extremely large scale and the incredibly small scale, experiments can become increasingly difficult to perform. Therefore, a scale model experiment for these kinds of physics can be useful as long as it is relatively easy to access and control. This is the idea of quantum simulation, where one system exactly mimics another when they are subjected to equivalent physical scenarios. Topological systems, those with a property that is stabilized by their specific configuration, are often quite challenging to experimentally probe. Therefore, test bed systems have been widely and successfully applied to studying topology in an effort to advance the field, which promises enticing applications such as robust quantum technology. Our particular scheme uses a novel type of ""synthetic"" dimension for atoms of a Bose–Einstein condensate, which we call a momentum-space lattice, to emulate one-dimensional lattice models. The idea of a synthetic dimension or synthetic lattices is a useful construct to describe how the particle dynamics can evolve in degrees of freedom that are not directly related to transport in real space. This can relate to population dynamics in an internal degree of freedom, such as spin, or, in our case, in the dynamics of free-particle-like atomic momentum states. We expose a Bose–Einstein condensate to two laser fields that are capable of driving transitions between different momentum states of the condensate atoms through a two-photon Bragg diffraction process. Neighboring states in the free-particle dispersion relation are characterized by a unique momenta and energy spacing such that we can independently control every aspect of their coupling. In this thesis, we report on the design and construction of an apparatus for the production of 87Rb Bose–Einstein condensates as well as the partial construction of an apparatus for the production of 39K Bose–Einstein condensates. We also discuss the experimental realization of, and theoretical background for, our momentum-space lattice. Using the momentum-space lattice, we perform experiments on not only topological systems, but also on driven artificial spins featuring chaotic dynamics, as well as faster-than-adiabatic quantum state control and preparation. Our measurements of topological systems are based in the 1D Su–Schrieffer–Heeger model, where we report on the direct observation of the localized, solitonic edge state created by the topology inherent in the system. Through the addition of exactly tunable tunneling disorder, we observe the destruction of the band topology due to disorder at a random-singlet transition. Furthermore, we drive a trivial system into a topological phase through the addition of disorder, observing a counter-intuitive phenomenon known as the topological Anderson insulator for the first time. We go on to report on our driven, tunable spin experiment, where we probe effective spin ""squeezing"" (in the absence of any entanglement in our noninteracting system), the out-of-time-ordered correlation function, and chaos. And finally, we use our momentum-space lattice to demonstrate a technique which allows a system to be transformed faster than the adiabatic limit with no diabatic transitions in a highly non-trivial regime. Due to the versatility of the momentum-space lattice, the results detailed here have a broad impact. Our observations of topological phenomena in elementary and canonical 1D systems are some of the first experimental verifications of long-standing predictions. Further, our studies of the interplay between classical chaos and quantum systems may help shed light on this complex and rich problem. And from a more ""applied science"" perspective, our demonstrations of generalized counter-diabatic techniques are widely beneficial and applicable to many systems where adiabaticity is challenged."
Abstract The Su–Schrieffer–Heeger (SSH) model perhaps is the easiest and the most basic model for topological excitations. Many variations and extensions of the SSH model have been proposed and explored to better understand both fundamental and novel aspects of topological physics. The SSH4 model has been proposed theoretically as an extended SSH model with higher dimension (the internal dimension changes from two to four). It has been proposed that the winding number in this system can be determined through a higher-dimensional extension of the mean chiral displacement measurement, however, this has not yet been verified in experiment. Here, we report the realization of this model with ultracold atoms in a momentum lattice. We verify the winding number through measurement of the mean chiral displacement in a system with higher internal dimension, we map out the topological phase transition in this system, and we confirm the topological edge state by observation of the quench dynamics when atoms are initially prepared at the system boundary.
We propose to simulate and detect quantum anomalous Hall phase with ultracold atoms in a one-dimensional optical lattice, with the other synthetic dimension being realized by modulating spin-orbit coupling. We show that the system manifests a topologically nontrivial phase with two chiral edge states which can be readily detected in this synthetic two-dimensional system. Moreover, it is interesting that at the phase transition point there is a flat energy band and this system can also be in a topologically nontrivial phase with two Fermi zero modes existing at the boundaries by considering the synthetic dimension as a modulated parameter. We also show how to measure these topological phases experimentally in ultracold atoms. Another model with a random Rashba and Dresselhaus spin-orbit coupling strength is also found to exhibit topological nontrivial phase, and the impact of the disorder to the system is revealed.
We study the interplay of spontaneous symmetry breaking and topological properties in interacting one-dimensional models. We solve these models using bozonization and identify topologically non-trivial phases by counting the additional degeneracy (affiliated with the edge modes) of a finite-size system relative to the infinite one. We find even if the mean-field solution is topological, this may not be true when it arises from spontaneous symmetry breaking, including in the Su-Schrieffer-Heeger (SSH) model. This implies that the original SSH model, as presented by Su, Schrieffer, and Heeger, is topologically trivial, as opposed to its mean-field version. A spinful version, on the other hand, does exhibit a topologically non-trivial phase. In that state, both mean-field solutions are topologically non-trivial and correspond to non-interacting SSH chains in the opposite phases with the winding number ν=1ν=1ν=1. We show that this phase is protected by a chiral symmetry, similar to the non-interacting phases.
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.
Quasi-one-dimensional or ladder models serve as a bridge between one-dimensional and two-dimensional models, providing a simplified framework for incorporating intricate physics. In this study, we extend the concept of the momentum-state lattice by introducing couplings between next-nearest-neighbor sites throughout the entire momentum chain to simulate the staggered Su-Schrieffer-Heeger (SSH) ladder model. This ladder model is a direct extension of the well-known SSH model, recognized as a fundamental progression in modeling systems. Through this model, we explore compelling physical phenomena and successfully replicate the topological phase diagram of the Kitaev-Majorana chain. Utilizing time-averaged mean displacements of wave packets as a metric, we ascertain the topological invariants of the model. By varying staggered dimerization and interchain hopping parameters, we map out the topological phase diagram of the staggered SSH ladder, elucidating the presence of three distinct phases classified by their winding numbers: W = + 1 , 0, and − 1 , respectively.