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arXiv:1805.11299 · Physical review. B./Physical review. B

Emergent Haldane phase in an alternating-bond Z3 parafermion chain

Shunyao Zhang, Hong-Ze Xu, Yue-Xin Huang, Guang‐Can Guo, Zheng-Wei Zhou, Ming Gong

Published Oct 1, 2019• 0 citations
arXivHTMLPDF

Abstract

The Haldane phase represents one of the most important symmetry-protected states in modern physics. This state can be realized using spin-1 and $\mathrm{spin}\ensuremath{-}\frac{1}{2}$ Heisenberg models and bosonic particles. Here we explore the emergent Haldane phase in an alternating bond ${\mathbb{Z}}_{3}$ parafermion chain, which is different from the previous proposals from fundamental statistics and symmetries. We show that this emergent phase can also be characterized by a modified long-range string order, as well as fourfold degeneracy in the ground-state energies and entanglement spectra. This phase is protected by both the charge conjugate and parity symmetry, and the edge modes are shown to satisfy parafermionic statistics in which braiding of the two edge modes yields a $\frac{2\ensuremath{\pi}}{3}$ phase. This model also supports rich phases, including a topological ferromagnetic parafermion (FP) phase, trivial paramagnetic parafermion phase, classical dimer phase, and gapless phase. The boundaries of the FP phase are shown to be gapless and critical with central charge $c=4/5$. Even in the topological FP phase, it is also characterized by long-range string order; thus we observe a drop of string order across the phase boundary between the FP phase and the Haldane phase. This work opens a new way for finding of exotic topological phases with parafermions.

DOI
10.1103/physrevb.100.165102

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The Haldane phase represents one of the most important symmetry-protected states in modern physics. This state can be realized using spin-1 and spin\ensuremath−12\mathrm{spin}\ensuremath{-}\frac{1}{2}spin\ensuremath−21​ Heisenberg models and bosonic particles. Here we explore the emergent Haldane phase in an alternating bond Z3{\mathbb{Z}}_{3}Z3​ parafermion chain, which is different from the previous proposals from fundamental statistics and symmetries. We show that this emergent phase can also be characterized by a modified long-range string order, as well as fourfold degeneracy in the ground-state energies and entanglement spectra. This phase is protected by both the charge conjugate and parity symmetry, and the edge modes are shown to satisfy parafermionic statistics in which braiding of the two edge modes yields a 2\ensuremathπ3\frac{2\ensuremath{\pi}}{3}32\ensuremathπ​ phase. This model also supports rich phases, including a topological ferromagnetic parafermion (FP) phase, trivial paramagnetic parafermion phase, classical dimer phase, and gapless phase. The boundaries of the FP phase are shown to be gapless and critical with central charge c=4/5c=4/5c=4/5. Even in the topological FP phase, it is also characterized by long-range string order; thus we observe a drop of string order across the phase boundary between the FP phase and the Haldane phase. This work opens a new way for finding of exotic topological phases with parafermions.
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Topological orders and associated topological protected excitations with non-Abelian statistics have been widely explored in various platforms. The Z3{\mathbb{Z}}_{3}Z3​ parafermions are regarded as the most natural generation of the Majorana fermions to realize these topological orders. Here we investigate the topological phase and emergent Z2{\mathbb{Z}}_{2}Z2​ spin phases in an extended parafermion chain. This model exhibits a rich variety of phases, including not only topological ferromagnetic phase, which supports non-Abelian anyon excitation, but also spin-fluid, dimer, and chiral phases from the emergent Z2{\mathbb{Z}}_{2}Z2​ spin model. These phases correspond to various metallic and insulator phases with different orders realized by parafermions. We generalize the measurement tools in Z2{\mathbb{Z}}_{2}Z2​ spin models to fully characterize these phases and map out the corresponding phase diagram. Surprisingly, all the phase boundaries finally merge to a single supercritical (SC) point with infinite degeneracy. The phases surrounding this SC point can be regarded as phases under different types of symmetry breakings. This criteria may suggest a general principle to search these SC points in the other models. Our phase diagram also reveals the asymptotic behavior of the parafermion towards canonical fermions. Regarding the rather generality of emergent phenomena in parafermion models, this approach opens a wide range of intriguing applications in investigating the various exotic phases using parafermion models.
We study the topological classification of parafermionic chains in the presence of a modified time-reversal symmetry that satisfies T2=1{\mathcal{T}}^{2}=1T2=1. Such chains can be realized in one-dimensional structures embedded in fractionalized two-dimensional states of matter, e.g., at the edges of a fractional quantum spin Hall system, where counterpropagating modes may be gapped either by backscattering or by coupling to a superconductor. In the absence of any additional symmetries, a chain of Zm{\mathbb{Z}}_{m}Zm​ parafermions can belong to one of several distinct phases. We find that when the modified time-reversal symmetry is imposed, the classification becomes richer. If mmm is odd, each of the phases splits into two subclasses. We identify the symmetry-protected phase as a Haldane phase that carries a Kramers doublet at each end. When mmm is even, each phase splits into four subclasses. The origin of this split is in the emergent Majorana fermions associated with even values of mmm. We demonstrate the appearance of such emergent Majorana zero modes in a system where the constituent particles are either fermions or bosons.
Symmetry protected topological (SPT) phase is a generalization of topological insulator (TI). Different from the intrinsic topological phase, e.g., the fractional quantum hall (FQH) phase, SPT phase is only distinguishable from a trivial disordered phase when certain symmetry is preserved. Indeed, SPT phase has a long history in 1D, and it has been shown that the well known Haldane phase of S=1 Heisenberg chain belongs to this class. However, in higher dimensions, most of the previous studies focus on free electron systems. Until very recently, it was realized that SPT phase also exists in interacting boson/spin systems in higher dimensions. In this talk, I will show that there is an interesting duality map between intrinsic topological phase and SPT phase. I will focus on a simple model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. The duality provides us a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z2 gauge field and then show that the flux excitations have different braiding statistics from that of a usual paramagnet. In addition, these braiding statistics directly implies the existence of protected edge modes. The duality map also provides us a complementary way to derive topological terms of SPT phases, which classify different SPT phases in bosonic systems. Finally, I will discuss the possible generalization for interacting fermion/electron systems.
We study different phases of the one-dimensional bond-alternating spin-1/21/21/2 Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as σ→−σ\sigma\rightarrow-\sigmaσ→−σ), parity (permutation of the chain about a specific bond), and dihedral (π\piπ-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-1/21/21/2 model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.
Symmetry-protected topological phases have fundamentally changed our understanding of quantum matter. An archetypal example of such a quantum phase of matter is the Haldane phase, containing the spin-1 Heisenberg chain. The intrinsic quantum nature of such phases, however, often makes it challenging to study them using classical means. Here, we use trapped-ion qutrits to natively engineer spin-1 chains within the Haldane phase. Using a scalable deterministic procedure to prepare the Affleck-Kennedy-Lieb-Tasaki (AKLT) state within the Haldane phase, we study the topological features of this system on a qudit quantum processor. Notably, we verify the long-range string order of the state, despite its short-range correlations, and observe spin fractionalization of the physical spin-1 particles into effective qubits at the chain edges, a defining feature of this system. The native realization of Haldane physics on a qudit quantum processor and the scalable preparation procedures open the door to the efficient exploration of a wide range of systems beyond spin-1/2.