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arXiv:1906.07090 · Physical Review Research

Topological spin excitations in Harper-Heisenberg spin chains

J. L. Lado, Oded Zilberberg

Published Oct 4, 2019• 35 citations
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Abstract

Topological phases of matter can appear through geometrical or spatial frustration, leading to spectral gaps with topological in-gap boundary modes. This paper explores signatures of topological modes in the excitation spectra of a many-body system. The full excitation spectrum of spin chains is explored numerically using a combination of tensor network algorithms with the Kernel Polynomial method.

DOI
10.1103/physrevresearch.1.033009

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Topological phases of matter can appear through geometrical or spatial frustration, leading to spectral gaps with topological in-gap boundary modes. This paper explores signatures of topological modes in the excitation spectra of a many-body system. The full excitation spectrum of spin chains is explored numerically using a combination of tensor network algorithms with the Kernel Polynomial method.
Spectral functions of non-Hermitian Hamiltonians can reveal the existence of topologically nontrivial line gaps and the associated topological edge modes. However, the computation of spectral functions in a non-Hermitian many-body system remains an open challenge. Here, we put forward a numerical approach to compute spectral functions of a non-Hermitian many-body Hamiltonian based on the kernel polynomial method and the matrix-product state formalism. We show that the local spectral functions computed with our algorithm reveal topological spin excitations in a non-Hermitian spin model, faithfully reflecting the nontrivial line gap topology in a many-body model. We further show that the algorithm works in the presence of the non-Hermitian skin effect. Our method offers an efficient way to compute local spectral functions in non-Hermitian many-body systems with tensor networks, allowing us to characterize line gap topology in non-Hermitian quantum many-body models.
Many-body spin systems represent a paradigmatic platform for the realization of emergent states of matter in a strongly interacting regime. Spin models are commonly studied in one-dimensional periodic chains, whose lattice constant is on the order of the interatomic distance. However, in cold atomic setups or functionalized twisted van der Waals heterostructures, long-range modulations of the spin physics can be engineered. Here we show that such superlattice modulations in a many-body spin Hamiltonian can give rise to observable topological boundary modes in the excitation spectrum of the spin chain. In the case of an XY spin-1/2 chain, these boundary modes stem from a mathematical correspondence with the chiral edge modes of a two-dimensional quantum Hall state. Our results show that the addition of many-body interactions does not close some of the topological gaps in the excitation spectrum, and the topological boundary modes visibly persist in the isotropic Heisenberg limit. These observations carry through when the spin moment is increased and a large-spin limit of the phenomenon is established. Our results show that such spin superlattices provide a promising route to observe many-body topological boundary effects in cold atomic setups and functionalized twisted van der Waals materials.
Parafermions are elusive fractional excitations potentially emerging in fractional quantum Hallsuperconductor junctions and represent one of the major milestones in fractional quantum matter.However, generic models of parafermions are not analytically solvable, and understanding their topological modes is a bigger challenge than conventional Majorana modes.Here, by using a combination of tensor network and kernel polynomial techniques, we demonstrate the emergence of zero modes and finite energy excitations in many-body parafermion chains.We show the appearance of zero-energy modes in the many-body spectral function at the edge of a topological parafermion chain, their relation with the topological degeneracy of the system, and we compare their physics with the Majorana bound states of topological superconductors.We demonstrate the robustness of parafermion topological modes with respect to a variety of perturbations, and we show how weakly coupled parafermion chains give rise to in-gap excitations.Our results exemplify the versatility of tensor network methods for studying dynamical excitations of interacting parafermion chains and highlight the robustness of topological modes in parafermion models.
The topological classification of a series of frustration-induced spin-gap phases in the spin-1/2 ferromagnetic--antiferromagnetic alternating Heisenberg chain with next-nearest-neighbour interaction reported in J. Phys. Soc. Jpn. 82, 064703 (2013) is confirmed using two kinds of entanglement spectra defined by different divisions of the whole chain. For the numerical calculation, the iDMRG method is used. The results are consistent with the valence bond solid picture proposed in the previous paper.
Many magnetic materials are predicted to exhibit bosonic topological edge modes in their excitation spectra, because of the nontrivial topology of their magnon, triplon, or other quasiparticle band structures. However, there is a discrepancy between theory prediction and experimental observation, which suggests some underlying mechanism that intrinsically suppresses the expected experimental signatures, like the thermal Hall current. Many-body interactions that are not accounted for in the noninteracting quasiparticle picture are most often identified as the reason for the absence of the topological edge modes. Here we report persistent bosonic edge modes at the boundaries of a ladder quantum paramagnet with gapped triplon excitations in the presence of the full many-body interaction. We use tensor network methods to resolve topological edge modes in the time-dependent spin-spin correlations and the dynamical structure factor, which is directly accessible experimentally. We further show that signatures of these edge modes survive even when the noninteracting quasiparticle theory breaks down; we discuss the topological phase diagram of the model, demonstrate the fractionalization of its low-lying excitations, and propose potential material candidates.
Competing interactions in quantum magnets lead to a variety of emergent states, including ordered phases, nematic magnets, and quantum spin liquids. Among them, topological quantum magnets represent a promising platform to create topological excitations protected by the bulk many-body excitation gap. Here we establish the phase diagram of a breathing frustrated antiferromagnetic J 1 − J 2 -Heisenberg model, featuring both ordered states and a higher-order topological quantum magnet state. Using variational many-body methods based on neural network quantum states and tensor networks, we determine the existence of a first-order phase transition between stripe order and the topological quantum magnet and the second-order phase transition between the Néel order and quantum magnet phase, further corroborated by calculations of the many-body gap. Using an auxiliary fermion parton formalism, we show the emergence of topological spinon corner modes stemming from the breathing order parameter of the parent Heisenberg model. Our results establish the breathing frustrated square lattice Heisenberg model as a paradigmatic system to engineer topological quantum magnetism, as recently realized in Ti lattices at MgO.