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arXiv:2412.03793 · Scientific Reports

Low-energy physics and finite-size effects in nonuniform parafermion chains

Mohammad Mahdi Nasiri Fatmehsari, Mohammad-Sadegh Vaezi

Published Apr 3, 2025• 0 citations
arXivHTMLPDF

Abstract

The nonuniform [Formula: see text] symmetric Kitaev chain, comprising alternating topological and normal regions, hosts localized states known as edge-zero modes (EZMs) at its interfaces. These EZMs can pair to form qubits that are resilient to quantum decoherence, a feature expected to extend to higher symmetric chains, i.e., parafermion chains. However, finite-size effects may impact this ideal picture. Diagnosing these effects requires first a thorough understanding of the low-energy physics where EZMs may emerge. Previous studies have largely focused on uniform chains, with nonuniform cases inferred from these results. While recent work [Narozhny, Sci. Rep. 7, 1447 (2017)] provides an insightful analytical solution for a nonuniform [Formula: see text] chain with two topological regions separated by a normal one, its complexity limits its applicability to chains with more regions or higher symmetries. Here, we present a new approach based on decimating the highest-energy terms, facilitating the scalable analysis of [Formula: see text] chains with any number of regions. We provide analytical results for both [Formula: see text] and [Formula: see text] chains, supported by numerical findings, and identify the critical lengths necessary to preserve well-separated EZMs.

DOI
10.1038/s41598-025-94657-z

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The nonuniform [Formula: see text] symmetric Kitaev chain, comprising alternating topological and normal regions, hosts localized states known as edge-zero modes (EZMs) at its interfaces. These EZMs can pair to form qubits that are resilient to quantum decoherence, a feature expected to extend to higher symmetric chains, i.e., parafermion chains. However, finite-size effects may impact this ideal picture. Diagnosing these effects requires first a thorough understanding of the low-energy physics where EZMs may emerge. Previous studies have largely focused on uniform chains, with nonuniform cases inferred from these results. While recent work [Narozhny, Sci. Rep. 7, 1447 (2017)] provides an insightful analytical solution for a nonuniform [Formula: see text] chain with two topological regions separated by a normal one, its complexity limits its applicability to chains with more regions or higher symmetries. Here, we present a new approach based on decimating the highest-energy terms, facilitating the scalable analysis of [Formula: see text] chains with any number of regions. We provide analytical results for both [Formula: see text] and [Formula: see text] chains, supported by numerical findings, and identify the critical lengths necessary to preserve well-separated EZMs.
A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the “trivial” and “topological” regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes – the nonuniform Ising-Kitaev chain, containing two “topological” regions separated by a “trivial” region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are “delocalized” between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the “nearest-neighbor” form of the effective low-energy Hamiltonian.
Strongly interacting fermionic chains obeying a ZnZ_nZn​ symmetry can host topological phases with highly degenerate ground-state manifolds. The different ground states are connected by strongly bound fractionalized edge modes called parafermionic zero modes. While exactly preserved under time evolution at zero temperature, interactions lead to a destabilization of these modes at finite energy densities, and, given a generic Hamiltonian, obtaining the lifetime of these modes is often impossible as it requires knowledge of the full spectrum of the Hamiltonian or long time evolutions. Here, we propose a novel quantum-inspired algorithm that can be used to variationally search for these almost strong zero modes in generic parafermionic chain models. We demonstrate analytically and numerically that our ansatz faithfully reproduces the exact results and finds strongly bound parafermionic operators in the non-integrable Kitaev-Hubbard chain and chiral Potts model. Our results further allow us to verify the existence of a true strong zero mode in the symmetrical limit of the chiral Potts model for Z3Z_3Z3​-parafermions.
A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the "trivial" and "topological" regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes - the nonuniform Ising-Kitaev chain, containing two "topological" regions separated by a "trivial" region. Solving the model exactly, I show that for generic values of the model parameters the four zero modes are localized at the four interface points of the chain. In the special case where additional inversion symmetry is present, the Majorana zero modes are "delocalized" between two interface points. In both cases, the low-energy sector of the theory can be formulated in terms of the localized Majorana fermions, but the couplings between some of them are independent of their respective separations: the exact solution does not support the "nearest-neighbor" form of the effective low-energy Hamiltonian.
Abstract In the original Kitaev chain, Majorana zero modes (MZMs) at opposite edges have the same localization length. Here it is shown that a Hermitian finite-range extended Kitaev chain can instead support asymmetrically localized MZMs with different left and right localization lengths. Such asymmetry is excluded when all pairing amplitudes share a common phase, so the effect requires a genuinely extended model beyond the nearest-neighbor Kitaev chain. For the minimal next-nearest-neighbor case, the asymmetry originates from the inequivalence of the left and right boundary equations, which leads to different selections of decaying bulk solutions and yields a necessary-and-sufficient condition for its occurrence. A transfer-matrix reformulation further shows that the asymmetry is robust against sufficiently weak disorder preserving the block-triangular structure. For finite chains, the hybridization-induced energy splitting is exponentially small and controlled by the shorter localization length.
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.
Interacting fermionic chains exhibit extended regions of topological degeneracy of their ground states as a result of the presence of Majorana or parafermionic zero modes localized at the edges. In the opposite limit of infinite temperature, the corresponding nonintegrable spin chains, obtained via generalized Jordan-Wigner mapping, are known to host so-called almost strong zero modes, which are long-lived with respect to any bulk excitations. Here we study the fairly unexplored territory that bridges these two extreme cases of zero and infinite temperature. We blend two established techniques for states, the Lanczos series expansion and a tensor network ansatz, uplifting them to the level of operator algebra. This allows us to efficiently simulate large system sizes for arbitrarily long timescales and to extract the temperature-dependent decay rates. We observe that for the Kitaev-Hubbard model, the decay rate of the edge mode depends exponentially on the inverse temperature β , and on an effective energy scale Δ eff that is greater than the thermodynamic gap of the system Δ .