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arXiv:2501.11121 · Physical Review Research

Almost strong zero modes at finite temperature

Niklas Tausendpfund, Aditi Mitra, Matteo Rizzi

Published Jun 11, 2025• 8 citations
arXivHTMLPDF

Abstract

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 .

DOI
10.1103/physrevresearch.7.023245

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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 .
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.
Topological phases in superconductors are of great interest due to their potential use for topological quantum computing. They are usually characterized by quasiparticles located at the edges of the superconductor. We investigate how long-range interactions affect the properties of the well-known Kitaev chain, which is solved self-consistently with the mean-field approximation. We find that for weak interactions, the Majorana edge modes have finite energy that decay algebraically to zero with the long-range power law. Strong long-range interactions are a special case due to their slow decay. The solution of the thermodynamic limit without finite-size effects is a local minimum to this problem, but with a larger ground-state energy in contrast to the discussed solution. Our results show the difference between the self-consistent and the Kitaev approach, which has clear constraints on long-range interactions.
I will discuss how edge spins in certain one-dimensional spin chains retain memory of their initial state for very long times, even with interactions and at infinite temperature. These long coherence times do not require disorder or integrability. I will show how this is a consequence of an "almost" strong edge zero mode that almost commutes with the Hamiltonian, and explain how to compute this operator explicitly with computer-aided algebra. I will examine the effect of this almost strong zero mode on the behaviour of a wide variety of one-dimensional systems analytically, from the ground state to infinite temperature, and from finite size to the thermodynamic limit. I will also examine the effect of the almost strong zero mode on the eigenstate spectrum and the matrix elements of the edge spin, and show that despite the long coherence time of the edge spin, the almost strong zero mode does not induce degeneracies in the entire spectrum, unlike its counterpart exact strong zero mode in non-interacting systems. At special values of the couplings, resonances dramatically limit the coherence time. I will explain how these resonances are related to physical processes which flip the edge spin for no change in the energy of the system, and calculate the coupling strengths for which they appear. I will propose an experimental test of the almost strong zero mode and the long coherence time of the edge spin in a trapped ion chain which is feasible to implement with current technology. Lastly, I will explain how these almost strong edge zero modes in a system with symmetry-protected topological order can be used to form a boundary qubit with both long decoherence and dephasing times.
The topological properties of the Kitaev chain model with extended-range interactions are investigated, focusing on cases where the topological winding number is preserved. We assume that the pairing and hopping terms decay algebraically in space with exponents ααα and βββ, respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.
The stability and dynamics of almost strong zero and $\ensuremath{\pi}$ modes in weakly nonintegrable Floquet spin chains are investigated. Such modes can also be viewed as localized Majorana modes at the edge of a topological superconductor. Perturbation theory in the strength of integrability breaking interaction Jz{J}_{z}Jz​ is employed to estimate the decay rates of these modes, and compared to decay rates obtained from exact diagonalization. The structure of the perturbation theory and thus the lifetime of the modes is governed by the conservation of quasienergy modulo $2\ensuremath{\pi}/T$, where TTT is the period of the Floquet system. If there are $4n\ensuremath{-}1$ bulk excitations whose quasienergies add up to zero (or $\ensuremath{\pi}/T$ for a $\ensuremath{\pi}$ mode), one obtains a decay channel of the Majorana mode with a decay rate proportional to Jz2n{J}_{z}^{2n}Jz2n​. Thus the lifetime is sensitively controlled by the width of the single-particle Floquet bands. For regimes where the decay rates are quadratic in Jz{J}_{z}Jz​, an analytic expression for the decay rate in terms of an infinite temperature autocorrelation function of the integrable model is derived, and shown to agree well with exact diagonalization.
Krylov subspace methods quantify operator growth in quantum many-body systems through Lanczos coefficients that encode how operators spread under time evolution. Although these diagnostics were originally motivated by questions of chaos and integrability, quadratic fermionic Hamiltonians are often expected to exhibit trivial Lanczos structure. Here we show that, in the long-range Kitaev chain, Lanczos coefficients generated from local boundary operators sharply diagnose whether the lowest excitation gap is controlled by boundary-localized or bulk-extended modes. We introduce the KrylovKrylovKrylov staggeringstaggeringstaggering parameterparameterparameter for the Lanczos coefficients. In the short-range Kitaev chain with balanced hopping and pairing, we derive analytically for arbitrary system size (valid in the thermodynamic limit) and show that this quantity is exactly constant and its sign cleanly distinguishes the topological phase with Majorana edge modes from the trivial phase. Away from that limit, long-range couplings and pairing-hopping imbalance deform the simple flat structure and analytical control is lost, nevertheless, we show that the sign pattern of the diagnostic still tracks whether the lowest excitation gap is controlled by boundary modes or by bulk excitations. These results are enabled by an exact single-particle operator Lanczos algorithm, as derived in this work, which reduces the recursion from exponentially large operator space to a finite-dimensional linear problem and achieves machine precision for chains of hundreds of sites. Krylov diagnostics thus emerge as practical probes of boundary-versus-bulk low-energy physics in topological superconductors with broken U(1)U(1)U(1) symmetry and algebraically decaying couplings.