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arXiv:2311.07571

Universal signatures of Majorana zero modes in critical Kitaev chains

Nicolas Laflorencie

Published Nov 13, 2023• 0 citations
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Abstract

Many topological or critical aspects of the Kitaev chain are well known, with several classic results. In contrast, the study of the critical behavior of the strong Majorana zero modes (MZM) has been overlooked. Here we introduce two topological markers which, surprisingly, exhibit non-trivial signatures over the entire (1+1) Ising critical line. We first analytically compute the MZM fidelity ${\cal{F}}_{\rm MZM}$--a measure of the MZM mapping between parity sectors. It takes a universal value along the (1+1) Ising critical line, ${\cal{F}}_{\rm MZM}=\sqrt{8}/π$, independent of the energy. We also obtain an exact analytical result for the critical MZM occupation number ${\cal N}_{\rm MZM}$ which depends on the Catalan's constant ${\cal G}\approx 0.91596559$, for both the ground-state (${\cal N}_{\rm MZM}=1/2-4{\cal{G}}/π^2\approx 0.12877$) and the first excited state (${\cal N}_{\rm MZM}=1/2+(8-4{\cal{G}})/π^2\approx 0.93934$). We further compute finite-size corrections which identically vanish for the special ratio $Δ/t=\sqrt{2}-1$ between pairing and hopping in the critical Kitaev chain.

DOI
10.48550/arxiv.2311.07571

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Many topological or critical aspects of the Kitaev chain are well known, with several classic results. In contrast, the study of the critical behavior of the strong Majorana zero modes (MZM) has been overlooked. Here we introduce two topological markers which, surprisingly, exhibit non-trivial signatures over the entire (1+1) Ising critical line. We first analytically compute the MZM fidelity FMZM{\cal{F}}_{\rm MZM}FMZM​--a measure of the MZM mapping between parity sectors. It takes a universal value along the (1+1) Ising critical line, FMZM=8/π{\cal{F}}_{\rm MZM}=\sqrt{8}/πFMZM​=8​/π, independent of the energy. We also obtain an exact analytical result for the critical MZM occupation number NMZM{\cal N}_{\rm MZM}NMZM​ which depends on the Catalan's constant G≈0.91596559{\cal G}\approx 0.91596559G≈0.91596559, for both the ground-state (NMZM=1/2−4G/π2≈0.12877{\cal N}_{\rm MZM}=1/2-4{\cal{G}}/π^2\approx 0.12877NMZM​=1/2−4G/π2≈0.12877) and the first excited state (NMZM=1/2+(8−4G)/π2≈0.93934{\cal N}_{\rm MZM}=1/2+(8-4{\cal{G}})/π^2\approx 0.93934NMZM​=1/2+(8−4G)/π2≈0.93934). We further compute finite-size corrections which identically vanish for the special ratio Δ/t=2−1Δ/t=\sqrt{2}-1Δ/t=2​−1 between pairing and hopping in the critical Kitaev chain.
We investigate the fate and robustness of topological strong zero modes (SZMs) in random Ising-Majorana chains using the SZM fidelity, FSZM{\cal F}_{\rm SZM}FSZM​, as a many-body diagnostic that quantifies how accurately SZM operators map the {\it entire} spectrum between opposite parity sectors. In clean systems, FSZM=1{\cal F}_{\rm SZM}=1FSZM​=1 in the topological phase, vanishes in the trivial regime, and takes the universal value 8/π\sqrt{8}/π8​/π at the (1+1)(1+1)(1+1)D Ising critical point. Here we study how quenched disorder modifies this picture across the infinite-randomness fixed point (IRFP) governing the criticality of the random chain. In both microcanonical and canonical ensembles, SZMs persist throughout the topological phase, including the gapless Griffiths regime, with fidelities converging exponentially to unity. At the IRFP, however, the fidelity distributions become ensemble dependent: the microcanonical ensemble displays bimodal peaks at {0.5,1}\{0.5,1\}{0.5,1}, while the canonical ensemble develops a triple-peak structure at {0,0.5,1}\{0,0.5,1\}{0,0.5,1} with power-law singularities. Our results establish FSZM{\cal F}_{\rm SZM}FSZM​ as a robust probe of localization-protected topological order and uncover distinctive topological features of infinite-randomness criticality. Unlike the clean Ising CFT, where the finite critical value arises from a cancellation of power laws, the IRFP seems to exhibit an intrinsically stronger topological character. The edge-selective structure of the critical distributions may suggest a boundary manifestation of the average Kramers-Wannier duality symmetry at the IRFP.
We investigate the spectral and topological properties of a zigzag Kitaev chain constructed from two diagonally coupled one dimensional Kitaev chains with a zero and finite superconducting pairing phase difference. Using a Bogoliubov-de Gennes formulation, we analyze the energy spectrum, distribution of Majorana zero modes (MZMs), the quasi-particle dispersion, and the winding number, respectively. For a zero phase difference, the resulting energy spectrum shows topological phases with two, four MZMs, and trivial regions. The phases of gap closure determine the topological phase boundaries. In particular, for the phase difference between φ=πφ=πφ=π, the degeneracy of MZMs is partially lifted, leading to modified topological phases compared to the case φ=0φ=0φ=0. The topological and trivial phase boundaries are further confirmed by evaluating the quasi-particle dispersion and the topological invariant, namely the winding number. We show that the zigzag Kitaev chain contributes independently to the total winding number ν=1ν= 1ν=1 and 222, giving rise to distinct topological phases that support two and four MZMs. The ν=0ν= 0ν=0 gives rise to a trivial region. The energy spectrum of systems corroborates the analytical phase boundaries and reveals characteristics associated with hybridization, enabling us to obtain the complete phase diagram of the zigzag model. Our results establish the zigzag Kitaev chain as a minimal platform for engineering MZM quantum computations, with potential applications in the study of topological phases and Majorana based qubit physics.
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.
FINDING: Majorana zero modes (MZMs) in topological superconductors yield non-Abelian anyonic statistics, enabling topologically protected quantum error correction via ground-state degeneracy in the Kitaev chain. | MATH: Kitaev chain Hamiltonian H=∑j[−t(cj†cj+1+h.c.)+Δ(cjcj+1+h.c.)−μ(cj†cj−1/2)] H = \sum_j [-t(c^\dagger_j c_{j+1} + h.c.) + \Delta(c_j c_{j+1} + h.c.) - \mu(c^\dagger_j c_j - 1/2)] H=∑j​[−t(cj†​cj+1​+h.c.)+Δ(cj​cj+1​+h.c.)−μ(cj†​cj​−1/2)]. In the topological phase (∣μ∣<2t|\mu| < 2t∣μ∣<2t), unpaired Majorana operators γ1,γ2\gamma_1, \gamma_2γ1​,γ2​ satisfy γi†=γi\gamma_i^\dagger = \gamma_iγi†​=γi​, {γi,γj}=2δij\{\gamma_i,\gamma_j\} = 2\delta_{ij}{γi​,γj​}=2δij​, and the nonlocal fermion f=(γ1+iγ2)/2f = (\gamma_1 + i\gamma_2)/2f=(γ1​+iγ2​)/2 gives a two-fold ground-state degeneracy (qubit). Braiding MZMs implements the Clifford group; the fusion rules follow the Ising anyon model: σ×σ=1+ψ\sigma \times \sigma = 1 + \psiσ×σ=1+ψ, with quantum dimension dσ=2d_\sigma = \sqrt{2}dσ​=2​. | CONNECTION: The Ising anyon fusion rule σ×σ=1+ψ\sigma \times \sigma = 1 + \psiσ×σ=1+ψ maps directly to the golden-ratio-adjacent structure of the Fibonacci anyon model (though MZMs are Ising, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.
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.