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arXiv:2006.03026 · The European Physical Journal Plus

Effects of geometric frustration in Kitaev chains

Alfonso Maiellaro, Francesco Romeo, R. Citro

Published Jun 1, 2021• 12 citations
arXivHTMLPDF

Abstract

Abstract We study the topological phase transitions of a Kitaev chain frustrated by the addition of a single long-range hopping. In order to study the topological properties of the resulting legged-ring geometry (Kitaev tie model), we generalize the transfer matrix approach through which the emergence of Majorana edge modes is analyzed. We find that geometric frustration gives rise to a topological phase diagram in which non-trivial phases alternate with trivial ones at varying the range of the hopping and the chemical potential. Robustness to disorder of non-trivial phases is also proven. Moreover, geometric frustration effects persist when translational invariance is restored by considering a multiple-tie system. These findings shed light on an entire class of experimentally realizable topological systems with long-range couplings.

DOI
10.1140/epjp/s13360-021-01592-9

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Abstract We study the topological phase transitions of a Kitaev chain frustrated by the addition of a single long-range hopping. In order to study the topological properties of the resulting legged-ring geometry (Kitaev tie model), we generalize the transfer matrix approach through which the emergence of Majorana edge modes is analyzed. We find that geometric frustration gives rise to a topological phase diagram in which non-trivial phases alternate with trivial ones at varying the range of the hopping and the chemical potential. Robustness to disorder of non-trivial phases is also proven. Moreover, geometric frustration effects persist when translational invariance is restored by considering a multiple-tie system. These findings shed light on an entire class of experimentally realizable topological systems with long-range couplings.
Long-range interactions can fundamentally alter properties in gapped topological phases such as emergent massive edge modes. However, recent research has shifted attention to topological nontrivial critical points or phases, and it is natural to explore how long-range interactions influence them. In this work, we investigate the topological behavior at the quantum critical point of extended Kitaev chains with long-range interactions, which can be derived from the critical Ising model via the Jordan-Wigner transformation in the short-range limit. Specifically, we analytically find that the critical edge modes at the critical point remain stable against long-range interactions. More importantly, we observe that these critical edge modes remain massless even when long- range interactions become very strong. As a by-product, we numerically find that the critical behavior of the long-range model belongs to the free-Majorana-fermion universality class, which is entirely different from the long-range universality class in the usual long-range spin models. Our work could shed new light on the interplay between long-range interactions (frustrated) and the gapless topological phases of matter.
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.
We investigate emergent topological gapless phases in the square-lattice Kitaev model with additional hopping terms. In the presence of nearest-neighbor hopping only, the model is known to exhibit gapless phases with two topological gapless points. When the strength of the newly added next-nearest-neighbor hopping is smaller than a certain value, qualitatively the same phase diagram persists. We find that further increase of the extra hopping results in a new topological phase with four gapless points. We construct a phase diagram to clarify the regions of emergent topological gapless phases as well as topologically trivial ones in the space of the chemical potential and the next-nearest-neighbor hopping strength. We examine the evolution of the gapless phases in the energy dispersions of the bulk as the chemical potential varies. The topological properties of the gapless phases are characterized by the winding numbers of the present gapless points. We also consider the ribbon geometry to examine the corresponding topological edge states. It is revealed that Majorana-fermion edge modes exist as flat bands in topological gapless phases. We also perform the analytical calculation as to Majorana-fermion zero-energy modes and discuss its implications on the numerical results.
We consider the Kitaev chain model with finite and infinite range in the hopping and pairing parameters, looking in particular at the appearance of Majorana zero-energy modes and massive edge modes. We study the system both in the presence and in the absence of time-reversal symmetry, by means of topological invariants and exact diagonalization, disclosing very rich phase diagrams. In particular, for extended hopping and pairing terms, we can get as many Majorana modes at each end of the chain as the neighbors involved in the couplings. Finally, we generalize the transfer matrix approach useful to calculate the zero-energy Majorana states at the edges for a generic number of coupled neighbors.
We investigate the topological properties of a dimerized Kitaev chain with long-range couplings, including the intercell hopping and superconducting pairing terms. It is found that once the intercell hopping term appears, the size of the energy gap, the proportion of topological phases, and the topological phase transition can be modulated. The notable result is that it leads to a new Kitaev-like phase featured by the twofold-degenerated Majorana zero-energy edge states. Next in the presence of the intercell superconducting pairing term, this Majorana phase can be magnified efficiently. Our findings provide new ideas to realize the twofold degenerated Majorana modes based on the intercell hopping and superconducting pairing terms of the dimerized Kitaev chain.
We investigate the topological properties of a dimerized Kitaev chain with long-range interactions, including the intercell hopping and superconducting pairing terms. It is found that even only when the intercell hopping term appears, the size of the energy gap, the proportion of topological phases, and the topological phase transition can be modulated. The notable result is that they lead to a new Kitaev-like phase featured by the twofold-degenerated Majorana zero-energy edge states. Next in the presence of the intercell superconducting pairing term, this kind of Majorana phase can be magnified. This work provides new proposals to realize the twofold degenerated Majorana modes based on the intercell hopping and superconducting pairing terms of the dimerized Kitaev chain.