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arXiv:2006.00092 · Physical review. B./Physical review. B

Edge states of the long-range Kitaev chain: An analytical study

Simon B. Jäger, Luca Dell’Anna, Giovanna Morigi

Published Jul 27, 2020• 40 citations
arXivHTMLPDF

Abstract

We analyze the properties of the edge states of the one-dimensional Kitaev model with long-range anisotropic pairing and tunneling. Tunneling and pairing are assumed to decay algebraically with exponents $\ensuremath{\alpha}$ and $\ensuremath{\beta}$, respectively, and $\ensuremath{\alpha},\ensuremath{\beta}>1$. We determine analytically the decay of the edge modes. We show that the decay is exponential for $\ensuremath{\alpha}=\ensuremath{\beta}$ and when the coefficients scaling tunneling and pairing terms are equal. Otherwise, the decay is exponential at sufficiently short distances and then algebraic at the asymptotics. We show that the exponent of the algebraic tail is determined by the smallest exponent between $\ensuremath{\alpha}$ and $\ensuremath{\beta}$. Our predictions are in agreement with numerical results found by exact diagonalization and reported in the literature.

DOI
10.1103/physrevb.102.035152

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We analyze the properties of the edge states of the one-dimensional Kitaev model with long-range anisotropic pairing and tunneling. Tunneling and pairing are assumed to decay algebraically with exponents \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathβ\ensuremath{\beta}\ensuremathβ, respectively, and \ensuremathα,\ensuremathβ>1\ensuremath{\alpha},\ensuremath{\beta}>1\ensuremathα,\ensuremathβ>1. We determine analytically the decay of the edge modes. We show that the decay is exponential for \ensuremathα=\ensuremathβ\ensuremath{\alpha}=\ensuremath{\beta}\ensuremathα=\ensuremathβ and when the coefficients scaling tunneling and pairing terms are equal. Otherwise, the decay is exponential at sufficiently short distances and then algebraic at the asymptotics. We show that the exponent of the algebraic tail is determined by the smallest exponent between \ensuremathα\ensuremath{\alpha}\ensuremathα and \ensuremathβ\ensuremath{\beta}\ensuremathβ. Our predictions are in agreement with numerical results found by exact diagonalization and reported in the literature.
We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance r as 1 / r α , as well as a related class of fermionic Hamiltonians that generalize the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents α , based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for α ≲ 1 , while connected correlation functions can decay with a hybrid exponential and power-law behavior, with a power that is α -dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every α . Along the critical lines, for all models breaking of conformal symmetry is argued at low enough α . For the fermionic models we show that the edge modes, massless for α ≳ 1 , can acquire a mass for α < 1 . The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.
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We analyze the behavior of edge states in long-range (LR) interacting systems. In terms of lattice model Hamiltonian with the LR coupling, we determine analytically the condition of existence of edge states within the transfer matrix method (TMM). The expressions we obtain are general and hold for any choice of the LR hopping. The reason why edge states can appear is the transfer matrix in the bulk different from that in the boundary layers. Our predictions are in good agreement with numerical results by exact diagonalization. Our result is helpful in solving novel edge states in one- and two-dimensional (2D) superconductors with LR hopping and pairing.
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