We study the parity-and time-reversal (PT ) symmetric non-Hermitian Su-Schrieffer-Heeger (SSH) model with two conjugated imaginary potentials ±iγ at two end sites. The SSH model is known as one of the simplest two-band topological models which has topologically trivial and nontrivial phases. We find that the non-Hermitian terms can lead to different effects on the properties of the eigenvalues spectrum in topologically trivial and nontrivial phases. In the topologically trivial phase, the system undergos an abrupt transition from unbroken PT -symmetry region to spontaneously broken PT -symmetry region at a certain γc, and a second transition occurs at another transition point γ c ′ when further increasing the strength of the imaginary potential γ. But in the topologically nontrivial phase, the zero-mode edge states become unstable for arbitrary nonzero γ and the PT -symmetry of the system is spontaneously broken, which is characterized by the emergence of a pair of conjugated imaginary modes.

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High-density genetic map provides an essential framework for accurate and efficient genome assembly and QTL fine mapping. Construction of high-density genetic maps appears more feasible since the advent of next-generation sequencing (NGS), which eases SNP discovery and high-throughput genotyping of large population. In this research, a high-density genetic map of cucumber (Cucumis sativus L.) was successfully constructed across an F2 population by a recently developed Specific Length Amplified Fragment sequencing (SLAF-seq) method. In total, 18.69 GB of data containing 93,460,000 paired-end reads were obtained after preprocessing. The average sequencing depth was 44.92 in the D8 (female parent), 42.16 in the Jin5-508 (male parent), and 5.01 in each progeny. 79,092 high-quality SLAFs were detected, of which 6784 SLAFs were polymorphic, and 1892 of the polymorphic markers met the requirements for constructing genetic map. The genetic map spanned 845.87 cm with an average genetic distance of 0.45 cm. It is a reliable linkage map for fine mapping and molecular breeding of cucumber for its high marker density and well-ordered markers.

Cucumber is an agriculturally and economically important vegetable crop worldwide. Fruit flesh thickness is an important trait for cucumber and also a central determinant of yield, yet little is known about the underlying mechanism of this trait. In this study, bulked segregant analysis (BSA) combined with specific length amplified fragment sequencing (SLAF-seq) was applied to finely map the gene that underlies fruit flesh thickness in cucumber. A 0.19-Mb-long quantitative trait locus on chromosome 2 controlling fruit flesh thickness (QTL fft2.1) was identified and further confirmed by simple sequence repeat (SSR) marker-based classical QTL mapping in 138 F2 individuals. Gene prediction of this 0.19-Mb region identified 20 genes. Quantitative RT-PCR revealed higher expression levels of Csa2 M058670.1 (SET domain protein-lysine methyltransferase) in D8 (thick fruit flesh parent) compared with that in XUE1 (thin fruit flesh parent) during fruit development. Sequence alignment analysis of Csa2M058670.1 from thick and thin fruit flesh cucumber lines revealed a 4-bp deletion mutation in the promoter region of this candidate gene, which may result in the loss of Csa2M058670.1 activation in thin fruit flesh lines. The data presented herein suggest that Csa2M058670.1 is a possible candidate gene for controlling flesh thickness in cucumber.

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