随着纳米科技的飞速发展，人类对器件的微型化需求日益增长。二维材料因其独特的力学、电学和热学特性，成为实现纳米器件的理想选择。从石墨烯到过渡金属硫族化合物，二维材料展现出优异的物理性能和广阔的应用前景。在这一研究背景下，深入理解二维材料的热力学行为变得尤为重要。在二维材料的研究中，Kosterlitz-Thouless (KT) 相变是一个关键的理论概念。这一相变机制由Kosterlitz和Thouless在上世纪70年代提出，揭示了二维体系中拓扑缺陷对相变行为的重要影响。人们在超导薄膜中观测到这种由无序到“准长程序”的特殊相变。这种拓扑相变，是二维体系中涡旋由束缚态到游离态转变所导致的，被称作KT相变。KT相变不仅存在于超导薄膜中，在二维晶体材料中也表现出独特的相变特征，表现为旋错对的解耦现象。当达到临界温度时，出现独立旋错，系统由短程有序变为无序。旋错作为晶体中的拓扑缺陷，其行为直接影响材料的力学和热学性质。探索这些系统中的KT相变行为成为了一个重要的研究方向。本文拟针对二维弯曲系统，探讨弯曲系统中KT相变的可能性。

碳纳米管具有独特的电子结构和优异的物理性质。碳纳米管中会产生拓扑缺陷，这些缺陷显著影响其力学、电学和热学等性质。显然拓扑缺陷相互作用的研究对碳纳米管的应用具有重要意义，因此本文拟研究柱面系统中缺陷的缺陷能。柱面系统中的格林函数不收敛是该研究的难点。本文探究了一种寻找共形变换的方法，找到了合适的共形变换解决了格林函数发散问题，给出了柱面系统的旋错缺陷能，并探寻了纳米管中旋错的相互作用特性。

至今为止，碳纳米管的熔化温度在实验上仍无法准确测定，且相关数值模拟有较大差异。考虑到KT相变是一种二维拓扑相变，其核心机制可以看做是正反旋错对的解耦。因此，碳纳米管作为典型的柱面系统是研究弯曲二维系统中KT相变的理想平台，可以应用柱面系统KT相变来分析碳纳米管的熔化温度。所以本文研究了柱面系统中的KT相变，进而应用建立的柱面系统KT相变理论，对碳纳米管的相变行为进行了深入研究。通过计算含孤立旋错的柱面的自由能，得到了柱面发生KT相变的相变温度，明确了相变温度与柱面半径的依赖关系。通过理论计算，本研究成功预测了碳纳米管的KT相变温度，从而在理论上求出了实验上难以测定的碳纳米管熔化温度。发现碳纳米管的熔化温度随管径增大而增大，当管径趋于无穷大时趋近于石墨烯的熔化温度。再者，本文从理论上预测了手征性对碳纳米管熔化温度没有影响。最后，通过数值模拟，验证了理论计算的正确性。

With the rapid development of nanotechnology, the demand for device miniaturization continues to grow. Two-dimensional (2D) materials have emerged as ideal candidates for nanodevices due to their unique mechanical, electrical, and thermal properties. From graphene to transition metal dichalcogenides, these materials demonstrate exceptional physical characteristics and broad application prospects. Against this research backdrop, gaining a profound understanding of the thermodynamic behavior of 2D materials has become particularly crucial. In the study of 2D materials, the Kosterlitz-Thouless (KT) transition represents a pivotal theoretical concept. First proposed by Kosterlitz and Thouless in the 1970s, this phase transition mechanism reveals the significant influence of topological defects on phase behavior in two-dimensional systems. Experimental observations of this special disorder-to-"quasi-long-range order" transition have been made in superconducting thin films. This topological phase transition, caused by the transition of vortices from bound states to free states in 2D systems, is termed the KT phase transition. The KT transition not only exists in superconducting films but also manifests unique phase transition characteristics in 2D crystalline materials, particularly through the decoupling of dislocation pairs. When reaching the critical temperature, independent dislocations emerge, transforming the system from short-range order to disorder. As topological defects in crystals, the behavior of dislocations directly affects the mechanical and thermal properties of materials. Investigating KT transition phenomena in these systems has become an important research direction. This paper aims to explore the possibility of KT phase transitions in curved two-dimensional systems, specifically focusing on geometrically deformed configurations.

Carbon nanotubes exhibit unique electronic structures and exceptional physical properties. Topological defects can arise in carbon nanotubes, significantly influencing their mechanical, electrical, and thermal behaviors. Research on the interactions of these topological defects is evidently critical for advancing carbon nanotube applications. This study focuses on investigating the defect energy of dislocations in cylindrical systems. A key challenge lies in the divergence of Green’s functions within such geometries. To address this issue, the work explores a method to identify suitable conformal transformations, resolving the divergence problem of Green’s functions. The dislocation energy in cylindrical systems is analytically derived, and the interaction characteristics of dislocations in nanotubes are systematically explored.

To date, the experimental determination of the melting temperature of carbon nanotubes remains challenging, and numerical simulations show significant discrepancies. Considering that the KT transition is a two-dimensional topological phase transition whose core mechanism involves the decoupling of oppositely charged dislocation pairs, carbon nanotubes as prototypical cylindrical systems serve as an ideal platform for studying KT transitions in curved two-dimensional geometries. This work leverages the framework of KT transitions in cylindrical systems to analyze the melting temperature of carbon nanotubes. The study investigates KT transitions in cylindrical systems and applies the developed theoretical framework to conduct an in-depth exploration of phase transition behavior in carbon nanotubes. By calculating the free energy of cylindrical systems containing isolated dislocations, the phase transition temperature for KT behavior in these systems is derived, clarifying its dependence on the cylinder radius. Through theoretical modeling, this research successfully predicts the KT transition temperature of carbon nanotubes, thereby theoretically estimating their experimentally elusive melting temperature. The results reveal that the melting temperature of carbon nanotubes increases with tube diameter and asymptotically approaches that of graphene as the diameter tends to infinity. Furthermore, the study theoretically predicts that chirality has no discernible impact on the melting temperature of carbon nanotubes. Finally, numerical simulations are employed to validate the accuracy of the theoretical calculations.

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