紧束缚超对称格点场论中的孤子
Solitons in lattice field theories via tight-binding supersymmetry
摘要：
无反射势在构造经典动力系统的精确解、各种大N场论的非微扰解以及超导理论中与Bogoliubov-de Gennes方程密切相关的孤立子解方面发挥着重要作用。这些解决方案依赖于逆散射方法，它减少了这些看似无关的问题，以确定辅助一维量子散射问题的无反射势。有几种方法可以构造这些势，其中之一是量子力学超对称性（SUSY）。在本文中，最近的实验平台的动机，我们推广这个框架，以发展相互作用场论的晶格版本中的孤子理论。我们的分析取决于使用的跟踪身份，这是连接潜在的运动方程的散射数据的关系。对于离散薛定谔算子，这样的迹恒等式是众所周知的，我们推导出一组新的恒等式的离散狄拉克算子。然后，我们使用这些在晶格Gross-Neveu和手性Gross-Neveu（Nambu-Jona-Lasinio）模型表明，晶格孤子对应于无反射的潜力与离散散射问题。这些模型等价于晶格超导体的平均场论，因而具有重要意义。为了明确地构造这些孤子，我们将超对称量子力学推广到紧束缚模型。我们表明，存在一个矩阵变换，映射一个紧束缚模型的等谱一个共享相同的结构和散射特性。相应的孤子解决方案都调制跳跃和现场的潜力，我们明确计算拓扑和非拓扑孤子解决方案，以及在上述模型的束缚态谱。
英文摘要：
Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems, non-perturbative solutions of various large-N field theories, and closely related solitonic solutions to the Bogoliubov-de Gennes equations in the theory of superconductivity. These solutions rely on the inverse scattering method, which reduces these seemingly unrelated problems to identifying reflectionless potentials of an auxiliary one-dimensional quantum scattering problem. There are several ways of constructing these potentials, one of which is quantum mechanical supersymmetry (SUSY). In this paper, motivated by recent experimental platforms, we generalize this framework to develop a theory of solitons in lattice versions of interacting field theories. Our analysis hinges on the use of trace identities, which are relations connecting the potential of an equation of motion to the scattering data. For a discrete Schrödinger operator, such trace identities are well known, and we derive a new set of identities for the discrete Dirac operator. We then use these in a lattice Gross-Neveu and chiral Gross-Neveu (Nambu-Jona-Lasinio) model to show that lattice solitons correspond to reflectionless potentials associated with the discrete scattering problem. These models are of significance as they are equivalent to a mean-field theory of a lattice superconductor. To explicitly construct these solitons, we generalize supersymmetric quantum mechanics to tight-binding models. We show that a matrix transformation exists that maps a tight-binding model to an isospectral one which shares the same structure and scattering properties. The corresponding soliton solutions have both modulated hopping and onsite potential; we explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models.
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