Solitons in lattice field theories via tight-binding supersymmetry
Abstract Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems (such as the Korteweg-de Vries equation), non-perturbative solutions of various large-N field theories (such as the Gross-Neveu model), and closely related solitonic solutions to...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-07-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP07(2021)055 |
| _version_ | 1818601096922267648 |
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| author | Shankar Balasubramanian Abu Patoary Victor Galitski |
| author_facet | Shankar Balasubramanian Abu Patoary Victor Galitski |
| author_sort | Shankar Balasubramanian |
| collection | DOAJ |
| description | Abstract Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems (such as the Korteweg-de Vries equation), non-perturbative solutions of various large-N field theories (such as the Gross-Neveu model), 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 lattice solitons. We first briefly review the classical inverse scattering method in the continuum limit, focusing on the Korteweg-de Vries (KdV) equation and SU(N) Gross-Neveu model in the large N limit. We then generalize this methodology to 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 had been known as far back as Toda; however, we derive a new set of identities for the discrete Dirac operator. We then use these identities 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, the former of which has no analogue in the continuum limit. We explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models. |
| first_indexed | 2024-12-16T12:45:57Z |
| format | Article |
| id | doaj.art-e3c6e2bbe33d44629ee9375a26194c16 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-16T12:45:57Z |
| publishDate | 2021-07-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-e3c6e2bbe33d44629ee9375a26194c162022-12-21T22:31:17ZengSpringerOpenJournal of High Energy Physics1029-84792021-07-012021714710.1007/JHEP07(2021)055Solitons in lattice field theories via tight-binding supersymmetryShankar Balasubramanian0Abu Patoary1Victor Galitski2Center for Theoretical Physics, Massachusetts Institute of TechnologyJoint Quantum Institute, University of MarylandJoint Quantum Institute, University of MarylandAbstract Reflectionless potentials play an important role in constructing exact solutions to classical dynamical systems (such as the Korteweg-de Vries equation), non-perturbative solutions of various large-N field theories (such as the Gross-Neveu model), 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 lattice solitons. We first briefly review the classical inverse scattering method in the continuum limit, focusing on the Korteweg-de Vries (KdV) equation and SU(N) Gross-Neveu model in the large N limit. We then generalize this methodology to 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 had been known as far back as Toda; however, we derive a new set of identities for the discrete Dirac operator. We then use these identities 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, the former of which has no analogue in the continuum limit. We explicitly compute both topological and non-topological soliton solutions as well as bound state spectra in the aforementioned models.https://doi.org/10.1007/JHEP07(2021)055Lattice Integrable ModelsSolitons Monopoles and InstantonsLattice Quantum Field Theory |
| spellingShingle | Shankar Balasubramanian Abu Patoary Victor Galitski Solitons in lattice field theories via tight-binding supersymmetry Journal of High Energy Physics Lattice Integrable Models Solitons Monopoles and Instantons Lattice Quantum Field Theory |
| title | Solitons in lattice field theories via tight-binding supersymmetry |
| title_full | Solitons in lattice field theories via tight-binding supersymmetry |
| title_fullStr | Solitons in lattice field theories via tight-binding supersymmetry |
| title_full_unstemmed | Solitons in lattice field theories via tight-binding supersymmetry |
| title_short | Solitons in lattice field theories via tight-binding supersymmetry |
| title_sort | solitons in lattice field theories via tight binding supersymmetry |
| topic | Lattice Integrable Models Solitons Monopoles and Instantons Lattice Quantum Field Theory |
| url | https://doi.org/10.1007/JHEP07(2021)055 |
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