[
    {
        "id": "authors:ytn34-gaf55",
        "collection": "authors",
        "collection_id": "ytn34-gaf55",
        "cite_using_url": "https://authors.library.caltech.edu/records/ytn34-gaf55",
        "type": "article",
        "title": "The non-Hermitian skin effect with three-dimensional long-range coupling",
        "author": [
            {
                "family_name": "Ammari",
                "given_name": "Habib",
                "orcid": "0000-0001-7278-4877"
            },
            {
                "family_name": "Barandun",
                "given_name": "Silvio",
                "orcid": "0000-0003-1499-4352"
            },
            {
                "family_name": "Cao",
                "given_name": "Jinghao",
                "orcid": "0000-0002-5381-218X",
                "clpid": "Cao-Jinghao"
            },
            {
                "family_name": "Davies",
                "given_name": "Bryn",
                "orcid": "0000-0001-8108-2316"
            },
            {
                "family_name": "Hiltunen",
                "given_name": "Erik Orvehed",
                "orcid": "0000-0003-2891-9396"
            },
            {
                "family_name": "Liu",
                "given_name": "Ping",
                "orcid": "0000-0002-7857-7040"
            }
        ],
        "abstract": "<p>We study the non-Hermitian skin effect in a three-dimensional system of finitely many subwavelength resonators with an imaginary gauge potential. We introduce a discrete approximation of the eigenmodes and eigenfrequencies of the system in terms of the eigenvectors and eigenvalues of the so-called gauge capacitance matrix &nbsp;<span class=\"jsx-3242141698\"><span class=\"katex\"><span class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathcal\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">N</span></span></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">&gamma;</span></span></span></span><span class=\"vlist-s\"></span></span></span></span></span></span></span></span></span>, which is a dense matrix due to long-range interactions in the system. Based on translational invariance of this matrix and the decay of its off-diagonal entries, we prove the condensation of the eigenmodes at one edge of the structure by showing the exponential decay of its pseudo-eigenvectors. In particular, we consider a range k -approximation to keep the long-range interaction to a certain extent, thus obtaining a&nbsp; k -banded gauge capacitance matrix <span class=\"jsx-3242141698\"><span class=\"katex\"><span class=\"katex-html\"><span class=\"base\"><span class=\"mord\"><span class=\"mord mathcal\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">N</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\">k</span></span></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">&gamma;</span></span></span></span><span class=\"vlist-s\"></span></span></span></span></span></span></span></span></span>. Using techniques for Toeplitz matrices and operators, we establish the exponential decay of the pseudo-eigenvectors of \\<span class=\"mord mathcal\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">N</span><span class=\"mpunct mtight\">,</span><span class=\"mord mathnormal mtight\">k</span></span></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">&gamma;</span></span></span></span><span class=\"vlist-s\"></span></span></span></span> and demonstrate that they approximate those of the gauge capacitance matrix <span class=\"mord mathcal\">C</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\"><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">N</span></span></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">&gamma;</span></span></span></span><span class=\"vlist-s\"></span></span></span></span>&nbsp;well. Our results are numerically verified. In particular, we show that long-range interactions affect only the first eigenmodes in the system. As a result, a tridiagonal approximation of the gauge capacitance matrix, similar to the nearest-neighbour approximation in quantum mechanics, provides a good approximation for the higher modes.</p>",
        "doi": "10.4171/jems/1685",
        "issn": "1435-9855",
        "publisher": "European Mathematical Society - EMS - Publishing House GmbH",
        "publication": "Journal of the European Mathematical Society",
        "publication_date": "2025-08-04"
    }
]