{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,14]],"date-time":"2026-11-14T02:43:40Z","timestamp":1794624220234,"version":"4.9.1"},"reference-count":25,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,11,19]],"date-time":"2018-11-19T00:00:00Z","timestamp":1542585600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2018,11,19]],"date-time":"2018-11-19T00:00:00Z","timestamp":1542585600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["npj Quant Mater"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>It is notoriously difficult to make quantitative theoretical predictions of the superconducting transition temperature, <jats:italic>T<\/jats:italic><jats:sub><jats:italic>c<\/jats:italic><\/jats:sub>, either from first-principles or even from a knowledge of normal state properties. Ultimately, this reflects the fact that the energy scales involved in the superconducting state are extremely small in natural units, and that <jats:italic>T<\/jats:italic><jats:sub><jats:italic>c<\/jats:italic><\/jats:sub> depends exponentially on a subtle interplay between different interactions so that small uncertainties in microscopic processes can lead to order one effects on <jats:italic>T<\/jats:italic><jats:sub><jats:italic>c<\/jats:italic><\/jats:sub>. However, in some circumstances, it may be possible to determine (approximate) bounds on <jats:italic>T<\/jats:italic><jats:sub><jats:italic>c<\/jats:italic><\/jats:sub>. Here we propose such a bound for the conventional phonon-mediated mechanism of pairing with strongly retarded interactions, i.e. in the case in which <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\hbar \\bar \\omega \\ll E_F$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow>\n                        <mml:mi>\u210f<\/mml:mi>\n                        <mml:mover>\n                          <mml:mrow>\n                            <mml:mi>\u03c9<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mo>\u00af<\/mml:mo>\n                        <\/mml:mover>\n                        <mml:mo>\u226a<\/mml:mo>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>F<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:mrow>\n                    <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\bar \\omega$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mover>\n                        <mml:mrow>\n                          <mml:mi>\u03c9<\/mml:mi>\n                        <\/mml:mrow>\n                        <mml:mo>\u00af<\/mml:mo>\n                      <\/mml:mover>\n                    <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is an appropriate characteristic phonon frequency and <jats:italic>E<\/jats:italic><jats:sub><jats:italic>F<\/jats:italic><\/jats:sub> is the Fermi energy. Specifically, drawing on both empirical results (shown in Fig. 2 below) and recent results<jats:sup>1<\/jats:sup> of determinant quantum Monte Carlo (DQMC) studies of the paradigmatic Holstein model, we propose that <jats:inline-formula><jats:alternatives><jats:tex-math>$$k_BT_c \\le A_{\\max }\\hbar \\bar \\omega$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>B<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi>T<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>c<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                        <mml:mo>\u2264<\/mml:mo>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>max<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                        <mml:mi>\u210f<\/mml:mi>\n                        <mml:mover>\n                          <mml:mrow>\n                            <mml:mi>\u03c9<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mo>\u00af<\/mml:mo>\n                        <\/mml:mover>\n                      <\/mml:mrow>\n                    <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:italic>A<\/jats:italic><jats:sub>max<\/jats:sub> is a dimensionless number of order one that we estimate to be <jats:italic>A<\/jats:italic><jats:sub>max<\/jats:sub>\u2009\u2248\u20091\/10.<\/jats:p>","DOI":"10.1038\/s41535-018-0133-0","type":"journal-article","created":{"date-parts":[[2018,11,12]],"date-time":"2018-11-12T13:44:32Z","timestamp":1542030272000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":54,"title":["A bound on the superconducting transition temperature"],"prefix":"10.1038","volume":"3","author":[{"given":"I.","family":"Esterlis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S. 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