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We first employ a phenomenological field theory to qualitatively determine the phase diagram. Furthermore, we generalize a previously established alternating pattern of superconductor and tunneling regions, coupled to fractional quantum Hall edge states, to map the problem onto a topological Josephson junction chain involving lattice parafermions. Using density matrix renormalization group simulations, we establish a phase diagram composed of Mott insulating phases and two different Luttinger liquids whose fundamental excitations carry charges 2e and 2\n                    <jats:italic>e<\/jats:italic>\n                    \/3, respectively. In agreement with analytical considerations using conformal field theory, we numerically find transitions of Berezinskii\u2013Kosterlitz\u2013Thouless (BKT) type as well as a continuous\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${{\\mathbb{Z}}}_{3}\\times U(1)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow>\n                                <mml:mi>Z<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:mn>3<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>U<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mrow>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    second-order phase transition characterized by central charge\n                    <jats:italic>c<\/jats:italic>\n                    \u2009=\u20099\/5. We finally extract information about a possible ground state degeneracy and comment on the stability of parafermionic edge states in the presence of fluctuations. These theoretical foundations can be expected to be of practical importance for gate-defined FCI-SC heterostructures in moir\u00e9 materials, in which broad superconducting transitions indicative of strong order parameter fluctuations were observed.\n                  <\/jats:p>","DOI":"10.1038\/s41535-026-00897-1","type":"journal-article","created":{"date-parts":[[2026,5,28]],"date-time":"2026-05-28T06:36:21Z","timestamp":1779950181000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Phases of quasi-one-dimensional fractional quantum anomalous Hall -- superconductor heterostructures"],"prefix":"10.1038","volume":"11","author":[{"given":"Steffen","family":"Bollmann","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andreas","family":"Haller","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jukka I.","family":"V\u00e4yrynen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas L.","family":"Schmidt","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Elio J.","family":"K\u00f6nig","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,28]]},"reference":[{"key":"897_CR1","doi-asserted-by":"publisher","first-page":"1083","DOI":"10.1103\/RevModPhys.80.1083","volume":"80","author":"C Nayak","year":"2008","unstructured":"Nayak, C., Simon, S. 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