* (58) Garcia M., Herrero A., Najarro F., Lennon D.J., Urbaneja M., 2014, ApJ, 788, 64
  * (59) Gilkis A., Vink J.S., Eldridge J.J., Tout C.A. 2019. MNRAS. 486:4451-4462
  * (60) Gilkis A., Shenar T., Ramachandran V., et al. 2021. MNRAS. 503:1884-1896
  * (61) Glatzel W., Kiriakidis M., Fricke K.J. 1993. MNRAS. 262:L7-L11
  * (62) Gormaz-Matamala A.C., Curé M., Hillier D.J., et al. 2021. arXiv e-prints
  * (63) Götberg Y., de Mink S.E., McQuinn M., et al. 2020. A&A. 634:134
  * (64) Gräfener G., Koesterke L., Hamann W.-R. 2002. A&A. 387:244-257
  * (65) Gräfener G., Hamann W.-R. 2005. A&A. 432:633-645
  * (66) Gräfener G., Hamann W.-R. 2008. A&A. 482:945-960
  * (67) Gräfener G., Owocki S.P., Vink J.S. 2012. A&A. 538:40
  * (68) Gräfener G., Vink J.S. 2015. A&A. 578:L2
  * (69) Gräfener G., Vink J.S. 2016. MNRAS. 455:112-126
  * (70) Gräfener G., Owocki S.P., Grassitelli L., Langer N. 2017. A&A. 608:34
  * (71) Grassitelli L., Langer N., Grin N.J., et al. 2018. A&A. 614:86
  * (72) Grassitelli L., Langer N., Mackey J., et al. 2021. A&A. 647:99
  * (73) Groh J.H. 2014. A&A. 572:L11
  * (74) Groh J.H., Hillier D.J., Damineli A., et al. 2009. ApJ. 698:1698-1720
  * (75) Groh J.H., Vink J.S. 2011. A&A. 531: L10
  * (76) Groh J.H., Meynet G., Ekström S., Georgy C. 2014. A&A. 564:30
  * (77) Gvaramadze V.V., Langer N., Mackey J. 2012. MNRAS. 427:L50-L54
  * (78) Hainich R., Rühling U., Todt H., et al. 2014. A&A. 565:27
  * (79) Hainich R., Pasemann D., Todt H., et al. 2015. A&A. 581:21
  * (80) ainich R., Ramachandran V., Shenar T. et al. 2019. A&A. 621:85
  * (81) Hamann W.-R., Koesterke L., Wessolowski U. 1995. A&A. 299:151
  * (82) Hamann W.-R., Feldmeier A., Oskinova L.M. 2008. Clumping in Hot-Star Winds
  * (83) Hamann W.-R., Gräfener G., Liermann A., et al. 2019. A&A. 625:57
  * (84) Harries T.J. 2000. MNRAS. 315:722-734
  * (85) Harries T.J., Howarth I.D., Evans C.J. 2002. MNRAS. 337:341-355
  * (86) Hawcroft C., Sana H., Mahy L., et al. 2021. arXiv e-prints.:arXiv:2108.08340
  * (87) Heger A., Jeannin L., Langer N., Baraffe I. 1997. A&A. 327:224-230
  * (88) Henney W. J., Arthur S. J. 2019. MNRAS. 486:3423-3433
  * (89) Hennicker L., Puls J., Kee N. D., Sundqvist J. O. 2018. A&A. 616:140
  * (90) Higgins E. R., Vink J. S. 2019. A&A. 622:50
  * (91) Hillier D.J. 1991. A&A. 247:455
  * (92) Hillier D.J. 2020. Galaxies. 8:60
  * (93) Hillier D.J., Miller D.L. 1998. ApJ. 496:407
  * (94) Hirschi R. 2015. Very Massive Stars in the Local Universe. 412:157
  * (95) Humphreys R.M., Davidson K. 1979. ApJ. 232:409-420
  * (96) Humphreys R.M., Davidson K. 1994. PASP. 106:1025
  * (97) Iglesias C.A., Rogers F.J. 1996. ApJ. 464:943
  * (98) Ignace R., Cassinelli J.P., Bjorkman J.E. 1996. ApJ. 459:671
  * (99) Inserra C., Sullivan M., Angus C.R., et al. 2021. MNRAS. 504:2535-2549
  * (100) Ishii M., Ueno M., Kato M. 1999. PASJ. 51:417-424
  * (101) de Jager C., Nieuwenhuijzen H., van der Hucht K.A. 1988. A&AS. 72:259-289
  * (102) Jiang Y.-F., Cantiello M., Bildsten, L., Quataert E., Blaes O. 2015. ApJ. 813:74
  * (103) Johnston C. 2021. ArXiv. 2107.09075
  * (104) Justham S., Podsiadlowski P., Vink J.S. 2014. ApJ. 796:121
  * (105) Kalari V.M., Vink J.S., Dufton P.L., Fraser M. 2018. A&A. 618:17
  * (106) Kasen D., Bildsten L. 2010. ApJ. 717:245-249
  * (107) Kee N.D., Sundqvist J.O., Decin L., de Koter A., Sana H. 2021. A&A. 646:180
  * (108) Keszthelyi Z., Puls J., Wade, G.A. 2017. A&A. 598:4
  * (109) Klein R.I., Castor J.I. 1978. ApJ. 220:902-923
  * (110) Kobulnicky H.A., Kiminki D.C., Lundquist M.J., et al. 2014. ApJS. 213:34
  * (111) Kobulnicky H.A., Chick W.T., Povich M.S. 2019. AJ. 158:73
  * (112) Köhler K., Langer N., de Koter A., et al. 2015. A&A. 573:71
  * (113) Kotak R., Vink J.S. 2006. A&A. 460:L5-L8
  * (114) de Koter A., Heap S.R., Hubeny I. 1997. ApJ. 477:792
  * (115) Koumpia E., Oudmaijer R.D., Graham V., et al. 2020. A&A. 635:183
  * (116) Krtička J., Kubát J. 2009. MNRAS. 394:2065-2079
  * (117) Krtička J., Kubát J. 2017. A&A. 606:31
  * (118) Krtička J., Feldmeier A. 2021. A&A. 648:79
  * (119) Krtička J., Kubát J., Krtičková I. 2021. A&A. 647:28
  * (120) Kudritzki R.P. 2002. ApJ. 577: 389
  * (121) Kudritzki R.P., Pauldrach A., Puls J. 1987. A&A. 173:293-298
  * (122) Kudritzki R.P., Lennon D.J., Puls J. 1995. Science with the VLT. 246
  * (123) Lagae C., Driessen F. A., Hennicker L., Kee N. D., Sundqvist J. O. 2021. A&A. 648:94
  * (124) Lamers H.J.G.L.M., Morton D.C. 1976. ApJS. 32:715-736
  * (125) Lamers H.J.G.L.M., Leitherer C. 1993. ApJ. 412:771
  * (126) Lamers H.J.G.L.M., Snow T.P., Lindholm D.M. 1995. ApJ. 455:269
  * (127) Langer N. 1998. A&A. 329:551-558
  * (128) Langer N. 2012. ARA&A. 50:107
  * (129) Langer N., Norman C. A., de Koter A., et al. 2007. A&A. 475:L19-L23
  * (130) Leitherer C. 2020. Galaxies. 8:13
  * (131) Levesque E.M., Massey P., Olsen K.A.G. et al. 2005. ApJ. 628:973-985
  * (132) Li Q.-K., Cassinelli J.P., Brown J.C., Ignace R. 2009. Research in A&A. 9:558-576
  * (133) Limongi M., Chieffi A. 2018. ApJS. 237:13
  * (134) Lobel A., Dupree A. K., Stefanik R. P., et al. 2003. ApJ. 583:923-954
  * (135) Lucy L.B. 2010. A&A. 512:33
  * (136) Lucy L.B., Abbott D.C. 1993. ApJ. 405:738
  * (137) Lucy L.B., Solomon P.M. 1970. ApJ. 159:879
  * (138) Maeder A., Meynet G. 2000. ARA&A. 38:143-190
  * (139) Macfarlane J.J., Cassinelli J.P., Welsh B.Y., et al. 1991. ApJ. 380:564
  * (140) Mahy L., Damerdji Y., Gosset E., et al. 2017. A&A. 607:96
  * (141) Markova N., Puls J. 2008. A&A. 478:823
  * (142) Massey P., McNeill R.T., Olsen K. A. G., et al. 2007. AJ. 134:2474-2503
  * (143) van Marle A.J., Owocki S.P., Shaviv N.J. 2009. MNRAS. 394:595-604
  * (144) Martins F. 2015. Very Massive Stars in the Local Universe. 412:9
  * (145) Martins F., Hillier D. J., Paumard T., et al. 2008. A&A. 478:219-233
  * (146) Martins F., Schaerer D., Hillier D.J., et al. 2005. A&A. 441:735
  * (147) Mauerhan J.C., Smith N., Filippenko A.V. et al. 2013. MNRAS. 430:1801-1810
  * (148) Meynet G., Maeder A., Schaller G., Schaerer D., Charbonnel C. 1994. A&AS. 103:97-105
  * (149) Moffat A.F.J., Robert C., 1994. ApJ. 421:310
  * (150) Mokiem M.R., de Koter A., Vink J. S., et al. 2007. A&A. 473:603-614
  * (151) Moriya T.J., Mazzali P.A., Pian E. 2020. MNRAS. 491:1384-1390
  * (152) Morford J.C., Fenech D.M., Prinja R.K., et al. 2020. A&A. 637:64
  * (153) Morton D.C. 1967. ApJ. 150:535
  * (154) Muijres L.E., de Koter A., Vink J.S., et al., 2011. A&A. 526:32
  * (155) Muijres L.E., Vink J.S., de Koter A., Müller P.E., Langer N. 2012. A&A. 537:37
  * (156) Müller P.E., Vink J.S. 2008. A&A. 492:493
  * (157) Müller P.E., Vink J.S. 2014. A&A. 564:57
  * (158) Najarro F., Hanson M.M., Puls J. 2011. A&A. 535:32
  * (159) Nieuwenhuijzen H., de Jager C. 1990. A&A. 231:134-136
  * (160) Noebauer U.M., Sim S.A. 2019. Living Reviews in Computational Astrophysics. 5:1
  * (161) Nugis T., Lamers H.J.G.L.M. 2000. A&A. 360:227-244
  * (162) Nugis T., Lamers H.J.G.L.M. 2002. A&A. 389:162-179
  * (163) Ofek E.O., Sullivan M., Shaviv N.J., et al. 2014. ApJ. 789:104
  * (164) Oskinova L.M., Hamann W.-R., Feldmeier A. 2007. A&A. 476:1331
  * (165) Owocki S.P. 2015. Very Massive Stars in the Local Universe. 412:113
  * (166) Owocki S.P., Rybicki G.B. 1984. ApJ. 284:337-350
  * (167) Owocki S.P., Castor J.I., Rybicki G.B. 1988. ApJ. 335:914
  * (168) Owocki S.P., Cohen D.H. 2006. ApJ. 648:565-571
  * (169) Owocki S.P., Gayley K.G., Shaviv N.J. 2004. ApJ. 616:525-541
  * (170) Panagia N., Felli M. 1975. A&A. 39:1-5
  * (171) Parker E.N. 1958. ApJ. 128:664
  * (172) Pastorello A., Smartt S.J., Mattila S., et al. 2007. Nature. 447:829-832
  * (173) Pauldrach A.W.A., Puls, J. 1990. A&A. 237: 409
  * (174) Pauldrach A.W.A., Puls J., Kudritzki R.P. 1986. A&A. 164:86-100
  * (175) Pauldrach A.W.A., Vanbeveren D., Hoffmann T. L. 2012. A&A. 538:75
  * (176) Petit V., Keszthelyi Z., MacInnis R., et al. 2017. MNRAS. 466:1052-1060
  * (177) Petrenz P., Puls J. 2000. A&A. 358:956-992
  * (178) Petrov B., Vink J.S., Gräfener G. 2014. A&A. 565:62
  * (179) Petrov B., Vink J.S., Gräfener G. 2016. MNRAS. 458:1999
  * (180) Poniatowski L.G., Sundqvist J.O., Kee N.D., et al. 2021. A&A. 647:151
  * (181) Prinja R.K., Barlow M.J., Howarth I.D. 1990. ApJ. 361:60
  * (182) Puls J., Kudritzki R.P., Herrero A., et al. 1996. A&A. 305:171
  * (183) Puls J., Kudritzki R.P., Santolaya-Rey A.E., et al. 1998. Properties of Hot Stars. 131:245
  * (184) Puls J., Springmann U., Lennon M. 2000. A&AS. 141:23-64
  * (185) Puls J., Markova N., Scuderi S., et al. 2006. A&A. 454:625-651
  * (186) Puls J., Vink J.S., Najarro F. 2008. A&ARv. 16:209
  * (187) Puls J., Najarro F., Sundqvist J.O., Sen K. 2020. A&A. 642:172
  * (188) Quataert E., Shiode J. 2012. MNRAS. 423:L92-L96
  * (189) Ramachandran V., Hamann W.-R., Oskinova L.M., et al. 2019. A&A. 625:104
  * (190) Ramírez-Agudelo, O.H., Sana, H., de Koter, A., et al. 2017. A&A. 600:81
  * (191) Renzo M., Ott C.D., Shore S.N., de Mink S.E. 2017. A&A. 603:118
  * (192) Richardson N.D., Mehner A. 2018. RNAAS. 2:121
  * (193) Ro S., Matzner C.D. 2016. ApJ. 821:109
  * (194) Robert C., Moffat A.F.J., Bastien P., Drissen L., St. -Louis N. 1989. ApJ. 347:1034
  * (195) Rodrigues C. V., Magalhães A. M. 2000. ApJ. 540:412-421
  * (196) Rubio-Díez M.M., Sundqvist J.O., Najarro F., et al. 2021. arXiv e-prints.:arXiv:2108.11734
  * (197) Sabhahit G.N., Vink J.S., Higgins E.R., Sander A.A.C. 2021. MNRAS. 506:4473-4487
  * (198) Sana H., de Koter A., de Mink S. E., et al. 2013. A&A. 550:107
  * (199) Sana H., Le Bouquin J.-B., Lacour S., et al. 2014. ApJS. 215:15
  * (200) Sander A.A.C., Hamann W.-R., Todt H., Hainich R., Shenar T. 2017. A&A. 603:86
  * (201) Sander A.A.C., Hamann W.-R., Todt H., et al. 2019. A&A. 621:92
  * (202) Sander A.A.C., Vink J.S., Hamann W.-R. 2020. MNRAS. 491:4406
  * (203) Sander A.A.C., Vink J.S. 2020. MNRAS. 499:873-892
  * (204) Sanyal D., Grassitelli L., Langer N., Bestenlehner J. M. 2015. A&A. 580:20
  * (205) Schmutz W., Abbott D.C., Russell R.S., Hamann W.-R., Wessolowski U. 1990. ApJ. 355:255
  * (206) Schneider F.R.N., Ramírez-Agudelo O.H., Tramper F., et al. 2018. A&A. 618:73
  * (207) Shaviv N.J. 2000. ApJL. 532:L137
  * (208) Shenar T., Hainich R., Todt H., et al. 2016. A&A. 591:22
  * (209) Shenar T., Gilkis A., Vink J. S., Sana H., Sander A. A. C. 2020. A&A. 634:79
  * (210) Shiode J. H., Quataert E. 2014. ApJ. 780:96
  * (211) Simón-Díaz S. 2020. Reviews in Frontiers of Modern Astrophysics.:155-187
  * (212) Smartt S.J. 2009. ARA&A. 47:63-106
  * (213) Smith N. 2014. ARA&A. 52:487-528
  * (214) Smith N., Gehrz R.D., Hinz P.M., et al. 2003. AJ. 125:1458-1466
  * (215) Smith N., Li W., Foley R.J., et al. 2007. ApJ. 666:1116-1128
  * (216) Spera M., Mapelli M., Giacobbo N., et al. 2019. MNRAS. 485:889-907
  * (217) Sun N.C., Maund J.R., Crowther P.A. 2020. MNRAS. 497:5118-5135
  * (218) Sundqvist J.O., Puls J. 2018. A&A. 619:59
  * (219) Sundqvist J.O., Owocki S.P., Puls J. 2018. A&A. 611:17
  * (220) Sundqvist J.O., Puls J., Owocki S.P. 2014. A&A. 568:59
  * (221) Šurlan B., Hamann W.-R., Aret A., et al. 2013. A&A. 559:130
  * (222) Tanikawa A., Kinugawa T., Yoshida T., Hijikawa K., Umeda H. 2021. MNRAS. 505:2170-2176
  * (223) Tramper F., Sana H., de Koter A., Kaper L., Ramírez-Agudelo O.H. 2014. A&A. 572:36
  * (224) Tramper F., Straal S. M., Sanyal D., et al. 2015. A&A. 581:110
  * (225) Trundle C., Kotak R., Vink J. S., Meikle W. P. S. 2008. A&A. 483:L47-L50
  * (226) ud-Doula A., Owocki S.P. 2002. ApJ. 576:413-428
  * (227) Vink J.S. 2000. Ph.D. Thesis. University of Utrecht, the Netherlands
  * (228) Vink J.S. 2017. A&A. 607:L8
  * (229) Vink J.S. 2018. A&A. 615:119
  * (230) Vink J.S., de Koter A., Lamers H.J.G.L.M. 1999. A&A. 345:109
  * (231) Vink J.S., de Koter A., Lamers H.J.G.L.M. 2000. A&A. 362:295
  * (232) Vink J.S., de Koter A., Lamers H.J.G.L.M. 2001. A&A. 369:574
  * (233) Vink J.S., de Koter, A. 2002. A&A. 393:543
  * (234) Vink J.S., de Koter A. 2005. A&A. 442:587-596
  * (235) Vink J.S., Brott I., Gräfener G., et al. 2010. A&A. 512:L7
  * (236) Vink J.S., Muijres L.E., Anthonisse B., et al. 2011. A&A. 531:132
  * (237) Vink J.S., Gräfener G. 2012. ApJL. 751:34
  * (238) Vink J.S., Higgins E.R., Sander A.A.C., Sabhahit G.N. 2021. MNRAS. 504:146-154
  * (239) Vink J.S., Sander A.A.C. 2021. MNRAS. 504:2051-2061
  * (240) Wright A.E., Barlow M.J. 1975. MNRAS. 170:41-51
  * (241) Woosley S.E. 1993. ApJ. 405:273
  * (242) Woosley S.E., Heger A. 2015. Very Massive Stars in the Local Universe. 412:199
  * (243) Wu S., Fuller J. 2021. ApJ. 906:3
  * (244) Yoon S.-C., Cantiello M. 2010. ApJL. 717:L62-L65
  * (245) Yoshida T., Umeda H., Maeda K., Ishii T. 2016. MNRAS. 457:351-361
  * (246) Yusof N., Hirschi R., Meynet G., et al. 2013. MNRAS. 433:1114-1132

##  8 Appendix: Mass-loss Recipes

An oft-used combination of mass-loss recipes for massive stars is the dutch
Recipe in MESA. It consists of a theoretical Vink et al. (2000, 2001) for hot
OB stars, an empirical recipe of de Jager et al. (1988) for cooler supergiants
below 10 kK, and an empirical recipe by Nugis & Lamers (2000) for WR stars,
with a simple definition that $Y$ is larger than 0.4 and $T_{\rm eff}$ is
above 10 kK.

For OB stars the most-oft used mass-loss recipe is that from Monte Carlo
models of Vink et al. (2000) on either side of the BS-Jump.

$\displaystyle{\rm log}~{}\dot{M}$ $\displaystyle=$
$\displaystyle~{}-~{}6.697~{}(\pm 0.061)$ (15)
$\displaystyle~{}+~{}2.194~{}(\pm 0.021)~{}{\rm log}(L_{*}/{10^{5}})$
$\displaystyle~{}-~{}1.313~{}(\pm 0.046)~{}{\rm log}(M_{*}/30)$
$\displaystyle~{}-~{}1.226~{}(\pm 0.037)~{}{\rm
log}\left(\frac{v_{\infty}/v_{\rm esc}}{2.0}\right)$
$\displaystyle~{}+~{}0.933~{}(\pm 0.064)~{}{\rm log}(\mbox{$T_{\rm
eff}$}/40000)$ $\displaystyle~{}-~{}10.92~{}(\pm 0.90)~{}\\{{\rm
log}(\mbox{$T_{\rm eff}$}/40000)\\}^{2}$ $\displaystyle~{}{\rm
for}~{}27~{}500<\mbox{$T_{\rm eff}$}\leq 50~{}000{\rm K}$

where $\dot{M}$ is in $M_{\odot}$ ${\rm yr}^{-1}$, $L_{*}$ and $M_{*}$ are in
solar units and $T_{\rm eff}$ is in Kelvin. In this range $v_{\infty}/v_{\rm
esc}$ = 2.6. Equation 15 predicts the calculated mass-loss rates of the 180
models with a root-mean-square accuracy of 0.061 dex. The second range
(roughly the range of the B-type supergiants) is taken from $T_{\rm eff}$
models below between 22 500 K:

$\displaystyle{\rm log}~{}\dot{M}$ $\displaystyle=$
$\displaystyle~{}-~{}6.688~{}(\pm 0.080)$ (16)
$\displaystyle~{}+~{}2.210~{}(\pm 0.031)~{}{\rm log}(L_{*}/{10^{5}})$
$\displaystyle~{}-~{}1.339~{}(\pm 0.068)~{}{\rm log}(M_{*}/30)$
$\displaystyle~{}-~{}1.601~{}(\pm 0.055)~{}{\rm
log}\left(\frac{v_{\infty}/v_{\rm esc}}{2.0}\right)$
$\displaystyle~{}+~{}1.07~{}(\pm 0.10)~{}{\rm log}(\mbox{$T_{\rm
eff}$}/20000)$ $\displaystyle~{}{\rm for}~{}12~{}500<\mbox{$T_{\rm eff}$}\leq
22~{}500{\rm K}$

where again $\dot{M}$ is in $M_{\odot}$ ${\rm yr}^{-1}$, $L_{*}$ and $M_{*}$
are in solar units and $T_{\rm eff}$ is in Kelvin. In this range
$v_{\infty}/v_{\rm esc}$ = 1.3. Vink et al. (2001) added a $Z$-dependence to
this recipe and as Fe sets the mass-loss rate, this is mostly Fe-dependent
mass loss. Using locally consistent mass-loss rates, Vink & Sander (2021)
derived a shallower $\dot{M}$ \- $Z$ dependence than in the original Vink et
al. (2001) recipe. Renzo et al. (2017) added a unitless $\eta$ parameter – of
order unity – to study the effects of weaker winds in a similar vein to the
Reimers mass-loss law for cool stars.

Björklund et al. (2021) constructed a modified wind momentum relation based on
CMF computations as:

$\log\left(\mbox{$\dot{M}v_{\infty}R_{*}^{0.5}$}\right)=-1.55+0.46\log\left(\frac{Z_{\ast}}{Z_{\odot}}\right)+\\\
\left[2.07-0.73\log\left(\frac{Z_{\ast}}{Z_{\odot}}\right)\right]\log\left(\frac{L_{\ast}}{10^{6}L_{\odot}}\right)$
(17)

and

$\log(\mbox{$\dot{M}$})=-5.55+0.79\log\left(\frac{Z_{\ast}}{Z_{\odot}}\right)+\\\
\left[2.16-0.32\log\left(\frac{Z_{\ast}}{Z_{\odot}}\right)\right]\log\left(\frac{L_{\ast}}{10^{6}L_{\odot}}\right),$
(18)

with the wind momentum rate in units of $M_{\odot}$ yr-1 km s-1
$R_{\odot}^{0.5}$ and the mass-loss rate in units of $M_{\odot}$ yr-1. When
the complete model-grid sample is considered, these fitted relations give mean
values that agree with the original simulations to within 10%. Note that there
is no $T_{\rm eff}$ dependence in this mass-loss recipe.

On the empirical side, de Jager et al. (1988) collected mass-loss rates
determined for 271 Galactic stars of spectral types O through M. The mass-loss
rates could be reproduced by a single formula dependent on $T_{\rm eff}$ and
$L$:

$\log\mbox{$\dot{M}$}=-8.158+1.769\log L-1.676\log\mbox{$T_{\rm eff}$}\,.$
(19)

Here the $T_{\rm eff}$ dependence is opposite to that in the Vink et al.
(2000) recipe, which means it is not meaningful to compare these recipes in an
$\dot{M}$-$L$ plot without accounting for the multi-variate – and in this case
even oppositely directed with regards to $T_{\rm eff}$ – behaviour of
$\dot{M}$. Note that the original de Jager formulation did not account for
stellar mass, and an improved relation was proposed by Nieuwenhuizen et al.
(1990).

For WR stars the oft-used empirical relation of Nugis & Lamers (2000) is:

$\displaystyle{\rm log}~{}\dot{M}$ $\displaystyle=$
$\displaystyle~{}-~{}11.0~{}+~{}1.29~{}(\pm 0.14)~{}{\rm log}L$ (20)
$\displaystyle~{}+~{}1.73~{}(\pm 0.42)~{}{\rm log}Y~{}+~{}0.47~{}(\pm
0.09)~{}{\rm log}Z$

For both WN and WC stars, and with with a standard deviation of 0.19 dex. Note
that this $Z$ dependence depends on the total self-enriched $Z$ and represents
a very different $Z$ dependence from the Fe-dependent mass-loss rate of Vink &
de Koter (2005) and Sander & Vink (2020):

$\displaystyle\log\dot{M}=a\cdot\log\left[-\log\left(1-\Gamma_{\rm
e}\right)\right]-\log(2)\cdot\left(\frac{\Gamma_{\rm e,b}}{\Gamma_{\rm
e}}\right)^{c}+\log\dot{M}_{\rm off}$ (21)

with coefficients

$\displaystyle a=2.932(\pm 0.016)$ (22) $\displaystyle\Gamma_{\rm
e,b}=-0.324(\pm 0.011)\cdot\log(Z/Z_{\odot})+0.244(\pm 0.010)$ (23)
$\displaystyle c=-0.44(\pm 1.09)\cdot\log(Z/Z_{\odot})+9.15(\pm 0.96)$ (24)
$\displaystyle\log\dot{M}_{\rm off}=0.23(\pm
0.04)\cdot\log(Z/Z_{\odot})-2.61(\pm 0.03)$ (25)

For stripped He stars with masses and luminosities below the classical WR
regime the only predictions available are the Monte Carlo computations by Vink
(2017):

$\log\mbox{$\dot{M}$}~{}=~{}-13.3~{}+~{}1.36\log(L/\mbox{$L_{\odot}$})~{}+~{}0.61\log(Z/\mbox{$Z_{\odot}$})$
(26)

with a fitting error of $\sigma$ = 0.11. The application of this formula is
only recommended for lower-mass stripped stars, and not for classical WR
stars. This stripped-star formula could be used as a lower bound to a
classical WR recipe such as that of Sander & Vink (2020).


# Pairing and Pair Breaking by Gauge Fluctuations in Bilayer Composite Fermion
Metals

Haoyun Deng  Luis Mendoza  Current Address: Fermi National Accelerator
Laboratory, P.O. Box 500, Batavia, IL 60510, USA  N.E. Bonesteel  Department
of Physics and National High Magnetic Field Laboratory, Florida State
University, Tallahassee, FL 32310, USA

###### Abstract

We study interlayer pairing of composite fermions in the total $\nu=1/2+1/2$
quantum Hall bilayer as a possible framework for understanding the
experimentally observed transition from a compressible state at large layer
spacing to a bilayer quantum Hall state at small layer spacing. We consider a
model in which the effective interlayer composite fermion pairing interaction
mediated by the Chern-Simons gauge fields in the two layers is singular with
both attractive (out-of-phase) and repulsive (in-phase) components diverging
at low frequency. If only the more singular attractive interaction is included
the pairing gap obtained by solving the gap equation is proportional to the
inverse of the layer spacing squared. In the so-called local approximation, we
find that when the less singular repulsive interactions are also included the
pairing gap still falls off as inverse layer spacing squared, consistent with
recent analyses, but is strongly suppressed to a degree that may account for
the fact that this predicted inverse square dependence is not observed
experimentally. The analytically obtained local approximation solutions are
then used as a starting point to numerically iterate the full gap equation to
assess the validity of the approximation in this limit.

##  I Introduction

The total $\nu=1$ quantum Hall bilayer consists of two parallel two-
dimensional electron gases separated by a distance $d$ and placed in a
perpendicular magnetic field $B$ such that the total electron density of the
two layers is that of a single filled Landau level. For the symmetrically
doped case, each layer has Landau level filling fraction $\nu=1/2$, and, if
interlayer electron tunneling can be ignored, the only coupling between layers
is through the interlayer Coulomb repulsion. The strength of this interlayer
repulsion, relative to the intralayer Coulomb repulsion, is set by the
dimensionless ratio $d/l_{0}$, where $l_{0}=(\hbar c/(eB))^{1/2}$ is the
magnetic length.

In the limit of small $d/l_{0}$ this system forms an incompressible bilayer
quantum Hall state in which electrons develop spontaneous interlayer phase
coherence [1]. This state can be viewed as an exciton condensate formed by
electron-hole pairs in the two layers [2, 3]. In the absence of interlayer
tunneling, the transition to this state is predicted to occur at finite
temperature via a Kosterlitz-Thouless transition. In the opposite limit of
large $d/l_{0}$, the two layers decouple and the system is presumably well
described in terms of two separate $\nu=1/2$ composite fermion metals. In
these compressible states, composite fermions, electrons attached to two
fictitious (Chern-Simons) flux quanta [4, 5, 6, 7], move in zero effective
magnetic field and are predicted to experience a fluctuating effective gauge
field. While there has been some speculation that the correct low-energy
description of the resulting compressible state is in terms of Dirac particles
[8], we take the view here that even if this is the case, the effective
Halperin, Lee, Read description is still essentially correct, provided it is
applied appropriately.

It has been argued recently that the appearance of the electron-hole exciton
condensate in this system may be equivalent to the formation of an interlayer
$p$-wave composite fermion superconductor [9], a type of interlayer pairing of
composite fermions first studied in [10]. This viewpoint has gained
experimental support in the recent work of [11] in which quantum Hall states
in graphene double layers separated by atomically thin insulators were
studied. In this work, through measurement of the Coulomb drag responses (both
longitudinal and transverse drag resistivities) and counterflow responses of
the graphene layers at total filling fraction $\nu=1$ as a function of
temperature, a temperature vs. $d/l_{0}$ phase diagram was determined. This
phase diagram showed that in the limit of small $d/l_{0}$, there is indeed a
finite temperature phase transition into a quantum Hall state, and for
temperatures above this transition there is a wide temperature range in which
interlayer pairs have formed (as evidenced by the counterflow response), but
the system does not have true long-range order and hence does not show the
quantum Hall effect (as observed in the drag response). This behavior is
characteristic of a crossover from a regime in which tightly bound pairs form
above the Bose-Einstein condensation temperature for these pairs, to a regime,
with increasing $d/l_{0}$ in which the pairs appear to form at temperatures
just above the critical temperature, as expected for more conventional weak
coupling BCS pairing, and so is at least consistent with the notion of
interlayer pairing.

A full random-phase approximation (RPA) calculation of the effective
interlayer pairing interaction for composite fermions in this system [12]
shows that within this approximation the leading pairing instability is in the
$p$-wave channel, and so may correspond to the experimentally observed quantum
Hall state as suggested in [9]. More recently this result has also been
obtained using a modified RPA which accounts for the renormalization of the
composite fermion mass required by the lowest Landau level projection [13].
This latter work also provides evidence that this composite fermion pairing
approach is equivalent to a third viewpoint (in addition to the exciton
condensate view) in which particle-hole symmetry of the half-filled Landau
level is exploited and the bilayer system is viewed in terms of pairing of
“composite electrons” in one layer with “composite holes” in the second layer
[11, 14]. All this, along with mounting numerical evidence [15, 16, 17] that
suggests these approaches are all equivalent and that interlayer pairing of
composite fermions of the type originally studied in [10] may indeed be an
appropriate framework for studying the transition from bilayer composite
fermion metal to bilayer quantum Hall state, motivates the present work.

In this paper, we analyze the effect of both pairing and pair-breaking gauge
fluctuations on bilayer pairing as a function of layer spacing. We focus on
the balanced case and take the point of view that the system can be described
for large layer spacing as two composite fermion metals whose low-energy
physics is described by the Halperin-Lee-Read theory, with appropriately
renormalized effective mass. We keep only the most singular interactions in
the interlayer Cooper channel which have the same form in all angular momentum
channels, and treat the less singular terms, which presumably determine the
pairing symmetry of the gap, phenomenologically. Here we are not focused on
determining the pairing channel with the leading instability, something which
is addressed by more detailed RPA calculations [12, 14, 13], but rather on the
question of how strong the pairing is, to what extent gauge fluctuation
pairing might play a role in it, and how it depends on layer spacing when both
pairing and pair breaking contributions are taken into account. The primary
tool we use for our analysis is the so-called local approximation applied to
the frequency-dependent $T=0$ gap equation. Within this approximation, we
derive analytic results for the combined effect of pairing and pair breaking
in this system.

The paper is organized as follows. In Section II we review the Halperin, Lee,
Read description of the the $\nu=1/2+1/2$ bilayer and present our model
frequency-dependent interaction. Section III then presents a simple example of
the local approximation applied to the $T=0$ gap equation due to both the
singular pairing and pair-breaking gauge fluctuations. In Section IV we focus
on the large layer spacing limit, obtaining analytic expressions for the $T=0$
gap, as well as the significance of pair-breaking effects. We find that the
latter are characterized by a single “pair breaking” parameter and are
actually quite large for the parameters relevant to the bilayer system.
Section V then describes our results for what happens as the short-range,
nonsingular interaction strength is varied, and the crossover from the weak
gauge pairing we associate with the large layer spacing limit, to the strong
BCS pairing due to short-range attraction in some angular momentum channel. In
Sections IV and V we also compare the results of the local approximation to
exact solutions of the gap equation obtained numerically, verifying the
essential validity of the approximation. Finally, in Section VI we review the
results of the paper and present our conclusions.

##  II Pairing and Pair-breaking Gauge Fluctuations

In what follows we assume that the Halperin, Lee, Read description of the
$\nu=1/2$ single layer state adequately captures the essential low-energy
physics of the system. In this approach, the system is described with a
Euclidean-time action for the $\nu=1/2+1/2$ bilayer which, at temperature $T$,
has the form $S=\int_{0}^{\beta}d\tau\int d^{2}r{\cal L}({\bf r},\tau)$ where
$\beta=(k_{B}T)^{-1}$ and the Lagrangian density is ${\cal L}={\cal
L}_{0}+{\cal L}_{int}+{\cal L}_{CS}$ with

$\displaystyle{\cal
L}_{0}=\sum_{\alpha=\uparrow,\downarrow}\overline{\psi}_{\alpha}\Bigl{(}\partial_{\tau}-ia_{0}^{\alpha}-\frac{1}{2m^{*}}(\nabla-i{\bf
a}^{\alpha})^{2}\Bigr{)}\psi_{\alpha},$ (1) ${\cal
L}_{int}=\frac{1}{2}\sum_{\alpha,\alpha^{\prime}=\uparrow,\downarrow}\int
d^{2}r^{\prime}~{}\delta\rho_{\alpha}({\bf
r},\tau)V_{\alpha\alpha^{\prime}}({\bf r}-{\bf
r}^{\prime})\delta\rho_{\alpha}^{\prime}({\bf r}^{\prime},\tau),$ (2)

where $\uparrow,\downarrow$ are pseudospin labels for the layers,

$\displaystyle V_{\uparrow\uparrow}({\bf r})=V_{\downarrow\downarrow}({\bf
r})$ $\displaystyle=$ $\displaystyle\frac{e^{2}}{\epsilon r},$ (3)
$\displaystyle V_{\uparrow\downarrow}({\bf r})=V_{\downarrow\uparrow}({\bf
r})$ $\displaystyle=$ $\displaystyle\frac{e^{2}}{\epsilon\sqrt{r^{2}+d^{2}}},$
(4)

are the interlayer and intralayer Coulomb repulsion, respectively, and

$\displaystyle{\cal
L}_{CS}=-\frac{i}{2\pi\lambda}\sum_{\alpha=\uparrow,\downarrow}a_{0}^{\alpha}{\hat{\bf
z}}\cdot\left(\nabla\times\left({\bf a}^{\alpha}+e{\bf
A}_{ext}\right)\right),$ (5)

is the Chern-Simons action in the Coulomb gauge. Here $\psi_{\alpha}$ is the
composite fermion field in layer $\alpha=\uparrow,\downarrow$, ${\bf A}_{ext}$
is the vector potential for the external applied magnetic field ${\bf
B}=\nabla\times{\bf A}_{ext}={\bf\hat{z}}2\pi\tilde{\phi}n/e$ where $n$ is the
electron density in each layer, $\tilde{\phi}$ is the number of flux quanta
attached to each composite fermion ($\tilde{\phi}=2$ for the case total
$\nu=1=1/2+1/2$ considered here), $m^{*}$ is the effective mass of the
composite fermions, and $(a_{0}^{\alpha},{\bf a}^{\alpha})$ is the Chern-
Simons gauge field seen by composite fermions in layer $\alpha$. ${\cal
L}_{CS}$ is a Chern-Simons term in the Coulomb gauge for which
$\nabla\cdot{\bf a}^{\alpha}=0$. The gauge field degrees of freedom in each
layer then consist entirely of the time component, $a^{\alpha}_{0}$, and
transverse component, $a^{\alpha}_{1}({\bf
q},\tau)={\bf{\hat{z}}}\cdot({\bf{\hat{q}}}\times{\bf a}^{\alpha}({\bf
q},\tau))$, of the Chern-Simons gauge fields.

The partition function for this system is then ${\cal
Z}=\int\prod_{\alpha=\uparrow,\downarrow}D\psi_{\alpha}Da^{\alpha}_{0}Da^{\alpha}_{1}e^{-S}$.
Integrating out $a^{\alpha}_{0}$ enforces the constraint

$\displaystyle\delta\rho_{\alpha}=\frac{1}{2\pi\tilde{\phi}}{\bf\hat{z}}\cdot\nabla\times{\bf
a}^{\alpha},$ (6)

which describes attaching $\tilde{\phi}$ quanta of Chern-Simons flux to the
composite fermions. Using (6) the density fluctuations in (2) can be replaced
with transverse gauge fields and the resulting action is purely quadratic in
the fermions. The RPA is then carried out by integrating out these fermions
and expanding the resulting effective action to second order in the gauge
fields. The result of this procedure is an effective action
$S_{RPA}=S^{+}_{RPA}+S^{-}_{RPA}$, where

$S^{\pm}_{RPA}=\frac{1}{2}\sum_{{\omega_{n},{\bf
q}\atop{\mu,\nu=0,1}}}{a^{\pm}_{\mu}}^{*}({\bf q},i\omega_{n}){{{\cal
D}^{\pm}}}^{-1}_{\mu\nu}({\bf q},i\omega_{n};\Phi)a^{\pm}_{\nu}({\bf
q},i\omega_{n}).$ (7)

Here

$\displaystyle a^{+}_{\mu}$ $\displaystyle=$
$\displaystyle(a^{\uparrow}_{\mu}+a^{\downarrow}_{\mu})/\sqrt{2},$ (8)
$\displaystyle a^{-}_{\mu}$ $\displaystyle=$
$\displaystyle(a^{\uparrow}_{\mu}-a^{\downarrow}_{\mu})/\sqrt{2},$ (9)

where $\omega_{n}$ are bosonic Matsubara frequencies, $a^{\pm}_{0}({\bf
q},i\omega_{n})$ and $a^{\pm}_{1}({\bf
q},i\omega_{n})={\bf{\hat{z}}}\cdot({\bf\hat{q}}\times{\bf a}^{\pm}({\bf
q},i\omega_{n}))$ are, respectively, the time and transverse components of the
gauge fields, and ${{\cal D}^{\pm}}_{\mu\nu}^{-1}$ is the inverse of the gauge
propagator matrix.

The result for the gauge field propagators ${\cal D}_{\mu\nu}^{\pm}(q,\omega)$
can be computed analytically for all $q$ and $\omega$ within the RPA [18, 12].
Here, as in [19, 10], we focus on the low-energy long wavelength fluctuations.
In this limit, the dominant interactions are mediated by the in-phase and out-
of-phase transverse gauge fluctuations [19]. For $\tilde{\phi}=2$ and in the
large layer spacing limit the result is,

$\displaystyle{\cal D}^{-}_{11}(q,\omega)$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\frac{e^{2}}{4\pi\epsilon}dq^{2}+\frac{1}{4\pi
l_{0}}\frac{|\omega|}{q}},$ (10) $\displaystyle{\cal D}^{+}_{11}(q,\omega)$
$\displaystyle\simeq$
$\displaystyle\frac{1}{\frac{e^{2}}{4\pi\epsilon}q+\frac{1}{4\pi
l_{0}}\frac{|\omega|}{q}},$ (11)

The out-of-phase gauge fluctuations are seen to be more singular than the in-
phase gauge fluctuations. This is a consequence of the fact that the
transverse gauge field fluctuations are directly tied to density fluctuations,
and out-of-phase density fluctuations are stronger than in-phase density
fluctuations due to the interlayer Coulomb repulsion.

In addition to being more singular, the out-of-phase fluctuations mediate a
strong attractive interaction in the interlayer Cooper channel for composite
fermions. This is because the out-of-phase gauge fields couple to composite
fermions in the two layers as if they had opposite charge. The in-phase
fluctuations, on the other hand, while less singular, are repulsive in the
Cooper channel and so suppress pairing.

Here we are concerned with the net interlayer pairing mediated by these gauge
fluctuations, and in particular the interplay between the singular pairing and
pair-breaking terms. To analyze this we use the frequency-dependent couplings
associated with the most singular gauge field contributions described above
[19, 20],

$\displaystyle\lambda_{\pm}(\omega)=\left\langle\left|\frac{{\bf
k}\times\hat{\bf k}^{\prime}}{m^{*}}\right|^{2}D^{\pm}_{11}(|{\bf k}-{\bf
k}^{\prime}|,\omega)e^{il\theta_{{\bf k},{\bf k^{\prime}}}}\right\rangle_{{\bf
k},{\bf k}^{\prime}\in{\rm FS}}$ (12)

where the angle brackets indicate a Fermi surface average over ${\bf k}$ and
${\bf k}^{\prime}$. Note that here, and all that follows, we work at $T=0$ and
have analytically continued to imaginary frequency. In the low-energy long
wavelength limit, and in the limit of large layer spacing, one finds (l.s.t. =
less singular terms),

$\displaystyle\lambda_{-}(\omega)$ $\displaystyle=$
$\displaystyle\frac{8}{3\sqrt{3}}\frac{1}{\beta^{2/3}}\left(\frac{l_{0}}{d}\right)^{2/3}\left|\frac{\omega_{0}}{\omega}\right|^{1/3}+{\rm
l.s.t.},$ (13) $\displaystyle\lambda_{+}(\omega)$ $\displaystyle=$
$\displaystyle\frac{2}{\pi}\frac{1}{\beta}\ln\left|\frac{\omega_{0}}{\omega}\right|+{\rm
l.s.t.},$ (14)

where $\omega_{0}=e^{2}/(\epsilon l_{0})$ sets the typical Coulomb energy
scale of the problem within the lowest Landau level, and
$\beta=e^{2}l_{0}m^{*}/\epsilon$ is a dimensionless parameter which we take to
be of order 1 to account for the required renormalization of the composite
fermion mass to a scale set by the Coulomb repulsion, with
$m^{*}\sim\epsilon/(e^{2}l_{0})$.

Here the integer $l$ appearing in the $e^{il\theta_{{\bf k},{\bf
k}^{\prime}}}$ factor in (12), where $\theta_{{\bf k},{\bf k}^{\prime}}$ is
the angle between the vectors ${\bf k}$ and ${\bf k}^{\prime}$, is the angular
momentum pairing channel. The leading singular contributions to $\lambda_{+}$
and $\lambda_{-}$ are independent of $l$ due to the fact that the gauge field
mediated interactions are singular at small $q$. All the $l$ dependence in the
couplings appears in the less singular terms.

##  III Gap Equation and Local Approximation

The simplest approach to the essential physics of this bilayer pairing is to
solve the $T=0$ frequency-dependent gap equation for this problem (again, here
$\omega$ and $\Omega$ are imaginary frequencies),

$\displaystyle\Delta(\omega)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d\Omega\lambda(\omega-\Omega)\frac{\Delta(\Omega)}{\sqrt{\Omega^{2}+|\Delta(\Omega)|^{2}}}.$
(15)

Here $\lambda(\omega)$ is the frequency-dependent effective interaction,
including contributions described in Section II. Precisely this problem was
studied numerically in [20]. Here we focus on understanding the nature of the
pairing and the interplay between the pairing and pair-breaking gauge
fluctuations. To this end, it is useful to have approximate analytic
solutions, which can provide some physical insight into the behavior of the
system, in addition to exact numerical solutions.

Specifically, we are interested in solving (15) for the case

$\displaystyle\lambda(\omega)$ $\displaystyle=$
$\displaystyle(\lambda_{-}(\omega)-\lambda_{+}(\omega)-V_{0})\Theta(|\omega|-\Lambda),$
(16)

where

$\displaystyle\lambda_{-}(\omega)=\alpha_{-}\Bigl{|}\frac{\omega_{0}}{\omega}\Bigr{|}^{\gamma},$
(17)
$\displaystyle\lambda_{+}(\omega)=\alpha_{+}\log\Bigl{|}\frac{\omega_{0}}{\omega}\Bigr{|}.$
(18)

We take the cutoff $\Lambda$ to be $\omega_{0}$ and view the parameters
$\alpha_{-}$, $\alpha_{+}$ and $V_{0}$ as effective parameters. From the RPA
result described in Sec. II we see that for the bilayer we have $\gamma=1/3$,
and in the large layer spacing limit $\alpha_{-}\simeq(l_{0}/d)^{2/3}$,
$\alpha_{+}\simeq 1$, and $\omega_{0}\simeq e^{2}/(\epsilon l_{0})$ [19]. Here
$V_{0}$ is a nonsingular term that represents the short-range interaction and
which will depend on the angular momentum channel. We make no attempt to
calculate $V_{0}$ from first principles here, as in [12] and [13], but rather
treat it as a semi-phenomenological parameter. The rest of the paper focuses
on finding approximate and exact solutions of (15) for the interaction (16)
and determining how these solutions depend on $\alpha_{-},\alpha_{+},$ and
$V_{0}$.

We now apply the local approximation, as described in [21], to this pairing
problem. The first step in this approximation is to linearize the gap
equation,

$\displaystyle\Delta(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{|\Omega|>|\Delta(0)|}d\Omega\lambda(\omega-\Omega)\frac{\Delta(\Omega)}{|\Omega|}.$
(19)

Taylor expanding $\lambda(\Omega-\omega)$ inside the integral in the following
way,

$\displaystyle\lambda(\Omega-\omega)=\left\\{\begin{array}[]{cc}\lambda(\omega)-\Omega\lambda^{\prime}(\omega)+\cdots,&|\Omega|<|\omega|\\\
\lambda(\Omega)-\omega\lambda^{\prime}(\Omega)+\cdots,&|\Omega|>|\omega|\end{array}\right.$
(22)

and keeping just the first term, we obtain a linear integral equation, which
upon successive derivatives with respect to $\omega$ can be shown to be
mathematically equivalent to the following second-order linear differential
equation,

$\displaystyle\frac{d}{d\omega}\left(\frac{\Delta^{\prime}(\omega)}{\lambda(\omega)}\right)-\frac{\Delta(\omega)}{\omega}=0,$
(23)

with boundary conditions

$\displaystyle\Delta^{\prime}(\omega)\Bigl{|}_{\omega=\Delta_{0}}$
$\displaystyle=$ $\displaystyle 0,$ (24)

and

$\displaystyle\frac{d}{d\omega}\left(\frac{\Delta(\omega)}{\lambda(\omega)}\right)\Bigl{|}_{\omega=\Lambda}=0.$
(25)

This second-order linear differential equation can be converted into a
nonlinear first-order differential equation by introducing the $V$ function,

$\displaystyle
V(\omega)=-\lambda^{\prime}(\omega)\frac{\Delta(\omega)}{\Delta^{\prime}(\omega)}.$
(26)

If we further introduce a flow parameter,

$\displaystyle l=\ln\frac{\Lambda}{\omega},$ (27)

the differential equation can be converted to the following,

$\displaystyle\frac{dV}{dl}=-\frac{d\lambda}{dl}-V^{2}.$ (28)

The UV boundary condition becomes,

$\displaystyle V(\Lambda)=-\lambda(\Lambda),$ (29)

and the IR boundary condition becomes,

$\displaystyle\lim_{\omega\rightarrow\Delta(0)}V(\omega)=-\infty,$ (30)

where here $\Delta(0)$ is the zero frequency gap.

Figure 1: (Color online) $V,\lambda_{-}$ flow diagrams for the case
$\alpha_{+}=0.2$ and $\alpha_{+}=0.4$. In each diagram the top red trajectory
corresponds to the large layer spacing limit solution, the lower red
trajectory corresponds to the crossover from gauge pairing to BCS pairing, as
defined in the text, and the solid black curves connect $V=+\sqrt{\alpha_{+}}$
with $V=-\sqrt{\alpha_{+}}$ at $\lambda_{-}=0$, following the curve
$V^{2}=\alpha_{+}-\gamma\lambda_{-}$, along which $\frac{dV}{dl}=0$.

The frequency-dependent gap function $\Delta(\omega)$ is then simply given by
the following,

$\displaystyle\ln\Delta=-\int\frac{\lambda^{\prime}}{V}d\omega,$ (31)

where here we keep the right-hand side as an indefinite integral because the
integration constant is just a multiplicative constant in $\Delta$.

For the case
$\lambda(\omega)=\lambda_{-}(\omega)-\alpha_{+}\log\left|\frac{\omega}{\omega_{0}}\right|-V_{0}$
the flow equation can be written

$\displaystyle\frac{dV}{dl}=-\gamma\lambda_{-}+\alpha_{+}-V^{2}.$ (32)

This equation is analogous to the renormalization group equations given in [9]
where the non-Fermi liquid pairing renormalization group introduced in [22]
was applied to the bilayer system.

To further this connection, we supplement (32) with the equation,

$\displaystyle\frac{d\lambda_{-}}{dl}=\gamma\lambda_{-},$ (33)

which follows directly from the definition of $\lambda_{-}$. Equations (32)
and (33) then form a system of coupled first-order differential equations for
which the solutions can be visualized by flow diagrams in the $V,\lambda_{-}$
plane. These diagrams, two of which are shown in Fig. 1, allow us to analyze
the solutions of this equation in ways that are not immediately apparent
starting with the gap equation (15). For that reason, in what follows we will
focus first on the local approximation. Later we will compare the exact
numerical result with the local approximation results in order to verify that,
while there are some quantitative differences, the local approximation appears
to capture the qualitative behavior of this problem well.

Figure 2: (Color online). Log of the ratio of zero frequency gap with pair
breaking to that without pair breaking as a function of the pair breaking
parameter $\sqrt{\alpha_{+}}/\gamma^{2}$ for different values of $\gamma$.
Local approximation results are shown for $\gamma=1/3$ (green line) and 1/6
(blue line), as well as the limit $\gamma\rightarrow 0$ (dashed line). Exact
numerical results for $\gamma=1/3$ are also shown.

Before proceeding we note some features of this flow diagram which will play
important roles in what follows. The UV boundary condition

$\displaystyle V(\omega_{0})=-\alpha_{-}+V_{0};\ \ \
\lambda_{-}(\omega_{0})=\alpha_{-},$ (34)

gives us a starting point in the flow diagram corresponding to the point where
the flow parameter is $l=\ln\frac{\omega_{0}}{\omega}=0$. We see that in the
limit of large layer spacing, for which $\alpha_{-}\ll 1$, provided
$V_{0}>-\sqrt{\alpha_{+}}$, the solution always flows to the trajectory which
starts at the point $(V,\lambda_{-})=(+\sqrt{\alpha_{+}},0)$. This gives a
universal form for $V(\omega)$ in this limit, the details of which are
discussed in Sec. IV.

We see that if $\alpha_{-}$ is kept small, but $V_{0}$ is varied in such a way
that it becomes more and more negative, eventually, once
$V_{0}<-\sqrt{\alpha_{+}}$, there will be an abrupt crossover to a pairing
phase with a much bigger energy gap. We view this crossover as being between a
“gauge pairing” phase, in which the pairing interaction is dominated by long-
wavelength low-frequency gauge fluctuations, to a paired state that is much
closer to a conventional BCS superconductor with the pairing due primarily to
a short-range nonsingular attractive $V_{0}$. It is natural to take the flow
trajectory which starts at $(V,\lambda_{-})=(-\sqrt{\alpha_{+}},0)$ as the
crossover point, for which $V(\omega)$ flows initially to $0$, just touching
this value, before flowing to $-\infty$ indicating the formation of a paired
state. This crossover is further discussed in Sec. V.

Aside from providing a link to the renormalization group approach, one
appealing feature of the local approximation is that it allows us to obtain
essentially analytic results for $\Delta(\omega)$ and its dependence on
$\alpha_{-}$, $\alpha_{+}$, and $V_{0}$. In addition to being of value in
their own right, these approximate $\Delta(\omega)$ solutions can be used as
starting points to iterate the exact gap equation, which we find then rapidly
converges. By performing such numerical iteration we are able to compare the
analytic local approximation results with the results of actually solving the
gap equation.

##  IV Large Layer Spacing Limit

In this section, we examine the limit of large layer spacing more closely. As
noted above, in this limit the coupling constant $\alpha_{-}<<1$, and the flow
in Fig. 1 begins asymptotically close to the $\lambda_{-}=0$ line. In the
limit of large layer spacing, we expect the coupling $V_{0}$ to be very small
in magnitude, but we see here that if $V_{0}<-\sqrt{\alpha_{+}}$, the flow
changes drastically, reflecting the crossover from gauge pairing to more BCS-
like pairing (discussed in the next section).

Local Approximation Exact

Figure 3: (Color online). $\Delta(\omega)$ vs. $\omega$ for
$\alpha_{+}=0,0.2,0.4$. Results are shown for the local approximation, as well
as the exact solution of the gap equation obtained by iterating the gap
equation, starting with the local approximation solution. Both the local
approximation and exact results show the sign change in $\Delta(\omega)$ when
$\alpha_{+}>0$, characteristic of the combination of pairing and pair breaking
with different frequency dependencies as discussed in the text.

This $V_{0}=-\sqrt{\alpha_{+}}$ unstable fixed point for $\alpha_{-}=0$
corresponds to the phase transition between the unpaired and paired state as a
function of $V_{0}$ when $\alpha_{-}=0$. In this case, we are simply analyzing
the problem of BCS pairing due to an attractive nonsingular $V_{0}$
interaction, in the presence of marginally singular pair breaking due to
$\lambda_{+}$. This singular pair breaking leads to a qualitative change in
the usual BCS pairing problem. A well-known feature of BCS pairing is that it
occurs for any attractive interaction strength. But here, because of the
singular pair breaking due to the in-phase gauge fields, there is a critical
interaction strength needed, as shown earlier in [22, 23], with the flow
equations appearing here closely related to those in [22].

As $\alpha_{-}$ becomes finite, this phase transition becomes a crossover.
Again, if the layer spacing is large we expect $V_{0}$ to be negligibly small,
but as the layers are brought together the variation of $V_{0}$ can lead to a
crossover, which may correspond to the experimental observation of a gap
turning on continuously as $d/l_{0}$ is decreased below a critical value
($\sim 1.6$ in experiment). First we consider the $\alpha_{-}<<0$ limit, and
assume that $V_{0}>-\sqrt{\alpha_{+}}$. In this case, the flow diagram makes
clear that the solution flows to the trajectory that starts at the point
$\lambda_{-}=0$, $V_{0}=+\sqrt{\alpha_{+}}$. We therefore continue to see
scaling behavior even with $\alpha_{+}$ present. For the universal scaling
curve, one can find $V(\omega)$ and use it to analytically obtain the zero
frequency gap $\Delta(0)$, as well as $\Delta(\omega)$ using (31). The
procedure for finding $V(\omega)$ for this case is described in the Appendix.

To address the question of how effective the in-phase gauge fluctuations are
for pair breaking, we show in Fig. 2 the ratio of the zero frequency gap with
$\alpha_{+}$ present to that in the absence of pair breaking for different
values of $\gamma$ within the local approximation, as well as numerical
results for the $\gamma=1/3$ case. It is apparent from this figure and is
shown explicitly in the Appendix, that in the limit of small $\gamma$, the
zero frequency gap has the form

$\displaystyle\Delta(0)=e^{(-2.566\cdots)\frac{\sqrt{\alpha_{+}}}{\gamma^{2}}}\Delta_{\alpha_{+}=0}(0)$
(35)

where

$\displaystyle\Delta_{(\alpha_{+}=0)}(0)=\left(\frac{{0.6917\cdots}}{\gamma}\right)^{1/\gamma}\alpha_{-}^{1/\gamma}\omega_{0}$
(36)

is the zero frequency gap in the absence of pair breaking.

From these expressions, we see that in the absence of pair breaking, for which
$\alpha_{+}=0$, for $\gamma=1/3$ and $\alpha_{\sim}(l_{0}/d)^{2/3}$ we have
$\Delta_{(\alpha+=0)}(0)\sim\omega_{0}(l_{0}/d)^{2}$, which is the well-known
result that $\Delta(0)$ is proportional to inverse layer spacing squared for
this case [10]. When $\alpha_{+}$ is now turned on, we can identify a pair-
breaking parameter $\sqrt{\alpha_{+}}/\gamma^{2}$, which serves as a
dimensionless measure of the effect of the pair breaking due to these out-of-
phase gauge fluctuations. This result makes physical sense and reflects the
fact that for smaller $\gamma$, and hence less singular pairing fluctuations,
the pair breaking is more effective. It further shows for the physically
relevant case of $\alpha_{+}\sim 1$ and $\gamma=1/3$ pair breaking is quite
strong. The observation that this pair breaking is so strong, despite the fact
that the pair-breaking fluctuations are less singular than the pairing
fluctuations, may account for the fact that the predicted inverse layer
spacing squared dependence of the energy gap is not seen experimentally.
While, as a matter of principle, for a perfectly clean sample at low enough
temperature, a paired state may always form even at large layer spacing, in
practice disorder will suppress the transition once the layer spacing is large
enough that the pairing gap becomes comparable to the disorder width.

Figure 3 shows the frequency-dependent gap $\Delta(\omega)$ for various values
of $\alpha_{+}$ in the small $\alpha_{-}$ limit. Results are shown both for
the local approximation and the exact numerical solution, obtained by using
the local approximation as a starting solution and iterating the gap equation.
The results show that the effect of the singular pair-breaking term on the
frequency dependence of the energy gap is significant. It leads to a change in
sign in $\Delta(\omega)$ which, in the local approximation, occurs when the
$V$ trajectory on the flow diagram passes through 0. Such a sign change is
similar to that which occurs in other pairing systems with a combination of
attractive and repulsive contributions to the effective coupling, each having
different frequency dependencies — in our case “fast” pair-breaking in-phase
fluctuations and “slow” pairing out-of-phase fluctuations. The possible
physical consequences of this sign change have recently been discussed in [24]
(see also [25]) in the context of taking Coulomb repulsion into account in
conventional BCS theory [26] (in that case the Coulomb repulsion is “fast” and
the pairing due to phonons “slow”). This zero in $\Delta(\omega)$, corresponds
to a dynamical vortex in its analytic structure on the imaginary axis of the
type discussed in [24]. As in that case, upon analytic continuation, this
implies the real and imaginary parts of the gap must (in general separately)
change sign for some finite frequencies on the real axis. While this vanishing
of the real and imaginary parts of the gap may be experimentally observable
for electronic superconductors as discussed in [24], it is less clear how this
could be observed in the present case given that the particles undergoing
pairing are composite fermions which are highly nonlocal objects when viewed
in terms of physical electrons. Nonetheless, we believe the appearance of a
dynamical vortex in this pairing problem can be viewed as a “singular” version
of that discussed in [24, 25].

##  V Crossover from gauge pairing to BCS pairing

The significant effect of in-phase pair breaking discussed above suggests the
following scenario for the formation of an observable bilayer quantum Hall
state using the model interaction (16). The nonsingular parameter $V_{0}$,
which unlike $\alpha_{+}$ and $\alpha_{-}$ depends on the angular momentum
channel $l$, can be viewed as a semi-phenomenological parameter that describes
the short-range part of the interlayer interaction and can be used to drive a
crossover from the gauge pairing ($V_{0}$ irrelevant) regime to BCS ($V_{0}$
driven) pairing regime. We identify the crossover from these two different
families of flow trajectories as the trajectory that starts at the point
$V_{0}=-\sqrt{\alpha_{+}}$, $\lambda_{-}=0$ and just touches the $V=0$ axis
without crossing it (bottom red flow in both diagrams shown in Fig. 1).

Figure 4: (Color online) $V,\lambda_{-}$ flow diagrams for the case
$\alpha_{+}=0.5$ and $\gamma=1/3$. The green arrow shows initial states for
the case keeping $\alpha_{-}=0.3$ and varying $V_{0}$ so that it passes
through the crossover from the gauge pairing regime, in which pairing is
dominated by long-wavelength low energy gauge fluctuations, and strongly
suppressed by pair-breaking fluctuations, to BCS pairing in which pairing is
dominated by an attractive nonsingular pairing interaction.

Local Approximation Exact

Figure 5: (Color online). $\Delta(0)$ vs. $V_{0}$ for $\alpha_{+}=0.5$ and
$\alpha_{-}=0.3$ for the local approximation result obtained analytically and
for the exact solution obtained numerically. The local approximation results
correspond to the trajectory shown in green in Fig. 4

As the layers are brought together the coupling $\alpha_{-}$ becomes larger,
and we can imagine that the parameter $V_{0}$ becomes negative enough, i.e.
provides strong enough nonsingular pairing, to drive this crossover. We
further assume that the leading instability occurs within the appropriate
$p$-wave channel, as suggested by more detailed RPA calculations [12, 13]. It
is natural to suppose that, within the framework of the model discussed here,
this is precisely what happens when the transition from bilayer composite
fermion metal to paired state is observed in experiment.

Local Approximation Exact

Figure 6: (Color online). $\Delta(\omega)$ vs. $\omega$ for
$\alpha_{+}=0.2,0.4$ at the crossover from gauge paring to BCS pairing.
Results are shown for the local approximation, as well as the exact solution
of the gap equation obtained by iterating the gap equation, starting with the
local approximation solution. All results are for $\alpha_{-}=.1$. For the
local approximation results $V_{0}=-.252657$ for $\alpha_{+}=.2$ and
$V_{0}=-.493164$ for $\alpha_{+}=.4$. For the exact results $V_{0}=-.28$ for
$\alpha_{+}=.2$ and $V_{0}=-.6316$ for $\alpha_{+}=.4$

An example of such a crossover is sketched in the $\alpha_{+}=0.5$ flow
diagram shown in Fig. 4, with the initial flow conditions $\alpha_{-}=0.3$ and
$V_{0}$ being tuned through the crossover from gauge pairing to BCS pairing
(green arrow). The dependence of the corresponding energy gap $\Delta(0)$ on
$V_{0}$ is shown in Fig. 5, both for the local approximation and for the
numerically obtained exact solutions of the gap equation. Both solutions show
a steep increase in $\Delta(0)$ as the crossover occurs, though the value of
$V_{0}$ for the crossover is, of course, different for the approximate and
exact solutions.

We can take the crossover value for $V_{0}$ to be that value for which the
trajectory in the flow diagram just touches the $V=0$ axis. It is for this
value that the sign change in the gap function just disappears, signaling a
crossover from strongly suppressed gauge pairing to more BCS-like pairing,
which, as $V_{0}$ becomes more negative, will lead to a substantial gap
persisting all the way up to the cutoff $\omega_{0}$.

Within the local approximation, this crossover value of $V_{0}$ corresponds to
the trajectory which starts at $\lambda_{-}=0$ and $V_{0}=-\sqrt{\alpha_{+}}$.
These solutions can be found analytically within the local approximation (see
Appendix) and then used following (31) to find the “crossover” form of
$\Delta(\omega)$, and are shown for the local approximation in Fig. 6. Note
that these gap functions no longer change sign, but simply touch the
$\Delta=0$ axis at some finite imaginary frequency. Results are also shown in
Fig. 6 for the numerically obtained exact solution of the gap equation, where
we again take our criteria for the crossover value of $V_{0}$ to be that which
corresponds to $\Delta(\omega)$ just touches zero, without changing sign. For
the exact solution, it was necessary to adjust $V_{0}$ and $\alpha_{-}$, since
the critical value of the transition is not determined analytically by the
local approximation. Nonetheless, we see qualitative agreement between the
crossover gap functions for both the local approximation and the gap equation,
suggesting that the local approximation captures the essential behavior of the
solutions of (15) for the model interaction (16).

##  VI Conclusions

In this paper, we have analyzed the solution to the $T=0$ gap equation for a
model frequency-dependent interaction meant to describe the total
$\nu=1/2+1/2$ bilayer quantum Hall state starting from the limit of large
layer spacing. The effective interaction contains a singular attractive
interlayer pairing interaction, a less singular repulsive interlayer pairing
interaction, and a non-singular coupling which we expect to depend on details
of the system and also determine the pairing channel. Using the local
approximation we obtain essentially analytic approximate solutions of the gap
equation, illustrating the very strong pair-breaking effect of the less
singular in-phase gauge fluctuations. We then argue that within this
interlayer pairing scenario, the experimentally observed transition from two
compressible bilayers to an incompressible bilayer quantum Hall state may
reflect a crossover from a gauge pairing regime to a more conventional BCS-
type pairing.

###### Acknowledgements.

HD acknowledges useful discussions with Wai-Ga Ho. The National High Magnetic
Field Laboratory is supported by the National Science Foundation through
NSF/DMR1644779 and by the State of Florida. *

##  Appendix A Flow Equation Solutions

###  A.1 Large Layer Spacing Limit

Equations (32) and (33) can be combined to give a single nonlinear first-order
differential equation

$\displaystyle\gamma\lambda_{-}\frac{dV}{d\lambda_{-}}=-\lambda_{-}+\alpha_{+}-V^{2}.$
(37)

This equation can be solved analytically with the result,

$V(\lambda_{-})=\frac{\sqrt{\gamma\lambda_{-}}\left(cJ_{-\frac{2\sqrt{\alpha_{+}}}{\gamma}-1}\left(\frac{2\sqrt{\lambda_{-}}}{\sqrt{\gamma}}\right)-cJ_{1-\frac{2\sqrt{\alpha_{+}}}{\gamma}}\left(\frac{2\sqrt{\lambda_{-}}}{\sqrt{\gamma}}\right)+J_{\frac{2\sqrt{\alpha_{+}}}{\gamma}-1}\left(\frac{2\sqrt{\lambda_{-}}}{\sqrt{\gamma}}\right)-J_{\frac{2\sqrt{\alpha_{+}}}{\gamma}+1}\left(\frac{2\sqrt{\lambda_{-}}}{\sqrt{\gamma}}\right)\right)}{2\left(cJ_{-\frac{2\sqrt{\alpha_{+}}}{\gamma}}\left(\frac{2\sqrt{\lambda_{-}}}{\sqrt{\gamma}}\right)+J_{\frac{2\sqrt{\alpha_{+}}}{\gamma}}\left(\frac{2\sqrt{\lambda_{-}}}{\sqrt{\gamma}}\right)\right)}$
(38)

where $c$ is an integration constant to be fixed by the UV boundary condition
$V(\lambda_{=}0)=-\alpha_{-}+V_{0}$. The value of $\lambda_{-}$ for which $V$
then goes to $-\infty$ then corresponds to a value of $\omega$ equal to the
zero frequency gap $\Delta(0)$.

It is apparent from the flow diagrams (Fig. 1) that, for the gauge pairing
case, in the limit of large layer spacing with $\alpha_{\ll}1$ with
$V_{0}>-\sqrt{\alpha_{+}}$, the solutions flow to the scaling solution flowing
from the point $V=\sqrt{\alpha_{+}}$, $\lambda_{=}0$. This solution
corresponds to (38) with $c=0$,

$V_{lls}(\lambda_{-})=\frac{\sqrt{\gamma\lambda_{-}}\left(J_{\frac{2\sqrt{\alpha_{+}}}{\gamma}-1}\left(\frac{2\sqrt{\lambda_{-}}}{\sqrt{\gamma}}\right)-J_{\frac{2\sqrt{\alpha_{+}}}{\gamma}+1}\left(\frac{2\sqrt{\lambda_{-}}}{\sqrt{\gamma}}\right)\right)}{2\left(J_{\frac{2\sqrt{\alpha_{+}}}{\gamma}}\left(\frac{2\sqrt{\lambda_{-}}}{\sqrt{\gamma}}\right)\right)}$
(39)

The energy gap $\Delta(0)$ is then determined by finding where
$V_{lls}\rightarrow-\infty$ as $\omega$ is decreased from $\omega_{0}$ to $0$,
which occurs when the Bessel function in the denominator has its first zero,
i.e., when,

$\displaystyle
2\sqrt{\frac{\lambda_{-}}{\gamma}}=J_{2\sqrt{\alpha_{+}}/\gamma,1}$ (40)

The result for $\Delta(0)$ can then be expressed as follows

$\displaystyle\Delta(0)=\left(\frac{j_{0,1}}{j_{2\sqrt{\alpha_{+}}/\gamma,1}}\right)^{2/\gamma}\Delta(0;0)$
(41)

where

$\displaystyle\Delta_{\alpha_{+}=0}(0)=\frac{1}{j_{0,1}^{2/\gamma}}\left(\frac{4}{\gamma}\right)^{1/\gamma}=\left(\frac{C}{\gamma}\right)^{1/\gamma}\alpha_{-}^{1/\gamma}\omega_{0}.$
(42)

where

$\displaystyle C=\frac{4}{j_{0,1}^{2}}=0.69166\cdots$ (43)

The multiplicative factor which suppresses the gap compared to its
$\alpha_{+}=0$ value is

$\displaystyle\frac{\Delta(0;\alpha_{+})}{\Delta(0;0)}=\left(\frac{j_{0,1}}{j_{2\sqrt{\alpha_{+}}/\gamma,1}}\right)^{2/\gamma}$
(44)

To get a simpler expression that captures the relevant behavior of this
function, consider the limit where $\gamma\rightarrow 0$ while
$\sqrt{\alpha_{+}}/\gamma^{2}$ is held fixed. In this limit, we can Taylor
expand the quantity in parenthesis in (44), to find

$\displaystyle\frac{j_{0,1}}{j_{2\sqrt{\alpha_{+}}/\gamma},1}=1-D\frac{\sqrt{\alpha_{+}}}{\gamma^{2}}\frac{\gamma}{2}+\cdots$
(45)

where

$\displaystyle D=4\frac{d}{dx}\ln j_{x,0}\Bigr{|}_{x=0}=2.566\cdots$ (46)

We then have, (again, keeping $\sqrt{\alpha_{+}}/\gamma^{2}$ fixed),

$\displaystyle\lim_{\gamma\rightarrow
0}\frac{\Delta(0;\alpha_{+})}{\Delta(0;0)}$ $\displaystyle=$
$\displaystyle\lim_{\gamma\rightarrow
0}\left(\frac{j_{0,1}}{j_{2\sqrt{\alpha_{+}}/\gamma,1}}\right)^{2/\gamma}=\lim_{\gamma\rightarrow
0}\left(1-D\frac{\sqrt{\alpha_{+}}}{\gamma^{2}}\frac{\gamma}{2}\right)^{2/\gamma}=\exp(-D\frac{\sqrt{\alpha_{+}}}{\gamma^{2}})$
(47)

Finally, the frequency-dependent gap function $\Delta(\omega)$ can be obtained
using (31) with the overall scale set by $\Delta(0)$.

###  A.2 Gauge Paring to BCS Crossover

The crossover trajectory corresponds to the solution for which $V$ just
touches the $V=0$ axis, without changing sign (and hence the energy gap
doesn’t change sign). The value of $c$ for this case is easily obtained by
setting $V(\lambda_{-}=\alpha_{+}/\gamma)=0$, which fixes the trajectory to
include the point in the flow diagram where the $\frac{dV}{dl}=0$ curve (shown
in black in Fig. 1) intersects the $V=0$ axis. The result is

$\displaystyle
c(\alpha_{+},\gamma)=\frac{J_{\frac{2\sqrt{\alpha_{+}}}{\gamma}+1}\left(\frac{2\sqrt{\alpha_{+}}}{\gamma}\right)-J_{\frac{2\sqrt{\alpha_{+}}}{\gamma}-1}\left(\frac{2\sqrt{\alpha_{+}}}{\gamma}\right)}{J_{-\frac{2\sqrt{\alpha_{+}}}{\gamma}-1}\left(\frac{2\sqrt{\alpha_{+}}}{\gamma}\right)-J_{1-\frac{2\sqrt{\alpha_{+}}}{\gamma}}\left(\frac{2\sqrt{\alpha_{+}}}{\gamma}\right)}.$
(48)

Plugging this value of $c$ back into the expression for $V$ gives $V(\omega)$
for this case which again can be used to determine the corresponding
$\Delta(\omega)$ using (31), which just touches the $\omega$ axis, as shown in
Fig. 6.

## References

  * Moon  _et al._ [1995] K. Moon, H. Mori, K. Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and S.-C. Zhang, Phys. Rev. B 51, 5138 (1995).
  * Das Sarma and Pinczuk [1997] S. Das Sarma and A. Pinczuk, eds.,  _Perspectives in Quantum Hall Effects_ (Wiley, New York, 1997).
  * Eisenstein and MacDonald [2004] J. Eisenstein and A. MacDonald, Nature 432, 691 (2004).
  * Jain [1989a] J. K. Jain, Phys. Rev. Lett. 63, 199 (1989a).
  * Jain [1989b] J. K. Jain, Phys. Rev. B 40, 8079 (1989b).
  * Jain [2007] J. K. Jain,  _Composite fermions_ (Cambridge University Press, 2007).
  * Halperin  _et al._ [1993] B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993).
  * Son [2015] D. T. Son, Phys. Rev. X 5, 031027 (2015).
  * Sodemann  _et al._ [2017] I. Sodemann, I. Kimchi, C. Wang, and T. Senthil, Phys. Rev. B 95, 085135 (2017).
  * Bonesteel  _et al._ [1996] N. E. Bonesteel, I. A. McDonald, and C. Nayak, Phys. Rev. Lett. 77, 3009 (1996).
  * Liu  _et al._ [2022] X. Liu, J. I. A. Li, K. Watanabe, T. Taniguchi, J. Hone, B. I. Halperin, P. Kim, and C. R. Dean, Science 375, 205 (2022).
  * Isobe and Fu [2017] H. Isobe and L. Fu, Phys. Rev. Lett. 118, 166401 (2017).
  * Lotrič and Simon [2023] T. Lotrič and S. H. Simon, arXiv preprint arXiv:2309.08329 (2023).
  * Rüegg  _et al._ [2023] L. Rüegg, G. Chaudhary, and R.-J. Slager, arXiv preprint arXiv:2303.10212 (2023).
  * Möller  _et al._ [2008] G. Möller, S. H. Simon, and E. H. Rezayi, Phys. Rev. Lett. 101, 176803 (2008).
  * Möller  _et al._ [2009] G. Möller, S. H. Simon, and E. H. Rezayi, Phys. Rev. B 79, 125106 (2009).
  * Wagner  _et al._ [2021] G. Wagner, D. X. Nguyen, S. H. Simon, and B. I. Halperin, Phys. Rev. Lett. 127, 246803 (2021).
  * Cipri and Bonesteel [2014] R. Cipri and N. E. Bonesteel, Phys. Rev. B 89, 085109 (2014).
  * Bonesteel [1993] N. E. Bonesteel, Phys. Rev. B 48, 11484 (1993).
  * Wang  _et al._ [2014] Z. Wang, I. Mandal, S. B. Chung, and S. Chakravarty, Annals of Physics 351, 727 (2014).
  * Wang  _et al._ [2017] H. Wang, S. Raghu, and G. Torroba, Phys. Rev. B 95, 165137 (2017).
  * Metlitski  _et al._ [2015] M. A. Metlitski, D. F. Mross, S. Sachdev, and T. Senthil, Phys. Rev. B 91, 115111 (2015).
  * Bonesteel [1999] N. E. Bonesteel, Phys. Rev. Lett. 82, 984 (1999).
  * Christensen and Chubukov [2021] M. H. Christensen and A. V. Chubukov, Phys. Rev. B 104, L140501 (2021).
  * Pimenov and Chubukov [2022] D. Pimenov and A. V. Chubukov, Annals of Physics 447, 169049 (2022).
  * Morel and Anderson [1962] P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 (1962).