Counting Majorana bound states using complex momenta

Recently, the connection between Majorana fermions bound to the defects in arbitrary dimensions, and complex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov–de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, 110, 67005). Based on this understanding, a for...

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Бібліографічні деталі
Дата:	2016
Автор:	Mandal, I.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 2016
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/156221
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Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Counting Majorana bound states using complex momenta / I. Mandal // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33703: 1–21. — Бібліогр.: 56 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-156221
record_format	dspace
spelling	irk-123456789-1562212019-06-19T01:25:26Z Counting Majorana bound states using complex momenta Mandal, I. Recently, the connection between Majorana fermions bound to the defects in arbitrary dimensions, and complex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov–de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, 110, 67005). Based on this understanding, a formula has been proposed to count the number (n) of the zero energy Majorana bound states, which is related to the topological phase of the system. In this paper, we provide a proof of the counting formula and we apply this formula to a variety of 1d and 2d models belonging to the classes BDI, DIII and D. We show that we can successfully chart out the topological phase diagrams. Studying these examples also enables us to explicitly observe the correspondence between these complex momentum solutions in the Fourier space, and the localized Majorana fermion wavefunctions in the position space. Finally, we corroborate the fact that for systems with a chiral symmetry, these solutions are the so-called “exceptional points”, where two or more eigenvalues of the complexified Hamiltonian coalesce. Нещодавно (EPL, 2015, 110, 67005) було встановлено зв’язок мiж фермiонами Майорани, зв’язаними з дефектами у довiльнiй вимiрностi, i комплексними iмпульсними коренями детермiнанта вiдповiдного об’ємного гамiльтонiану Боголюбова-де Жена. Базуючись на цьому розумiннi, запропоновано формулу для пiдрахунку числа (n) зв’язаних станiв Майорани з нульовою енергiєю, якi пов’язанi з топологiчною фазою системи. В цiй статтi дається вивiд формули пiдрахунку, яка застосовується до низки 1d i 2d моделей, що належать до класiв BDI, DIII i D. Показано, як можна успiшно побудувати топологiчнi фазовi дiаграми. Вивчення даних прикладiв дозволяє явно спостерiгати вiдповiднiсть мiж цими комплексними розв’язками для iмпульсу в Фур’є просторi i локалiзованими хвильовими функцiями фермiонiв Майорани в позицiйному просторi. Накiнець, пiдтверджено факт, що для систем з хiральною симетрiєю цi розв’язки є так званими “винятковими точками”, де два чи бiльше власних значень ускладненого гамiльтонiана зливаються. 2016 Article Counting Majorana bound states using complex momenta / I. Mandal // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33703: 1–21. — Бібліогр.: 56 назв. — англ. 1607-324X PACS: 73.20.-r, 74.78.Na, 03.65.Vf DOI:10.5488/CMP.19.33703 arXiv:1503.06804 http://dspace.nbuv.gov.ua/handle/123456789/156221 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Recently, the connection between Majorana fermions bound to the defects in arbitrary dimensions, and complex momentum roots of the vanishing determinant of the corresponding bulk Bogoliubov–de Gennes (BdG) Hamiltonian, has been established (EPL, 2015, 110, 67005). Based on this understanding, a formula has been proposed to count the number (n) of the zero energy Majorana bound states, which is related to the topological phase of the system. In this paper, we provide a proof of the counting formula and we apply this formula to a variety of 1d and 2d models belonging to the classes BDI, DIII and D. We show that we can successfully chart out the topological phase diagrams. Studying these examples also enables us to explicitly observe the correspondence between these complex momentum solutions in the Fourier space, and the localized Majorana fermion wavefunctions in the position space. Finally, we corroborate the fact that for systems with a chiral symmetry, these solutions are the so-called “exceptional points”, where two or more eigenvalues of the complexified Hamiltonian coalesce.
format	Article
author	Mandal, I.
spellingShingle	Mandal, I. Counting Majorana bound states using complex momenta Condensed Matter Physics
author_facet	Mandal, I.
author_sort	Mandal, I.
title	Counting Majorana bound states using complex momenta
title_short	Counting Majorana bound states using complex momenta
title_full	Counting Majorana bound states using complex momenta
title_fullStr	Counting Majorana bound states using complex momenta
title_full_unstemmed	Counting Majorana bound states using complex momenta
title_sort	counting majorana bound states using complex momenta
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/156221
citation_txt	Counting Majorana bound states using complex momenta / I. Mandal // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33703: 1–21. — Бібліогр.: 56 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT mandali countingmajoranaboundstatesusingcomplexmomenta
first_indexed	2023-05-20T17:49:12Z
last_indexed	2023-05-20T17:49:12Z
_version_	1796154165840314368

Counting Majorana bound states using complex momenta

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