We show that for closed finite-sized systems with an odd number of real fermionic modes, even in the presence of many-body interactions, there are always at least two fermionic operators that commute with the Hamiltonian. There is a zero mode corresponding to the total Majorana operator, as shown by Akhmerov [Phys. Rev. B 82, 020509 (2010)], as well as additional linearly independent zero modes, one of which is (1) continuously connected to the Majorana mode solution in the noninteracting limit, and (2) less prone to decoherence when the system is opened to contact with an infinite bath. We also show that in the idealized situation where there are two or more well-separated zero modes each associated with a finite number of interacting fermions at a localized vortex, these modes have non-Abelian Ising statistics under braiding. Furthermore the algebra of the zero mode operators makes them useful for fermionic quantum computation [S. B. Bravii and A. Y. Kitaev, Ann. Phys. 298, 210 (2002)].
Abstract I explicitly construct a strong zero mode in the XYZ chain or, equivalently, Majorana wires coupled via a four-fermion interaction. The strong zero mode is an operator that pairs states in different symmetry sectors, resulting in identical spectra up to exponentially small finite-size corrections. Such pairing occurs in the Ising/Majorana fermion chain and possibly in strongly disordered many-body localized phases. The proof here shows that the strong zero mode occurs in a clean interacting system, and that it possesses some remarkable structure—despite being a rather elaborate operator, it squares to the identity. Eigenstate phase transitions separate regions with different strong zero modes.
Strong Zero Modes (SZMs) are (approximately) conserved quantities that result in (approximate) double degeneracies in the entire spectra of certain Hamiltonians, with the Majorana zero mode of the transverse-field Ising chain being a primary example. In this work, we discover via a systematic search that many examples of SZMs can be understood as symmetries in the commutant algebra framework, which reveals novel algebraic structures hidden in Hamiltonians with well-known SZMs, including the transverse-field Ising chain. Our findings unify the understanding of different examples of SZMs in the literature, demystify their connections to ground state phases of matter, and reveal novel symmetries in simple models, such as exact quasilocal
U(1) symmetries that sometimes accompany the SZMs such as in the spin-1/2 XY model for certain parameter values. Moreover, while analytically tractable SZMs have mostly been demonstrated only for non-interacting or integrable models, the algebraic structures revealed in this work can be exploited to construct integrability-breaking interactions that exactly preserve these SZMs. Such non-integrable models are expected to show more clear dynamical signatures of SZMs without the interference of other conserved quantities that appear in integrable models, and we discuss many examples, including those of novel hydrodynamic modes associated with such symmetries for some special parameter values. We also show that while this commutant understanding extends to the non-interacting limit of the celebrated Fendley SZM in the spin-1/2 XYZ chain, the SZM in the interacting case cannot be understood in this framework. This suggests that there are two types of SZMs -- those that survive integrability breaking and those that do not. We close by using this commutant understanding to construct an alternate proof of the Fendley SZM, which might be of independent interest.
Abstract For the many-body-localized phase of random Majorana models, a general strong disorder real-space renormalization procedure known as RSRG-X (Pekker et al 2014 Phys. Rev . X 4 011052) is described to produce the whole set of excited states, via the iterative construction of the local integrals of motion (LIOMs). The RG rules are then explicitly derived for arbitrary quadratic Hamiltonians (free-fermions models) and for the Kitaev chain with local interactions involving even numbers of consecutive Majorana fermions. The emphasis is put on the advantages of the Majorana language over the usual quantum spin language to formulate unified RSRG-X rules.
Abstract We report the emergence of exact zero modes (EZMs) in junctions of two, three and four short interacting Majorana wires, equivalent to a chain with an impurity bond, Y - and X - junctions respectively. These EZMs arise from incommensurate short-range correlations and manifest as exact level crossings between in-gap states upon continuously tuning the interaction strength. In a junction of two chains the EZMs and parity switching appear as soon as the coupling across a junction is positive. This conclusion has a broad implication including the existence of EZM in quench- disordered systems. Remarkably, for junctions with multiple chains the in-gap states group up into sets of parity pairs–pairs of quasi-degenerate states with opposite parity attributed to the effective interaction of the outer edges of the junction. In the Y -junction with four in-gap states, we report EZMs not only within each parity pair but also between them. We demonstrate that this is attributed to an additional Majorana fermion localized at the center of the junction and protected by fermion parity symmetry. Therefore, coupling between the Majorana fermions at the outer edges of the junction is mediated by the central one. We argue that this is a generic feature of junctions with an odd number of arms. Surprisingly, in the X -junction we identify eight in-gap states arising from two Majorana degrees of freedom localized at the junction center. In contrast to the Y -junction, however, we show that the emergence of Majorana modes at the center of the X -junction is not topologically protected. As a result, the system parameters can be tuned such that the Majorana modes are expelled from the junction center and remain localized exclusively at the outer edges of the system.
In the topological phase of
p-wave superconductors, zero-energy Majorana quasiparticle excitations can be well defined in the presence of local density-density interactions. Here, we examine this phenomenon from the perspective of matrix representations of the commutator $\mathcal{H}=[H,\ifmmode\bullet\else\textbullet\fi{}]$, with the aim of characterizing the multiparticle content of the many-body Majorana mode. To do this we show that, for quadratic fermionic systems,
H can always be decomposed into subblocks that act as multiparticle generalizations of the Bogoliubov--de Gennes/Majorana forms that encode single-particle excitations. In this picture, density-density-like interactions will break this exact excitation-number symmetry, coupling different subblocks and lifting degeneracies so that the eigenoperators of the commutator
H take the form of individual eigenstate transitions $|\phantom{\rule{0.16em}{0ex}}n\ensuremath{\rangle}\ensuremath{\langle}m\phantom{\rule{0.16em}{0ex}}|$. However, the Majorana mode is special in that zero-energy transitions are not destroyed by local interactions and it becomes possible to define many-body Majoranas as the odd-parity zero-energy solutions of
H that minimize their excitation number. This idea forms the basis for an algorithm which is used to characterize the multiparticle excitation content of the Majorana zero modes of the one-dimensional
p-wave lattice model. We find that the multiparticle content of the Majorana zero-mode operators is significant even at modest interaction strengths. This has important consequences for the stability of Majorana-based qubits when they are coupled to a heat bath. We will also discuss how these findings differ from previous work regarding the structure of the many-body Majorana operators and point out that this should affect how certain experimental features are interpreted.