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Bogoliubov-De Gennes Formalism for Tracking Free Majoranas on Ultra Cold Fermi Gases

A.A. Pérez-Losada, Karen Rodríguez, A. Argüelles

Published Oct 2, 2018• 0 citations
Publisher

Abstract

The collective excitations of a system, particularly at low temperatures, behave as quasi-particles with properties that may differ from their constituents.In particular, there are exotic excitations called Majorana fermions or zero-energy modes modes, which have the special characteristic of being their own antiparticle.In that sense, this work seeks tracing these excitations in an artificial arrangement of a nanowire simulated through a one-dimensional chain.In that direction, the Fourier transform is used with the purpose of bringing the Hamiltonian from position to momentum space.Subsequently, it is proposed to diagonalize the system using the Bogoliubov-de Gennes formalism.In this way, we obtain the phase diagram displaying the set of parameters for which zero energy modes are stabilized.

DOI
10.7149/opa.51.3.50313

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The collective excitations of a system, particularly at low temperatures, behave as quasi-particles with properties that may differ from their constituents.In particular, there are exotic excitations called Majorana fermions or zero-energy modes modes, which have the special characteristic of being their own antiparticle.In that sense, this work seeks tracing these excitations in an artificial arrangement of a nanowire simulated through a one-dimensional chain.In that direction, the Fourier transform is used with the purpose of bringing the Hamiltonian from position to momentum space.Subsequently, it is proposed to diagonalize the system using the Bogoliubov-de Gennes formalism.In this way, we obtain the phase diagram displaying the set of parameters for which zero energy modes are stabilized.
In the topological phase of ppp-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\mathcal{H}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\mathcal{H}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\mathcal{H}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 ppp-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.
The simplest continuum model of a one-dimensional noninteracting superconducting fermionic symmetry-protected topological (SPT) phase is studied in great detail using analytical methods. In a first step, we present a full exact diagonalization of the fermionic Bogoliubov--de Gennes Hamiltonian for a system of finite length and with open boundaries. In particular, we derive exact analytical expressions for the Majorana zero modes emerging in the topologically nontrivial phase, revealing their spatial localization, their transformation properties under symmetry operations, and the exact finite-size energy splitting of the associated quasidegenerate ground states. We then proceed to analyze the model via exact operator bosonization in both open and closed geometries. In the closed wire geometry, we demonstrate fermion parity switching from twisting boundary conditions in the topologically nontrivial phase. For the open wire, on the other hand, we first take a semiclassical approach employing the Mathieu equation to study the two quasidegenerate ground states as well as their energy splitting at finite system sizes. We then finally derive the exact forms of the Majorana zero modes in the bosonic language using vertex-algebra techniques. These modes are verified to be in exact agreement with the results obtained from the fermionic description. The complementary viewpoints provided by the fermionic and bosonic formulations of the superconducting SPT phase are reconciled, allowing us to provide a complete and exact account of how Majorana zero modes manifest in a bosonized description of an SPT phase.
Abstract By solving the Bogoliubov–de Gennes equations analytically, we derive the fermionic zero-modes satisfying the Majorana property that exist in vortices of a two-dimensional s-wave Fermi superfluid with spin–orbit coupling and Zeeman spin-splitting. The Majorana zero-mode becomes normalisable and exponentially localised to the vicinity of the vortex core when the superfluid is topologically non-trivial. We calculate the energy splitting due to Majorana hybridisation and identify that the s-wave Majorana vortices obey non-Abelian statistics.
It is quite common that several different phases exist simultaneously in a system of trapped quantum gases of ultra-cold atoms. One example is the strongly interacting Fermi gas with two imbalanced spin species, which has received a great deal of attention owing to the possible occurrence of exotic superfluid phases. By using novel numerical techniques and algorithms, we self-consistently solve the Bogoliubov–de Gennes equations, which describe Fermi superfluids in the mean-field framework. From this study, we investigate the novel phases of spin-imbalanced Fermi gases and examine the validity of local density approximation (LDA), which is often invoked in the extraction of bulk properties from experimental measurements within trapped systems. We show how the validity of the LDA is affected by the trapping geometry, the number of atoms and the spin imbalance.
One promising avenue to study one-dimensional (1D) topological phases is to realize them in synthetic materials such as cold atomic gases. Intriguingly, it is possible to realize Majorana boundary modes in a 1D number-conserving system consisting of two fermionic chains coupled only by pair-hopping processes [C. V. Kraus et al., Phys. Rev. Lett. 111, 173004 (2013)]. It is commonly believed that significant interchain single-particle tunneling necessarily destroys these Majorana modes, as it spoils the Z2{\mathbb{Z}}_{2}Z2​ fermion-parity symmetry that protects them. In this Rapid Communication, we present a mechanism to overcome this obstacle, by piercing a (synthetic) magnetic $\ensuremath{\pi}$ flux through each plaquette of the Fermi ladder. Using bosonization, we show that in this case there exists an exact leg-interchange symmetry that is robust to interchain hopping, and acts as fermion parity at long wavelengths. We utilize density matrix renormalization group and exact diagonalization to verify that the resulting model exhibits Majorana boundary modes up to large single-particle tunnelings, comparable to the intrachain hopping strength. Our work highlights the unusual impacts of different topologically trivial band structures on these interaction-driven topological phases, and identifies a distinct route to stabilizing Majorana boundary modes in 1D fermionic ladders.
Exact self-consistent soliton dynamics based on the Bogoliubov--de Gennes (BdG) formalism in unconventional Fermi superfluids/superconductors possessing an SU(d)\text{SU}(d)SU(d)-symmetric two-body interaction is presented. The derivation is based on the ansatz having the similar form as the Gelfand-Levitan-Marchenko equation in the inverse scattering theory. Our solutions can be regarded as a multicomponent generalization of the solutions recently derived by Dunne and Thies [Phys. Rev. Lett. 111, 121602 (2013)]. We also propose superpositions of occupation states, which make it possible to realize various filling rates even in one-flavor systems, and include Dirac and Majorana fermions. The soliton solutions in the d=2d=2d=2 systems, which describe the mixture of singlet sss-wave and triplet ppp-wave superfluids, exhibit a variety of phenomena such as rotating polar phases by soliton spins, SU(2)-DHN breathers, Majorana triplet states, $s\ensuremath{-}p$ mixed dynamics, and so on. These solutions are illustrated by animations, where order parameters are visualized by spherical harmonic functions. The full formulation of the BdG theory is also supported, and the double-counting problem of BdG eigenstates and NNN-flavor generalization are discussed.