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arXiv:2311.07359 · Physical review. B./Physical review. B

Electronic ladder model harboring Z 4 parafermions

Botond Osváth, Gergely Barcza, Örs Legeza, Balázs Dóra, László Oroszlány

Published Aug 23, 2024• 1 citation
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Abstract

Parafermions are anyons with the potential for realizing nonlocal qubits that are resilient to local perturbations. Compared with Majorana zero modes, braiding of parafermions implements an extended set of topologically protected quantum gates. This, however, comes at the price that parafermionic zero modes cannot be realized in the absence of strong interactions, posing a challenge for their theoretical depiction. In this paper, we construct a simple lattice model for interacting spinful electrons with parafermionic zero-energy modes. The explicit microscopic nature of the considered model highlights realization avenues for these exotic excitations in recently fabricated quantum dot arrays. By density matrix renormalization group calculations, we identify a broad range of parameters, with well-localized zero modes, whose parafermionic nature is substantiated by their unique $8\ensuremath{\pi}$ periodic Josephson spectrum.

DOI
10.1103/physrevb.110.085304

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Parafermions are anyons with the potential for realizing nonlocal qubits that are resilient to local perturbations. Compared with Majorana zero modes, braiding of parafermions implements an extended set of topologically protected quantum gates. This, however, comes at the price that parafermionic zero modes cannot be realized in the absence of strong interactions, posing a challenge for their theoretical depiction. In this paper, we construct a simple lattice model for interacting spinful electrons with parafermionic zero-energy modes. The explicit microscopic nature of the considered model highlights realization avenues for these exotic excitations in recently fabricated quantum dot arrays. By density matrix renormalization group calculations, we identify a broad range of parameters, with well-localized zero modes, whose parafermionic nature is substantiated by their unique 8\ensuremathπ8\ensuremath{\pi}8\ensuremathπ periodic Josephson spectrum.
Parafermion modes are non-Abelian anyons which were introduced as ZN{\mathbb{Z}}_{N}ZN​ generalizations of Z2{\mathbb{Z}}_{2}Z2​ Majorana states. In particular, Z3{\mathbb{Z}}_{3}Z3​ parafermions can be used to produce Fibonacci anyons, laying a path towards universal topological quantum computation. Due to their fractional nature, much of the theoretical work on Z3{\mathbb{Z}}_{3}Z3​ parafermions has relied on bosonization methods or parafermionic quasiparticles. In this paper, we introduce a representation of Z3{\mathbb{Z}}_{3}Z3​ parafermions in terms of purely fermionic models. We establish the equivalency of a family of lattice fermionic models written in the basis of the $t\ensuremath{-}J$ model with a Kitaev-like chain supporting free Z3{\mathbb{Z}}_{3}Z3​ parafermionic modes at its ends. By using density matrix renormalization group calculations, we are able to characterize the topological phase transition and study the effect of local operators (doping and magnetic fields) on the spatial localization of the parafermionic modes and their stability. Moreover, we discuss the necessary ingredients towards realizing Z3{\mathbb{Z}}_{3}Z3​ parafermions in strongly interacting electronic systems.
Shuttling of Z4\mathbb{Z}_4Z4​ parafermions in an electronic ladder model
Parafermions with non-Abelian statistics have been proposed as a promising platform for quantum computation, potentially enabling a broader set of topologically protected gates than Majorana fermions. The experimental and theoretical exploration of these exotic quasiparticles remains challenging, as their stability is linked to strong electron-electron interactions. A key step toward practical applications is the controlled shuttling of parafermionic modes, which is required for implementing geometric braiding operations. In the present work, we investigate the real-time dynamics of the elementary shuttling process by applying a combination of the density matrix renormalization group and the time-dependent variational principle approaches. We analyze the transport of Z4\mathbb{Z}_4Z4​ parafermion edge states and assess the corresponding adiabatic speed limit under experimentally relevant conditions.
Parafermionic zero modes, Zn{\mathbb{Z}}_{n}Zn​-symmetric generalizations of the well-known Z2{\mathbb{Z}}_{2}Z2​ Majorana zero modes, can emerge as edge states in topologically nontrivial strongly correlated systems displaying fractionalized excitations. In this paper, we investigate how signatures of parafermionic zero modes can be detected by its effects on the properties of a quantum dot tunnel-coupled to a system hosting such states. Concretely, we consider a strongly correlated one-dimensional fermionic model supporting Z4{\mathbb{Z}}_{4}Z4​ parafermionic zero modes coupled to an interacting quantum dot at one of its ends. By using a combination of density matrix renormalization group calculations and analytical approaches, we show that the dot's zero-energy spectral function and average occupation numbers can be used to distinguish between trivial, Z4{\mathbb{Z}}_{4}Z4​ and $2\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ phases of the system. The present work opens the prospect of using quantum dots as detection tools to probe nontrivial topological phases in strongly correlated systems.
Parafermion zero modes are generalizations of Majorana modes that underlie comparatively rich non-Abelian-anyon properties. We introduce exact mappings that connect parafermion chains, which can emerge in two-dimensional fractionalized media, to strictly one-dimensional fermionic systems. In particular, we show that parafermion zero modes in the former setting translate into symmetry-enriched Majorana modes that intertwine with a bulk order parameter---yielding braiding and fusion properties that are impossible in standard Majorana platforms. Fusion characteristics of symmetry-enriched Majorana modes are directly inherited from the associated parafermion setup and can be probed via two kinds of anomalous pumping cycles that we construct. Most notably, our mappings relate Z4{\mathbb{Z}}_{4}Z4​ parafermions to conventional electrons with time-reversal symmetry. In this case, one of our pumping protocols entails fairly minimal experimental requirements: Cycling a weakly correlated wire between a trivial phase and time-reversal-invariant topological superconducting state produces an edge magnetization with quadrupled periodicity. Our work highlights new avenues for exploring beyond-Majorana physics in experimentally relevant one-dimensional electronic platforms, including proximitized ferromagnetic chains.
Parafermions are non-Abelian anyons which generalize Majorana fermions and hold great promise for topological quantum computation. We study the braiding of Z2n{\mathbb{Z}}_{2n}Z2n​ parafermions which have been predicted to emerge as localized zero modes in fractional quantum Hall systems at filling factor $\ensuremath{\nu}=1/n$ (nnn odd). Using a combination of bosonization and refermionization, we calculate the energy splitting as a function of distance and chemical potential for a pair of parafermions separated by a gapped region. Braiding of parafermions in quantum Hall edge states can be implemented by repeated fusion and nucleation of parafermion pairs. We simulate the conventional braiding protocol of parafermions numerically, taking into account the finite separation and finite chemical potential. We show that a nonzero chemical potential poses challenges for the adiabaticity of the braiding process because it leads to accidental crossings in the spectrum. To remedy this, we propose an improved braiding protocol which avoids those degeneracies.
Because of potential relevance to topological quantum information processing, we introduce and study the self-dual family of representations of the braid group. Self-dual representations are physically well motivated and provide a natural generalization of the Majorana and Gaussian representations, which appear as particular instances. To show that self-dual representations admit a particle interpretation, we introduce and describe in second quantization a family of particle species with $p=2,3,\ensuremath{\cdots}$ exclusion and $\ensuremath{\theta}=2\ensuremath{\pi}/p$ exchange statistics. We call these anyons Fock parafermions, because they are the particles naturally associated to the parafermionic zero-energy modes, potentially realizable in mesoscopic arrays of fractional topological insulators. Josephson junctions modeled with Fock parafermions may display a $2\ensuremath{\pi}p$ periodic relation between the Josephson current and the phase difference across the junction. Self-dual representations can be realized in terms of local quadratic combinations of either parafermions or Fock parafermions, an important requisite for physical implementation of quantum logic gates. The second-quantization description of Fock parafermions entails the concept of Fock algebra, i.e., a Fock space endowed with a statistical multiplication that captures and logically correlates these anyons' exclusion and exchange statistics. As a consequence normal ordering continues to be a well-defined operation.