We present evidence for Majorana edge states in a number conserving theory describing a system of spinless fermions on two wires that are coupled by pair hopping. Our analysis is based on a combination of a qualitative low energy approach and numerical techniques using the density matrix renormalization group. In addition, we discuss an experimental realization of pair-hopping interactions in cold atom gases confined in optical lattices.
We present evidence for Majorana edge states in a number conserving theory describing a system of spinless fermions on two wires that are coupled by pair hopping. Our analysis is based on a combination of a qualitative low energy approach and numerical techniques using the density matrix renormalization group. In addition, we discuss an experimental realization of pair-hopping interactions in cold atom gases confined in optical lattices.
The remarkable properties and potential applications of Majorana fermions have led to considerable efforts in recent years to realize topological matters that host these excitations. For a number conserving system, there have been a few proposals, using either coupled chain models or a multi-component system with spin–orbit coupling (SOC), to create number fluctuation of fermion pairs in achieving Majorana fermions. In this work, we show that Majorana edge states can occur in a spinless Fermi gas in 1D lattices with tunable p -wave interaction. This is facilitated by the conversion between a pair of (open-channel) fermions and a (close channel) boson, thereby allowing the number fluctuation of fermion pairs in a single chain. This scheme requires neither SOC nor a multi-chain setup and can be implemented easily. Using the density-matrix-renormalization-group method, we have identified the Majorana phase in a wide range of parameter regimes as well as its associated phase transitions. The topological nature of the Majorana phase manifests itself in a strong edge–edge correlation in an open chain that is robust against disorder, as well as in a non-trivial winding number in the bulk generated by using twisted boundary condition. It is also shown that the Majorana phase in this system can be stable against atom losses due to few-body collisions on the same site, and can be easily identified from the fermion momentum distribution. These results pave the way for probing the intriguing Majorana physics in a simple and stable cold atoms system.
We devise a number-conserving scheme for the realization of Majorana zero modes in an interacting fermionic ladder coupled by Aharonov-Bohm cages. The latter provide an efficient mechanism to cancel single-particle hopping by destructive interference. The crucial parity symmetry in each wire is thus encoded in the geometry of the setup, in particular, its translation invariance. A generic nearest-neighbor interaction generates the desired correlated hopping of pairs. We exhibit the presence of an extended topological region in parameter space, first in a simplified effective model via bosonization techniques, and subsequently in a larger parameter regime with matrix-product-states numerical simulations. We demonstrate the adiabatic connection to previous models, including exactly solvable ones, and we briefly comment on possible experimental realizations in synthetic quantum platforms, like cold atomic samples.
Topological superconductors are believed to host exotic quasiparticle excitations known as Majorana zero modes (MZMs), with much of the evidence based on BCS mean-field theory. The direct application of mean-field arguments is tenuous in finite, isolated systems relevant in some experiments. Here, we develop a new correlation-based method for identifying MZMs in interacting, number-conserving systems. Using the density matrix renormalization group, we study fermion number-conserving models with long-range interactions, which under periodic boundary conditions exhibit robust topological and nontopological superconductivity, tuned by the strength of interaction [Ortiz et al., Phys. Rev. Lett. 113, 267002 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.267002]. We find evidence that, on the topological side, Majorana edge modes appear in open chains, manifesting as the vanishing of the energy splitting between odd- and even-parity ground states with increasing system size. Additionally, off-diagonal two-point correlation functions show nonlocal, parity-dependent edge effects. These correlations reveal the spatial structure of Majorana modes in the many-body wave function. We show that the correlation diagnostic applies broadly, including to short-range interacting models, where topological superconductivity is more fragile due to the absence of a bulk excitation gap.
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 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.
In this Letter we present, in a number conserving framework, a model of interacting fermions in a two-wire geometry supporting nonlocal zero-energy Majorana-like edge excitations. The model has an exactly solvable line, on varying the density of fermions, described by a topologically nontrivial ground state wave function. Away from the exactly solvable line we study the system by means of the numerical density matrix renormalization group. We characterize its topological properties through the explicit calculation of a degenerate entanglement spectrum and of the braiding operators which are exponentially localized at the edges. Furthermore, we establish the presence of a gap in its single particle spectrum while the Hamiltonian is gapless, and compute the correlations between the edge modes as well as the superfluid correlations. The topological phase covers a sizable portion of the phase diagram, the solvable line being one of its boundaries.
In this Proceedings we report some recent results on Majorana quasi-particles published in Phys. Rev. Lett. 118, 200404 (2017). We show how angular momentum conservation can stabilise a symmetry-protected quasi-topological phase of matter supporting Majorana quasi-particles as edge modes in one-dimensional cold atom gases. We investigate a number-conserving four-species Hubbard model in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional reservoirs. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated by density-matrix-renormalization-group simulations. Our results pave the way toward the observation of Majorana edge modes with alkaline-earth-like fermions in optical lattices, where all basic ingredients for our recipe - spin-orbit coupling and strong inter-orbital interactions - have been experimentally realized over the last two years.