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arXiv:1711.11064

Majorana Zero Modes in Synthetic Dimensions

Or Golan, Eran Sela, Kirill Shtengel

Published Nov 29, 2017• 0 citations
arXivHTMLPDF

Abstract

Recent experimental advances in the field of cold atoms led to the development of novel techniques for producing synthetic dimensions and synthetic magnetic fields, thus greatly expanding the utility of cold atomic systems for exploring exotic states of matter. In this paper we investigate the possibility of using experimentally tunable interactions in such systems to mimic the physics of Majorana chains, currently a subject of intense research. Crucially to our proposal, the interactions, which are local in space, appear non-local in the synthetic dimension. We use this fact to induce coupling between counter-propagating edge modes in the quantum Hall regime. For the case of attractive interactions in a system composed of two tunneling-coupled chains, we find a gapless quasi-topological phase with a doubly-degenerate ground state. While the total number of particles in the system is kept fixed, this phase is characterized by strong fluctuations of the pair number in each chain. Each ground state is characterized by the parity of the total particle number in each chain, similar to Majorana wires. However, in our system this degeneracy persists for periodic boundary conditions. For open boundary conditions there is a small splitting of this degeneracy due to the single-particle hopping at the edges. We show how subjecting the system to additional synthetic flux or asymmetric potentials on the two chains can be used to control this nonlocal qubit. We propose experimental probes for testing the nonlocal nature of such a qubit and measuring its state.

DOI
10.48550/arxiv.1711.11064

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Recent experimental advances in the field of cold atoms led to the development of novel techniques for producing synthetic dimensions and synthetic magnetic fields, thus greatly expanding the utility of cold atomic systems for exploring exotic states of matter. In this paper we investigate the possibility of using experimentally tunable interactions in such systems to mimic the physics of Majorana chains, currently a subject of intense research. Crucially to our proposal, the interactions, which are local in space, appear non-local in the synthetic dimension. We use this fact to induce coupling between counter-propagating edge modes in the quantum Hall regime. For the case of attractive interactions in a system composed of two tunneling-coupled chains, we find a gapless quasi-topological phase with a doubly-degenerate ground state. While the total number of particles in the system is kept fixed, this phase is characterized by strong fluctuations of the pair number in each chain. Each ground state is characterized by the parity of the total particle number in each chain, similar to Majorana wires. However, in our system this degeneracy persists for periodic boundary conditions. For open boundary conditions there is a small splitting of this degeneracy due to the single-particle hopping at the edges. We show how subjecting the system to additional synthetic flux or asymmetric potentials on the two chains can be used to control this nonlocal qubit. We propose experimental probes for testing the nonlocal nature of such a qubit and measuring its state.
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.
We introduce a tunable synthetic-dimension platform for realizing Kitaev-chain physics with high degree of control over Majorana zero modes. It is based on a generic Landau-quantized two dimensional electron system coupled to the magnetic flux of a superconducting LC circuit. The structured vector potential of a superconducting LC inductor induces attractive interactions between electron angular-momentum states at the lowest Landau level. These states serve as a synthetic dimension for the coveted fermionic Kitaev chain, with Majorana zero modes existing at the boundaries of the angular-momentum lattice. The crucial advantage of this proposal is the possibility of a robust, nonlocal readout and control of the Majorana states by a LC resonator. The platform relies on mature circuit QED and semiconductor technologies and provides a promising pathway to topological quantum computing.
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.
Majorana zero modes are the hallmark of topological superconductivity. In one-dimensional systems, these zero modes are usually introduced in the context of gapped, mean-field models that do not conserve particle number, such as the Kitaev chain. By non-locally encoding a conventional fermion across spatially separated Majorana zero modes, these systems become inherently immune to local decoherence. In this work, we show that signatures of Majorana edge physics persist in a number-conserving, gapless Luttinger liquid of spinless fermions with short-range attractive interactions. We identify the two-point correlator as a sharp diagnostic, revealing an edge-to-edge revival whose sign depends on the fermion-number parity. This revival is robust in the thermodynamic limit, and persists in the excited states of the system and at different fillings. A simple particle-hole ansatz for the ground state of the system with an odd number of fermions captures the physics of the system for a wide range of interaction strengths, interpolating between the free-fermion limit and the strongly interacting Majorana regime. Finally, we propose a concrete protocol to realize this model with ultracold dipolar molecules or atoms in an optical lattice, and to detect the revival via beam-splitter interferometry, opening an experimental route to Majorana physics beyond the conventional gapped-superconductor paradigm.
A key hallmark of quantum Hall physics is the existence of topological chiral states at the system boundary. Signatures of these edge states have been experimentally observed in cold atoms by using different approaches, including notably that of “synthetic dimensions” in which internal states are coupled together and reinterpreted as sites along an artificial spatial dimension. However, previous atomic synthetic dimension implementations have been limited to relatively small system sizes with inflexible boundaries. In this paper, we propose instead how to use a synthetic dimension of atomic trap states to observe chiral edge states in a large quantum Hall system with a tunable edge. We present numerical simulations for relevant experimental parameters, showing how this scheme may be used to probe the properties and robustness of the edge states to defects. Our work opens the way for future experiments in topological physics with synthetic dimensions, while also providing new ways to manipulate and control highly excited trap states.
Topological phases of matter that depend for their existence on interactions are fundamentally interesting and potentially useful as platforms for future quantum computers. Despite the multitude of theoretical proposals, the only interaction-enabled topological phase experimentally observed is the fractional quantum Hall liquid. To help identify other systems that can give rise to such phases, we present in this work a detailed study of the effect of interactions on Majorana zero modes bound to vortices in a superconducting surface of a three-dimensional topological insulator. This system is of interest because, as was recently pointed out, it can be tuned into the regime of strong interactions. We start with a zero-dimensional system suggesting an experimental realization of the interaction-induced Z8{\mathbb{Z}}_{8}Z8​ ground-state periodicity previously discussed by Fidkowski and Kitaev [Phys. Rev. B 81, 134509 (2010); Phys. Rev. B 83, 075103 (2011)] . We argue that the periodicity is experimentally observable using a tunnel probe. We then focus on interaction-enabled crystalline topological phases that can be built with the Majoranas in a vortex lattice in higher dimensions. In one dimension, we identify an interesting exactly solvable model which is related to a previously discussed one that exhibits an interaction-enabled topological phase. We study these models using analytical techniques, exact numerical diagonalization, and density matrix renormalization group. Our results confirm the existence of the interaction-enabled topological phase and clarify the nature of the quantum phase transition that leads to it. We finish with a discussion of models in dimensions 2 and 3 that produce similar interaction-enabled topological phases.