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.
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.
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.
The realization and manipulation of topological Majorana zero modes (MZMs) have been pivotal in both theoretical and experimental research. In this paper, we present a symmetry enhanced topological model that supports stable, localized MZMs in a one-dimensional interacting fermionic system. The MZMs in our model, induced by particle-number conserving interactions, are fully decoupled from the chain's interior and remain degenerate regardless of chain length, allowing for simulations in extremely short fermionic chains. The topological phase transition is driven by changes in coupling parameters, transitioning to a trivial phase without affecting the MZMs' independence or degeneracy, only their topological protection. Additionally, local symmetries in the model enhance MZMs' stability and reveal unique odd-even effects across different chain lengths.
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.
We study a system of interacting spinless fermions in one dimension that, in the absence of interactions, reduces to the Kitaev chain [Kitaev, Phys. Usp. 44, 131 (2001)]. In the noninteracting case, a signal of topological order appears as zero-energy modes localized near the edges. We show that the exact ground states can be obtained analytically even in the presence of nearest-neighbor repulsive interactions when the on-site (chemical) potential is tuned to a particular function of the other parameters. As with the noninteracting case, the obtained ground states are twofold degenerate and differ in fermionic parity. We prove the uniqueness of the obtained ground states and show that they can be continuously deformed to the ground states of the noninteracting Kitaev chain without gap closing. We also demonstrate explicitly that there exists a set of operators each of which maps one of the ground states to the other with opposite fermionic parity. These operators can be thought of as an interacting generalization of Majorana edge zero modes.
Abstract A sign of topological order in a gapped one-dimensional quantum chain is the existence of edge zero modes. These occur in the Z 2-invariant Ising/Majorana chain, where they can be understood using free-fermion techniques. Here I discuss their presence in spin chains with Z n symmetry, and prove that for appropriate couplings they are exact, even in this strongly interacting system. These modes are naturally expressed in terms of parafermions, generalizations of fermions to the Z n case. I show that parafermionic edge zero modes do not occur in the usual ferromagnetic and antiferromagnetic cases, but rather only when the interactions are chiral, so that spatial-parity and time-reversal symmetries are broken.
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.