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arXiv:2607.06751 · Physical review. B./Physical review. B

Majorana physics in a Luttinger liquid with attractive interactions

Francesco Debortoli, Nitya Cuzzuol, Luca Barbiero, Fabian Grusdt

cond-mat.quant-gascond-mat.str-elquant-phSubmitted Jul 7, 2026• Announced Jul 9, 2026• 0 citations

Abstract

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.

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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.

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.

We revisit the low-energy physics of one-dimensional spinless fermion liquids, showing that with sufficiently strong interactions the conventional Luttinger liquid can give way to a strong-pairing phase. While the density fluctuations in both phases are described by a gapless Luttinger liquid, single fermion excitations are gapped only in the strong-pairing phase. Smooth spatial Interfaces between the two phases lead to topological degeneracies in the ground-state and low-energy spectrum of the density fluctuations. Using a concrete microscopic model, with both single-particle and pair hopping, we show that the strong-pairing state is established through emergence of a new low-energy fermionic mode. We characterize the two phases with numerical calculations using the density matrix renormalization group. In particular we find enhancement of the central charge from

c=1

in the two Luttinger liquid phases to

c=3/2

at the critical point, which gives direct evidence for an emergent critical Majorana mode. Finally, we confirm the existence of topological degeneracies in the low-energy phonon spectrum, associated with spatial interfaces between the two phases.

We show that a one dimensional ultra-cold Fermi gas with Rashba-like spin orbit coupling, a Zeeman field and intrinsic attractive interactions exhibits a novel topological superfluid state, which forms in spite of total number conservation and the absence of a single particle gap. Majorana zero modes are localized to the interface between a topological region in the middle of the trap and trivial regions at its wings. Unlike the realization of a topological superconductor in proximity coupled nano-wires, the Majorana modes do not carry a quantum number associated with the total fermion parity. Instead, the topological degeneracy is protected by an emergent

Z_2

symmetry present only at low energies. We discuss the experimental implications of the novel zero modes, as manifest for example in the response to modulation of a local potential near the position of the Majorana bound states. For the range of interaction strength corresponding to Luttinger parameter

12

, on the other hand the zero-mode can be detected as a sharp low frequency response at an energy which generically scales with system size as

1/L^{K/2}

and is therefore parametrically separated from the phonons.

The ground state behavior of the spinless fermion chain with an interaction between fermions at neighboring sites is studied for free open boundaries. For the strong enough repulsion boundary gapless states (bound state of Majorana operators from opposite sites of the chain) can exist inside the gap for bulk excitations, i.e., in the topological insulator regime. We propose to use those Majorana zero modes as topological qubits, similar to the ones in one-dimensional topological superconductors. Possible physical realizations of the considered model are discussed.

The Hubbard and spinless fermion chains are paradigms of strongly correlated systems, very well understood using the Bethe ansatz, density matrix renormalization group (DMRG), and field theory/renormalization group (RG) methods. They have been applied to one-dimensional materials and have provided important insights for understanding higher-dimensional cases. Recently, an interacting fermion model has been introduced, with possible applications to topological materials. It has a single Majorana fermion operator on each lattice site and interactions with the shortest possible range that involve four sites. We present a thorough analysis of the phase diagram of this model in one dimension using field-theory/RG and DMRG methods. It includes a gapped supersymmetric region and a gapless phase with coexisting Luttinger liquid and Ising degrees of freedom. In addition to a first-order transition, three critical points occur: tricritical Ising, Lifshitz, and a generalization of the commensurate-incommensurate transition. We also survey various gapped phases of the system that arise when the translation symmetry is broken by dimerization and find both trivial and topological phases with 0, 1, and 2 Majorana zero modes bound to the edges of the chain with open boundary conditions.

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.