🔬 condensed matter

Topology meets superconductivity in a one-dimensional $t-J$ model of magnetic atoms

This paper proposes a realistic experimental setup using ultracold magnetic lanthanide atoms in a one-dimensional optical lattice to realize a tunable $t-J$ model, where analytical and numerical studies reveal the coexistence of topology and superconductivity in an exotic topological triplet superconductor phase, along with a practical protocol for its detection.

Original authors: Leonardo Bellinato Giacomelli, Thomas Bland, Louis Lafforgue, Francesca Ferlaino, Manfred J. Mark, Luca Barbiero

Published 2026-07-20

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Leonardo Bellinato Giacomelli, Thomas Bland, Louis Lafforgue, Francesca Ferlaino, Manfred J. Mark, Luca Barbiero

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the tiniest building blocks of matter, called atoms, don't just sit still but dance to a complex rhythm. In the realm of quantum physics, scientists study how these atoms interact, especially when they are "fermions"—a specific type of particle that refuses to share the same space, much like introverts at a crowded party who need their own personal bubble. When these particles are cooled down to temperatures near absolute zero and trapped in a grid of light called an optical lattice, they form a playground for discovering new states of matter. Two of the most exciting phenomena in this playground are superconductivity, where electricity flows with zero resistance (like a frictionless slide), and topology, which describes shapes that stay the same even if you stretch or twist them (like a coffee mug and a donut both having one hole). For decades, scientists have tried to build a perfect model to understand how these particles behave when they push and pull on each other, but the math has been incredibly difficult, often forcing researchers to choose between simplicity and realism.

This paper takes a giant leap forward by designing a new, highly flexible model using magnetic atoms like erbium and dysprosium. Instead of the usual rigid rules, the researchers show how to create a setup where the atoms can hop between grid spots, interact with their neighbors, and even form pairs in ways that were previously impossible to simulate. By combining advanced math with powerful computer simulations, they discovered that this new setup doesn't just mimic known behaviors; it reveals a brand-new, exotic state of matter where superconductivity and topology coexist. It's like finding a new flavor of ice cream that is simultaneously chocolate and vanilla, but in a way that creates a whole new texture. The authors suggest that with current technology, we could actually build this system in a lab to see these strange quantum dances in action, opening a door to understanding some of the universe's most mysterious materials.

The Dance of Magnetic Atoms

Think of the atoms in this study as dancers on a one-dimensional stage (a single line of light). Usually, when physicists try to model how these dancers interact, they are stuck with a "strict teacher" scenario: the dancers can't stand on top of each other (no double occupancy), and their interactions are limited. This paper introduces a new set of rules using magnetic atoms that have huge magnetic moments, acting like tiny bar magnets. Because these magnets are so strong, the researchers can tune the interactions independently. They can tell the dancers how fast to hop to the next spot, how strongly they should repel or attract each other if they land on the same spot, and how they should spin and flip their magnetic orientation when they are near each other.

The researchers derived a new version of a famous model called the t–J model. In the old versions, the "spin-flip" interaction (where two dancers swap their magnetic directions) was weak and tied to other forces. In this new magnetic setup, the spin-flip interaction becomes incredibly strong—about 100 times stronger than the usual magnetic forces—while the other rules stay flexible. This allows the system to explore regimes where two atoms can sit on the same spot (double occupancy) if the attraction is strong enough, a scenario that was previously off-limits in these models.

The New Quantum States

By running detailed computer simulations (using a method called DMRG) and analytical math, the team mapped out a "phase diagram," which is like a weather map for quantum matter. They found that by adjusting the strength of the attraction between atoms and the strength of the spin-flipping, the system settles into seven distinct phases. Some of these are familiar, like a "Luttinger Liquid" (a fluid of electrons that behaves differently than normal water) or a "Luther-Emery Liquid" (a state with a gap in its energy spectrum).

However, the most exciting discoveries are the new, exotic states:

Extended Singlet Superconductor (ESS): In this phase, atoms pair up with their neighbors to form "singlets" (a specific type of magnetic bond) and flow without resistance. This happens even when the atoms are repelling each other, which is a surprising twist.
Local Singlet Superconductor (LSS): Here, the pairing happens right on the same spot. If the attraction is strong enough, two atoms with opposite spins sit on the same lattice site and form a superconducting pair.
Topological Liquid (TL): This is a state where the system has a "gap" (a forbidden energy zone) in its bulk but hosts special, protected states at the edges. It's like a solid block of ice that is frozen inside but has a slippery, protected layer on the surface that never melts.
Luttinger Triplet Superconductor (LTS): A gapless state where atoms pair up in a "triplet" configuration (a different magnetic alignment than the singlet), flowing freely.

The Grand Prize: Topological Triplet Superconductor

The crown jewel of this research is the Topological Triplet Superconductor (TTS). This is a state where two of the most fascinating concepts in physics—superconductivity and topology—merge into one.

In this state, the atoms form triplet pairs (superconductivity) while simultaneously maintaining a topological order that protects the edges of the system. The researchers found that this happens when there is a mix of strong spin-flipping interactions and a small, but finite, amount of double occupancy (atoms sharing a spot). It's a delicate balance, like a tightrope walker who is also juggling. The simulations show that this state is robust, surviving across a wide range of densities and interaction strengths.

Crucially, the paper notes that this state is driven by competing interactions (the push and pull of the magnetic forces) and does not rely on the presence of "Majorana fermions" (a specific type of exotic particle often hunted in topological superconductors). Instead, the topology arises naturally from the interactions of the atoms themselves. The authors also point out that, unlike many other theoretical proposals, the number of particles in this system is strictly conserved, which might make it easier to create in a real lab.

How to Build and See It

The paper doesn't just stay in the realm of theory; it offers a practical recipe for building this system. The authors propose using ultracold atoms of erbium or dysprosium trapped in a one-dimensional optical lattice.

Preparation: They suggest loading the atoms into a grid of light, using lasers to isolate specific spin states (like picking out dancers wearing red or blue shirts), and then tuning the magnetic field to set the interaction strengths.
Detection: To see these phases, they propose using a "quantum gas microscope." This is a high-powered camera that can take pictures of individual atoms on the grid. By looking at how the atoms are arranged and measuring their magnetic spins at the edges of the line, scientists can detect the "edge magnetization" and the specific patterns of pairing that signal the presence of the Topological Triplet Superconductor.

The simulations suggest that the temperatures required to see these effects are within reach of current experiments (around $T/t \lesssim 0.2$ ), making this a very realistic goal for the near future.

Why It Matters

This research is significant because it breaks the limitations of previous models. For a long time, scientists had to choose between models that were mathematically simple but physically unrealistic, or models that were realistic but too complex to solve. This paper shows that by using magnetic lanthanide atoms, we can have our cake and eat it too: a model that is both flexible and rich in physics.

The discovery of the Topological Triplet Superconductor suggests that we might be able to create materials that conduct electricity perfectly while also having topological protection, which could be a stepping stone toward more stable quantum computers. While the paper does not claim to have built this yet, it provides a clear, step-by-step guide on how to do it, turning a theoretical curiosity into a tangible experimental target. The authors emphasize that this setup offers a powerful new route to understanding strongly interacting quantum matter, potentially unlocking secrets of high-temperature superconductivity and other complex phenomena that have puzzled physicists for decades.

Problem Statement
Strongly interacting fermionic systems are central to understanding complex quantum phases such as magnetism, superconductivity, and topology. However, the intrinsic complexity of these many-body regimes often renders them inaccessible to both analytical theory and current experimental platforms. While ultracold atoms in optical lattices have successfully simulated various states of matter, the realization of the $t-J$ model—a paradigmatic effective Hamiltonian derived from the Fermi-Hubbard model—has been largely restricted to the strongly repulsive regime where double occupancy is energetically suppressed. In standard implementations using alkaline atoms, the magnetic spin-spin interaction ( $J$ ) is necessarily isotropic (preserving SU(2) symmetry) and its magnitude is constrained to be much smaller than the hopping amplitude ( $J \ll t$ ), scaling as $t^2/|U|$ . Furthermore, existing dipolar realizations remain limited to regimes with very large $U/t$ , preventing access to intermediate interaction strengths and the formation of significant double occupancy. Consequently, regimes where topology and superconductivity might coexist, particularly those driven by anisotropic spin-flip interactions and attractive onsite interactions, remain experimentally unexplored.

Methodology
The authors propose a realistic and flexible experimental setup utilizing ultracold magnetic lanthanide atoms (Erbium or Dysprosium) trapped in a one-dimensional optical lattice. Leveraging the large magnetic moments of these atoms, they derive a generalized fermionic $t-J$ model where the hopping amplitude ( $t$ ), onsite interaction ( $U$ ), and spin-spin couplings ( $J_z, J_\perp$ ) are independently tunable.

Model Derivation: The system is described by an effective Hamiltonian derived from the microscopic Fermi-Hubbard model with dipole-dipole interactions (DDI). By employing spin-selective engineering of the quadratic light shift, the authors isolate the $m_F = \pm 1/2$ states, reducing the high-spin manifold to an effective spin-1/2 system. Crucially, the DDI contributes to the onsite interaction $U$ , and the large spin of lanthanides enhances the transverse spin-flip interaction $J_\perp$ by approximately two orders of magnitude compared to the longitudinal Ising term $J_z$ . This allows for the explicit breaking of SU(2) spin-rotational symmetry and the exploration of regimes where $J_\perp \gg t$ and where double occupancy is energetically permitted.
Analytical Approach: In the low-energy regime ( $U, J_\perp \ll t$ ), the authors employ bosonization techniques to map the microscopic Hamiltonian onto a Sine-Gordon model with decoupled charge and spin sectors. Renormalization group (RG) analysis is used to determine the relevance of interaction terms, the pinning of bosonic fields, and the resulting Luttinger parameters ( $K_c$ ) and spin gaps ( $\Delta_s$ ).
Numerical Approach: To explore the full phase diagram, including strongly interacting non-perturbative regimes, the authors utilize the Density Matrix Renormalization Group (DMRG) method. They analyze ground state properties, correlation functions (parity, string, charge, and spin density waves), edge magnetization, and the entanglement spectrum to identify distinct quantum phases.
Experimental Protocol: The paper outlines a specific protocol for state preparation and detection using current quantum gas microscopy techniques. This includes adiabatic loading into a 1D lattice, local addressing to control density and spin populations, and spin-resolved imaging to measure local magnetization and density correlations.

Key Contributions and Results
The study uncovers a rich phase diagram featuring seven distinct many-body quantum phases, including several exotic states where topology and superconductivity coexist:

Extended and Local Singlet Superconductors (ESS/LSS): In regimes with repulsive or attractive onsite interactions combined with specific spin-flip couplings, the system exhibits dominant singlet pairing correlations. The ESS phase involves pairs occupying adjacent sites, while the LSS phase (enabled by attractive $U$ ) involves onsite pairing.
Topological Liquid (TL): For positive $J_\perp/t$ and repulsive $U$ , the system enters a symmetry-protected topological (SPT) phase characterized by a finite spin gap, long-range string order, and protected edge states with fractional spin accumulation.
Luttinger Triplet Superconductor (LTS): In the gapless regime with large positive $J_\perp/t$ , the system supports dominant triplet superconducting correlations.
Topological Triplet Superconductor (TTS): A novel finding of this work is the coexistence of topological order and triplet superconductivity in the regime of attractive onsite interaction ( $U < 0$ ) and positive $J_\perp$ . This phase is characterized by a finite spin gap, long-range string order, edge magnetization, and dominant triplet pairing correlations. The TTS phase is driven by strong spin-flip interactions and a finite density of doubly occupied sites.

The phase diagram demonstrates that these phases are robust across a wide range of interaction strengths and particle densities. Notably, the topological phases (TL and TTS) are absent in the limit of infinite repulsive interaction ( $U/t \to \infty$ ), highlighting the necessity of tunable onsite interactions and the allowance of double occupancy.

Significance
The paper claims that this work establishes an alternative and powerful route for understanding strongly interacting fermionic quantum matter. By utilizing magnetic lanthanide atoms, the proposed setup overcomes the limitations of previous $t-J$ implementations, offering independent control over hopping, onsite interaction, and anisotropic spin-spin couplings.

The significance of the results lies in three main areas:

Accessibility: The required temperatures for observing these phases (below the spin gap, $T/t \lesssim 0.2$ ) are within reach of current experimental platforms, unlike the extremely low temperatures required for 2D Fermi-Hubbard superconductivity.
Novelty of Phases: The work provides a concrete scheme to realize and probe symmetry-protected topological phases in itinerant systems (where particle motion is allowed), going beyond the static SPT insulators previously observed.
Topological Superconductivity: The discovery of the Topological Triplet Superconductor (TTS) represents a significant novelty. Unlike the Kitaev chain, this phase is induced purely by competing interacting processes ( $U$ and $J_\perp$ ) without requiring external superconducting proximity effects or Majorana fermions. Furthermore, the strict conservation of particle number in this model potentially makes its experimental realization less challenging than other topological superconducting proposals.

The authors conclude that their derived protocols for state preparation and detection pave the way for the experimental realization of these exotic states, offering a deeper understanding of the interplay between topology and superconductivity in strongly correlated systems.

The Dance of Magnetic Atoms

The New Quantum States

The Grand Prize: Topological Triplet Superconductor

How to Build and See It

Why It Matters

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