Ladder Lattices Reveal Robust Insulating Phases of Matter

Researchers are increasingly focused on understanding strongly correlated quantum systems, and a new study details the Mott-insulating phases within the Bose-Hubbard model on quasi-one-dimensional ladder lattices. Lorenzo Carfora, from the Department of Physics and SUPA at the University of Strathclyde, alongside Callum W. Duncan of Aegiq Ltd., and colleagues, have mapped the phase diagram of this model, revealing the surprising persistence of the rung-Mott insulator phase even with considerable on-site interactions. This collaborative work, also involving Stefan Kuhr and Peter Kirton from the University of Strathclyde, demonstrates how these phases can be distinguished via measurable variances, potentially accessible with current experimental techniques like -gas microscopy. Significantly, the findings suggest that these Mott-insulating behaviours arise from the mapping of one-dimensional states onto higher-dimensional systems, offering insights into the behaviour of complex quantum materials and potentially informing the design of novel quantum technologies.

At interactions exceeding 10% of the hopping energy, a distinct insulating state emerges in this ultracold atom system. This surprising durability of the Mott-insulating phase challenges conventional understanding of atomic behaviour in confined lattices. By understanding these phases unlocks potential control over matter at the quantum level, paving the way for advanced materials design.

Scientists have long understood that the arrangement of atoms within a material dictates its properties. Yet predicting behaviour in strongly interacting quantum systems remains a considerable challenge. Recent work focuses on the Bose-Hubbard model, a fundamental description of bosons arranged on a lattice. Investigations into this model using ladder-like lattices reveal a surprising persistence of an insulating phase known as the rung-Mott insulator (RMI).

This phase continues to exist even with considerable interactions between bosons, and calculations now define the boundary between the RMI and a superfluid phase, extending this understanding to the thermodynamic limit. To establish the precise conditions for this RMI phase has proven difficult, particularly when considering the softcore regime where bosons are not strictly confined to single lattice sites.

Scientists have now calculated the phase diagram of the Bose-Hubbard model on a half-filled ladder lattice, accounting for finite on-site interactions. Here, this detailed analysis clarifies the conditions under which the RMI phase remains stable and provides a means to distinguish it from the superfluid phase using measurable quantities. Such microscopes allow scientists to image individual atoms, offering a powerful tool for verifying theoretical predictions.

The principles governing the RMI phase are not limited to simple ladder geometries, as analogous insulating phases appear in other quasi-one-dimensional lattice structures, suggesting a broader connection between lattice connectivity, filling fraction, and the emergence of insulating behaviour. In turn, this effort has implications for designing novel quantum materials, as it leverages the ability to control dimensionality and geometry in ultracold atom experiments, providing an ideal platform to explore complex quantum phenomena.

Hamiltonian formulation and phase boundary determination for bosonic ladder systems

Meanwhile, this effort underpinned a Bose-Hubbard model, describing bosons arranged on a two-leg ladder lattice. Calculations proceeded by defining a Hamiltonian incorporating both on-site interaction terms and hopping integrals between lattice sites, with the hopping matrix, J. Detailing the rate of boson movement across the lattice. The Hamiltonian allowed investigation of the system’s ground state properties and phase transitions.

A detailed analysis of the phase diagram delineated the boundary between superfluid and rung-Mott insulator phases, focusing on half-filling. Where each lattice site contains, on average, one boson. Characterising these phases necessitated the identification of measurable quantities accessible through experimentation — as a result, number and parity variances were selected as key observables. As these directly reflect the quantum state of the bosons and can be determined using a quantum-gas microscope.

At the same time, the effort broadened its scope to include quasi-one-dimensional lattices beyond the simple ladder geometry to assess the generality of these findings. By considering staggered two-dimensional and triangular lattices, researchers aimed to understand how lattice connectivity influences the emergence and stability of insulating phases. Researchers mapped one-dimensional structures onto higher dimensional systems, driven by reductions in hopping rates. In turn, the calculations accounted for finite on-site interactions, providing a more realistic description of the system’s behaviour in experimentally achievable regimes.

Superfluid to rung-Mott insulator transition boundaries defined by extrapolated numerical data

Numerical data reveals the critical on-site interaction, Uc/J, converges to a finite fixed value as J⊥/J approaches infinity. An analytical approximation for large J⊥ describing the superfluid (SF)-to-rung-Mott insulator (RMI) critical boundary is given by J⊥,c/J ≈ 0.235 exp ” π 4(0.457) p Uc/2U 1D c −1 # , where U 1D c = 3.25J represents the critical point for the Bose-Hubbard model on a one-dimensional chain at commensurate filling.

Fitting extrapolated numerical data with this equation yields fitting constants of A = 0.235 and B = 0.457. Meanwhile, this demonstrates good agreement across the entire phase diagram, even when J⊥ nears J. At the same time, this expression indicates the critical hopping rate, J⊥,c, diverges when the on-site interaction approaches 2U 1D c. No phase transition is possible for lower interactions.

The RMI phase persists to finite interaction strength, with the phase boundary determined through careful analysis. Calculations of the on-site and rung number variances, κ and κrung, provide means to distinguish between phases accessible via a quantum-gas microscope. At higher interaction strengths, κ approaches 1/4, mirroring the reduction of on-site population to either 0 or 1, while κrung diminishes towards zero.

For J⊥ approaching zero, the RMI phase exhibits a slow gap opening, with ∆E/J remaining below 10−2 before substantial growth occurs. When considering parity projections, the average on-site parity variance, σ, increases towards 0.1/4 as U increases. Aligning with the half-filling condition in the RMI phase. At the same time, σrung decays alongside κrung, mirroring the behaviour observed for a one-dimensional chain.

Differences between κ and κrung, when compared to σ and σrung, arise from contributions of states with highly populated sites, increasing κ and decreasing σ due to the inclusion of additional even-parity states. The difference between κrung and σrung is directly related to the average density correlation along the rungs, with an increase indicating more doubly occupied rungs.

Rung-Mott insulator persistence and superfluidity boundaries in Bose-Hubbard ladder lattices

Understanding how interactions between particles dictate the behaviour of complex quantum systems remains a central challenge in condensed matter physics — this latest work on the Bose-Hubbard model, specifically examining ladder lattices. Offers a refined picture of how these interactions give rise to distinct phases of matter, and calculations demonstrate the persistence of a rung-Mott insulator phase, a state where particles are localized, even with considerable interaction strength. Define the boundary where this phase gives way to superfluidity.

The true power of this project resides in its connection to experimental verification, as researchers have identified measurable quantities, number and parity variances, providing a clear pathway for confirming these theoretical predictions using quantum-gas microscopes. This investigation extends the understanding of quasi-one-dimensional lattices, revealing how lattice connectivity influences phase boundaries, unlike previous studies focusing on simpler systems.

Once these boundaries are better understood, the implications extend beyond fundamental physics, potentially informing the design of new materials with tailored quantum properties. It is important to acknowledge that these calculations represent a specific model, and real materials invariably present additional complexities. The precise control needed to create and observe these phases in experiments remains a significant hurdle.

This effort opens avenues for exploring analogous phases in other lattice structures, and for investigating how these phases respond to external stimuli. This represents a step towards bridging the gap between theoretical prediction and experimental realisation, and may eventually contribute to the development of quantum technologies.