Functional Renormalization Group Advances Understanding of 2D Layered Quantum Materials

Scientists are tackling the complex behaviour of electrons in layered quantum materials, a crucial step towards designing next-generation electronic devices. AmusementLennart Klebl (Universität Würzburg) and Dante M. Kennes (RWTH Aachen University and Max Planck Institute for the Structure and Dynamics of Matter) et al present a novel application of the functional renormalization group to efficiently model interactions occurring at the surfaces and interfaces of these three-dimensional systems. Their research, focusing on a stack of two-dimensional layers with strong electron correlations, reveals a surprising interplay between established two-dimensional physics and the effects of interlayer coupling. Notably, the team discovered that intermediate coupling strengths can give rise to exotic states, including chiral spin-bond order, potentially unlocking new avenues for manipulating and controlling quantum materials.

Surface Functional Renormalisation Group for 3D Materials offers

Scientists have extended the two-dimensional functional renormalization group to efficiently analyse interactions occurring on the surface or at interfaces of three-dimensional materials. The team achieved a breakthrough in modelling strongly correlated electron systems by developing a ‘surface FRG’ method, enabling the study of how interactions within a two-dimensional layer influence the overall three-dimensional system. This innovative approach focuses on the renormalization of interactions specifically on the surface, simplifying calculations while retaining crucial physics relevant to layered quantum materials and heterostructures. Researchers applied this method to a semi-infinite stack of two-dimensional square lattices, incorporating a Hubbard interaction on the surface layer and an alternating interlayer coupling, a configuration akin to the Su-Schrieffer-Heeger model.
The study reveals how strongly correlated states, initially present in the decoupled two-dimensional Hubbard model on the surface, evolve when subjected to this SSH-like interlayer coupling. For a significant portion of the phase diagram, the behaviour of the two-dimensional system dominates, with antiferromagnetic, d-wave superconducting, and ferromagnetic correlations being central. However, for intermediate interlayer couplings, a fascinating transition occurs: the superconducting state at intermediate interaction strengths splits into two distinct regimes, separated by a narrow region exhibiting incommensurate spin-density-wave and spin-bond order. This separation potentially enables the realization of chiral spin-bond order, a highly sought-after state in condensed matter physics.

Experiments show that the functional renormalization group, when adapted for surface interactions, provides a robust and unbiased characterization of ordering tendencies in these systems. The researchers formulated their approach as a diagrammatic technique, integrating out high-energy degrees of freedom while accounting for interchannel feedback, a crucial aspect for accurate modelling. By focusing on the static four-point vertex and employing spin rotational and translational invariance, they significantly reduced the computational complexity without sacrificing essential physics. This advancement allows for efficient investigations into strong correlations arising from surface interactions, utilising surface Green’s functions as input for the FRG calculations.

The work opens new avenues for understanding and predicting the behaviour of complex layered materials, particularly those with strong electron correlations. By accurately modelling the interplay between surface and bulk properties, this research establishes a powerful tool for exploring novel quantum phenomena and designing materials with tailored electronic properties. The potential for realizing chiral spin-bond order, revealed through these calculations, highlights the exciting possibilities for future materials discovery and technological applications in areas such as spintronics and quantum computing . This breakthrough demonstrates the power of FRG techniques when adapted to address the unique challenges posed by surface and interface physics in three-dimensional systems.

Surface Functional Renormalisation Group for Layered Systems offers

Scientists extended the two-dimensional functional renormalization group (FRG) to efficiently address interactions occurring on the surface or at interfaces of three-dimensional systems. This work pioneered a ‘surface FRG’ variant, specifically designed to analyse interactions confined to a single layer embedded within a larger three-dimensional structure. The researchers investigated a semi-infinite stack of two-dimensional square lattices, incorporating a Hubbard interaction on the surface layer and an alternating interlayer coupling reminiscent of a Su-Schrieffer-Heeger (SSH) chain. This setup allowed them to explore how strongly correlated states, initially decoupled on the surface, evolve when subjected to this SSH-like interlayer coupling.

The study employed a momentum space FRG technique, initiating the renormalization of the two-particle interaction solely within the outermost layer, where a Hubbard-U term was included. Surface Green’s functions, incorporating the semi-infinite system’s characteristics, were recursively generated from the bulk couplings, providing efficient input for the FRG calculations. The team focused on the static four-point vertex, representing the two-particle interaction, discarding higher-order vertices and self-energy feedback to manage computational complexity. Spin rotational invariance and in-plane translational invariance were leveraged, encoding the effects of out-of-plane hopping within the propagators via an exact self-energy contribution.

Experiments utilized a diagrammatic flow equation for the four-point vertex, as depicted in Figure 0.2, grouping diagrams into particle-particle, direct particle-hole, and crossed particle-hole channels. The flow equation, expressed as dVΛ/dΛ = d/dΛ(P−1PΛ + C−1CΛ + D−1DΛ), governs the renormalization of the interaction vertex. Integral equations (2-4) detail the evolution of channel-specific contributions, integrating over the Brillouin zone with a normalization factor of 1. The researchers initiated the flow at an infinite scale (Λ = ∞), where fluctuations are suppressed, and solved the differential equation down to Λ = 0, recovering the fully interacting description.

This innovative approach enables the identification of ordered phases by monitoring the divergence of the four-point vertex at a finite critical scale, ΛC. The team defined channel-projections of the four-point vertex (equations 5-7) to isolate contributions from particle-particle, crossed particle-hole, and direct particle-hole channels, facilitating detailed analysis of the system’s behaviour. By. Experiments revealed that for a significant portion of the parameter space, the physics remains dominated by the decoupled two-dimensional Hubbard model, exhibiting antiferromagnetic, superconducting, and ferromagnetic correlations.

The team measured the evolution of these correlations as the interlayer hopping parameters were varied, providing insights into how dimensionality impacts electronic behaviour. Results demonstrate that at intermediate interlayer couplings, the superconducting state undergoes a fascinating transformation, splitting into two distinct regimes separated by a narrow region of incommensurate spin-density and spin-bond order. Data shows this region enables the potential realization of chiral spin-bond order, a state with intriguing magnetic properties and potential applications in spintronics. The researchers calculated the flow of the four-point vertex function, a key quantity in FRG, to track the emergence of these ordered phases, stopping the flow when the maximum element of the vertex exceeded a critical value.

Information on the ordered phase was then obtained from the vertex at this critical scale, ΛC. Tests prove the accuracy of the method by employing a static limit and disregarding self-energy feedback, focusing on the renormalization of the static four-point vertex while maintaining spin rotational and translational invariance. Measurements confirm the use of a sharp frequency cutoff is crucial for evaluating the loop integrals, simplifying the calculations without compromising accuracy, specifically, the team only needed to evaluate the free fermionic Green’s function, G0, for a single Λ during each step of solving the FRG equations. The breakthrough delivers a powerful tool for investigating surface and interface phenomena in strongly correlated materials.

Scientists constructed a three-dimensional Hubbard-SSH model to demonstrate the FRG’s capabilities, visualising a semi-infinite stack of two-dimensional layers coupled via alternating hopping amplitudes, v and w. The team measured the kinetic energy within each layer using a next-nearest-neighbor tight-binding model, defining the hopping amplitudes as t for nearest-neighbor sites and t′ for next-nearest-neighbor sites. The resulting Hamiltonian, H0, incorporates both the two-dimensional layers and the interlayer coupling, providing a realistic representation of the system’s electronic structure and enabling detailed analysis of surface interactions. This work paves the way for exploring novel quantum phases and functionalities in layered materials.

Surface Coupling Modifies 2D Correlated States significantly

Scientists have extended the two-dimensional functional renormalization group to efficiently examine interactions occurring on surfaces or at interfaces within three-dimensional systems. Researchers applied this method to a semi-infinite stack of two-dimensional square lattices, incorporating a Hubbard interaction on the surface and an alternating interlayer coupling. Their investigation explored how strongly correlated states, initially present in the decoupled two-dimensional Hubbard model on the surface, evolve when subjected to this interlayer coupling. For much of the explored phase diagram, the behaviour of the two-dimensional system dominates, with antiferromagnetic, superconducting, and ferromagnetic correlations being prominent.

However, at intermediate interlayer couplings, the superconducting state splits into two distinct regions, separated by a small area exhibiting incommensurate spin-density wave and spin-bond order, potentially enabling chiral spin-bond order. A d-wave symmetric superconducting solution was consistently observed, and the study identified an extended antiferromagnetic region alongside superconducting and ferromagnetic phases, mirroring the monolayer limit. Notably, an incommensurate spin-density wave order emerged in the intermediate coupling regime, potentially leading to complex chiral spin-bond order. This work demonstrates a novel application of momentum-space functional renormalization group to study strongly correlated electrons on surfaces, successfully modelling a layered square lattice system.

The findings reveal that interlayer coupling can induce novel phases, including incommensurate spin-density wave order separating superconducting regions, and suggest the possibility of chiral spin-bond order, a non-collinear magnetic state. The authors acknowledge limitations related to the model’s simplicity, specifically the exclusion of longer-ranged interactions and frequency-dependent self-energies. Future research directions include scrutinizing the chiral spin-density-wave ordering via mean-field decoupling, incorporating longer-range interactions to explore charge-bond order, and including frequency-dependent self-energies to analyse quasiparticle weight evolution, potentially extending the methodology to more realistic models like Weyl semimetals and topological insulators, and investigating phonon-induced interactions to understand surface superconductivity.