Beenakker and Colleagues Develop Tangent-Fermion Lattice for Chiral Fermion Realization

Scientists at University Leide, in collaboration with Ghent University and University of Cambridge, have developed a new lattice formulation that addresses critical obstacles to realising symmetric mass generation in one-dimensional systems. This phenomenon describes the acquisition of mass by fermions, fundamental particles like electrons, without the need for spontaneous symmetry breaking, a cornerstone of the Standard Model of particle physics. V. A. Zakharov and colleagues have successfully navigated the challenges of fermion doubling, a common issue arising from chiral discretizations in lattice field theory, and the perturbative irrelevance of the interactions required to induce this mass generation. Their approach, employing a tangent-fermion lattice and a Hubbard-type interaction, offers a significant step towards understanding strongly correlated fermion systems and their emergent properties.

Relevant interactions enabled via tangent-fermion lattices and density-density interactions

A central difficulty in realising the Wang and Wen 3-4-5-0 model, a minimal one-dimensional setting for symmetric mass generation, lies in the inherent weakness of the interaction needed to ‘gap’ the fermions, that is, to create an energy gap separating the ground state from excited states. This interaction, a six-fermion term, is typically ‘perturbatively irrelevant’, meaning its effect diminishes at low energies and cannot generate a stable mass gap. The scaling dimension of the 3-4-5-0 interaction, conventionally 5, was decreased to 5K, effectively creating a ‘relevant’ interaction when K is less than 2/5. This reduction is crucial, as a relevant interaction grows stronger at low energies, allowing it to dominate the system’s behaviour and induce the desired mass generation. Previously, achieving such a relevant interaction proved impossible due to its inherent perturbative irrelevance, hindering the direct lattice realisation of the model.

This breakthrough, realised by researchers at Ghent University and the University of Cambridge, hinges on a novel combination of a tangent-fermion lattice and a Hubbard-type density-density interaction. The tangent-fermion lattice, a specific discretization scheme, was employed to create a single chiral branch of fermions, effectively avoiding the problem of fermion doubling. Fermion doubling arises in lattice formulations when the discretization introduces unwanted, spurious fermion species, ‘mirror particles’, that complicate the physics and obscure the desired effects. By carefully constructing the lattice, the team eliminated these unwanted degrees of freedom, ensuring a clean realisation of the one-dimensional model. The Hubbard-type interaction, a density-density interaction, describes the repulsion between electrons and is commonly used to model strongly correlated electron systems. By tuning the strength of this interaction, represented by the parameter K, the researchers were able to manipulate the scaling dimension of the six-fermion term and achieve relevance when K is less than 2/5. Density-matrix renormalization group (DMRG) calculations, a powerful numerical technique for studying strongly correlated systems, confirmed the opening of an excitation gap, a key indicator of symmetric mass generation, without a degenerate ground state. This confirms the stability of the generated mass gap and the absence of other competing phases.

The team’s manipulation of the density-density interaction’s strength, represented by the parameter K, enabled this reduction in scaling dimension. Precise tuning allowed relevance when K is less than 2/5, circumventing the usual perturbative irrelevance that has previously hindered such models. This approach offers a controlled environment for studying interacting fermions and could inform the development of more complex models, potentially revealing new insights into quantum systems. The DMRG calculations provide strong evidence for the emergence of a gapped state, demonstrating that the fermions have indeed acquired a mass through interaction without breaking any underlying symmetries. This is a significant result, as it provides a concrete example of symmetric mass generation in a tractable model.

Realising symmetric mass generation in a constrained one-dimensional model

Symmetric mass generation presents a compelling alternative to traditional understandings of particle mass acquisition, sidestepping the need for spontaneous symmetry breaking, a mechanism that relies on the system choosing a particular ground state from a degenerate manifold. In many conventional models, particles acquire mass through interactions with the Higgs field, which breaks electroweak symmetry. Symmetric mass generation, however, offers a different pathway, where mass emerges directly from the interactions between fermions without altering fundamental symmetries. Direct realisation of the Wang and Wen 3-4-5-0 model within a one-dimensional system, utilising a tangent-fermion lattice and Hubbard-type interaction, addresses a long-standing challenge in theoretical physics. However, this achievement highlights a key tension. The model’s reliance on a strictly one-dimensional lattice and a limited parameter range, specifically where K is less than 2/5, raises questions about its scalability and relevance to more complex, realistic systems.

The one-dimensional nature of the model is a significant simplification, as real materials are typically three-dimensional. While one-dimensional systems can exhibit fascinating physics, their behaviour often differs significantly from that of higher-dimensional systems. Furthermore, the requirement that K be less than 2/5 restricts the range of accessible parameters and may not be representative of the interactions found in real materials. Despite these restricted conditions, the findings provide a strong testbed for exploring more intricate models and potentially understanding similar effects in materials with stronger interactions and multiple dimensions. The demonstration of symmetric mass generation, where particles gain mass without altering fundamental symmetries, offers a new foundation for exploring complex quantum phenomena and could lead to insights into the behaviour of strongly correlated electron systems, such as high-temperature superconductors and topological insulators. Combining a tangent-fermion lattice with a Hubbard-type density-density interaction allowed the team to successfully demonstrate this phenomenon in a simplified, one-dimensional model. Manipulation of the interaction’s strength, governed by the parameter K, was key to achieving a ‘relevant’ interaction, essential for observing this effect. This research establishes a viable route to symmetric mass generation, a process where particles acquire mass through interaction without disrupting fundamental symmetries. Future work will focus on extending this approach to higher dimensions and exploring the possibility of realising symmetric mass generation in more realistic materials.

The researchers successfully demonstrated symmetric mass generation within a one-dimensional model, achieving a gap in particle excitation without spontaneous symmetry breaking. This is significant because it provides a minimal setting to study how particles can gain mass through interaction, differing from traditional mechanisms. Using a tangent-fermion lattice and a Hubbard-type interaction, controlled by the parameter K, they showed this effect occurred when K was less than 2/5. The authors intend to extend this work to higher dimensions and investigate potential realisation in more complex materials.