Researchers are increasingly focused on understanding the relationship between the topology of the Fermi surface and phase transitions in condensed matter physics. Gennady Y. Chitov from the Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, and colleagues present a detailed analysis of this connection, utilising the exactly solvable Hatsugai-Kohmoto model to explore the interplay between Fermi surface topology, the Luttinger theorem, and the emergence of both metallic and insulating states. This work, conducted in collaboration with researchers across multiple institutions, significantly advances the theory of Fermi surface topology by introducing the concept of a critical manifold determined by Lee-Yang zeros and a corresponding order parameter based on the volume of the Fermi sea. By applying homology theory to analyse the Fermi surface topology, the study establishes a robust universality class governing metal-metal and metal-insulator transitions, demonstrating its resilience to interactions and specific model parameters, provided critical points remain non-degenerate, and confirming the Landau Fermi liquid behaviour of gapless states.
The study confirms the validity of the Luttinger theorem within the HK model, demonstrating that the order parameter, defined as the volume of the Fermi sea, accurately describes transitions between metals and insulators, including Lifshitz and van Hove transitions which involve changes in the electronic band structure.
These gapless phases are identified as conventional Landau Fermi liquids, characterised by well-defined quasiparticles. Beyond the standard approach of continuous order parameters, the research applies homology theory, a branch of mathematics dealing with shapes and their properties, to analyse these Fermi surface transitions as critical points of a Morse function.
To quantify changes in Fermi surface topology, the Euler characteristic, a topological invariant describing the ‘holes’ in a shape, is calculated for each phase of the HK model. The findings suggest that this Fermi surface topology universality class remains robust even with interactions and variations in model details, provided the critical points are non-degenerate.
The Euler characteristic was calculated for each phase of the HK model, revealing discrete shifts corresponding to critical points, specifically differing between phases, indicating a quantifiable change in the Fermi surface’s fundamental structure. The study details that transitions between gapless phases, specifically between phases M and M2, represent Landau Fermi liquids characterised by one and two Fermi surfaces, respectively. These transitions, identified as Lifshitz transitions, were previously unaddressed in literature concerning the HK model, and the establishment of these phases as conventional metals is supported by their distinct Fermi surface configurations.
Furthermore, the research demonstrates that the order parameter and the Fermi surface topology universality class accurately describe transitions between metallic phases and insulating phases, both band and Mott insulators, revealing that the Lifshitz and van Hove transitions belong to the same universality class as these metal-insulator transitions. The distribution function, nkσ, is defined by an equation yielding values of 0, 0.5, or 1, depending on the relationship between the energy scale ε and the chemical potential μ.
At zero temperature, this function exhibits a step-like behaviour, transitioning sharply between these values at specific energy levels, μ = ε ± U/2. This research offers a new framework for understanding metal-insulator transitions and could inform the design of materials with tailored electronic properties. The ability to predict and control these transitions is crucial for advancements in superconductivity, magnetism, and next-generation electronic devices.
Scientists have long sought a unified framework to understand the subtle transitions between different states of matter, and this work offers a compelling advance by linking the topology of the Fermi surface, the Luttinger theorem, and homology theory into a cohesive picture. The power of this approach lies in its ability to characterise phase transitions not simply by identifying an order parameter, but by meticulously mapping the geometry of the Fermi surface itself.
Viewing critical points as arising from the topology of this surface, and quantifying that topology using tools borrowed from homology, provides a new lens through which to examine the emergence of insulating states and the behaviour of electrons in these materials. While the current analysis focuses on a simplified model system, the demonstrated robustness of this “Fermi surface topology universality class” suggests broader applicability. Limitations remain, including extending this framework to genuinely disordered or three-dimensional materials, and directly linking these topological features to macroscopic properties, such as superconductivity or magnetism, requires further investigation.
👉 More information
🗞 Topology of the Fermi surface and universality of the metal-metal and metal-insulator transitions: dd-dimensional Hatsugai-Kohmoto model as an example
🧠 ArXiv: https://arxiv.org/abs/2602.13050