🔬 mesoscale physics

The Crossover from Ordinary to Higher-Order van Hove Singularity in a Honeycomb System: A Parquet Renormalization Group Analysis

This paper employs parquet renormalization group analysis to investigate the crossover from ordinary to higher-order van Hove singularities in honeycomb systems like bilayer and trilayer graphene, revealing how the merging of Fermi pockets and the resulting changes in density of states drive complex electronic ordering tendencies through the evolution, disappearance, and reemergence of renormalization group fixed points.

Original authors: Yueh-Chen Lee, Dmitry V. Chichinadze, Andrey V. Chubukov

Published 2026-07-17

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Yueh-Chen Lee, Dmitry V. Chichinadze, Andrey V. Chubukov

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 the electrons in a solid material not as tiny, lonely marbles, but as a bustling crowd at a massive concert. Sometimes, this crowd is spread out evenly, but other times, they all rush to a specific spot, creating a "singularity"—a point where the density of people (or electrons) goes wild. In the world of quantum materials, these crowded spots are called van Hove singularities. Think of them as the "hype zones" of the electron world. When electrons pile up here, they get very excited and start doing weird, collective things, like forming superconductors (materials that conduct electricity with zero resistance) or magnets.

Usually, these hype zones are "ordinary," meaning the crowd gathers in a predictable, standard way. But scientists have discovered that under certain conditions, the crowd can gather in an even more extreme, "higher-order" way, where the density spikes much more violently. The big question is: what happens when a material shifts from the ordinary hype zone to this extreme one? Does the crowd just get a little more excited, or does the entire concert change its genre? Understanding this crossover is crucial because it tells us how to engineer materials that can switch between different powerful states, like turning a magnet into a superconductor with a simple tweak.

This paper investigates exactly that shift. The researchers looked at a specific type of electronic system found in special forms of graphene (a material made of carbon atoms arranged in a honeycomb pattern) when it is subjected to an electric field. They wanted to see what happens as the system transitions from having three separate, small "pockets" of electrons (an ordinary singularity) to having them all merge into one giant, chaotic pocket (a higher-order singularity).

To figure this out, the authors used a sophisticated mathematical tool called Parquet Renormalization Group (pRG). You can think of this method as a high-speed simulation that tracks how the electrons' interactions change as you zoom in closer and closer to the "hype zone." Instead of just looking at the electrons, the simulation tracks the "rules of engagement" (called couplings) between them. As the system changes from the ordinary state to the higher-order state, these rules evolve, and the simulation reveals which "dance" the electrons will choose to do next. Will they pair up to become superconductors? Will they line up to become magnets? Or will they split into different groups?

The study found that the transition is far more dramatic and complex than a simple slide from one state to another. As the system moves from the ordinary to the higher-order singularity, the "dance floor" of possible electron behaviors undergoes a topological makeover. The authors discovered that the mathematical "fixed points"—which act like the destinations or attractors that determine the final state of the electrons—don't just move; they appear, disappear, and even annihilate each other in pairs.

Specifically, the paper shows that in the ordinary state, the electrons are most likely to form a "pair density wave" (a specific type of superconducting pattern) or a "spin-triplet superconductivity." However, as the system shifts toward the higher-order state, the rules change completely. The preferred states become ferromagnetism (where all spins point the same way), a mix of spin-singlet and spin-triplet superconductivity, or "valley polarization" (where electrons choose a specific valley in the material's structure).

The authors mapped out exactly how this evolution happens. They found that as they tuned a parameter (represented by the Greek letter $\gamma$ , which controls the shape of the electron energy landscape), the phase diagram—the map of which state wins—underwent a series of sudden, topological shifts. For instance, a stable "destination" for the electrons might suddenly vanish into infinity and reappear on the other side of the map, or two destinations might crash into each other and disappear, forcing the electrons to find a new path. These mathematical "births" and "deaths" of fixed points are what cause the material's behavior to flip from one type of order to another.

In short, the paper suggests that the journey between an ordinary and a higher-order van Hove singularity is not a smooth road but a landscape of sudden, dramatic changes. The electronic orders don't just evolve gradually; they are reshaped by the creation and destruction of the mathematical structures that govern them. This provides a detailed blueprint for how complex quantum materials might behave as they are tuned, offering a deeper understanding of why some materials suddenly switch from being magnetic to superconducting when their internal structure is slightly altered.

Technical Summary: The Crossover from Ordinary to Higher Order van Hove Singularity in a Honeycomb System

Problem Statement
This paper investigates the interplay between various electronic ordering tendencies (such as superconductivity, magnetism, and density waves) in a honeycomb lattice system as it transitions from an Ordinary van Hove Singularity (OVHS) to a Higher Order van Hove Singularity (HOVHS). The physical motivation stems from systems like Bernal bilayer graphene (BBG) and rhombohedral trilayer graphene (RTG) under a displacement field. In these materials, the Fermi surface topology changes with doping: at low doping, there are three small, spin-degenerate Fermi pockets near each Dirac point ( $K$ and $K'$ ); at higher doping, these merge into a single pocket. This Lifshitz-type transition is accompanied by a van Hove singularity (VHS). Depending on the displacement field strength, the system exhibits either three separate OVHS points or a single HOVHS point where the density of states (DOS) diverges with a power law rather than a logarithm. The central problem is to understand how the competition between different electronic instabilities evolves as the system is tuned between these two distinct VHS limits.

Methodology
The authors employ a Parquet Renormalization Group (pRG) analysis, a technique well-suited for weakly to moderately correlated metallic systems where multiple scattering channels (particle-particle and particle-hole) exhibit logarithmic or power-law singularities.

Model Construction:
- The system is modeled as fermions on a honeycomb lattice with two independent patches near the $K$ and $K'$ valleys.
- A single tunable parameter, $\gamma$ ( $0 \le \gamma \le 1$ ), interpolates the dispersion relation between the OVHS limit ( $\gamma=1$ , quadratic dispersion $k^2$ ) and the HOVHS limit ( $\gamma=0$ , cubic dispersion $k^3$ ). The dispersion is given by $\epsilon^{(\pm)}_k = \gamma k^2 \cos 2\theta \pm \sqrt{1-\gamma^2} k^3 \cos 3\theta$ .
- The Hamiltonian includes four-fermion interactions allowed by symmetry, characterized by three coupling constants: $g_1$ (inter-valley exchange), $g_2$ (inter-valley density-density), and $g_4$ (intra-valley density-density).
Renormalization Group Procedure:
- Stage 1 (Coupling Flow): The authors derive and solve coupled differential equations for the running couplings ( $g_i$ ). The flow is driven by polarization bubbles ( $\Pi$ ) in particle-particle and particle-hole channels. The authors define the RG scale $L$ based on the inter-valley particle-particle bubble $\Pi_{pp}(0)$ . For intermediate $\gamma$ , these bubbles are evaluated numerically and fitted to analytical functions.
- Stage 2 (Order Parameter Flow): Infinitesimal bare order parameters ( $\Delta_j$ ) are introduced. Their flow equations are solved using the running couplings from Stage 1 to determine the critical exponents ( $\beta_i$ ). The leading instability corresponds to the order parameter with the largest exponent.
- Fixed Point Analysis: The evolution of the phase diagram is analyzed by tracking the fixed points (stable, unstable, and saddle points) and fixed lines of the RG flow in the space of coupling ratios ( $x_2 = g_2/g_1, x_4 = g_4/g_1$ ).

Key Contributions and Results

Phase Diagram Evolution:
The study maps the evolution of the phase diagram as $\gamma$ decreases from 1 (OVHS) to 0 (HOVHS).
- OVHS Limit ( $\gamma=1$ ): The phase diagram features three regions: an asymptotically free Fermi gas (FG), spin-triplet superconductivity (tSC), and pair-density wave (PDW).
- HOVHS Limit ( $\gamma=0$ ): The phase diagram is entirely ordered. The leading states are ferromagnetism (FM), superconductivity (degenerate between spin-triplet and spin-singlet), and valley polarization (VP), which is degenerate with spin-density wave (SDW) and charge-density wave (CDW).
- Crossover: The transition is not smooth; it involves topological changes in the RG flow. As $\gamma$ decreases, the boundaries between phases rotate, and new phases emerge. Specifically, the PDW order is replaced by VP order, and the tSC and sSC states become degenerate at small $\gamma$ .
Topological Nature of the Crossover:
The paper identifies that the changes in the phase diagram are driven by the disappearance, re-emergence, and pair production/annihilation of fixed points in the RG flow.
- As $\gamma$ drops slightly below 1, a half-stable fixed point ( $P_3$ ) moves to infinity and re-emerges as a stable fixed point.
- Further reduction leads to the annihilation of fixed points ( $P_3$ and $P_4$ ), altering the flow direction in the upper phase diagram.
- At $\gamma \approx 0.14$ , the stable fixed point $P_1$ moves to infinity and re-emerges as a half-stable point, changing the flow in the left part of the diagram.
- Finally, at $\gamma \approx 0.09$ , a new pair of fixed points ( $P_5, P_6$ ) is created, establishing the flow structure characteristic of the HOVHS limit.
Specific Instability Transitions:
- The boundary between tSC and PDW rotates and eventually becomes vertical, replacing the FG region with tSC.
- The PDW order is superseded by Valley Polarization (VP).
- At the HOVHS limit, the system exhibits a degeneracy between spin-triplet and spin-singlet superconductivity, and between VP and SDW/CDW, a feature absent in the OVHS limit.

Significance and Claims
The authors claim that their work provides a systematic understanding of how electronic ordering tendencies evolve across the Lifshitz transition from OVHS to HOVHS. They demonstrate that the crossover is governed by the topological restructuring of the RG flow's fixed points.

The paper modestly notes that while the pRG approach for HOVHS is less rigorous than for OVHS (because non-logarithmic terms in perturbation theory become comparable to logarithmic ones in the power-law divergence regime), previous comparisons with direct perturbation theory suggest the quantitative differences are small. Therefore, the pRG method is justified for capturing the qualitative evolution of instabilities in these systems. The results offer a theoretical framework for interpreting experimental data in BBG and RTG, particularly regarding the competition between superconductivity, magnetism, and valley polarization near van Hove singularities. The model specifically captures orders preserving $C_3$ rotational symmetry; it does not detect nematic symmetry breaking between the three pockets, which the authors note requires stronger spin or valley orders to develop in experiments.

Technical Summary: The Crossover from Ordinary to Higher Order van Hove Singularity in a Honeycomb System

More like this