🔬 condensed matter

Quantum critical fan and emergent relativistic symmetry of two-dimensional Dirac semimetals

This paper employs a non-perturbative field-theory approach to map the finite-temperature phase diagram of the chiral Ising Gross-Neveu-Yukawa model, demonstrating emergent relativistic symmetry at the zero-temperature quantum critical point, a classical two-dimensional Ising transition at finite temperature, and characterizing the scaling properties of the resulting quantum critical fan.

Original authors: Friederike Ihssen, Bilal Hawashin, Mireia Tolosa-Simeón, Michael M. Scherer

Published 2026-07-22

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CC BY 4.0

Original authors: Friederike Ihssen, Bilal Hawashin, Mireia Tolosa-Simeón, Michael M. Scherer

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 a world where the rules of physics change depending on how fast you move or how hot things get. In the realm of quantum materials, scientists study "Dirac semimetals"—a special type of crystal where electrons behave less like tiny billiard balls and more like ghostly, massless particles zipping around at near-light speeds. These materials are fascinating because they sit right on a knife-edge between being a conductor (letting electricity flow) and an insulator (blocking it). This knife-edge is called a "quantum critical point." It's like a tightrope walker balancing perfectly between two states; even a tiny nudge, like a change in temperature or pressure, can send them tumbling into one state or the other.

What makes this even more magical is that near this balancing point, the electrons seem to forget they are in a solid crystal and start acting as if they are in a universe governed by Einstein's relativity, where space and time are perfectly linked. However, real materials are messy. They have different speeds for different things, and they are never perfectly cold. Scientists have long wondered: Does this "relativistic magic" actually happen in real, warm materials, or is it just a trick of the math? Understanding this is crucial because if we can control these materials, we might build super-fast, super-efficient electronics or even new types of quantum computers.

This paper takes a deep dive into that question using a powerful mathematical tool called the "functional renormalization group." Think of this tool as a high-tech microscope that lets researchers zoom in and out of the material's behavior, watching how the electrons and their interactions change as they look at smaller and smaller scales. The authors focused on a specific model, the "chiral Ising model," which describes a material where electrons can spontaneously organize themselves, breaking a symmetry and turning the material into an insulator.

The researchers simulated what happens when they start with a material where the electrons and the organizing forces move at different speeds (a realistic scenario) and then watched what happened as they cooled the system down to absolute zero or warmed it up. They found that even though the material started out "lopsided" with different speeds, as it approached the critical tipping point, the electrons and the organizing forces magically synchronized their speeds. The system spontaneously developed a perfect "relativistic symmetry," just as the theories predicted. It's as if a group of runners starting at different paces suddenly found a rhythm and began running in perfect unison as they approached the finish line.

The team also mapped out what happens when the material isn't perfectly cold. They discovered that even at finite temperatures, the material holds onto its ordered state up to a specific critical temperature, where it undergoes a classic phase transition (like water freezing into ice) that follows the same rules as a simple 2D magnet. They drew a detailed "phase diagram"—a map of the material's behavior—showing a "quantum critical fan." This fan is a wedge-shaped region above the critical point where the material's properties are dominated by the quantum critical point, scaling in a very specific, predictable way.

Crucially, the authors didn't just guess; they used advanced numerical methods to solve complex equations that describe the entire "potential energy landscape" of the system. Their simulations confirmed that the "relativistic symmetry" emerges on both sides of the transition, even in the region where the material has already turned into an insulator and developed a "mass gap" (a barrier that stops the electrons from moving freely). They also tracked how the "quasiparticle weight"—a measure of how well-defined the electron's identity is—changes, finding that at absolute zero, the electrons lose their individual identity near the critical point, while at higher temperatures, they behave differently depending on whether you look at their movement through space or through time.

In short, this paper provides a unified, computer-simulated framework that proves these exotic quantum materials do indeed develop a beautiful, symmetric order near their critical points, even when they start out imperfect and are kept at real-world temperatures. It bridges the gap between abstract theory and the messy reality of experimental materials, offering a clear roadmap for understanding how these quantum phases form and behave.

Technical Summary: Quantum Critical Fan and Emergent Relativistic Symmetry of Two-Dimensional Dirac Semimetals

Problem Statement
Two-dimensional Dirac semimetals near a quantum critical point (QCP) are theoretically described by Gross–Neveu–Yukawa (GNY) models. While the zero-temperature critical behavior of these systems is well-studied, a comprehensive understanding of finite-temperature effects remains necessary, particularly in light of recent experimental observations of semimetal-to-insulator transitions in tunable van-der-Waals heterostructures (e.g., twisted tetralayer tungsten diselenide). A key theoretical challenge is that real materials possess distinct bare velocities for Dirac fermions ( $v_\psi$ ) and order-parameter fluctuations ( $v_\phi$ ), explicitly breaking relativistic symmetry at the microscopic scale. Theoretical frameworks must determine if and how relativistic symmetry emerges near the QCP and how this influences the phase diagram, including the extent of the quantum critical fan (QCF) and the behavior of quasiparticle weights at finite temperatures.

Methodology
The authors employ a non-perturbative functional renormalization group (fRG) approach to study the chiral Ising model, a specific GNY model featuring a single real scalar field breaking a discrete chiral $Z_2$ symmetry. The study utilizes the following technical components:

Model: A 2+1-dimensional Euclidean field theory with eight-component Dirac fermions coupled to a scalar order parameter via a Yukawa interaction. The model explicitly includes scale-dependent velocities $v_{\psi,k}$ and $v_{\phi,k}$ , allowing for the tracking of velocity renormalization.
Regulators: Two regulator schemes are employed. A covariant regulator is used to demonstrate the emergence of relativistic symmetry at $T=0$ . A spatial regulator is utilized for finite-temperature calculations to enable analytical evaluation of Matsubara sums and to handle the explicit breaking of relativistic symmetry ( $v_\phi \neq v_\psi$ ) at intermediate scales.
Numerical Implementation: The authors resolve the fully field-dependent effective potential using the DiFfRG framework, which applies fluid dynamical methods to solve the highly non-linear partial differential flow equations. This allows for the treatment of the convexity of the effective action and the resolution of non-analytic structures (kinks) in the potential, which standard Taylor expansions cannot capture.
Initial Conditions: The UV cutoff is set to correspond to the moiré lattice scale. Initial velocities are chosen to reflect experimental estimates for twisted tetralayer WSe $_2$ ( $v_{\psi,\Lambda}/v_{\phi,\Lambda} \approx 1.35$ ), with the Yukawa coupling tuned to locate the QCP.

Key Contributions and Results

Emergent Relativistic Symmetry at $T=0$ :
The study confirms that relativistic symmetry emerges near the QCP even when starting with distinct bare velocities. The authors demonstrate that the subspace where $v_\psi = v_\phi$ is closed under the RG flow and that linear perturbations out of this subspace are irrelevant. This holds true on both the symmetric side and the symmetry-broken side (where a dynamical mass gap is generated). The velocities flow toward a common terminal value in the infrared, establishing $z=1$ at the fixed point.
Quantum Critical Exponents:
Using the spatial regulator and numerical integration, the authors extract critical exponents for the chiral Ising universality class in 2+1 dimensions. The results for the correlation length exponent ( $\nu \approx 1.008$ ) and anomalous dimensions ( $\eta_\phi \approx 0.748$ , $\eta_\psi \approx 0.031$ ) show excellent agreement with established results from perturbative field theory, conformal bootstrap, and quantum Monte Carlo simulations.
Finite-Temperature Transition and Precondensation:
The paper maps the phase diagram into the ordered ( $Z_2$ broken) and disordered phases. The finite-temperature transition is shown to belong to the classical two-dimensional Ising universality class, with extracted exponents ( $\nu \approx 1.02$ , $\eta \approx 0.236$ ) matching the exact 2D Ising values. The study identifies a "precondensation" regime where a finite order parameter develops at intermediate RG scales but vanishes at larger scales (in the thermodynamic limit) for temperatures slightly above $T_c$ . This regime signals the formation of locally ordered domains.
Quantum Critical Fan (QCF):
The authors determine the extent and scaling properties of the QCF in the presence of different bare velocities. They find an extended region above the QCP where the thermal correlation length scales as $\xi_T \propto T^{-1/z}$ with $z \approx 1$ . The boundaries of this fan follow the scaling law $T \propto |h - h_c|^{z\nu}$ . The study clarifies that this scaling region is cut off at high temperatures (approaching the UV cutoff) where universal behavior gives way to non-universal regimes.
Fermionic Quasiparticle Weight:
The behavior of the fermionic wave-function renormalization ( $Z_\psi$ ), acting as a proxy for quasiparticle weight, is analyzed. At $T=0$ , $Z_\psi$ vanishes in the infrared, indicating the absence of coherent quasiparticles. At finite $T$ , the temporal and spatial components of $Z_\psi$ exhibit distinct behaviors due to the dimensional reduction near the transition. The spatial component shows a pronounced suppression along the precondensation line, while the temporal component is less suppressed, leading to an increase in the effective Fermi velocity in this regime.

Significance and Claims
The paper claims to provide a unified field-theoretical framework that quantitatively describes both the universal quantum critical behavior and the finite-temperature critical behavior of the chiral Ising model, as well as the non-universal regions of the phase diagram.

It offers the first non-perturbative evidence for the emergence of relativistic symmetry on the symmetry-broken side of the transition.
It successfully benchmarks the fRG approach against established critical exponents using a sophisticated numerical treatment of the effective potential.
It systematically explores the QCF and quasiparticle weight in the presence of velocity mismatches, a scenario relevant to real materials like twisted WSe $_2$ .
The authors state that these developments prepare the ground for calculating thermodynamic observables and spectral functions for correlated Dirac semimetals, with the ultimate goal of attaching realistic energy scales (e.g., millikelvin or electronvolt) to theoretical predictions for specific materials.

The work does not propose new experimental setups but rather provides a theoretical tool to interpret existing experimental data and predict scaling behaviors in tunable van-der-Waals heterostructures. Future work is noted to include the investigation of secondary instabilities (e.g., superconductivity) via dynamical bosonization.

Technical Summary: Quantum Critical Fan and Emergent Relativistic Symmetry of Two-Dimensional Dirac Semimetals

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