🔬 mesoscale physics

Upper critical dimension for dirty Weyl semimetal-to-metal quantum phase transitions

This paper establishes that the disorder-driven Weyl semimetal-to-metal quantum phase transition possesses lower and upper critical dimensions of $d=2$ and $d=4$ , respectively, with a correlation length exponent of $\nu \approx 0.5$ for $d \geq 4$ and a dynamic scaling exponent of $z \approx d/2$ for $d \geq 3$ , as confirmed by both numerical simulations and field-theoretic renormalization group calculations.

Original authors: Yongtai Li, Bitan Roy

Published 2026-09-25

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CC BY 4.0

Original authors: Yongtai Li, Bitan Roy

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

Matter changes its fundamental nature when pushed to extremes. Just as water freezes into ice or boils into steam when temperature shifts, exotic materials can undergo a sudden transformation when other conditions are altered, such as the strength of a magnetic field or the amount of impurities mixed inside. These transformations, known as quantum phase transitions, happen at the coldest possible temperatures where thermal jitters vanish, leaving only the strange rules of quantum mechanics to dictate behavior. In many materials, adding disorder—like scattering electrons off random impurities—eventually turns a conductor into an insulator, a phenomenon well understood for decades. However, a newer class of materials called Weyl semimetals behaves differently. In these crystals, electrons move in a unique, straight-line fashion that makes them exceptionally robust against weak disorder. The big question for physicists has been to determine exactly how much disorder these materials can withstand before they finally collapse into a different state, and whether there is a specific limit to the number of dimensions in space where this collapse follows simple, predictable rules.

A team of researchers at Lehigh University has now mapped out these limits with high precision, revealing that the behavior of these materials depends critically on the number of spatial dimensions they occupy. By running extensive computer simulations on models of these crystals, the scientists explored what happens when disorder is gradually increased from a whisper to a roar. They found that in a two-dimensional world, even the tiniest amount of disorder is enough to destroy the unique properties of the material, turning it into a metal immediately. This establishes two dimensions as the lower boundary where the material can no longer survive as a pristine semimetal. However, the story changes dramatically in higher dimensions. In three dimensions and above, the material remains stable against weak disorder, only succumbing to a metallic state once the disorder crosses a specific, finite threshold. The researchers discovered that this transition is governed by a special upper limit: four dimensions. At and above this four-dimensional mark, the transition behaves in a simple, predictable way known as mean-field behavior, where the complex interactions between particles average out into a straightforward rule. Below four dimensions, the behavior is more complex and requires deeper analysis.

To reach these conclusions, the team constructed digital models of these crystals, ranging from two to six dimensions, and introduced random impurities to simulate disorder. They used a sophisticated computational technique to calculate how many electron states were available at zero energy, a key indicator of whether the material was acting as a semimetal or a metal. In the two-dimensional simulations, they observed that the material became metallic for any non-zero amount of disorder, with the number of available states growing in a specific exponential pattern. In contrast, for dimensions three through six, the material held its ground as a semimetal until the disorder strength reached a critical point. Beyond this point, the material abruptly shifted into a diffusive metal, where electrons move in a scattered, random walk rather than a straight line. The simulations allowed the team to pinpoint the exact disorder strength required to trigger this shift for each dimension, finding that the threshold increases as the number of dimensions grows.

The study also uncovered a precise mathematical relationship governing how the material behaves right at the moment of transition. The researchers found that a specific scaling exponent, which describes how the material's properties change as it approaches the critical point, is directly tied to the number of dimensions. Specifically, this exponent is always equal to half the number of dimensions. For instance, in three dimensions, the value is roughly 1.5, and in four dimensions, it is exactly 2. This relationship was confirmed not only by the computer simulations but also by advanced theoretical calculations that treat the material as a continuous field rather than a grid of points. These calculations showed that the relationship is exact, holding true for any dimension greater than two. Furthermore, the team determined that for dimensions four and higher, the critical exponent describing the correlation length—a measure of how far the influence of one electron extends to another—locks into a constant value of 0.5. This confirms that four dimensions is indeed the upper critical dimension, the point beyond which the transition becomes simple and follows standard statistical rules.

The findings provide a complete picture of how disorder drives these quantum phase transitions across different spatial dimensions. The researchers suggest that while the behavior in four and higher dimensions is now well understood and follows simple rules, the transition in three dimensions remains a more complex puzzle, with the critical exponent deviating from the simple mean-field value. This distinction is crucial because it tells scientists where they can use simplified theories to predict material behavior and where they must rely on more complex, detailed calculations. The work also highlights that the unique stability of Weyl semimetals is a fragile balance, preserved only in three dimensions and above, provided the disorder does not exceed a specific limit. By identifying the lower and upper critical dimensions, the study offers a clear framework for understanding these exotic materials, suggesting that future theoretical work can now focus on the specific complexities of three-dimensional systems while relying on established, simpler rules for higher dimensions. This clarity helps bridge the gap between abstract theory and the real-world behavior of materials that could one day power the next generation of electronic devices.

Technical Summary: Upper Critical Dimension for Dirty Weyl Semimetal-to-Metal Quantum Phase Transitions

Problem Statement
The paper investigates the disorder-driven quantum phase transition (QPT) between a ballistic Weyl semimetal (WSM) and a compressive diffusive metal (CDM). While the Anderson metal-to-insulator transition (AMIT) in noninteracting systems is well-established to have a lower critical dimension (LCD) of $d_\ell = 2$ but no finite upper critical dimension (UCD) (i.e., $d_u \to \infty$ ), the nature of the WSM-to-metal QPT remains less defined. Specifically, the existence and value of an upper critical dimension $d_u$ for this unconventional transition, where critical phenomena become mean-field or Gaussian, had not been rigorously established. The study aims to determine the LCD and UCD for this transition and characterize the associated critical exponents (correlation length exponent $\nu$ and dynamic scaling exponent $z$ ) across dimensions $d=2$ to $6$.

Methodology
The authors employ a dual approach combining extensive numerical simulations and analytical field-theoretic calculations:

Numerical Simulations:
- Model: A tight-binding model on a $d$ -dimensional hypercubic lattice supporting linearly dispersing gapless Weyl quasiparticles. Disorder is introduced via random pointlike charge impurities sampled from a box distribution $[-W/2, W/2]$ .
- Technique: The Kernel Polynomial Method (KPM) is used to compute the disorder-averaged density of states (ADOS), $\rho(E)$ .
- Parameters: Calculations are performed for spatial dimensions $d = 2, 3, 4, 5, 6$ . System sizes ( $L$ ) vary from $L=12$ ( $d=6$ ) to $L=2500$ ( $d=2$ ) due to computational constraints.
- Analysis: The transition is identified by the behavior of $\rho(0)$ (or the difference $\Delta\rho(0)$ between disordered and clean systems). The critical disorder strength $W_c$ , dynamic scaling exponent $z$ (derived from $\rho(E) \sim |E|^{d/z-1}$ at criticality), and correlation length exponent $\nu$ (derived from scaling in the metallic phase) are extracted via finite-size scaling and data collapse techniques.
Analytical Calculations:
- Self-Consistent Born Approximation (SCBA): Used to anticipate the stability of Weyl fermions and the existence of $d_\ell$ and $d_u$ by solving for the quasiparticle lifetime $\tau_{qp}$ .
- Renormalization Group (RG) Analysis: Two approaches are utilized:
  - An $\epsilon_d$ expansion ( $\epsilon_d = d-2$ ) around the lower critical dimension.
  - An alternative RG approach involving a disorder field with a non-analytic propagator (related to an $\epsilon_m$ expansion), which preserves a reduced $U(1)$ gauge symmetry. This approach is used to derive exact results for the dynamic exponent.

Key Contributions and Results

Lower Critical Dimension ( $d_\ell = 2$ ):
- Numerical results in $d=2$ show that $\rho(0)$ becomes finite for infinitesimal disorder, following a BCS-like scaling $\rho(0) \sim \exp(-\lambda/W)$ .
- SCBA analysis confirms a logarithmic divergence in the integral for quasiparticle lifetime, indicating that the WSM phase is unstable to any amount of disorder in $d=2$ . Thus, $d_\ell = 2$ .
Upper Critical Dimension ( $d_u = 4$ ):
- For $d \ge 3$ , the system remains a stable WSM ( $\rho(0)=0$ ) up to a finite critical disorder $W_c$ , beyond which it transitions to a CDM.
- Numerical extraction of the correlation length exponent $\nu$ yields $\nu \approx 1.0$ in $d=3$ , but $\nu \approx 0.5$ for $d=4, 5, 6$ .
- The locking of $\nu$ to the mean-field value of $0.5$ for $d \ge 4$ identifies $d_u = 4$ as the upper critical dimension.
- SCBA analysis in $d=4$ reveals a logarithmic correction to the self-consistent solution for $\tau_{qp}$ , signaling the violation of hyperscaling characteristic of the UCD.
Dynamic Scaling Exponent ( $z$ ):
- Numerical fitting of the ADOS at the critical point yields $z \approx d/2$ for all $d \ge 3$ (e.g., $z \approx 1.5$ in $d=3$ , $z \approx 2.0$ in $d=4$ ).
- The alternative RG approach, utilizing the Ward identity for the disorder field, proves that $z = d/2$ is an exact result for any $d > 2$ , independent of higher-order corrections.
Critical Exponents and Scaling:
- The paper establishes the scaling form $\rho(E) = \delta^{(d-z)\nu} G(|E|\delta^{-\nu z}, L^{1/\nu}\delta)$ , where $\delta = (W-W_c)/W_c$ .
- High-quality data collapses are achieved for $d=3$ through $d=6$ using the extracted values of $W_c$ , $z$ , and $\nu$ , validating the universality class.

Significance and Claims
The paper claims to establish that the disorder-driven Weyl semimetal-to-metal QPT is characterized by a lower critical dimension $d_\ell = 2$ and an upper critical dimension $d_u = 4$ . This finding is significant because it contrasts sharply with the AMIT, which lacks a finite UCD.

The authors assert that:

For $d \ge 4$ , the transition is governed by mean-field (Gaussian) criticality with $\nu = 0.5$ .
The dynamic exponent $z = d/2$ is exact for any $d > 2$ , a result derived from field-theoretic RG calculations that go beyond the limitations of the standard $\epsilon_d$ expansion.
These results justify the use of field-theoretic approaches (specifically $\epsilon_u$ expansions around $d=4$ and $\epsilon_d$ expansions around $d=2$ ) to study this QPT.
While the paper notes that the $\epsilon_d$ expansion fails to correctly predict $\nu$ above $d_u$ (a known limitation in critical phenomena), the $\epsilon_m$ expansion successfully predicts the exact $z$ and the mean-field $\nu$ for $d \ge 4$ .

The study concludes that WSM-to-metal QPTs can be systematically studied theoretically around both critical dimensions, providing a controlled framework for understanding disorder effects in topological semimetals. The authors note that demonstrating the logarithmic violation of scaling at $d=4$ numerically remains challenging with current lattice models but suggests SLAC fermions as a potential future platform for this verification.

Technical Summary: Upper Critical Dimension for Dirty Weyl Semimetal-to-Metal Quantum Phase Transitions

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