🔬 mesoscale physics

Hyperbolic lattices with mass disorder: Phases and phase transitions

This study investigates the impact of mass disorder on nearest-neighbor tight-binding models across three distinct hyperbolic lattices, revealing that while the Dirac liquid phase remains stable against weak disorder before undergoing sequential semimetal-to-metal and metal-to-insulator transitions, the other two systems exhibit only the latter transition, with critical exponents closely resembling those found in on-site potential disorder scenarios.

Original authors: Sheersh Sen, Christopher A. Leong, Bitan Roy

Published 2026-10-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Sheersh Sen, Christopher A. Leong, 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

In the vast landscape of materials science, researchers often look to the geometry of a material's internal structure to predict how electricity will flow through it. Imagine a grid where every point is connected to its neighbors; if this grid is flat, like a sheet of graph paper, the rules for electron movement are well understood. However, nature also allows for grids that curve, much like the surface of a saddle or a coral reef, where space itself is stretched and twisted. This curvature changes the fundamental rules of the game, creating exotic states of matter that cannot exist on flat surfaces. Scientists are particularly interested in how these curved structures behave when they are not perfect, but rather contain random imperfections or "disorder," such as atoms that are slightly heavier or lighter than their neighbors. Understanding this interplay between curved space and disorder is crucial because it reveals how electrons organize themselves in complex environments, potentially leading to new types of electronic devices or helping us understand the fundamental limits of conductivity.

A team of physicists has recently explored this question by simulating electrons moving on specific, highly curved grids known as hyperbolic lattices. These structures are built from repeating shapes, like polygons, that fit together in a way that forces the space to curve negatively. The researchers focused on three distinct types of these lattices, each with a different arrangement of connections. In a perfect, clean version of these lattices, the electrons behave in three very different ways depending on the geometry: in one type, they act like a "Dirac liquid," where the number of available energy states drops to zero at the center; in another, they form a standard "Fermi liquid" with a steady number of states; and in the third, they crowd into a "flat band," where the density of states explodes to infinity. The team wanted to see what happens when they introduce mass disorder, a specific type of imperfection where the atoms in the lattice have random, varying weights, effectively creating a bumpy landscape for the electrons to navigate.

Using powerful computer simulations, the researchers mapped out exactly how these systems respond to increasing levels of disorder. They found that the behavior of the electrons depends heavily on which type of curved lattice they inhabit. For the lattice that starts as a Dirac liquid, the system is surprisingly robust against weak disorder. The electrons remain in their semimetallic state, resisting the chaos. However, as the disorder increases to a moderate level, the system undergoes a sudden shift, transforming into a metal where electricity flows freely. If the disorder becomes even stronger, the electrons finally get trapped, and the material turns into an insulator, stopping the flow of electricity entirely. This means the Dirac liquid system passes through three distinct phases: a semimetal, a metal, and an insulator, separated by two specific tipping points.

In contrast, the other two types of lattices, which start as a Fermi liquid and a flat band, behave differently. These systems do not have a stable semimetallic phase to begin with. Instead, they remain metallic as disorder is introduced, only undergoing a single transition to become insulators when the disorder becomes very strong. This suggests that the initial shape of the curved space dictates the number of stable phases a material can support. The researchers also calculated the precise mathematical "exponents" that describe how the system changes near these tipping points. They found that these values are very similar to those seen in flat, two-dimensional materials when they are disturbed by random electrical potentials, indicating that the underlying physics of the transition is universal, regardless of whether the disorder comes from mass or electric charge.

One of the most striking findings is that the transition from metal to insulator in these curved systems is driven purely by the geometry of the space itself. In flat, two-dimensional materials, theory predicts that electrons should never fully stop conducting in the presence of this type of disorder. Yet, on these curved lattices, the electrons do eventually localize and stop moving, proving that the negative curvature of space acts as a powerful force that can trap electrons. The study also revealed that the transition to the insulating state might not follow a simple, smooth curve but could involve a more complex, abrupt change in behavior, a detail that requires further investigation.

The implications of these findings extend beyond theoretical physics. The researchers suggest that these predictions could be tested in the real world using classical wave systems, such as light traveling through specially designed photonic crystals. Since light behaves similarly to electrons in these mathematical models, scientists could build physical models of these curved, disordered lattices using glass or plastic structures with varying refractive indices. By observing how light scatters and localizes in these curved, imperfect structures, experimentalists could verify the theoretical predictions about phase transitions and the stability of different electronic states. This work opens a new window into understanding how the shape of space itself influences the flow of energy and matter, offering a glimpse into a world where geometry dictates the very possibility of conduction.

Technical Summary: Hyperbolic Lattices with Mass Disorder: Phases and Phase Transitions

Problem Statement
This study investigates the impact of mass disorder on the electronic states and density of states (DOS) of nearest-neighbor (NN) tight-binding models (TBMs) defined on bipartite hyperbolic lattices (HLs). Specifically, the authors examine plaquette-centered $\{10, 3\}$ , $\{8, 3\}$ , and $\{8, 4\}$ lattices embedded on a Poincaré disk with open boundary conditions. In the clean limit, these systems exhibit distinct ground states: the $\{10, 3\}$ lattice hosts a Dirac liquid (DL) with a power-law vanishing DOS at the band center; the $\{8, 3\}$ lattice supports a Fermi liquid (FL) with a finite DOS; and the $\{8, 4\}$ lattice displays a flat band (FB) with a diverging DOS. The central question is how these distinct phases respond to mass disorder, particularly given that the underlying spatial curvature of hyperbolic space fundamentally alters the scaling of the return probability for particles, potentially enabling Anderson metal-to-insulator transitions (AMIT) in symmetry classes where they are forbidden in Euclidean 2D lattices.

Methodology
The authors employ the Kernel Polynomial Method (KPM) to numerically compute both the average density of states (ADOS, $\rho_a(E)$ ) and the typical density of states (TDOS, $\rho_t(E)$ ) across varying strengths of mass disorder ( $W$ ).

Model: The system is modeled using a NN-TBM on bipartite HLs. Mass disorder is implemented by assigning random on-site potentials $\pm V(r_i)$ to the two complementary sublattices (A and B) within each unit cell. The magnitude of the potential is drawn from a box distribution $[-W/2, W/2]$ , while the signs are strictly opposite on the A and B sublattices. This specific disorder configuration breaks particle-hole and sublattice symmetries but preserves time-reversal symmetry, placing the system in the Wigner-Dyson orthogonal symmetry class AI.
System Size: Simulations are performed on large systems (exceeding $10^5$ sites) with total generations $n_{tot} = 7$ or $9$, ensuring the ratio of bulk to boundary sites approaches the thermodynamic limit.
Analysis: The study analyzes the scaling of DOS near zero energy to identify phase transitions. Critical exponents are extracted by fitting the scaling of $\rho_a(0)$ and $\rho_t(0)$ near the critical disorder strengths ( $W_c$ ). Both power-law and essential singularity forms are tested for the TDOS near the AMIT.

Key Contributions and Results

Phase Diagram of the Dirac Liquid ( $\{10, 3\}$ ): The $\{10, 3\}$ system, which is a semimetal in the clean limit, exhibits a rich phase diagram under mass disorder.
- Stability: The Dirac liquid remains stable against weak mass disorder, with both $\rho_a(0)$ and $\rho_t(0)$ remaining vanishingly small.
- Semimetal-to-Metal QPT: At a moderate critical disorder strength ( $W_{c,1} \approx 0.40 - 0.55$ ), the system undergoes a quantum phase transition (QPT) to a diffusive metal, where the DOS at zero energy becomes finite.
- Anderson Metal-to-Insulator Transition (AMIT): Upon further increasing the disorder to a stronger critical strength ( $W_{c,2}$ ), the system undergoes an AMIT. At this point, the TDOS ( $\rho_t(0)$ ) vanishes, indicating localization, while the ADOS may remain finite.
- Critical Exponents: The study extracts critical exponents for the semimetal-to-metal transition ( $\alpha \approx 0.99$ , $\beta_a \approx 1.78$ , $\beta_t \approx 1.02$ ) and the AMIT ( $\beta_A \approx 1.84$ for power-law scaling). The authors note that the TDOS near the AMIT may also follow an essential singularity form, yielding a higher critical disorder ( $W_{c,2} \approx 10.45$ ).
Phase Diagrams of Fermi Liquid and Flat Band ( $\{8, 3\}$ and $\{8, 4\}$ ): Unlike the Dirac liquid, the FL and FB systems do not exhibit a semimetal-to-metal transition. They remain metallic (finite DOS) under weak disorder and undergo a direct AMIT at a critical disorder strength ( $W_{c,2}$ ).
- For $\{8, 3\}$ , $W_{c,2} \approx 6.00$ (power-law) or $8.25$ (essential).
- For $\{8, 4\}$ , $W_{c,2} \approx 12.00$ (power-law) or $14.35$ (essential).
- The critical exponents for the AMIT in these systems ( $\beta_A \approx 1.80 - 1.96$ ) are found to be consistent with those in the $\{10, 3\}$ system and close to values found in systems with on-site potential disorder.
Role of Curvature and Symmetry: The results demonstrate that the finite spatial curvature of hyperbolic lattices provides an infrared cutoff that stabilizes a metallic phase against moderate disorder, a feature absent in 2D Euclidean lattices of symmetry class AI. The transition to the insulating phase is driven solely by the strength of the disorder overcoming this curvature-induced stability.

Significance and Claims
The paper claims to establish a comprehensive global phase diagram for electronic fluids on bipartite hyperbolic lattices subject to mass disorder.

Distinct Phase Behaviors: It highlights that while dirty hyperbolic Dirac systems support three distinct phases (semimetal, metal, and Anderson insulator) separated by two quantum critical points, disordered hyperbolic Fermi liquid and flat band systems support only two phases (metal and Anderson insulator) separated by a single critical point.
Universality: The critical exponents observed for both the semimetal-to-metal QPT and the AMIT are reported to be sufficiently close to those mediated by on-site potential disorder, suggesting a universality within symmetry class AI for these hyperbolic systems.
Experimental Relevance: The authors suggest that classical metamaterials capable of engineering hyperbolic lattices, particularly photonic lattices, serve as natural testbeds for these theoretical predictions. They propose that by randomly varying the refractive index in a staggered intra-unit cell pattern, one could observe the predicted semimetal-to-metal QPT and AMIT in photonic systems, phenomena not typically observed in 3D Euclidean disordered photonic lattices due to the transverse nature of light and the absence of AMIT in Euclidean class AI.

The study concludes that the interplay between the emergent bipartite nature of these lattices, the specific form of mass disorder, and the constant negative curvature of the space dictates a unique set of quantum phase transitions distinct from their Euclidean counterparts.

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