Hyperbolic lattices with mass disorder: Phases and phase transitions

arXiv:2610.02192 · cond-mat.dis-nn, cond-mat.mes-hall, cond-mat.stat-mech · Submitted 2026-10-01 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Hyperbolic lattices with mass disorder".

Mira: Nearest-neighbor tight-binding models (TBMs) on hyperbolic lattices (HLs) with mass disorder reveal distinct electronic phases and phase transitions, providing insights into how spatial curvature influences quantum materials.

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So, we're looking at this paper by Sen, Leong, and Roy called "Hyperbolic lattices with mass disorder: Phases and phase transitions." It seems they've looked closely at how putting mass disorder into these specific hyperbolic lattice structures affects the electronic states.

Mira: That’s right, Kai; the paper focuses on three specific hyperbolic lattices—the plaquette-centered ten three eight three and eight four types—and how they host distinct electronic phases like a Dirac liquid or a flat band in the clean limit <ref:2610.02192#pg0>.

Lev: From my side, I'm curious about what this means for experimental realization; if we can engineer these geometries on a chip or in an optical system, how feasible is it to measure these kinds of DOS changes?

Kai: Exactly, Lev; the paper dives into three different clean lattice types and shows they give us three very different starting points: a Dirac liquid with a power-law vanishing density of states near zero energy, a Fermi liquid with a finite density of states near the band center, and then flat bands that show a diverging DOS.

Mira: And what gets really interesting is how mass disorder changes these clean behaviors; they find that the hyperbolic Dirac liquid stays stable against weak mass disorder because both the average and typical density of states near zero energy are vanishingly small, specifically less than ten to the power of negative three <ref:2610.02192#pg1>.

Lev: So if we want to build a robust system that exhibits this Dirac liquid phase, what is the practical threshold for disorder strength before we see those DOS values become finite?

Kai: The paper pinpoints a critical disorder strength, W c,one around zero point five five where the semimetal-to-metal quantum phase transition happens, shifting both rho a(zero) and rho t(zero) from vanishingly small to finite values <ref:2610.02192#pg1>.

Mira: That transition point is where things get interesting because it marks the shift away from the stable Dirac liquid phase into a more metallic regime, which is what we usually want to explore in AI simulations for material design <ref:2610.02192#pg1>.

Lev: If we're designing error-correcting codes based on this physics, does that mean the transition itself might introduce new types of topological protection or instability?

Kai: Well, the paper suggests that beyond W c,one we see a further phase change into a diffusive metal via a quantum phase transition where both rho a(zero) and rho t(zero) become finite <ref:2610.02192#pg1>.

Mira: But they also found that at even stronger disorder, W c,two the system hits an Anderson metal-to-insulator transition where only rho t(zero) becomes zero <ref:2610.02192#pg1>.

Lev: That AMIT behavior is something we need to consider for hardware reliability; if we are aiming for robust quantum computation, knowing exactly when that localization happens is crucial for designing better error correction schemes <ref:2610.02192#pg2>.

Title and authors: Kai: The paper provides some very specific scaling exponents near these transitions, like the DOS exponent alpha being zero point nine nine plus or minus zero point zero one for the semimetal-to-metal QPT in the ten three system at W c,one of zero point five five <ref:2610.02192#pg2>.

Mira: Those exponents give us a mathematical fingerprint of how the DOS behaves as you approach that critical point, which is essential for any theoretical modeling we do for these systems <ref:2610.02192#pg3>.

Lev: From an error correction standpoint, understanding beta a and beta t, which they found to be one point seven eight plus or minus zero point zero five in the ten three system at W c,one tells us about the rate at which these order parameters grow as we move away from the critical point <ref:2610.02192#pg3>.

Kai: And for those who want to know about the Anderson metal-to-insulator transition near W c,two of seven point zero zero, they get a power-law scaling with an exponent beta A of one point eight four plus or minus zero point one five <ref:2610.02192#pg3>.

Mira: The paper also mentions that near the AMIT, there's an essential singularity in the typical density of states at zero energy, which is described by a linear fit between rho t(zero) - D and delta - one/two A when W c,two is ten point four five <ref:2610.02192#pg3>.

Lev: That distinction between power-law and essential singularity near the AMIT is vital for distinguishing different types of localization physics, which affects how we model the robustness of quantum states in real hardware <ref:2610.02192#pg3>.

Kai: The core concept they are pushing here is that the stability of these Dirac liquids against weak mass disorder comes from "the relativistic nature of quasiparticles living on a negative curved space with a constant curvature alpha " <ref:2610.02192#pg1>.

Mira: That spatial curvature is the key assumption underpinning their findings, suggesting that geometry itself imposes constraints that keep the system metallic under mild perturbations <ref:2610.02192#pg3>.

Lev: If we were to implement this on a physical platform, say using engineered photonic structures instead of traditional tight-binding models, how much do you think the curvature term needs to be precisely controlled?

Kai: That's a big question for the experimentalist; controlling that specific negative curvature in a lattice structure is something we have to figure out when we move from theory to actual building <ref:2610.02192#pg3>.

Mira: The paper notes a limitation: they compute the density of states using the kernel polynomial method after averaging over sixty independent disorder realizations, which means their results are based on that specific numerical procedure <ref:2610.02192#pg1>.

Lev: That reliance on averaging over sixty realizations is something I'd watch closely; if we could design an experiment that probes the typical density of states directly without relying on this kind of averaging, it would be a significant step for validating these results <ref:2610.02192#pg1>.

Title and authors: Kai: So, to wrap up this discussion on "Hyperbolic lattices with mass disorder: Phases and phase transitions," the paper shows how geometry dictates the clean phases and how disorder drives them through specific transition points characterized by scaling exponents like alpha = zero point nine nine plus or minus zero point zero one for the semimetal-to-metal QPT <ref:2610.02192#pg2>.

Mira: Essentially, the paper demonstrates that mass disorder doesn't just scramble things; it triggers a predictable sequence of phase transitions—from a stable Dirac liquid to a diffusive metal and eventually an Anderson insulator depending on how strong the disorder gets <ref:2610.02192#pg1>.

Lev: For those of us working on quantum error correction, this provides a roadmap for understanding exactly what kind of localization we might expect when designing systems with intrinsic curvature <ref:2610.02192#pg3>.

Kai: It’s clear that the interplay between the lattice geometry and the random potentials creates rich electronic phases that are highly sensitive to disorder strength <ref:2610.02192#pg3>.

Mira: The implications for AI in materials science, for instance, is huge because it suggests we can use topological features like spatial curvature as a fundamental parameter to predict whether a material will be conductive or insulating <ref:2610.02192#pg3>.

Lev: I think the most important implication is that it shows how structural topology can fundamentally alter the localization landscape, which is something we need to factor into our models for complex quantum hardware <ref:2610.02192#pg3>.

Kai: So, in summary of this paper, "Hyperbolic lattices with mass disorder: Phases and phase transitions" reveals distinct electronic signatures based on lattice structure and how disorder pushes the system across critical points defined by specific scaling behaviors <ref:2610.02192#pg2>.

Mira: It really solidifies the idea that the intrinsic geometry of a lattice, like being hyperbolic, imposes specific rules on how mass disorder influences emergent quantum phases <ref:2610.02192#pg3>.

Lev: We've seen how these transitions are characterized by precise critical exponents, which gives us concrete mathematical tools to work with when we try to map this physics onto any realizable quantum system <ref:2610.02192#pg3>.

Kai: It’s a lot of detail, but it shows a clear path for how researchers can start designing novel materials where the geometry itself is part of the control mechanism <ref:2610.02192#pg3>.

Mira: The future work mentioned in the paper seems to focus on extending these findings beyond just these three specific lattice types, which could open up a much broader area of investigation <ref:2610.02192#pg3>.

Lev: I hope future work explores how this framework can be applied to more complex, interacting many-body problems that are relevant to actual quantum computation challenges <ref:2610.02192#pg3>.

Kai: That sounds like the next logical step for experimental validation and theoretical refinement of these concepts <ref:2610.02192#pg3>.

The paper's summary: Kai: So, essentially, this paper shows that when you take these hyperbolic lattice structures and you introduce mass disorder—which is just random noise in the on-site energy—the system doesn't just get messed up; it actually undergoes a series of distinct physical transformations depending on how much noise there is.

Mira: That’s right, Kai; the core finding revolves around three specific types of hyperbolic lattices, and each one starts off in a very different electronic state before the disorder kicks in. They found that some lattices naturally host a Dirac liquid, others are Fermi liquids, and still others have flat bands in their clean state.

Lev: From my side, I'm interested in how those initial clean states dictate the subsequent behavior under perturbation; if you start with a Dirac liquid, what does that imply about the resilience of its electronic structure when disorder is added?

Kai: The paper tells us that the Dirac liquid phase is surprisingly stable against weak mass disorder because the quasiparticles behave in a way dictated by that negative curvature, which keeps their density of states vanishingly small initially.

Mira: Exactly, Lev; that stability comes from a specific geometric property—the relativistic nature of quasiparticles on this negatively curved space—which leads to very controlled scaling behaviors for the DOS near zero energy when the disorder is tiny.

Lev: If we're thinking about building quantum hardware, that stability against weak noise is important because it suggests a certain robustness in the system's fundamental electronic description, regardless of minor imperfections.

Kai: And then things get interesting when the disorder crosses a certain threshold; they identified a critical strength where that stable Dirac liquid starts to change into something else entirely through a quantum phase transition.

Mira: That transition point, W c,one is crucial because it signals the shift from a semimetal state to a diffusive metal where both the average and typical densities of states near zero energy start becoming finite values.

Lev: For error correction, that QPT boundary tells us exactly where the system's transport properties fundamentally change; if we can place our qubits in this region, we know precisely when they might start exhibiting different error scaling mechanisms.

Kai: But the paper doesn't stop there; they showed that if you push the disorder even further past a second threshold, W c,two you hit an Anderson metal-to-insulator transition where the typical density of states at zero energy drops to absolute zero, with only one component changing its behavior.

Mira: That AMIT behavior is quite telling because it suggests that in these curved systems, the localization isn't just due to standard on-site potential disorder; the spatial curvature itself plays a role in triggering that insulating phase under stronger conditions.

Lev: That distinction between disorder types is really something we need to model carefully when designing error correction protocols; we can’t just treat all localization as a single problem, because here it seems geometry provides an extra knob to turn.

Kai: So, the big picture here is that the shape of the lattice dictates your starting point, and then the strength of the disorder acts like a dial that pushes you through predictable phase transitions toward different types of metallic or insulating behavior <ref:2610.02192#pg1>.

Mira: It really hammers home how fundamental geometric properties, like those inherent in hyperbolic space, aren't just cosmetic details; they are active ingredients that govern the entire electronic phase diagram of the material <ref:2610.02192#pg3>.

Lev: This means when we think about designing new quantum devices, we have to start thinking less about a perfect lattice and more about how we can engineer or utilize specific curvatures to control where those transitions occur <ref:2610.02192#pg3>.

Kai: So, the next step for us is figuring out how to build something that can actually probe this curvature directly, because the theory is showing us a really rich landscape of possibilities <ref:2610.02192#pg3>.

The paper's improvements: Tom: So, moving past just describing the results, Kai and Mira are going over how this work suggests ways to push these ideas further than what they actually did in this study.

Kai: Mira, can you tell us what kind of extensions or future research directions the authors themselves pointed out? I'm interested in whether they suggest specific lattice types or disorder profiles that we should focus on next.

Mira: They definitely pointed toward extending this work beyond just those three specific lattice types, suggesting that exploring a wider variety of hyperbolic geometries could uncover even more interesting phase diagrams <ref:2610.02192#pg3>.

Lev: From an error correction standpoint, if they are looking at more complex geometries, does that mean the complexity of characterizing the resulting error syndromes will increase significantly?

Kai: Absolutely; incorporating a broader range of lattice topologies means we'll likely need entirely new ways to map those physical structures onto our existing quantum codes, which is a big challenge for experimentalists <ref:2610.02192#pg3>.

Mira: Exactly, and they also hinted that the next logical step involves applying this framework to more complex, interacting many-body problems where the environment isn't just simple mass disorder but something more dynamic <ref:2610.02192#pg3>.

Lev: That sounds like a good direction for us; if we can model those interactions, we might be able to better predict how noise affects the stability of logical states in a real quantum computer <ref:2610.02192#pg3>.

Kai: It's exciting because it shows that geometry is such a fundamental parameter; it implies that controlling the environment through spatial structure could become a major tool for engineering quantum materials, not just tuning parameters like magnetic fields <ref:2610.02192#pg3>.

Mira: Precisely, and this points toward using structural topology as a primary control mechanism to dictate whether a material ends up being conductive or insulating based on its underlying geometry <ref:2610.02192#pg3>.

Lev: So the implication for error correction is that we might need to start thinking about geometric codes, where the code structure itself leverages spatial curvature to enhance stability against noise rather than just relying on distance in a Euclidean lattice <ref:2610.02192#pg3>.

Kai: That’s a big shift in how we think about protecting quantum information; it suggests we need to look at the physical shape of our substrate as much as the way we wire the qubits <ref:2610.02192#pg3>.

Conclusion: Kai: To wrap things up, this paper on "Hyperbolic lattices with mass disorder: Phases and phase transitions" really shows us that geometry is a powerful lever in controlling quantum behavior <ref:2610.02192#pg3>.

Mira: It confirms that the specific spatial curvature of a lattice type dictates the fundamental starting electronic state before we even consider adding noise, which is a pretty deep concept to pin down <ref:2610.02192#pg3>.

Lev: From my perspective, seeing how geometry stabilizes those Dirac liquids against weak disorder gives us concrete rules for what kind of robustness we can expect when designing error correction codes on curved substrates <ref:2610.02192#pg3>.

Kai: It's a really solid foundation, and the idea that we can tune the material's properties by choosing its shape is something experimentalists need to focus on for future lab work <ref:2610.02192#pg3>.

Mira: Exactly, and the detailed scaling exponents they found for those transitions give us a mathematical language to predict exactly how sensitive these systems are to changes in disorder strength <ref:2610.02192#pg3>.

Lev: Those exponents are vital because they tell us the rate at which certain error parameters will grow as we approach those critical points, which is exactly what we need for designing better fault-tolerant architectures <ref:2610.02192#pg3>.

Kai: So, even though we haven't built anything yet, this paper gives us a very clear target: engineer lattices with specific curvatures to see if we can control the resulting metallic or insulating behavior <ref:2610.02192#pg3>.

Mira: It really emphasizes that understanding how geometric constraints influence many-body physics is crucial for predicting material properties <ref:2610.02192#pg3>.

Lev: We're looking forward to seeing how these geometric constraints apply to more complex, interacting systems in the future, which I think is where the real challenges for quantum error correction lie <ref:2610.02192#pg3>.

Kai: That sounds like a great topic for our next discussion; we need to figure out how to actually cool and measure these kinds of complex structures before we can really test these theoretical predictions <ref:2610.02192#pg3>.

Sheersh Sen, *Christopher A. Leong, *Bitan Roy

Centre for Condensed Matter Theory, Department of Physics, Indian Institute of Science · Department of Physics, Lehigh University

cond-mat.dis-nn, cond-mat.mes-hall, cond-mat.stat-mech

Submitted: 2026-10-01

Updated: 2026-10-01

Comments: 6 Pages and 6 Figures (Supplemental Materials as Ancillary File)

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 84/100

The gist: Nearest-neighbor tight-binding models (TBMs) on hyperbolic lattices (HLs) with mass disorder reveal distinct electronic phases and phase transitions, providing insights into how spatial curvature

Key concepts

Dirac Liquid (DL): This electronic state is characterized by a density of states that vanishes near zero energy, similar to massless relativistic particles. It is observed in plaquette-centered 10, 3 hyperbolic lattices and remains stable under weak mass disorder.
Fermi Liquid (FL): In this phase, the density of states near the band center remains finite. This behavior is seen in plaquette-centered 8, 3 hyperbolic lattices and contrasts with the Dirac liquid state.
Anderson Metal-to-Insulator Transition (AMIT): This transition occurs when disorder becomes very strong, causing the system to become an insulator. In this case, only the typical density of states at zero energy ($ ho_t(0)$) goes to zero, marking a fundamental change in electronic behavior.

Terminology

Phase Transitions — 10× in this paper · explained in 2 episodes
Typical Density of States — 6× in this paper
Robustness — 3× in this paper · explained in 5 episodes
alpha — 3× in this paper · explained in 1 episode
Anderson Metal-to-Insulator Transition (AMIT) — 2× in this paper · explained in 1 episode
Core — 2× in this paper · explained in 2 episodes
Antiferromagnetism — 1× in this paper · explained in 1 episode
Average Density of States — 1× in this paper
Symmetry Breaking — 1× in this paper · explained in 3 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

Nearest-neighbor tight-binding models (TBMs) on hyperbolic lattices (HLs) with mass disorder reveal distinct electronic phases and phase transitions, providing insights into how spatial curvature influences quantum materials. The study investigates three specific HL types—plaquette-centered 10, 3; 8, 3; and 8, 4—and demonstrates that the Dirac liquid phase remains stable against weak disorder while undergoing a semimetal-to-metal quantum phase transition (QPT) at moderate disorder.

Key Findings on Electronic Phases

The nearest-neighbor TBMs on these hyperbolic lattices exhibit different behaviors in the clean limit:

Plaquette-centered 10, 3 HLs display a power-law vanishing density of states (DOS) near zero energy, characterizing a Dirac liquid (DL).
Plaquette-centered 8, 3 HLs display a finite DOS near the band center, identifying them as Fermi liquids (FL).
Plaquette-centered 8, 4 HLs display a diverging DOS near the band center, characterizing flat bands (FB).

When mass disorder is introduced via random on-site potentials of opposite signs assigned to complementary sublattices within each unit cell, the system's stability changes:

** A hyperbolic DL remains stable for sufficiently weak mass disorder, where both ρa(0) and ρt(0) are vanishingly small (namely, ≲ 10−3). **

** Beyond a moderate disorder strength, denoted as Wc,1, the semimetallic system becomes a diffusive metal via a quantum phase transition (QPT), where both ρa(0) and ρt(0) become finite. **

** As disorder increases further to even stronger Wc,2, the system encounters an Anderson metal-to-insulator transition (AMIT), at which point only ρt(0) becomes zero. **

Disorder Realization and Numerical Methods

The mass disorder is realized by assigning a pair of random on-site potentials ±V (r), drawn from a box distribution [−W/2, W/2], to complementary sublattices A and B within each unit cell, with the values of W being independent and random across the system. To compute the density of states, two methods are employed:

Average Density of States (ADOS), defined as ρa(E) = 1/N Σ N j=1 δ (E − Ej), which is computed by averaging over 10 independent disorder realizations.
Typical Density of States (TDOS), defined as ρn t(E) = exp [1/p Σ p i=1 ln ρ i loc(E)], where the local DOS is computed using the kernel polynomial method (KPM) after averaging over 60 independent disorder realizations.

Critical Exponents and Scaling Behavior

The study extracts various critical exponents near the identified transitions:

** Near the semimetal-to-metal QPT in DL systems, the DOS exponent is found to be α = 0.99 ± 0.01 for Wc,1 = 0.55. **

** The order parameter exponents βa and βt are determined from scaling relations: ρa(0) ∼ δ βa and ρt(0) ∼ δ βt, yielding values like βa = 1.78 ± 0.05 for the QPT in the 10, 3 system. **

** Near the AMIT, power-law scaling of TDOS at E = 0 is analyzed as ρt(0) ∼ δA βA, yielding exponents such as βA = 1.84 ± 0.15 for the clean system's transition points. **

** For the AMIT in disordered systems, an essential singularity in TDOS near E = 0 is also observed, with critical disorder strengths (Wc,2) varying depending on whether a power-law or essential singularity is present. **

Symmetry Class and Physical Interpretation

The clean NN-TBM Hamiltonian belongs to the chiral orthogonal symmetry class BDI. The addition of mass disorder causes the system to lose the particle-hole (C) and sublattice (S) symmetries but preserves time-reversal (T), placing it in the WignerDyson orthogonal class AI. This loss of C and S symmetries is what permits the AMIT in 2D HLs, contrasting with Euclidean lattices where such transitions are forbidden in class AI. The stability of Dirac liquids against weak disorder is attributed to the relativistic nature of quasiparticles living on a negative curved space with a constant curvature a, which leads to specific scaling behaviors for the DOS near zero energy.

Improvements for AI systems

This scientific paper investigates the interplay between mass disorder and electronic phases (Dirac Liquid, Fermi Liquid, Flat Band) in hyperbolic lattices (HLs), drawing analogies to condensed matter physics.

As an AI researcher, I can derive several potential improvements by translating the physical principles described into computational models or algorithmic strategies for AI systems. The key takeaways are:

The existence of distinct phases (DL, FL, FB) based on geometric structure and disorder strength.
Quantum phase transitions (QPTs) driven by disorder strength, characterized by specific critical exponents near these transitions.
The role of spatial curvature in stabilizing metallic phases against localization (Anderson Metal-to-Insulator Transition, AMIT).

Here are the specific improvements for AI systems and what the improved system can do:

) Improved AI System Capabilities:

A system capable of modeling complex, disordered quantum many-body states with emergent symmetries and phase transitions on curved or non-Euclidean structures.
A system that can predict material behavior (electronic properties) based on structural topology (like the Schläfli symbol).
An AI framework that can distinguish between different types of disorder and their specific impact on emergent physics (e.g., differentiating mass disorder effects from potential disorder effects).

) Specific Improvements for AI Systems:

A system incorporating a Hyperbolic Geometry Module capable of mapping physical constraints or data structures onto hyperbolic spaces defined by the Schläfli symbol notation (e.g., relating lattice parameters to specific HL types like, or deriving effective Hamiltonians from curvature).
Implementation of a Disorder Sensitivity Engine that calculates critical transition points (like the semimetal-to-metal QPT at disorder strength Wc,1) and predicts the resulting phase change based on the input disorder profile (mass disorder).
A Phase Transition Classifier trained on simulated or derived DOS/TDOS scaling exponents to accurately classify complex electronic systems into DL, FL, or FB regimes under varying degrees of disorder.
A Localization Prediction Module specifically designed to predict the onset and nature (power-law vs. essential singularity) of Anderson Metal-to-Insulator Transitions (AMITs) in systems with intrinsic spatial curvature, rather than relying solely on Euclidean lattice models.

) What the Improved AI System Can Do:

Predict the electronic stability of novel materials designed on hyperbolic substrates, determining whether they will exhibit metallic conduction (DL/FL) or insulation (AMIT).
Design designer disorder profiles that intentionally induce specific quantum phase transitions (e.g., tuning W to precisely hit a QPT point where the system shifts from a semimetal to a metal).
Diagnose the underlying physical mechanism of electronic localization—determining if an insulator is due to standard on-site potential disorder or if it is uniquely driven by the spatial curvature of the lattice (as suggested by the paper's finding that curvature alone can trigger AMIT in 2D HLs).
Optimize quantum device architectures (like photonic lattices) to maximize wave transport efficiency by selecting optimal lattice geometries and disorder configurations that exploit relativistic or geometric effects for enhanced delocalization.

Abstract

Nearest-neighbor (NN) tight-binding models (TBMs) on plaquette-centered 10,3 (Schläfli symbol), 8,3, and 8,4 hyperbolic lattices on a Poincaré disk with open boundary conditions display a vanishing, a finite, and a diverging density of states (DOS) near zero energy (band center), respectively, yielding a Dirac liquid, a Fermi liquid, and a flat band. Emergent bipartite nature of these lattices within the framework of NN-TBMs, allows us to scrutinize the impact of mass disorder on the electronic states and DOS therein. From extensive numerical calculations of the average and typical DOS using the kernel polynomial method, we show that hyperbolic Dirac liquid remains stable against weak mass disorder, undergoing a semimetal-to-metal quantum phase transition at moderate disorder, followed by an Anderson metal-to-insulator transition at even stronger disorder. The remaining two systems display only the latter transition. Critical exponents near all these transitions are found to be close to the ones mediated by on-site potential disorder.