How does mass disorder drive phase transitions in hyperbolic lattices?

Three clean lattices, three different zero-energy responses

The study compares three hyperbolic lattices, labeled by their Schläfli symbols {10,3}, {8,3}, and {8,4}. Each is tiled on a Poincaré disk with open boundary conditions, and each is described by a nearest-neighbor tight-binding model in which hopping occurs only between two complementary sublattices, A and B. That bipartite structure is what makes a staggered on-site potential possible in the first place.

In the clean limit, the three lattices differ at the band center, meaning zero energy. The {10,3} lattice has a density of states that vanishes as a power law near zero energy, which the authors call a Dirac liquid. The {8,3} lattice has a finite density of states there, a Fermi liquid. The {8,4} lattice has a diverging density of states, a flat band. These are three different starting points, so the same disorder need not produce the same sequence of phases.

Mass disorder means opposite potentials on the two sublattices

The disorder in this paper is not a random potential applied identically to every site. Within each unit cell, the A and B sites receive potentials of equal magnitude and opposite sign, ±V(r), and the magnitude is drawn independently for each unit cell from a box distribution over [−W/2, W/2]. The parameter W therefore sets the strength of a staggered, sublattice-antisymmetric disorder.

This choice matters because of its symmetry effect. The clean model preserves time-reversal, particle-hole, and sublattice (chiral) symmetries, placing it in class BDI. Adding mass disorder removes the particle-hole and sublattice symmetries but keeps time-reversal symmetry, moving the system into the orthogonal class AI. The authors note that class AI does not permit an Anderson metal-to-insulator transition in two Euclidean dimensions, so any such transition found here must be tied to the hyperbolic geometry rather than to ordinary flat-space behavior.

Average and typical densities answer different questions

The paper tracks two quantities that are easy to conflate. The average density of states, ρa(E), is the disorder-averaged spectrum at energy E. The typical density of states, ρt(E), is built from the geometric mean of local densities of states over sites in a generation of the lattice, then averaged over generations. A finite average density of states can be carried by rare regions, while the typical value collapses toward zero; that is why the two are reported separately.

Both are computed with the kernel polynomial method, which expands the density of states in Chebyshev polynomials. The authors average the average density of states over 10 independent disorder realizations and the typical density of states over 60, and they check convergence with the number of Chebyshev moments. The systems are large, each containing more than 10^5 sites, with total generations of 7, 9, and 7 for {10,3}, {8,3}, and {8,4} respectively.

The Dirac liquid passes through two separate transitions

On the {10,3} lattice, both ρa(0) and ρt(0) stay vanishingly small, below roughly 10^(−3), up to a moderate disorder strength. Beyond that point the semimetal becomes a diffusive metal through a quantum phase transition, with both average and typical densities of states becoming finite at zero energy. The critical strengths extracted from the two quantities differ: Wc,1 ≈ 0.55 from the average density of states and Wc,1 ≈ 0.40 from the typical one. The authors attribute the discrepancy to the kernel polynomial method being strictly stable only at disorder strengths above about 1, and they state that the two estimates are expected to coincide otherwise.

At stronger disorder the system undergoes a second transition, the Anderson metal-to-insulator transition, at which only ρt(0) goes to zero while the average density of states remains finite. The critical strength here depends on which functional form is assumed for the vanishing typical density of states: a power-law fit gives Wc,2 ≈ 7.00, while an essential-singularity fit gives Wc,2 ≈ 10.45. The authors report good agreement with both forms and do not rule out the essential singularity, citing the large number of boundary sites that make the effective dimensionality infinite.

The Fermi liquid and flat band skip the first transition

The {8,3} and {8,4} lattices behave differently. Both display only the Anderson metal-to-insulator transition, with no preceding semimetal-to-metal step, because their clean zero-energy density of states is already finite or diverging rather than vanishing. Their phase diagrams therefore contain just a metal and an Anderson insulator separated by one quantum critical point.

The critical strengths again depend on the assumed singularity form. For {8,3}, a power-law fit gives Wc,2 ≈ 6.00 and an essential-singularity fit gives Wc,2 ≈ 8.25. For {8,4}, the corresponding values are Wc,2 ≈ 12.00 and Wc,2 ≈ 14.35. The {8,4} lattice also keeps a prominent density-of-states peak near zero energy up to moderate disorder, a residue of the clean flat band.

Critical exponents cluster near the on-site-disorder values

Near the semimetal-to-metal transition on {10,3}, the average density of states follows a power law in energy with exponent α = 0.99 ± 0.01. The order-parameter exponents extracted at zero energy are βa = 1.78 ± 0.05 for the average density of states and βt = 1.02 ± 0.03 for the typical one. The authors note that the distinct values of βa and βt suggest wavefunctions near this transition may be multifractal, a possibility they raise rather than establish.

Across the Anderson transition, the power-law fits give βA = 1.84 ± 0.15 on {10,3}, 1.96 ± 0.14 on {8,3}, and 1.80 ± 0.08 on {8,4}, values that agree within numerical error bars. The paper's stated conclusion is that these exponents are sufficiently close to those found when the same transitions are driven by on-site potential disorder, which is consistent with both disorder types placing the system in class AI.

Curvature, boundaries, and the photonic testbed

The authors argue that the metal-to-insulator transition exists here because the negative spatial curvature provides an infrared cutoff on the return probability of a particle, weakening the interference that would localize it. On a two-dimensional Euclidean lattice in class AI, no such transition is expected. The paper also states that its density-of-states scaling may depend on the boundary condition, open or periodic, but is independent of system size, since the bulk-to-boundary site ratios approach their thermodynamic values in these large lattices.

The proposed experimental setting is a classical metamaterial that supports wave dynamics, with photonic lattices named as the most promising candidate; the authors note that hyperbolic lattices have already been designed on such a platform. Their specific prediction is that a dirty photonic hyperbolic lattice with open boundary conditions and a refractive index varied in the staggered intra-unit-cell pattern could show both the semimetal-to-metal transition on {10,3} and the Anderson transition on all three lattices. This is a proposed testbed, not a reported measurement.

The lattice, not the disorder strength alone, sets the phase sequence

The most useful comparison in this paper is between the three lattices at the same kind of disorder. Mass disorder of the staggered sublattice form produces three stable phases, semimetal, metal, and Anderson insulator, only when the clean band center already has a vanishing density of states, as on {10,3}. Where the clean density of states is finite or diverging, as on {8,3} and {8,4}, the same disorder yields only metal and Anderson insulator.

A second distinction deserves equal weight: the reported critical strengths are not single numbers. For every Anderson transition, Wc,2 shifts substantially depending on whether the typical density of states is fitted with a power law or an essential singularity, and the paper explicitly leaves that question open. Readers comparing disorder thresholds across these lattices should therefore compare the assumed singularity form as well as the lattice, and should treat the {10,3} two-step sequence as the paper's central, lattice-specific finding.

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