Critical behavior of the 2D Ising model with quasi-periodic coupling modulation beyond the weak-disorder regime: strong modulation, non-analytic modulations, and non-Diophantine frequencies

Gallone and Mastropietro (Commun. Math. Phys. 2024) proved that the 2D Ising model with a weak quasi-periodic modulation of the nearest-neighbor couplings in both lattice directions retains the exact Onsager critical exponents, with no logarithmic corrections - the first rigorous verification of the Harris-Luck irrelevance criterion. Their theorem requires the modulation to be weak (amplitude below a threshold), analytic with exponentially decaying Fourier coefficients, and incommensurate with frequencies satisfying a Diophantine condition. The open problem is to determine the critical behavior outside this regime: whether a sharp amplitude threshold separates the Onsager class from the distinct quasi-periodic universality class suggested by numerical studies of strong quasi-periodic modulation, what the exponents and renormalization-group fixed-point structure of that class are, and whether Onsager universality survives non-analytic modulations or non-Diophantine frequencies (those failing every Diophantine condition, the Liouville numbers being the extreme case).

Background

The ferromagnetic Ising model on the two-dimensional square lattice with nearest-neighbor couplings was solved exactly by Onsager: at the critical inverse temperature the specific heat diverges logarithmically (critical exponent $\alpha=0$) and the correlation length diverges with exponent $\nu=1$. Universality - the robustness of these exponents under perturbations of the couplings - is a central question of statistical mechanics, and a natural deterministic perturbation is a quasi-periodic modulation of the couplings, motivated by quasicrystals and by cold-atom experiments with incommensurate optical lattices (standing-wave lattices whose wavelengths are irrationally related to the atomic spacing).

For random (independent) coupling disorder, the Harris criterion predicts that disorder is irrelevant when $\nu d/2>1$; in $d=2$ with $\nu=1$ the random case is marginal and subtle. For quasi-periodic modulation the corresponding prediction is the Harris-Luck criterion of Luck (1993): the modulation is irrelevant when the running average of its fluctuations, $\frac{1}{\xi}\sum_{x=0}^{\xi}\delta_x$, remains bounded and small on the scale $\xi$ of the correlation length, a condition governed by the wandering exponent of the modulation (the power-law rate at which partial sums of the modulation fluctuate); for analytic quasi-periodic functions the fluctuations are bounded, while for discrete aperiodic sequences they can grow logarithmically (marginal) or as a power (relevant).

Gallone and Mastropietro considered the Hamiltonian

$$H=-\sum_{\mathbf{x}\in\Lambda}\big[J^{(1)}_{\mathbf{x}}\sigma_{\mathbf{x}}\sigma_{\mathbf{x}+\mathbf{e}_1}+J^{(0)}_{\mathbf{x}}\sigma_{\mathbf{x}}\sigma_{\mathbf{x}+\mathbf{e}_0}\big],\qquad J^{(j)}_{\mathbf{x}}=\big(1+\lambda\,\phi^{(j)}(2\pi\omega_0 x_0+\theta_{j,0},\,2\pi\omega_1 x_1+\theta_{j,1})\big)J^{(j)},$$

where the modulation $\phi^{(j)}$ is analytic with exponentially decaying Fourier coefficients, $|\hat\phi^{(j)}_{\mathbf{n}}|\le A e^{-\eta|\mathbf{n}|}$, and the frequencies $\omega_0,\omega_1$ are irrational numbers satisfying a Diophantine condition, i.e. inequalities of the form $|2\pi\omega_j n|_T\ge c_j|n|^{-\rho_j}$ for all nonzero integers $n$, where $|\cdot|_T$ is the distance to the nearest multiple of $2\pi$. Such conditions exclude Liouville numbers - irrationals that admit rational approximations better than any inverse power of the denominator, for which the quantities $2\pi\omega n$ can be exponentially small (the small-divisor problem). The infinite-volume model is defined as the limit of periodic models whose period equals the denominator of the best rational approximant of the frequencies. Their main theorem states that for modulation amplitude $|\lambda|<\lambda_0$ small enough, the critical exponents of the specific heat and of the energy-energy correlations (correlations between bond-energy operators, whose scaling dimension governs the specific-heat singularity) are exactly the Onsager ones, with no logarithmic corrections; the disorder only renormalizes the velocities (the anisotropic rescaling of positions), the critical temperature, and multiplies correlations by quasi-periodic amplitudes. This was the first rigorous proof of the Harris-Luck criterion and the first rigorous universality result for the 2D Ising model with quasi-periodic disorder acting in both directions and for any choice of phases. The proof represents the Ising correlations as Gaussian Grassmann integrals (an exact fermionic rewriting via the dimer/Pfaffian representation) and controls the small divisors by a convergent multiscale renormalization group of KAM type (the techniques of Kolmogorov-Arnold-Moser theory for perturbed integrable Hamiltonian systems, adapted to fermionic integrals).

The theorem is explicitly perturbative and conditional, and its authors state that without the smallness, analyticity, or Diophantine assumptions a different behavior is expected. On the physics side, the one-dimensional quantum transverse-field Ising chain with quasi-periodic couplings - equivalent to a 2D classical Ising model with quasi-periodic modulation in a single (layered) direction - has been studied in detail: weak modulation is irrelevant, while strong modulation produces new, localized gapless phases and critical behavior intermediate between the clean and randomly disordered Ising transitions, i.e. a distinct universality class with exponents conjectured to be governed by the wandering coefficient of the modulation (Crowley, Chandran and Laumann 2018; Chandran and Laumann 2017; Crowley, Laumann and Chandran 2022; Agrawal, Gopalakrishnan and Vasseur 2020, who showed that generic quasi-periodic modulations flow to discrete self-similar sequences under a real-space renormalization group). These results are non-rigorous and concern the layered case; the genuinely bidirectional classical model at strong modulation has, to date, no dedicated study, and the associated lattice Dirac equation with quasi-periodic potential, which underlies the fermionic representation, has not been analyzed rigorously.

Problem Statement

Determine the critical behavior of the 2D Ising model with quasi-periodic modulation of the nearest-neighbor couplings, $J^{(j)}_{\mathbf{x}}=(1+\lambda\,\phi^{(j)}(2\pi\omega_0 x_0+\theta_{j,0},2\pi\omega_1 x_1+\theta_{j,1}))J^{(j)}$ (with the infinite-volume model defined as the limit of periodic approximants whose period is the denominator of the best rational approximant of the frequencies), outside the regime in which universality to the Onsager class (the exact exponents $\alpha=0$, $\nu=1$ of the homogeneous model) has been proven, namely beyond small amplitude $|\lambda|<\lambda_0$, analytic modulation with exponentially decaying Fourier coefficients, and Diophantine frequencies (frequencies $\omega_j$ satisfying $|2\pi\omega_j n|_T\ge c_j|n|^{-\rho_j}$ for all nonzero integers $n$, where $|\cdot|_T$ denotes the distance to the nearest multiple of $2\pi$). The overarching objective is a classification of the universality class of the quasi-periodic Ising transition as a function of the modulation amplitude $\lambda$, the regularity of the modulation $\phi$, and the arithmetic of the frequencies $\omega_0,\omega_1$, jointly expressed through the same observables used in the proven weak regime: the specific-heat singularity, the divergence of the correlation length, and the critical decay of the energy-energy correlations (correlations between bond-energy operators), including the presence or absence of logarithmic (wandering) corrections and the renormalization of velocities and critical temperature. Concretely: (i) Strong modulation - does there exist a sharp threshold amplitude separating the Onsager universality class from a distinct strong-quasi-periodic-disorder universality class, and if so what are the critical exponents and the renormalization-group fixed-point structure (including wandering or multifractal properties) of that class; or do the Onsager exponents persist at all amplitudes at which the transition survives? (ii) Regularity and arithmetic - (ii-a) does Onsager universality survive when the modulation is non-analytic or has only polynomially decaying Fourier coefficients, with the frequencies held Diophantine; and (ii-b) does Onsager universality survive when the frequencies are non-Diophantine, i.e. fail every Diophantine condition (the Liouville numbers, admitting rational approximations better than any inverse power of the denominator, being the extreme case), with the modulation held analytic with exponentially decaying Fourier coefficients? (iii) A related open analytic core discussed in the source: Gallone and Mastropietro note that the energy-energy correlations could, in principle, be deduced from a suitable lattice Dirac equation in a quasi-periodic potential - an equation which, as their paper states, has not been studied - where $\psi_{\mathbf{x}}$ is a two-component lattice spinor, $\sigma_1,\sigma_2,\sigma_3$ are the Pauli matrices, $m$ is the mass, $E$ is the spectral parameter, and $\phi^{(j)}_{\mathbf{x}}=\phi^{(j)}(2\pi\omega_0 x_0+\theta_{j,0},2\pi\omega_1 x_1+\theta_{j,1})$:

$$\sigma_2(\psi_{\mathbf{x}+\mathbf{e}_0}-\psi_{\mathbf{x}}+\lambda\phi^{(0)}_{\mathbf{x}}\psi_{\mathbf{x}})+\sigma_1(\psi_{\mathbf{x}+\mathbf{e}_1}-\psi_{\mathbf{x}}+\lambda\phi^{(1)}_{\mathbf{x}}\psi_{\mathbf{x}})+\mathrm{i}m\sigma_3\psi_{\mathbf{x}}=E\sigma_1\psi_{\mathbf{x}},$$

whose spectral and eigenfunction asymptotics would be expected to yield the correlation decay. This equation is a source-discussed analytic route toward the classification; resolving it would constitute partial progress on the open problem rather than, by itself, a complete answer. Facets (i), (ii-a) and (ii-b) are each independently sufficient as a complete answer to the problem, whereas facet (iii) counts only as partial progress.

References

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