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arXiv:2501.14134

Classical and Quantum Phase Transitions in Multiscale Media: Universality and Critical Exponents in the Fractional Ising Model

Lewis, Joshua M., Lincoln D. Carr

Published Jan 23, 2025• 0 citations
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Abstract

Until now multiscale quantum problems have appeared to be out of reach at the many-body level relevant to strongly correlated materials and current quantum information devices. In fact, they can be modeled with $q$-th order fractional derivatives, as we demonstrate in this work, treating classical and quantum phase transitions in a fractional Ising model for $0 < q \leq 2$ ($q = 2$ is the usual Ising model). We show that fractional derivatives not only enable continuous tuning of critical exponents such as $ν$, $δ$, and $η$, but also define the Hausdorff dimension $H_D$ of the system tied geometrically to the anomalous dimension $η$. We discover that for classical systems, $H_D$ is precisely equal to the fractional order $q$. In contrast, for quantum systems, $H_D$ deviates from this direct equivalence, scaling more gradually, driven by additional degrees of freedom introduced by quantum fluctuations. These results reveal how fractional derivatives fundamentally modify the fractal geometry of many-body interactions, directly influencing the universal symmetries of the system and overcoming traditional dimensional restrictions on phase transitions. Specifically, we find that for $q < 1$ in the classical regime and $q < 2$ in the quantum regime, fractional interactions allow phase transitions in one dimension. This work establishes fractional derivatives as a powerful tool for engineering critical behavior, offering new insights into the geometry of multiscale systems and opening avenues for exploring tunable quantum materials on NISQ devices.

DOI
10.48550/arxiv.2501.14134

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We study the quantum Ising model on the Sierpiński triangle, whose Hausdorff dimension is log⁡3/log⁡2≈1.585\log 3/ \log 2 \approx 1.585log3/log2≈1.585, and demonstrate that it undergoes second-order phase transition with scaling relations satisfied precisely. We also study the quantum 333-state Potts model on the Sierpiński triangle and find first-order phase transition, which is consistent with a prediction from εεε-expansion that the transition becomes first-order for D>1.3D > 1.3D>1.3. We then compute critical exponents of the Ising model on higher-dimensional Sierpiński pyramids with various Hausdorff dimension via Monte-Carlo simulations and real-space RG analysis for D∈[1,3]D\in[1,3]D∈[1,3]. We find that only the correlation length exponent ννν interpolates the values of integer-dimensional models. This implies that, contrary to a generally held belief, the universality class of quantum phase transition may not be uniquely determined by symmetry and spatial dimension of the system. This work initiates studies on quantum critical phenomena on graphs and networks which may be of significant importance in the context of quantum networks and communication.
Statistical methods are essential for understanding thermodynamic systems with many degrees of freedom. For systems in equilibrium, a very useful method is that of correlation functions, which establish a correlation between a field phi(x), which depends on the spatial position x, and the same field evaluated at another position, phi(x0). Fisher [Journal of Mathematical Physics 5, 944322 (1964)] introduced the autocorrelation function for fluctuations of the order parameter, which has been an important mathematical tool for understanding second-order phase transitions in equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent eta is introduced to correct the spatial behavior of the correlation function at T = Tc. In a recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that a modern fractional differential analysis is necessary for a complete description of the correlation function at Tc. In this study, we highlight the deep connection among scaling behavior, critical exponents, and fractal geometry. Our results provide a unified geometric interpretation of critical exponents and fractal dimensions, broadly applicable to thermodynamic phase transitions. However, the approach does not apply to topological phase transitions, which lack local order parameters and the associated scale-invariant fractal geometry. We verify its predictions for several cornerstone thermodynamic models: the Ising, Potts, XY, and Heisenberg systems.
Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.
Abstract Understanding phase transitions requires not only identifying order parameters but also characterizing how their correlations behave across scales. By quantifying how fluctuations at distinct spatial or temporal points are related, correlation functions reveal the structural organization of complex systems. In this work, we reexamine the theoretical foundations of these correlations in systems undergoing second-order phase transitions, with emphasis on the Ising model extended to non-integer spatial dimensions. We revisit the hypotesis that, at criticality, the equilibrium dynamics become effectively confined to the fractal edge of spin clusters and redo the analysis using fractional calculus. Within this framework, the fractal dimension that governs the correlations in that subspace is directly related to Fisher’s exponent η\etaη η , which quantifies the singular behavior of the correlation function near criticality. Importantly, this correlation fractal dimension is distinct from the fractal dimension associated with the order parameter. The fractional approach allows us to directly compute the correlation fractal dimension and to establish an explicit geometrical relation connecting the two fractal dimensions. Moreover, the formulation naturally extends to non-integer spatial dimensions, remaining valid below the upper critical dimension and yielding the correct value of Fisher’s exponent η\etaη η for a continuous spatial dimension d . Within this framework, we also provide empirical functions describing how the main critical exponents vary continuously as a function of the space dimension.
We study causality and criticality in a one-dimensional fractional multiscale transverse-field Ising model, where fractional derivatives generate long range interactions beyond the scope of standard power laws. Such fractional responses are common in classical systems including the anomalous stress-strain behaviour of viscoelastic polymers, Lévy-like contaminant transport in heterogeneous porous media, and the non-Debye dielectric relaxation of glassy dielectrics. Furthermore, these unique interactions can be implemented in current quantum information architectures, with intriguing consequences for the many-body dynamics. Using a truncated Jordan-Wigner approach, we show that in the long wavelength limit of the mean field, the dynamical critical exponent is set by the fractional order q as z=q/2z=q/2z=q/2. To probe genuine many-body dynamics, we apply matrix-product-state simulations with the time-dependent variational principle adapted to nonlocal couplings. Tracking the entanglement-entropy light cone and performing finite-size scaling of the many-body gap for $0