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SUMMARY:Fractional Multiscale Many-Body Quantum Materials: From Phase Tran
 sitions to Causality - Lincoln Carr (Colorado School of Mines)
DTSTART:20260227T141500Z
DTEND:20260227T151500Z
UID:TALK243955@talks.cam.ac.uk
CONTACT:Georg Maierhofer
DESCRIPTION:Fractional multiscale materials are common in classical and bi
 ological systems\, being in fact quite typical in disparate natural system
 s ranging from anomalous diffusion of pollutants in ground water systems t
 o faster than expected infection rates of catheters in hospital settings. 
  Until now multiscale quantum problems of this nature have appeared to be 
 out of reach at the many-body level relevant to strongly correlated materi
 als and current quantum information devices.\n\nIn fact\, they can be mode
 led with $q$-th order fractional derivatives\, as I demonstrate in the fir
 st part of this talk\, 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 tun
 ing of critical exponents such as $\\nu$\, $\\delta$\, and $\\eta$\, but a
 lso define the Hausdorff dimension $H_D$ of the system tied geometrically 
 to the anomalous dimension $\\eta$. 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 m
 ore gradually\, driven by additional degrees of freedom introduced by quan
 tum fluctuations. These results reveal how fractional derivatives fundamen
 tally modify the fractal geometry of many-body interactions\, directly inf
 luencing the universal symmetries of the system and overcoming traditional
  dimensional restrictions on phase transitions. Specifically\, we find tha
 t for $q < 1$ in the classical regime and $q < 2$ in the quantum regime\, 
 fractional interactions allow phase transitions in one dimension\, with BK
 T transitions in the $q = 1$ and $q = 2$ borderline cases\, respectively. 
 This work establishes fractional derivatives as a powerful tool for engine
 ering critical behavior\, offering new insights into the geometry of multi
 scale systems and opening avenues for exploring tunable quantum materials 
 on NISQ devices.\n\nIn the second part of this talk\, I will present the p
 ropagation of quantum information in a one-dimensional fractional transver
 se-field Ising model\, where Riesz fractional derivatives generate interac
 tions beyond the scope of standard power laws. Using matrix product states
  (MPS) and time-dependent variational principle (TDVP) methods adapted for
  nonlocal couplings\, we systematically vary the fractional order $q$ and 
 show that the dynamical critical exponent takes the form\n\\[\nz = \\frac{
 q}{2}.\n\\]\nThis finding directly links fractional interactions to a Lév
 y flight framework\, since the mean-square displacement of a classical Lé
 vy flight scales as $t ^{2/q}$\, mirroring the $t^{1/z}$ dependence of cor
 relation fronts in our spin chain. As a result\, the usual short-range lim
 it is recovered for $q \\leq 2$\, whereas $q > 2$ gives rise to a unique f
 rustration-driven regime that remains genuinely nonlocal and displays subl
 inear growth of entanglement and correlations. These observations illustra
 te how fractional derivatives unify short-range\, power-law\, and frustrat
 ed long-range interactions within a single framework\, offering a window i
 nto exotic phases and nonlocal critical phenomena.\n\nReferences: \n\n1. B
 ruce J. West\, “Colloquium: Fractional calculus view of complexity: A tu
 torial.” Reviews of Modern Physics 86\, 1169 (2014)\, https://doi.org/10
 .1103/RevModPhys.86.1169\n\n2. Mark J. Ablowitz\, Joel B. Been\, and Linco
 ln D. Carr\, “Fractional Integrable Nonlinear Soliton Equations\,” Phy
 s. Rev. Lett.\, v. 128\, p.184101 (2022)\n\n3. Joshua M. Lewis and Lincoln
  D. Carr\, “Exploring Multiscale Quantum Media: High-Precision Efficient
  Numerical Solution of the Fractional Schrödinger equation\, Eigenfunctio
 ns with Physical Potentials\, and Fractionally-Enhanced Quantum Tunneling\
 ,” J. Phys. A\, v. 58\, p. 175303 (2025)\n\n4. Joshua M. Lewis and Linco
 ln D. Carr\, “Classical and Quantum Phase Transitions in Multiscale Medi
 a: Universality and Critical Exponents in the Fractional Ising Model\,” 
 Phys. Rev. Lett.\, under review (2025)\, https://arxiv.org/abs/2501.14134 
 \n\n5. Joshua M. Lewis\, Zhexuan Gong\, and Lincoln D. Carr\, “Fractiona
 l Ising Model and Lévy Light Cones: Nonlocal Causality Constraints Beyond
  Power-Law Decays\,” Quantum Science and Technology\, under review (2025
 )\, https://arxiv.org/abs/2505.05645
LOCATION:Centre for Mathematical Sciences\, MR14
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