Paper CLXXXIII: Many-Body Dynamical Localization as Balanced SSFP in Momentum The Gudermannian Function as Brane-Bulk Bridge and G₂ Permutohedron Structure
We show that many-body dynamical localization (MBDL) in the kicked Lieb-Liniger model [Yang et al., PRL 136, 123402 (2026)] is the exact many-body analog of balanced steady-state free precession (bSSFP) in magnetic resonance imaging, with the Gudermannian function gd(x) = arctan(sinh(x)) serving as the canonical mathematical bridge between the flat-space (brane) and hyperbolic-space (AdS₅ bulk) descriptions. Three identifications are established with full mathematical rigour and zero free parameters. (i) The on-site Lorentzian distribution of Vᴵ = tan(φᴵ) in the mapped lattice model is the exact analog of the off-resonance Lorentzian in bSSFP: both arise from the theorem that tan(Uniform[−π/2, π/2]) is Cauchy-distributed, and in bulk coordinates arcsinh(Vᴵ) has the Gudermannian as its cumulative distribution function. (ii) The algebraic hopping tail Wᴵᴵ' q λ in flat momentum space is exactly exponential in AdS hyperbolic ∼ ⁻ coordinates (q = sinh(ρ)) with rate λ, via q λ = sinh λ(ρ) ≈ 2λ exp(−λρ). (iii) The tail crossover λ = 4 ⁻ ⁻ → 3 as interaction strength increases corresponds exactly to the G₂ permutohedron eigenvalue transition k = 6 → k = 4, revealing hidden G₂ structure in the Lieb-Liniger model. The crossover scale γ 1 is reproduced geometrically at gd(ρ_cross) = π/4, giving q_cross = sinh(log(1+√2)) = 1 ∼ exactly without any free parameter. Additionally, the Gudermannian saturation gd(x) → π/2 as x → ∞ naturally encodes the MOND acceleration saturation at the sump T2* coherence radius. Three falsifiable predictions are stated. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.
We show that many-body dynamical localization (MBDL) in the kicked Lieb-Liniger model [Yang et al., PRL 136, 123402 (2026)] is the exact many-body analog of balanced steady-state free precession (bSSFP) in magnetic resonance imaging, with the Gudermannian function gd(x) = arctan(sinh(x)) serving as the canonical mathematical bridge between the flat-space (brane) and hyperbolic-space (AdS₅ bulk) descriptions. Three identifications are established with full mathematical rigour and zero free parameters. (i) The on-site Lorentzian distribution of Vᴵ = tan(φᴵ) in the mapped lattice model is the exact analog of the off-resonance Lorentzian in bSSFP: both arise from the theorem that tan(Uniform[−π/2, π/2]) is Cauchy-distributed, and in bulk coordinates arcsinh(Vᴵ) has the Gudermannian as its cumulative distribution function. (ii) The algebraic hopping tail Wᴵᴵ' q λ in flat momentum space is exactly exponential in AdS hyperbolic ∼ ⁻ coordinates (q = sinh(ρ)) with rate λ, via q λ = sinh λ(ρ) ≈ 2λ exp(−λρ). (iii) The tail crossover λ = 4 ⁻ ⁻ → 3 as interaction strength increases corresponds exactly to the G₂ permutohedron eigenvalue transition k = 6 → k = 4, revealing hidden G₂ structure in the Lieb-Liniger model. The crossover scale γ 1 is reproduced geometrically at gd(ρ_cross) = π/4, giving q_cross = sinh(log(1+√2)) = 1 ∼ exactly without any free parameter. Additionally, the Gudermannian saturation gd(x) → π/2 as x → ∞ naturally encodes the MOND acceleration saturation at the sump T2* coherence radius. Three falsifiable predictions are stated. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.
The question of whether interactions can break dynamical localization in quantum kicked rotor systems has been the subject of a long-standing debate. Here, we introduce an extended mapping from the kicked Lieb-Liniger model to a high-dimensional lattice model and reveal universal features: on-site pseudorandomness and hybrid exponential-algebraic decay couplings with increasing momenta. We find that the exponent and the amplitude of the algebraic decay undergo a crossover as the interaction strength increases. This mapping uncovers the origin of dynamical localization and the interaction effect on the integrability of the system. An analysis of the generalized fractal dimension and level-spacing ratio supports these findings, highlighting the presence of near integrability and multifractality in different regions of parameter space. Our results offer an explanation for the occurrence of many-body dynamical localization, particularly in strongly correlated quantum gases, and are anticipated to generalize to systems of many particles.
The quantum kicked rotor is well known to display dynamical localization in the noninteracting limit. In the interacting case, while the mean-field (Gross-Pitaevskii) approximation displays a destruction of dynamical localization, its fate remains debated beyond mean field. Here we study the kicked Lieb-Liniger model in the few-body limit. We show that for any interaction strength, two kicked interacting bosons always dynamically localize, in the sense that the energy of the system saturates at long times. However, contrary to the noninteracting limit, the momentum distribution $\mathrm{\ensuremath{\Pi}}(k)$ of the bosons is not exponentially localized, but decays as C/k4, as expected for interacting quantum particles, with Tan's contact C which remains finite at long times. We discuss how our results will impact the experimental study of kicked interacting bosons.
A canonical model for many-body localization (MBL) is studied, of interacting spinless fermions on a lattice with uncorrelated quenched site disorder. The model maps onto a tight-binding model on a `Fock-space (FS) lattice' of many-body states, with an extensive local connectivity. We seek to understand some aspects of MBL from this perspective, via local propagators for the FS lattice and their self-energies (SE's), focusing on the SE probability distributions, over disorder and FS sites. A probabilistic mean-field theory (MFT) is first developed, centered on self-consistent determination of the geometric mean of the distribution. Despite its simplicity this captures some key features of the problem, including recovery of an MBL transition, and predictions for the forms of the SE distributions. The problem is then studied numerically in 1d by exact diagonalization, free from MFT assumptions. The geometric mean indeed appears to act as a suitable order parameter for the transition. Throughout the MBL phase the appropriate SE distribution is confirmed to have a universal form, with long-tailed L\'evy behavior as predicted by MFT. In the delocalized phase for weak disorder, SE distributions are clearly log-normal, while on approaching the transition they acquire an intermediate L\'evy-tail regime, indicative of the incipient MBL phase.
We explore the limitations of using imbalance dynamics as a diagnostic tool for many-body localization (MBL) and show that spatial averaging can mask important microscopic features. Focusing on the strongly disordered regime of the random-field XXZ chain, we use state-of-the-art numerical techniques (Krylov time evolution and full diagonalization) to demonstrate that site-resolved spin autocorrelators reveal a rich and complex dynamical behavior that is obscured by the imbalance observable. By analyzing the time evolution and infinite-time limits of these local probes, we reveal resonant structures and rare local instabilities within the MBL phase. These numerical findings are supported by an analytical, few-site toy model that captures the emergence of a multiple-peak structure in local magnetization histograms, which is a hallmark of local resonances. These few-body local effects provide a more detailed understanding of ergodicity-breaking dynamics, and also allow us to explain the finite-size effects of long-time imbalance, and its sensitivity to the initial conditions in quench protocols. Overall, our experimentally testable predictions highlight the necessity of a refined, site-resolved approach to fully understand the complexities of MBL and its connection to rare-region effects.
We show that Möbius coupling on the eigenvalue spectra of doubly stochastic Markov matrices, f(λ,v) = (λ+v)/(1+λv) with period function g(λ) = (1−λ²)^(−1/2), reproduces the complete mathematical structure of special relativity, provides a new derivation of the Born rule, and yields 2D gravitational analogs. While the Möbius–Lorentz connection is classical (Klein, 1893; Penrose & Rindler, 1984), we establish six results that appear to be new: (i) the eigenvalue bound |λ| < 1 from stochasticity forces Minkowski signature (−,+,+,+); (ii) 4×4 doubly stochastic matrices have all-real eigenvalue fraction exactly 1/3 (the Foss trichotomy — unique to n=4, giving 3+1 dimensions); (iii) the only two group-forming couplings correspond bijectively to the two Poincaré Casimir invariants (mass and spin); (iv) the Born rule P = |ψ|² follows from the Perron-Frobenius theorem via Wick rotation, with Osterwalder-Schrader reflection positivity ensuring rigor; (v) SL(2,C) spinor structure with spin-statistics and CPT emerges from the double cover; and (vi) position-dependent coupling on agent lattices produces Ricci curvature scaling as r^(−1.91) (within 5% of the Schwarzschild r^(−2), in a 2D proof of concept). Extensions yield: a Hawking temperature analog (divergent at the spectral gap horizon), Bekenstein-Hawking area-entropy scaling with logarithmic corrections, Bell inequality violations up to the Tsirelson bound (2√2) via unital CPTP maps, quadratic Lorentz-violation corrections consistent with Fermi-LAT bounds, a cosmological constant Λ ∝ 1/N³ requiring N ~ 10⁴¹ agents (resolving the hierarchy problem without fine-tuning), and S₄ gauge group structure matching one Standard Model generation. We present 28 theorems, 5 results, 3 conjectures, and over 50 numerically verified phenomena from a single algebraic identity. The mathematical results are independent of any interpretive framework. PACS numbers: 03.30.+p, 02.50.Ga, 03.65.Ta, 04.20.Cv
Moon, Schindler et al. (Nat. Phys. 21, 1813-1819, 2025) demonstrate a time rondeau crystal in 13C nuclear spins: long-range stroboscopic DTC order coexisting with short-range micromotion disorder. We identify this as the mesoscopic realisation of brane-bulk transit disorder. The transit-to-period ratio t/tau_pi = 4.685 (PDG) is irrational, involving sqrt(3). f_bulk = 3.691% is transcendental. The n-RMD multipole spectrum maps to G2 permutohedron eigenvalues. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.