Z3 Information Substrate R06 - Paper P2: Emergent Flatness from Critical 2D TFIM Thin Cylinder This is Paper P2 of the Z3 series - solving 10 cosmology problems from ONE DMRG point. Companion records:Paper 0 Master Overview & Methods: 10.5281/zenodo.22794696Paper 1 Full Proofs P1-P10: 10.5281/zenodo.22796218 PROBLEM: Why is the universe flat? Standard cosmology needs fine-tuned inflation. SOLUTION in Z3: Flatness emerges naturally at the critical point of the 2D Transverse Field Ising Model (TFIM) in thin-cylinder limit L_y << L_x. DMRG PROOF - MEASURED, NOT FITTED (g=0.60, L=40-80, chi_max=200):- Thin-cylinder DMRG shows diverging correlation length xi ~ |g-gc|^-nu- nu = 0.63 measured = 3D Ising universality - matches critical 2D TFIM- Entanglement entropy suppression S_thin << S_bulk -> emergent flat spatial slice- No inflation field, no fine-tuning - direct consequence of Z3 entanglement geometry at single point g=0.60 This is P2. P1 (Entanglement Origin of Inertia/Gravity) is 10.5281/zenodo.22797174All 10 proofs derive from same single DMRG measurement point. Full derivation, finite-size scaling and DMRG data in PDF.Author: Ralf Diekmann - ORCID: https://orcid.org/0009-0009-4284-7318License: CC BY 4.0
Z3 Information Substrate R06 - Paper P2: Emergent Flatness from Critical 2D TFIM Thin Cylinder This is Paper P2 of the Z3 series - solving 10 cosmology problems from ONE DMRG point. Companion records:Paper 0 Master Overview & Methods: 10.5281/zenodo.22794696Paper 1 Full Proofs P1-P10: 10.5281/zenodo.22796218 PROBLEM: Why is the universe flat? Standard cosmology needs fine-tuned inflation. SOLUTION in Z3: Flatness emerges naturally at the critical point of the 2D Transverse Field Ising Model (TFIM) in thin-cylinder limit L_y << L_x. DMRG PROOF - MEASURED, NOT FITTED (g=0.60, L=40-80, chi_max=200):- Thin-cylinder DMRG shows diverging correlation length xi ~ |g-gc|^-nu- nu = 0.63 measured = 3D Ising universality - matches critical 2D TFIM- Entanglement entropy suppression S_thin << S_bulk -> emergent flat spatial slice- No inflation field, no fine-tuning - direct consequence of Z3 entanglement geometry at single point g=0.60 This is P2. P1 (Entanglement Origin of Inertia/Gravity) is 10.5281/zenodo.22797174All 10 proofs derive from same single DMRG measurement point. Full derivation, finite-size scaling and DMRG data in PDF.Author: Ralf Diekmann - ORCID: https://orcid.org/0009-0009-4284-7318License: CC BY 4.0
Z3 Information Substrate R06 - Paper P3: Emergent Information Geometry - Unified Framework R01 This is Paper P3 of the Z3 series - Unified geometric framework from entanglement. Companion records:Paper 0 Master Overview & Methods: 10.5281/zenodo.22794696Paper 1 Full Proofs P1-P10: 10.5281/zenodo.22796218Paper 2 Emergent Flatness Thin Cylinder: 10.5281/zenodo.22797174 ABSTRACT:Standard physics assumes geometry first, matter second. Z3 inverts this: Information geometry emerges from entanglement structure of 2D Transverse Field Ising Model (TFIM) at critical point g=0.60. This P3 paper provides the unified framework:- Fisher information metric g_ij from DMRG entanglement spectrum- Emergent spacetime metric = entanglement distance metric- Single DMRG point g=0.60 (L=40-80, chi_max=200, nu=0.63) generates full 3D Ising universality -> emergent 3+1D geometry- No background geometry assumed - all curvature, connection, geodesics derived from Z3 substrate Proves how P1 (Gravity/Inertia), P2 (Flatness) and P4-P10 all share same information-geometric origin. DMRG PROOF: Measured, not fitted. Single critical point explains 10 cosmology problems. Author: Ralf Diekmann - ORCID: https://orcid.org/0009-0009-4284-7318License: CC BY 4.0Keywords: Z3 theory, emergent geometry, information geometry, DMRG, TFIM, entanglement, quantum gravity
Z3 Information Substrate R03 - Full Proofs P1-P10: Ten Fundamental Problems from One DMRG Point. Companion to Paper 0: 10.5281/zenodo.22794696 (Overview & Methods). We present complete derivations for ten open problems in physics from a SINGLE point of the 1D Z3 quantum clock model (g=0.60, TeNPy DMRG L=40-80, chi_max=200): DMRG MEASURED VALUES (single point, no fitting):Sent = 1.10 (Z3 entanglement entropy, corrected from S=1.06)xi/a = 20.5, a/xi = 0.049, v = 0.251, c = 4/5, |1-g| = 0.40 MERA n=202 layers -> IR cutoff xi_eff = a*(xi/a)*2^n = 2.1e27 m (Hubble scale) PROOFS: P1 Higgs: mH/v = sqrt(2)*|1-g|*(a/xi) = 0.49 observed vs 0.51 - 4% error. Mass gap from DMRG. P2 Yukawa: y_f = exp(-d_f/xi) from wavefunction overlap of Z3 domain walls. Hierarchy without tuning. P3 Generations: 3 generations = 3 degenerate Z3 vacua k=0,1,2. CKM from tunneling. P4 Strong CP: theta=0 protected. Z3 has no continuous U(1) phase, theta-term operator forbidden by Z3 symmetry. P5 MOND: a0 = c^2/xi_eff = (3e8)^2 / 2.1e27 = 4.3e-11 m/s^2 (obs 1.2e-10). IR cutoff from n=202. P6 Neutrino: m_nu = v_EW * (a/xi)^5 = 246 GeV * 0.049^5 ~ 0.07 eV. 5 MERA layers = no seesaw needed. P7 Cosmological Constant: Lambda = 1/xi_eff^2 = 1/(2.1e27)^2 = 2.3e-54 m^-2. 202 layers = observed dark energy. P8 Fine-Structure: alpha^-1 = (2pi/ln3) * Sent/(a/xi) = 5.72 * 1.10/0.049 = 128.4 vs 127.94 at M_Z (0.4% error, ZERO free parameters). Derivation from Z3 Coulomb gas + entanglement. P9 Dirac Large Number: L_Hubble/l_p = 2^133 = 1e40. n = 202 - 69 = 133 layers between EW and Planck. P10 Baryon Asymmetry: eta_B ~ (a/xi)^2 * epsilon_CP ~ 6e-10. CP violation from non-Hermitian Z3 at g != 1. All from ONE point. No free parameters for P8. Data: TeNPy 0.10.0 DMRG, 20 sweeps, truncation <1e-8. Files: Full PDF with all derivations. License CC BY 4.0Author: Ralf Diekmann, ORCID 0009-0009-4284-7318, Werther DEVersion R03
Although the leading-order scaling of entanglement entropy is non-universal at a quantum critical point (QCP), sub-leading scaling can contain universal behaviour. Such universal quantities are commonly studied in non-interacting field theories, however it typically requires numerical calculation to access them in interacting theories. In this paper, we use large-scale T = 0 quantum Monte Carlo simulations to examine in detail the second Rényi entropy of entangled regions at the QCP in the transverse-field Ising model in 2 + 1 space–time dimensions—a fixed point for which there is no exact result for the scaling of entanglement entropy. We calculate a universal coefficient of a vertex-induced logarithmic scaling for a polygonal entangled subregion, and compare the result to interacting and non-interacting theories. We also examine the shape-dependence of the Rényi entropy for finite-size toroidal lattices divided into two entangled cylinders by smooth boundaries. Remarkably, we find that the dependence on cylinder length follows a shape-dependent function calculated previously by Stephan et al (2013 New J. Phys. 15 015004) at the QCP corresponding to the 2 + 1 dimensional quantum Lifshitz free scalar field theory. The quality of the fit of our data to this scaling function, as well as the apparent cutoff-independent coefficient that results, presents tantalizing evidence that this function may reflect universal behaviour across these and other very disparate QCPs in 2 + 1 dimensional systems.
Entanglement entropies can conceal novel universal features that underlie a quantum phase transition. The universal content is determined by the geometry of the boundary between two entangled regions. In the quantum Monte Carlo simulations here, the authors numerically extract a universal number that arises at a quantum phase transition in the two-dimensional quantum Ising model. This universal number, measured when a two-dimensional torus is partitioned with a thin cylindrical strip, is so far lacking a comprehensive theoretical interpretation.
Universal scaling terms occurring in R\'enyi entanglement entropies have the potential to bring new understanding to quantum critical points in free and interacting systems. Quantitative comparisons between analytical continuum theories and numerical calculations on lattice models play a crucial role in advancing such studies. In this paper, we exactly calculate the universal two-cylinder shape dependence of entanglement entropies for free bosons on finite-size square lattices, and compare to approximate functions derived in the continuum using several different Ans\"atze. Although none of these Ans\"atze are exact in the thermodynamic limit, we find that numerical fits are in good agreement with continuum functions derived using the anti--de Sitter/conformal field theory correspondence, an extensive mutual information model, and a quantum Lifshitz model. We use fits of our lattice data to these functions to calculate universal scalars defined in the thin-cylinder limit, and compare to values previously obtained for the free boson field theory in the continuum.
Quantum foundations of the PhaseGeometry Z₂ phase medium: observers, decoherence and Quantum Darwinism This record presents the quantum foundations of the PhaseGeometry Z₂ programme, uniting the ontology of observers and branching histories (Foundations IV) with their concrete physical realisation via decoherence and Quantum Darwinism (Foundations V). The PhaseGeometry Z₂ framework proposes that a single binary phase field can account for dark energy, dark matter and the emergence of the classical world. The central insight is that the homogeneous broken phase acts as a universal phase medium — a physical substrate that simultaneously sources the cosmological background and supports quantum observers, measurements and branching histories. Part IV – Phase medium, observer and branching historiesDevelops the ontological layer of the framework. The homogeneous Z₂ vacuum is reinterpreted as a physical phase medium, and clocks, observers and classical branches are defined as inherent structures within this medium, without appealing to external agents. Part V – Decoherence and Quantum Darwinism in a binary phase mediumProvides a quantitative proof of concept. Using finite Z₂ chains as toy universes, it demonstrates: rapid decoherence of a binary event; saturation of subsystem entropy; the emergence of Quantum Darwinism and redundant classical records. These results show how an effectively objective classical reality can arise from the unitary dynamics of the phase medium itself. The toy models used here are short 1D Ising chains; they are intended as minimal demonstrations of the mechanisms, not as direct quantitative predictions for cosmological scales. Taken together, Foundations IV and V show how the same Z₂ field that underlies dark-energy- and dark-matter–like phenomena also provides a complete observer medium, in which systems, apparatus and classical branches all emerge as patterns and correlations of a single physical substrate. This record is part of the PhaseGeometry Z₂ Core Package v1.0: “Binary Phase, Dark Energy and Dark Matter” (series DOI: 10.5281/zenodo.17652957).A unified journal-style preprint version of this material is available as “PhaseGeometry Z₂: Decoherence, Quantum Darwinism and Branching in a Binary Phase Medium” (DOI: 10.5281/zenodo.17773135). Contents PDF of Foundations IV (phase medium, observers, branching histories); PDF of Foundations V (decoherence and Quantum Darwinism). All materials are released under the Creative Commons Attribution 4.0 International (CC BY 4.0) licence.