ROOT-FQPTMathematics & Physics

Fractal Quantum Phase Transitions (FQPT)

Formulates quantum criticality and phase transitions under scale-invariant fractal topological operators with multi-scale renormalization groups across discrete lattice levels.

ClassQuantum Lattice Fabric

Evidence TierTier 1 (Formal Root)

Model ProfileFORMAL_SPECIFICATION

AttributionIvan Pasev

Procedural PlateFormal procedural plate defining the bounded theoretical geometry and invariant obligations for this research fabric.

EPISTEMIC DEMARCATION // Tier 1 (Formal Root)

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Constitutional Demarcation: A Research Fabric is a bounded epistemological architecture. Structural formulation, digital execution, and proof-assistant replay declare local invariants; they do not constitute unverified physical validation.

Architectural Overview

Fractal Quantum Phase Transitions (FQPT) formulates zero-temperature quantum critical phenomena, ground-state fidelity susceptibility, and topological entanglement scaling on non-integer dimensional lattice systems.

By replacing standard Euclidean momentum integrals with spectral decimation measures on fractal manifolds, FQPT models the emergence of non-perturbative critical exponents and scale-invariant excitation spectra:

$${\mathcal{H}}_{FQPT}=-J\sum _{⟨i,j⟩\in \mathcal{E}(\mathcal{F})}{\sigma }_i^z{\sigma }_j^z-h\sum _{i\in \mathcal{V}(\mathcal{F})}{\sigma }_i^x-\lambda \sum _{\mathrm{\Delta }\in {\mathcal{C}}_3(\mathcal{F})}{\mathcal{O}}_{\mathrm{\Delta }}$$

Where the critical coupling ratio $(h/J{)}_c$ shifts predictably as a function of the spectral dimension $d_s$ and Hausdorff dimension $d_H$.

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Core Invariants

1. Topological Order Preservation: Bounded non-local string operators preserve ground-state degeneracy across scale transformations.
2. Entanglement Entropy Area-to-Fractal Law: Subsystem entanglement entropy obeys $S(A)\sim |\partial A{|}^{d_f-1}\log \xi$.
3. Universality Class Invariance: Critical exponents $(\nu ,z,\eta )$ remain invariant under irrelevant micro-scale perturbation operators.
4. Spectral Gap Non-Triviality: The excitation spectrum maintains a strictly non-vanishing or controlled power-law closing at criticality.

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Epistemic Boundary Notice

Condensed Matter & Statistical Physics Formulation

FQPT is an analytical and computational modeling architecture for theoretical condensed matter systems. It provides models for synthetic quantum simulators (e.g. Rydberg atom arrays, superconducting qubit lattices) and does not assert undocumented room-temperature macroscopic quantum states.

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Reciprocal Bridges & Related Research

• FQPT-Q Blueprint (RF-032) — The dedicated quantum simulator implementation framework.
• Fractal Quantum Field Theory (RF-007) — Relativistic continuum limit of fractal fields.
• Infinite Symmetry Principle (RF-011) — Algebraic representation theory under scale dilation.
• FQPT Whitepaper Reader — Analytical proofs and critical exponent tables.