The analytical determination of critical scaling in three-dimensional strongly coupled field theories is inherently restricted by the Borel non-summability of perturbative expansions. In this paper, we establish a symplectic isomorphism that maps the infrared dynamics of scalar quantum field theories onto the finite-strain bifurcation mechanics of hyperelastic continua. By reformulating the renormalization group flow as an effective spatial Hamiltonian evolution within a symplectic phase space, we identify the critical fixed point with the spectral degeneration of a continuous Riccati-Lyapunov dynamical system. The algebraic integrity of this geometric correspondence is rigorously verified against exactly solvable limits, recovering both the Gaussian free-field scaling and the localized Jackiw-Rebbi zero-mode of the continuum Su-Schrieffer-Heeger model. Applying this framework to the 3D Ising universality class, we construct a self-consistent algebraic scaling Ansatz for the anomalous dimension $η$. This derivation dispenses with arbitrary perturbative truncations, demonstrating instead that interaction-induced non-perturbative screening emerges as a strict consequence of the nonlinear self-consistency demanded by the continuous Algebraic Riccati Equation. The resulting rational scaling law functions as a geometric Padé approximant that asymptotically satisfies the unitarity bound limits, providing a mathematically transparent, first-principles macroscopic impedance analogue for the dynamic suppression of strong-coupling divergences.
The analytical determination of critical scaling in three-dimensional strongly coupled field theories is inherently restricted by the Borel non-summability of perturbative expansions. In this paper, we establish a symplectic isomorphism that maps the infrared dynamics of scalar quantum field theories onto the finite-strain bifurcation mechanics of hyperelastic continua. By reformulating the renormalization group flow as an effective spatial Hamiltonian evolution within a symplectic phase space, we identify the critical fixed point with the spectral degeneration of a continuous Riccati-Lyapunov dynamical system. The algebraic integrity of this geometric correspondence is rigorously verified against exactly solvable limits, recovering both the Gaussian free-field scaling and the localized Jackiw-Rebbi zero-mode of the continuum Su-Schrieffer-Heeger model. Applying this framework to the 3D Ising universality class, we construct a self-consistent algebraic scaling Ansatz for the anomalous dimension η. This derivation dispenses with arbitrary perturbative truncations, demonstrating instead that interaction-induced non-perturbative screening emerges as a strict consequence of the nonlinear self-consistency demanded by the continuous Algebraic Riccati Equation. The resulting rational scaling law functions as a geometric Padé approximant that asymptotically satisfies the unitarity bound limits, providing a mathematically transparent, first-principles macroscopic impedance analogue for the dynamic suppression of strong-coupling divergences.
The two-dimensional metallic quantum critical problem is a long-standing puzzle that is widely believed to hold the key to resolving ubiquitous non-Fermi liquid behavior in strongly correlated electronic systems. In this study, we present a non-perturbative renormalization group (RG) analysis of the metallic Ising-nematic quantum critical point in two dimensions, formulated directly around an intrinsically nonlocal infrared (IR) boson propagator. Rather than treating the anomalous dynamical critical exponent a as a fixed phenomenological parameter, we regard it as an intrinsic component of the fixed-point data to be determined from the internal consistency of the low-energy patch field theory under highly anisotropic scaling dimensions ([k0]=a+1, [kx]=2, [ky]=1). While the leading two-loop diagrammatics vanish identically due to kinematic pole configurations, our three-loop evaluation reveals a profound structural asymmetry between the sectors: the fermion self-energy and Yukawa vertex receive non-vanishing logarithmic corrections, whereas the corresponding bosonic counter-term remains strictly zero. Consequently, we find that no self-consistent, intersecting fixed-point solution for the exponent a exists within the three-loop truncation, failing to reproduce the physical value of a≈1.85 observed in quantum Monte Carlo simulations. We conjecture that the cross-linked topology of the four-loop boson self-energy diagrams is exactly marginal and yields the minimal, mandatory bosonic counter-term required to restore multi-sector self-consistency. Our framework establishes a rigid multi-loop matching scheme necessary to uniquely pin down the critical exponent, and uncovers a stable phase space for field anomalous dimensions.
Continuous phase transitions provide a natural setting for understanding universality. Near criticality, the correlation length diverges, microscopic details become increasingly irrelevant, and physical observables obey scaling laws characterized by universal critical exponents. Within the Functional Renormalization Group (FRG), this scale dependence is described through the Effective Average Action, which progressively incorporates fluctuations as the momentum scale is lowered. A first part of this thesis is developed within the Local Potential Approximation (LPA), where we study multicritical fixed points of scalar field theories. Their universal properties are compared with results obtained at higher orders in the derivative expansion. Particular attention is devoted to the relation between global fixed-point potentials and their local representations through finite polynomial truncations. We show that the convergence of such expansions is constrained by the analytic structure of the global solution, and that expanding around a suitable nonzero stationary point significantly improves the reconstruction of multicritical branches. Beyond the structure of individual fixed points, a central part of this thesis concerns RG irreversibility. In two dimensions, Zamolodchikov’s c-theorem describes the flow from the ultraviolet to the infrared through a monotonically decreasing c-function. Within the FRG framework, we develop a variational procedure to reconstruct RG trajectories between fixed points and integrate the corresponding c-function along them. The resulting quantity displays monotonic behaviour and becomes increasingly stable as the truncation order is raised, providing a manifestation of RG irreversibility within the approximation considered. Finally, close to the upper critical dimensions, the linearized FRG flow around the Gaussian fixed point is naturally organized in terms of Hermite polynomials. Using the epsilon-expansion, we derive a curvature-based candidate c-function and obtain an analytic explanation for the alternating curvature of successive multicritical fixed points and for their progressive accumulation around the Gaussian fixed point.
Abstract We establish a complete geometric reconstruction framework in which quantization, symplectic structure, and infrared curvature emerge from the discrete phase overlaps between low-energy eigenstates of the spectral vacuum Hessian H. This work provides a rigorous derivation of Planck's constant ℏ as a measurable geometric quantity rather than a fundamental postulate. Key Results: 1. Emergent Symplectic Structure: From discrete phase overlaps θₐᵦ = arg⟨ψₐ|ψᵦ⟩, we construct a closed, non-degenerate 2-form: ω = Σ_{⟨ab⟩∈𝒩} (dθₐᵦ ∧ dλₐ) / √(ΔₐΔᵦ) satisfying the symplectic condition dω = 0 and rank(ω) = 2n, providing the first principles derivation of Hamiltonian dynamics from spectral data. 2. Geometric Origin of ℏ: We prove that Planck's constant emerges as the minimal symplectic area of a phase-space cell in the reconstructed manifold: ℏ ≈ ∮_{∂Σ} θₐᵦ dλ ≡ Area_ω(cell) linking quantization to the topological density of the spectral graph. 3. Topological Phase Invariants: We demonstrate that the discrete phases θₐᵦ converge to the Berry curvature and the Aharonov-Bohm phase shift in the continuum limit, where the magnetic flux is recovered as a spectral holonomy: Φ_AB = ∮_Γ 𝒜_Berry · dl establishing the framework as a discrete U(1) gauge theory on eigenvalue space. 4. Infrared Curvature Scaling: We show that the spectral gap Δ of the vacuum Hessian dictates the large-scale (IR) curvature of the emergent manifold: R_curv ~ 1 / √Δ offering a natural resolution to the infrared divergence problem and linking the cosmological constant to spectral stability. 5. Phase-Space Crystallization: We introduce the concept of Phase-Space Crystallization: quantization as a phase transition from continuous (classical) to discrete (quantum) phase space, with the spectral gap Δ acting as the order parameter: Δ = 0 → ℏ_eff = 0 (liquid/classical) Δ > 0 → ℏ_eff > 0 (crystal/quantum) exhibiting critical scaling ℏ_eff ~ (τ - τ_c)^(1/4) near the transition, where quantum mechanics emerges as the 'solid phase' of spectral geometry. Verification and Implementation: The framework is verified via a Discrete Exterior Calculus (DEC) pipeline through 10-seed Monte Carlo simulations confirming: • Symplectic closure: |dω| < 10⁻¹⁴ (machine precision) • Phase coherence: ⟨|δθ|⟩ < 10⁻³ • ℏ_eff stability: σ/⟨ℏ_eff⟩ < 5% • IR residual: R_IR/R* = 1.02 ± 0.02 with full Python implementation provided for reproducibility. Significance: This work completes the SVM foundation by showing that: • Quantization is geometric, not axiomatic • Planck's constant ℏ is a derived quantity from phase overlap structure • Symplectic geometry emerges from Berry phase curvature • Gauge theory (U(1)) arises from spectral holonomy (Aharonov-Bohm equivalence) • Quantum mechanics is a 'crystalline phase' of spectral geometry thereby unifying quantum mechanics, gauge invariance, and emergent gravity under a single spectral principle, with no external postulates. Keywords: Spectral vacuum mechanism, phase geometry, Berry phase, Aharonov-Bohm effect, symplectic reconstruction, discrete exterior calculus, emergent quantization, Planck constant derivation, phase-space crystallization, infrared curvature, spectral gap Other works by the author on a related topic: [1] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXIX: Spectral Yang–Mills Dynamics: Field Equations, Running Coupling, and Confinement from Vacuum Geometry, Zenodo. DOI: 10.5281/zenodo.18624234 (2026). [2] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXX: Emergent Geometry from Spectral Flow, Zenodo. DOI: 10.5281/zenodo.18636847 (2026). [3] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXIV: Quantum Spectral Flow and Vacuum Stability, DOI: 10.5281/zenodo.18704453 (2026). [4] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXV: Spectral Strings and Emergent Worldsheet Dynamics, DOI: 10.5281/zenodo.18726006 (2026). [5] M. Nemirovsky, Spectral Vacuum Mechanism — Part XXXVI: Spectral Derivation of String Tension and Confinement from Stable Vacuum Structure, DOI: 10.5281/zenodo.18749131 (2026).
Abstract Chapter 6 describes how the perturbative renormalization group (RG) of quantum field theory has made it possible to derive universal properties of continuous macroscopic phase transitions. The RG initially emerged as a consequence of the necessity to cancel infinities that appear in the perturbative expansion (the renormalization procedure) and the possibility of defining the parameters of the renormalized theory at different momentum scales. Although the field theory RG is now understood to be an asymptotic form, it has made it possible to confirm the Wilson–Fisher fixed point and led to an understanding of universality for a large class of critical phenomena. In the framework of dimensional continuation, zeros of RG beta functions, which correspond to Wilson–Fisher’s fixed points, have made it possible to recover Wilson–Fisher’s epsilon expansion, proving scaling relations and calculating critical exponents. Series summation methods have then been used to generate precise values of exponents.
Strongly correlated quantum field theories present a formidable challenge to conventional theoretical methods, particularly those rooted in perturbative expansions. The breakdown of perturbation theory in regimes where interaction strengths are comparable to or exceed kinetic energies necessitates the development of non-perturbative approaches. This paper introduces and elaborates on a novel framework for diagrammatic renormalization specifically designed for strongly correlated quantum fields, moving beyond the limitations inherent in iterative or weak-coupling expansions. We explore how a systematic resummation of infinite classes of diagrams, guided by a functional renormalization group perspective, can capture the complex interplay of quantum fluctuations and emergent phenomena characteristic of these systems. Our approach leverages a self-consistent treatment of Green's functions and self-energies, allowing for the faithful description of spectral functions, phase transitions, and exotic ground states in a manner that traditional perturbation theory cannot. We discuss the theoretical underpinnings, computational strategies, and potential applications across diverse domains, including condensed matter physics, quantum chromodynamics, and ultracold atomic systems. The methodology aims to provide a robust, diagrammatically transparent pathway to understanding the non-trivial fixed points and scaling behaviors that govern strongly correlated phenomena, offering a powerful alternative where standard approximations fail.
We study the interplay between critical isotropic elasticity and classical Ising criticality using a functional renormalization group (FRG) approach which is implemented such that the volume is fixed during the entire renormalization group flow. For dimensions slightly smaller than four, we use a simple truncation of the FRG flow equations to recover the fixed points of the constrained Ising model: the Gaussian fixed point G, the Ising fixed point I, the renormalized Ising fixed point R, and the spherical fixed point S. We show that the fixed points R and S are both characterized by a finite anomalous dimension y_{*}<0 of strain fluctuations, implying that the energy dispersion of longitudinal acoustic phonons exhibits a nonanalytic momentum dependence proportional to k^{1-y_{*}/2} for small momentum k. We also derive and solve flow equations for the free energy at constant strain and compute stress-strain relations in the vicinity of the fixed points. As a result, we reaffirm that Ising criticality, controlled by the fixed point I, is preempted by a bulk instability. Beyond that, we find that the stress-strain relation at R and S remains linear to leading order (Hooke's law), as long as the interaction between strain and Ising fluctuations is sufficiently weak. However, the finite anomalous dimension of strain fluctuations y_{*} gives rise to nonanalytic corrections to Hooke's law.