A Renormalized Ginzburg-Landau Framework for Dimensional Crossover and Fluctuation Specific Heat in High-$T_c$ Superconductors in a Magnetic Field

Abstract

This paper presents a theoretical analysis of phase transitions and critical phenomena in high-temperature superconductors using a renormalized Ginzburg-Landau framework. Rather than assuming a conventional linear temperature dependence, we treat the quadratic coefficient as a self-consistent, Hartree-renormalized quantity determined by fluctuation-loop corrections. This renormalization regularizes mean-field divergences and yields a finite, dimensionality-dependent specific-heat anomaly near the transition. When an external magnetic field is applied, minimal coupling quantizes order-parameter fluctuations into discrete Landau levels, reducing the effective dimensionality via an effective spectral dimension. In this framework, the intrinsic fluctuation-coupling strength self-consistently determines the temperature width of the critical Ginzburg region, while the cyclotron energy of the quantized fluctuations uniquely establishes both the specific-heat peak position and the upper-critical-field crossover boundary. Comparison with $\text{YBa}_2\text{Cu}_3\text{O}_{7-δ}$ experimental data demonstrates that this approach captures the suppression of the sharp mean-field discontinuity and accurately reproduces the vortex-lattice topological structure under finite magnetic fields.