Induced phase transitions and spontaneous symmetry breaking based on the renormalized Ginzburg-Landau theory

TL;DR

The paper develops a renormalized Ginzburg-Landau framework to address phase transitions in low-dimensional systems by introducing a dimension-dependent correction $\Omega(d,T)$ to the quadratic coefficient, yielding $r(d,T)=r_0(T-T_c)+\Omega(d,T)$. A self-consistent KMDDH equation governs $\Omega(d,T)$, providing explicit forms of $r(d,T)$ for $d=1$–$4$ and revealing a lower critical dimension $d_c=2$ consistent with the Mermin-Wagner theorem. The work derives renormalized thermodynamic quantities, including susceptibility and a Gaussian-approximation specific heat, showing dimension-dependent behavior and potential suppression or enhancement of the specific heat jump at the renormalized transition. These findings offer a quantitative framework to understand fluctuation effects and dimensionality on phase transitions, with implications for low-dimensional superconductors and materials exhibiting competing order parameters.

Abstract

In this study, we present theoretical investigations of phase transitions and critical phenomena in materials through the lens of second-order Ginzburg-Landau theory, in conjunction with considerations of symmetry groups and thermal fluctuations. By addressing the residual effects after a renormalization process, a small number of macroscopic degrees of freedom can effectively replace the infinite number of microscopic degrees of freedom, emphasizing the significant role of dimensionality and the intrinsic characteristics of the system in understanding and analyzing transitions. We highlight several non-universal characteristics of continuous phase transitions near the transition temperature, including the non-monotonic relationship between the critical temperature and dimensionality, as well as the enhancement or disappearance of the specific heat jump in complex superconductors. While the resulting expressions for thermodynamic quantities are complex for one-dimensional systems, obeying Mermin-Wagner's theorem, they are considerably simplified for two-dimensional and three-dimensional systems.

Figures (6)

• Figure 1: Schematic illustrations of the evolution of the effective on-site potential corresponding to: a) non-breaking/Unbroken symmetry, b) proposed ${\mathbb{Z}_{2}}$-symmetric system with two possible states, and c) effective symmetry breaking/ Broken symmetry. This general framework identifies the mechanisms and provides the guideline for the phase transition mechanism.
• Figure 2: Schematic geometry representation of the evolution of the effective on-site potential with temperature, illustrating spontaneous symmetry breaking behavior at $T_{c}$ and the structural phase transition mechanism of the multistable systems.
• Figure 3: The plot illustrates the standard and renormalized quadratic coefficients for a one-dimensional system. It is assumed that $T_{c} = 5 K$ and $\eta$ = 0.8. The solid curve represents the correction term $\Omega_{1D}(T)$, the dotted curve shows the standard quadratic coefficient $r_0(T - T_c)$, and the dashed curve depicts the behavior of the renormalized quadratic coefficient, defined as $r(d, T) = r_0(T - T_c) + \Omega_{1D}(T)$. While the standard coefficient goes to zero at $T_c$, the renormalized quadratic coefficient asymptotically approaches zero, i.e. $T_{c_{1D}}$ = 0 is the critical temperature for one-dimensional systems in the renormalized theory.
• Figure 4: Profile of fluctuation-correction dependence of the two-dimensional critical temperature according to Eq. \ref{['eq12']} for $T_{c}$ = 2.5 K. Here, the scaled fluctuation correction ratio $\epsilon$ defines the relative temperature distance from $T_{c}$.
• Figure 5: Illustration of the susceptibility as a function of temperature in a four-dimensional system, considering various values of the scaled fluctuation-correction term, $\epsilon$. For a critical temperature of $T_{c}$ = 2.5 K, the condition $\epsilon < 1$ indicates the relative distance between the cancellation point and the divergence point.