We have been studying the statistical mechanics of one-dimensional Ginzburg-Landau fields for real, complex, and phase-only fields. Here, results for the specific heat, the order-parameter-order-parameter, and energy-density-energy-density correlation functions, will be presented. Formally, these solutions are of interest because they describe the behavior of systems which are nearly ordered but do not undergo sharp phase transitions. Physically, the real-field results may have application in some organic chain systems, while the complex field and phase-only fields are associated with superconducting strips and linear arrays of coupled weak links, respectively.
In an earlier paper the Ginzburg-Landau free energy functional was used to calculate the effect of thermodynamic fluctuations on the off-diagonal correlation function and we found no off-diagonal long-range order in one- and two-dimensional systems. It has been pointed out that, for a charged system, the use of the Ginzburg-Landau free energy functional is in error for arbitrary nonequilibrium values of the order parameter since the electrostatic energy of the charge fluctuations associated with an arbitrary order parameter is not included in the free energy functional. We have not succeeded in a direct generalization of the free energy functional so we are forced to proceed by inference from the generalized random phase approximation (RPA). We find that, for uncharged systems, the RPA gives a linearization of the results obtained earlier using the Ginzburg-Landau theory. For charged systems we find in the RPA results similar to those obtained for uncharged systems. From this we conclude that it is very likely that, as in uncharged systems, there will be no ODLRO in charged infinite one- and two-dimensional systems.
The free-energy density of a one-dimensional Ginzburg-Landau field with
n components ${\ensuremath{\varphi}}_{3}$ is identified with the quantum-mechanical ground-state energy of an
n-dimensional anharmonic oscillator. Normalizing the anharmonic term by ${(4n)}^{\ensuremath{-}1}$ and expanding in powers of ${n}^{\ensuremath{-}1}$, we find for the ground-state energy, $n{E}_{\ensuremath{\infty}}+{E}^{(1)}+{n}^{\ensuremath{-}1}{E}^{(2)}$, where ${E}_{\ensuremath{\infty}}=\frac{\ensuremath{\kappa}}{2}\ensuremath{-}\frac{1}{16{\ensuremath{\kappa}}^{2}}$, and ${E}^{(1)}=\frac{({\ensuremath{\kappa}}_{2}\ensuremath{-}2\ensuremath{\kappa})}{2}$ and ${E}^{(2)}=3{\ensuremath{\kappa}}_{2}^{\ensuremath{-}2}\ensuremath{-}6\ensuremath{\kappa}{\ensuremath{\kappa}}_{2}^{\ensuremath{-}3}\ensuremath{-}(\frac{11}{4}){\ensuremath{\kappa}}^{\ensuremath{-}1}{\ensuremath{\kappa}}_{2}^{\ensuremath{-}4}$ are the Hartree and first and second screening approximations, respectively. $\ensuremath{\kappa}$ and ${\ensuremath{\kappa}}_{2}={(4{\ensuremath{\kappa}}^{2}+{\ensuremath{\kappa}}^{\ensuremath{-}1})}^{\frac{1}{2}}$ are the inverse correlation lengths for ${\ensuremath{\varphi}}_{i}$ and $\ensuremath{\Sigma}{\ensuremath{\varphi}}_{i}^{2}$ respectively. The temperature is linearly related to the spring constant $\ensuremath{\tau}$, which in turn is connected with the correlation lengths by $\ensuremath{\tau}={\ensuremath{\kappa}}^{2}\ensuremath{-}\frac{1}{2\ensuremath{\kappa}}$. The analytic results are compared with exact numerical computations for
n=1,2,3,and4. The second screening correction modestly improves the accuracy of the approximation.
Using the functional-integral method applied to the BCS Hamiltonian we have calculated the partition function, off-diagonal correlation function, specific heat, and conductivity of a super-conductor in the transition region. The partition function is obtained in the form $\ensuremath{\int}D\ensuremath{\psi}{e}^{\ensuremath{-}\ensuremath{\beta}F[\ensuremath{\psi}]}$, where
F is of the Ginzburg-Landau form but with a time-dependent order parameter. An examination of the coefficient
b of the fourth-order term in
F, however, shows that
b is sharply peaked at zero frequency, thus justifying the use of a time-independent order parameter. We then apply a self-consistent mean-field approximation similar to that of Mar\ifmmode \check{c}\else \v{c}\fi{}elja to the fourth-order term. The self-consistency condition determines the mean-square fluctuation in the order parameter and a renormalized temperature shift in terms of which the nature of the phase transition (or lack of it) can be understood. We find that only in three dimensions does a phase transition strictly occur, at a slightly lowered transition temperature with critical indices $\ensuremath{\gamma}=2$, $\ensuremath{\nu}=1$. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for
T>0 and the mean value of the order parameter vanishes, indicating that the transition is completely suppressed in samples of reduced dimensionality. This behavior is reflected in the off-diagonal correlation function which manifests off-diagonal long-range order only in three dimensions. The specific heat in three dimensions is found to be finite at
Tc, with critical index $\ensuremath{\alpha}=\ensuremath{-}1$. In two, one, and zero dimensions the specific-heat transition is rounded relative to the BCS result. We find an anomalous factor of 2 in the specific-heat calculation deriving from our treatment of the fourth-order term in
F. As a result we conjecture that the Hartree approximation of Mar\ifmmode \check{c}\else \v{c}\fi{}elja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Mar\ifmmode \check{c}\else \v{c}\fi{}elja.
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.
From the exact solution of the Ising model by Onsager in 1944 up to that of the hard hexagon model by Baxter in 1980, the statistical mechanics of two-dimensional systems has been enriched by a number of exact results. One speaks (in quick manner) of exact models once a convenient mathematical expression has been obtained for a physical quantity such as the free energy, an order parameter or some correlation, or at the very least once their evaluation is reduced to a problem of classical analysis. Such solutions, often considered as singular curiosities upon their appearance, often have the interest of illustrating the principles and general theorems rigorously established in the framework of definitive theories, and also enabling the control of approximate or perturbative methods applicable to more realistic and complex models. In the theory of phase transitions, the Ising model and the results of Onsager and Yang have eminently played such a reference role. With the various vertex models, the methods of Lieb and Baxter have extended this role and the collection of critical exponents, providing new useful elements of comparison with extrapolation methods, and forcing a refinement of the notion of universality. Intimately linked to two-dimensional classical models (but of less interest for critical phenomena), one-dimensional quantum models such as the linear magnetic chain and Bethe's famous solution have certainly contributed to the understanding of fundamental excitations in many-body systems. One could also mention the physics of one-dimensional conductors.