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Physical review. B, Solid state

Critical Fluctuations in Superconductors: A Functional-Integral Approach

R. F. Hassing, John W. Wilkins

Published Mar 1, 1973• 26 citations
Publisher

Abstract

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 ${T}_{c}$, 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.

DOI
10.1103/physrevb.7.1890

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Journal of Mathematical Physics•Aug 1, 1967•25 citations
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∫D\ensuremathψe\ensuremath−\ensuremathβF[\ensuremathψ]\ensuremath{\int}D\ensuremath{\psi}{e}^{\ensuremath{-}\ensuremath{\beta}F[\ensuremath{\psi}]}\ensuremath∫D\ensuremathψe\ensuremath−\ensuremathβF[\ensuremathψ], where FFF is of the Ginzburg-Landau form but with a time-dependent order parameter. An examination of the coefficient bbb of the fourth-order term in FFF, however, shows that bbb 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 \ensuremathγ=2\ensuremath{\gamma}=2\ensuremathγ=2, \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for T>0T>0T>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{T}_{c}Tc​, with critical index \ensuremathα=\ensuremath−1\ensuremath{\alpha}=\ensuremath{-}1\ensuremathα=\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 FFF. 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.
The partition function for interacting bosons is derived as the coherent state path integral and then generalized to magnetic transitions. Phase transitions in the Ginzburg–Landau–Wilson theory for a fluctuating order parameter are discussed in Hartree's and Landau's approximations, and the fundamental limitation of perturbation theory near the critical point is exposed. The concept of upper critical dimension is introduced. Partition function for interacting bosons As a prototypical system with a continuous phase transition we will consider the system of interacting bosons. A well studied physical realization is provided by helium ( 4 He) with the pressure–temperature phase diagram as depicted in Fig. 2.1. Since the atoms of helium are light and interact via weak dipole–dipole interactions, due to quantum zero-point motion helium stays liquid down to the lowest temperatures, at not too high pressures. Instead of solidifying it suffers a continuous normal liquid–superfluid liquid transition at T c ≈ 2K, also called the λ-transition due to the characteristic form of the specific heat in Fig. 1.6. The λ-transition represents the best quantitatively understood critical point in nature. We have already quoted the specific heat exponent α = -0.0127 ± 0.0003, with the power-law behavior being observed over six decades of the reduced temperature! To achieve this accuracy the experiment had to be performed in the space shuttle so that the small variations in Tc along the height of the sample due to Earth's gravity would be minimized. At higher pressures He eventually solidifies, with the superfluid–solid and the normal liquid–solid phase transitions being discontinuous, the former being so rather weakly.
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We study the effect of critical fluctuations on the (B,T)(B,T)(B,T) phase diagram in extreme type-II superconductors in zero and finite magnetic field. In zero magnetic field the critical fluctuations are transverse phase fluctuations of the complex scalar Ginzburg-Landau order parameter, which when excited thermally will induce topological line defects in the form of closed vortex loops into the system. The distribution function D(p)D(p)D(p) of vortex loops of perimeter p changes from an exponential function $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}\mathrm{exp}[\ensuremath{-}\ensuremath{\varepsilon}{(T)p/k}_{B}T]$ to a power law distribution $D(p)\ensuremath{\sim}{p}^{\ensuremath{-}\ensuremath{\alpha}}$ at the zero-field critical temperature T=Tc.{T=T}_{c}.T=Tc​. We find that the long-wavelength vortex-line tension vanishes as $\ensuremath{\varepsilon}(T)\ensuremath{\sim}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{\gamma}};\ensuremath{\gamma}\ensuremath{\approx}1.45,$ as $\stackrel{\ensuremath{\rightarrow}}{T}{T}_{c}.$ At T=Tc,{T=T}_{c},T=Tc​, an extreme type-II superconductor suffers an unbinding of large vortex loops of order the system size. When this happens, the connectivity of the thermally excited vortex tangle of the system changes abruptly. The loss of phase stiffness in the Ginzburg-Landau order parameter, the anomaly in specific heat, the loss of vortex-line tension, and the change in the connectivity of the vortex tangle are all found at the same temperature, the critical temperature of the superconductor. At zero magnetic field, unbinding of vortex loops of order the system size can be phrased in terms of a global U(1)\mathrm{U}(1)U(1)-symmetry breaking involving a local complex disorder field which is dual to the order parameter of the usual Ginzburg-Landau theory. There is one parameter in the theory that controls the width of the critical region, and for the parameters we have used, we show that a vortex-loop unbinding gives a correct picture of the zero-field transition even in the presence of amplitude fluctuations. A key result is the extraction of the anomalous scaling dimension of the dual field directly from the statistics of the vortex-loop excitations of the Ginzburg-Landau theory in the phase-only approximation. A scaling analysis of the vortex lattice melting line is carried out, yielding two different scaling regimes, namely, a high-field scaling regime and a distinct low-field three-dimensional XY critical scaling regime. We also find indications of an abrupt change in the connectivity of the vortex tangle in the vortex liquid along a line TL(B),{T}_{L}(B),TL​(B), which at low enough fields appears to coincide with the vortex line lattice melting transition line within the resolution of our numerical calculations. We study the temperature at which this phenomenon takes place as a function of system size and shape. Our results show that this temperature decreases and appears to saturate with increasing system size, and is insensitive to aspect ratios of the systems on which the simulations are performed, for large enough systems.
If the zero-field transition in high-temperature superconductors such as ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ is a critical point in the universality class of the three-dimensional XY\mathrm{XY}XY model, then the general theory of critical phenomena predicts the existence of a critical region in which thermodynamic functions have a characteristic scaling form. We report an attempt to calculate the universal scaling function associated with the specific heat, for which experimental data have become available in recent years. Scaling behavior is extracted from a renormalization-group analysis, and the 1/N1/N1/N expansion is adopted as a means of approximation. The estimated scaling function is qualitatively similar to that observed experimentally, and also to the lowest-Landau-level scaling function used by some authors to provide an alternative interpretation of the same data. Unfortunately, the 1/N1/N1/N expansion is not sufficiently reliable at small values of N for a quantitative fit to be feasible.
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.