Static Path Approximation Research Articles

The specific heat of superconducting metallic grains in magnetic field

More and more studies indicate that the effects of quantum size and energy level statistics play a crucial role in the thermodynamic properties of ultrasmall metallic grains. This paper aims to investigate how they affect the specific heat of ultrasmall metallic grains in magnetic field. As the particle size decreases, fluctuation effects and the impact of energy level separation are becoming more and more important. The method of static path approximation (SPA) is adopted to handle the fluctuation effect. Random matrix theory (RMT) is adopted due to its successful description of the energy level of metal nanoparticles. The normalized specific heat of several typical temperatures and electron spins were taken in the calculation, and the results were analyzed. It was found that spin and the spin-orbit coupling affect the specific heat very obviously, and the suppressed high spin weakens the contribution of electrons to the heat capacity.

Disorder stabilized breached-pair phase in an s -wave superconductor

The breached-pair state wherein superconductivity coexists with magnetic polarization is known to exist as a ground state of an imbalanced Fermi system only under very fine-tuned conditions. Here, we show that an s-wave superconductor that is well described by a spin-selectively disordered attractive Hubbard model away from half filling has the breached-pair state as a ground-state without the need for such fine-tuning. The existence of this breached-pair phase is established by laying recourse to a Monte Carlo technique called static path approximation (SPA). Further, by using the SPA, we map out the entire phase diagram of the spin-selectively disordered attractive Hubbard model and show that apart from the breached-pair phase, the many-body system hosts the putative s-wave superconducting state and a polarized Fermi-liquid state.

Studying temperature dependence of pairing gap parameter in a nucleus as a small superconducting system

In this paper, we have taken the effect of small size of nucleus and static fluctuations into account in the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity calculations of [Formula: see text]Ti nucleus. Thermodynamic quantities of [Formula: see text]Ti have been extracted within the BCS model with the inclusion of the average value of the pairing gap square, extracted by the modified Ginzburg–Landau (MGL) method for small systems. Calculated values of the excitation energy and entropy within the MGL+BCS method improve the extracted results within the usual BCS model and show a smooth behavior around the critical temperature with a very good agreement with the semi-empirical values. The result of using MGL+BCS method for the heat capacity of [Formula: see text]Ti is compared with the corresponding semi-empirical values and the calculated values within the BCS, static path approximation (SPA) and Modified Pairing gap BCS (MPBCS) which is a method that was proposed in our previous publications. Both MGL+BCS and MPBCS avoid the discontinuity of the heat capacity curve, which is observed in the usual BCS method, and lead to an S-shaped curve with a good agreement with the semi-empirical results.

The effect of level statistics and fluctuations on the specific heat of small metallic superconducting grains

The thermodynamic properties of small metallic superconducting grains are affected by fluctuations in the order parameter and the level statistics, especially in the critical region. Here, the static path approximation was applied to deal with the fluctuations and the thermodynamic contribution of quasiparticles, and random matrix theory was applied to deal with level statistics. The specific heats were calculated numerically in the cases of even and odd number of electrons and different spins.

Studying thermal properties of 45Sc in static path approximation

We have extracted thermal quantities such as chemical potential and heat capacity of the nucleons system of 45Sc in the static path approximation. This method was used in order to take into account the pairing correlations between the nucleons and the fluctuations of the nucleus finite system. The single particle energies of the nucleons which were used in the calculations have been extracted by the use of the modified Nilsson potential. The results have been compared with the corresponding values calculated in the other microscopic models and the empirical values which were extracted by the use of the experimental data on nuclear level densities.

Influence of Disorder on Superconducting Correlations in Nanoparticles

We investigate how the interplay of quantum confinement and level broadening caused by disorder affects superconducting correlations in ultra-small metallic grains. We use the electron-phonon interaction-induced electron mass renormalization and the reduced static-path approximation of the BCS formalism to calculate the critical temperature as a function of the grain size. We show how the strong electron-impurity scattering additionally smears the peak structure in the electronic density of states of a metallic grain and imposes additional limits on the critical temperature under strong quantum confinement.

Thermodynamics of ultrasmall metallic grains in the presence of pairing and exchange correlations: Mesoscopic fluctuations

We study the mesoscopic fluctuations of thermodynamic observables in a nanosized metallic grain in which the single-particle dynamics are chaotic and the dimensionless Thouless conductance is large. Such a grain is modeled by the universal Hamiltonian describing the competition between exchange and pairing correlations. The exchange term is taken into account exactly by a spin-projection method, and the pairing term is treated in the static-path approximation together with small-amplitude quantal fluctuations around each static fluctuation of the pairing field. Odd-even particle-number effects induced by pairing correlations are included using a number-parity projection. We find that the exchange interaction shifts the number-parity effects in the heat capacity and spin susceptibility to lower temperatures. In the regime where the pairing gap is similar to or smaller than the single-particle mean level spacing, these number-parity effects are suppressed by exchange correlations, and the fluctuations of the spin susceptibility may be particularly large. However, for larger values of the pairing gap, the number-parity effects may be enhanced by exchange correlations.

Theoretical Description of the Superconducting State of Nanostructures at Intermediate Temperatures: A Combined Treatment of Collective Modes and Fluctuations

A rigorous treatment of the combined effect of thermal and quantum fluctuations in a zero-dimensional superconductor is considered one of the most relevant and still-unsolved problems in the theory of nanoscale superconductors. In this Letter, we notice that the divergences that plagued previous calculations are avoided by identifying and treating nonperturbatively a low-energy collective mode. In this way, we obtain for the first time closed expressions for the partition function and the superconducting order parameter which include both types of fluctuation and are valid at any temperature and to leading order in δ/Δ(0), where δ is the mean level spacing and Δ(0) is the bulk energy gap. Our results pave the way for a quantitative description of superconductivity in nanostructures at finite temperature and pairing in hot nuclei.

Particle-number conservation in static-path approximation for thermal superfluid systems

By applying particle-number projection to the static-path approximation (SPA), the heat capacity and the breakdown of pairing correlations are investigated in the thermally excited, superfluid systems $^{172}\mathrm{Yb}$, $^{94}\mathrm{Mo}$, and $^{56}\mathrm{Fe}$. For the heavy nucleus $^{172}\mathrm{Yb}$, the heat capacities in both the SPA and the number-projected SPA (NPSPA) exhibit an $\mathcal{S}$ shape; the difference between the SPA and NPSPA heat-capacity curves is not very large and the particle-number projection thereby enhances the $\mathcal{S}$ shape already seen in the SPA. The temperature at which the $\mathcal{S}$-shape of heat capacity curve occurs parallels the temperature of the breakdown of pairing correlations as indicated by the effective pairing gap. However, for the comparatively lighter nuclei $^{94}\mathrm{Mo}$ and $^{56}\mathrm{Fe}$, the SPA does not produce an $\mathcal{S}$-shaped heat capacity on its own; only after particle-number projection the $\mathcal{S}$ shape appears in the heat-capacity curve. For $^{94}\mathrm{Mo}$, we compare the NPSPA result with thermal odd-even mass differences, which are regarded as a direct measure of the pairing gap.

Description of thermal entanglement with the static path plus random-phase approximation

We discuss the application of the static path plus random-phase approximation $(\mathrm{SPA}+\mathrm{RPA})$ and the ensuing mean $\text{field}+\mathrm{RPA}$ treatment to the evaluation of entanglement in composite quantum systems at finite temperature. These methods involve just local diagonalizations and the determination of the generalized collective vibrational frequencies. As an illustration, we evaluate the pairwise entanglement in a fully connected $XXZ$ chain of $n$ spins at finite temperature in a transverse magnetic field $b$. It is shown that already the mean $\text{field}+\mathrm{RPA}$ provides an accurate analytic description of the concurrence below the mean field critical region $(\ensuremath{\mid}b\ensuremath{\mid}<{b}_{c})$, exact for large $n$, whereas the full $\mathrm{SPA}+\mathrm{RPA}$ is able to improve results for finite systems in the critical region. It is proven as well, that for $T>0$ weak entanglement also arises when the ground state is separable $(\ensuremath{\mid}b\ensuremath{\mid}>{b}_{c})$, with the limit temperature for pairwise entanglement exhibiting quite distinct regimes for $\ensuremath{\mid}b\ensuremath{\mid}<{b}_{c}$ and $\ensuremath{\mid}b\ensuremath{\mid}>{b}_{c}$.

Spin distribution of nuclear levels using the static path approximation with the random-phase approximation

We present a thermal and quantum-mechanical treatment of nuclear rotation using the formalism of the static path approximation plus the random-phase approximation. Naive perturbation theory fails because of the presence of zero-frequency modes resulting from dynamical symmetry breaking. Such modes lead to infrared divergences. We show that composite zero-frequency excitations are properly treated within the collective coordinate method. The resulting perturbation theory is free from infrared divergences. Without the assumption of individual random spin vectors, we derive microscopically the spin distribution of the level density. The moment of inertia is thereby related to the spin-cutoff parameter in the usual way. Explicit calculations are performed for $^{56}\mathrm{Fe}$; various thermal properties are discussed. In particular, we demonstrate that the increase of the moment of inertia with increasing temperature is correlated with the suppression of pairing correlations.

Contour integral method for thermal and quantal fluctuations

The partition function by means of the static path approximation (SPA) plus the random-phase approximation (RPA) treatment can be written as a contour integral form without solving the RPA equations for a separable interaction. This method is an efficient way to evaluate numerically the partition function for realistic calculations. As an illustration, we adopt the pairing model at finite temperature.

Quenching of pairing gap at finite temperature inW184

We extract the pairing gap in $^{184}\mathrm{W}$ at finite temperature for the first time from the experimental level densities of $^{183}\mathrm{W}$, $^{184}\mathrm{W}$, and $^{185}\mathrm{W}$ using the ``thermal'' odd-even mass difference. We found the quenching of the pairing gap near the critical temperature ${T}_{c}=0.47\phantom{\rule{0.3em}{0ex}}\text{MeV}$ in the BCS calculations. It is shown that the monopole pairing model with a deformed Woods-Saxon potential explains the reduction of the pairing correlation using the partition function with the number parity projection in the static path approximation plus random-phase approximation.

Effects of gap fluctuations on the pair-transfer correlation function in nanometer-scale superconducting grains

~Received 29 July 2002; revised manuscript received 26 November 2002; published 28 April 2003! We investigate the effects of gap fluctuations on the pair-transfer correlation and spectral functions in nanoscopic superconducting grains at finite temperature, by means of the correlated static path approximation ~CSPA!. The present approach is able to provide a reliable description of the lowest energy weighted moments of the spectral function of small samples in critical regions, improving both the random-phase approximation ~RPA! and the SPA predictions. The results confirm the persistence of pairing effects in the spectral function beyond the BCS critical temperatures and sizes, which is visible through the enhancement of the strength at low energies and the concomitant decrease in the normalized first energy moment. The role played by the zero and the imaginary quasiparticle RPA energies present in the CSPA is also discussed.

The partition function of an interacting many body system

Based on the path integral representation of the partition function of a many body system with separable two body interaction we propose a systematic extension of the perturbed static path approximation (PSPA) to lower temperatures. Thereby, special attention must be paid to instabilities of the classical mean field solution in functional space that cause divergencies within the conventional PSPA. As a result we develop an approximation applicable from high to very low temperatures. These findings are tested against exact results for the archetypical cases of a particle moving in a one dimensional double well and the exactly solvable Lipkin model. In particular, we obtain a very good approximation to the level density of the Lipkin model even at low thermal excitations. Our results may have potential applications in low temperature nuclear physics and mesoscopic systems, e.g. for gap fluctuations in nanoscale superconducting devices previously studied within a PSPA type of approximation. PACS: 5.30.-d, 24.60.-k, 21.10.Ma, 74.25.Bt

Magnetic behavior of nanoscopic superconductors

The magnetic behavior of nanometer-scale superconducting grains at finite temperature is investigated using the correlated static path approximation to the partition function. In these systems, the field penetration depth is much larger than the spatial dimension and Pauli paramagnetism becomes dominant. It is shown that deviations from both the bulk behavior and the BCS results for fixed number parity become significant due to gap fluctuations, which lead to appreciable pairing effects in the spin magnetization and susceptibility beyond the BCS critical fields or sizes. Differences between even and odd systems are also discussed.

Self-consistent quantal treatment of decay rates within the perturbed static path approximation

The framework of the perturbed static path approximation is used to calculate the partition function of a finite Fermi system from a Hamiltonian with a separable two body interaction. Therein, the collective degree of freedom is introduced in self-consistent fashion through a Hubbard-Stratonovich transformation. In this way, all transport coefficients that dominate the decay of a metastable system are defined and calculated microscopically. Otherwise the same formalism is applied as in the Caldeira-Leggett model to deduce the decay rate from the free energy above the so called crossover temperature T(0).

Nuclear structure at high excitation energies

The present paper deals with the investigation of hot GDR and quadrupole shapes of 106,120Sn isotopes as a function of temperature and spin utilizing cranked quadrupole-quadrupole model interaction hamlitonian in the linear response theory and static path approximation to the grand canonical partition function.

Thermal dependence of pairing correlations in small superconducting particles in a finite magnetic field

We investigate the thermal behavior of pairing correlations in nanoscale particles in the presence of a magnetic field. By computing the exact partition function in a small system, it is shown that at finite fields, these correlations may behave in a nonmonotonous way with increasing temperature in both the even and odd systems, leading to characteristic signatures of pairing effects in the magnetization, which are seen to subsist up to extremely small sizes. Results obtained with the static path plus random phase approximation in larger configuration spaces exhibit the same features. The peculiar thermal behavior of the BCS gap for finite fields and odd parity is also examined.

Thermal shape and orientation fluctuation corrections for the hot giant dipole resonance within the static path approximation

The $\ensuremath{\gamma}$-absorption cross section [i.e., giant dipole resonance (GDR)] of ${}^{106}\mathrm{Sn}$ is calculated at temperature $T=2$ MeV and angular momenta $J=40$ and 55 following the linear response theory at finite temperature and incorporating the thermal shape as well as orientation fluctuation corrections within the static path approximation approach. The orientation fluctuation corrections have been included for the first time in such a calculation, and are found to be rather unimportant. Then without the inclusion of these effects the GDR is calculated at several more temperature and spin values. We find that at $T=4$ MeV the GDR width $\ensuremath{\Gamma}$ shows an increase of only about 1.5 MeV compared to its value at $T=0.5$ MeV. However, at a fixed temperature $T=2$ MeV, the increase in the width at $J=69$ is by about 3 MeV compared to its value at $J=0.$ Finally, we have also compared the increase of $\ensuremath{\Gamma}$ at $T=2$ MeV as a function of J for ${}^{106}\mathrm{Sn}$ and ${}^{120}\mathrm{Sn}.$ The increase is steeper for the lighter isotope, as expected.