Thermodynamics of pairing in mesoscopic systems

Using numerical and analytical methods implemented for different models, we conduct a systematic study of the thermodynamic properties of pairing correlations in mesoscopic nuclear systems. Various quantities are calculated and analyzed using the exact solution of pairing. An in-depth comparison of canonical, grand canonical, and microcanonical ensembles is conducted. The nature of the pairing phase transition in a small system is of a particular interest. We discuss the onset of discontinuity in the thermodynamic variables, fluctuations, and evolution of zeros of the canonical and grand canonical partition functions in the complex plane. The behavior of the invariant correlational entropy is also studied in the transitional region of interest. The change in the character of the phase transition due to the presence of a magnetic field is discussed along with studies of superconducting thermodynamics.

Figure 1

(a) BCS pairing gap as a function of pairing strength for the two-level, half-occupied system with 20 particles. (b) Energy difference per particle between BCS and the exact result as a function of pairing strength for the same $N=20$ system and compared with results for a larger half-occupied two-level model containing 100 particles. The single-particle level spacing is chosen as the unit of energy.

Figure 2

(a) Free energy, (b) entropy, (c) energy, (d) energy fluctuations, (e) specific heat, and (f) order parameter of a ladder system with 12 levels and 12 particles as functions of temperature. The unit of energy is determined by the single-particle level spacing.

Figure 3

Contour plot of the order parameter as a function of pairing strength and temperature, for the half-occupied 12-level system. The line separates normal and paired regions based on the BCS equation.

Figure 4

(a) Entropy and (b) specific heat as a function of temperature, for an odd and even number of particles and $V=1$.

Figure 5

(a) Energy and (b) specific heat as a function of temperature, for $V=1$ and various numbers of particles $N=10$, 30, 50, and 100. The distance between single-particle levels is taken as the unit of energy.

Figure 6

Lowest branch of zeros computed for $N=100$ and $G=1.00$.

Figure 7

Evolution of DOZ and $C_v$ in the complex temperature plane $\mathcal{B}=\beta +i\tau$ for $N=100$ particles in the half-occupied two-level system. A number of poles near and exactly on the imaginary axis are of no interest to our discussion and are not shown. The poles for other systems are discussed in Ref. [2] and references therein, the interpretation and nature of branches is discussed in Ref. [62], and further in-depth exploration of the above model can be found in Ref. [25]. Temperature is expressed in the units of energy with one unit taken equal to the single-particle level spacing.

Figure 8

Parameter of the phase transition for the two-level system with $N=100$ particles as a function of pairing strength. (a) α vs $V$; (b) transition angle ν vs $V$.

Figure 9

Thermodynamic properties of the ladder system with 12 levels, 12 particles, and $V=1.00$ as functions of temperature. The grand canonical ensemble is compared with the canonical. (a) Fluctuation in the number of particles in the grand canonical ensemble; (b) entropy; (c) specific heat; (d) excitation energy. The single-particle level spacing is chosen as the unit of energy.

Figure 10

Excitation energy difference between canonical and grand canonical statistical ensembles as a function of temperature. (a) Ladder system model; (b) two-level model. The distance between single-particle levels is taken as the unit of energy.

Figure 11

Distribution of zeros (DOZ) of grand canonical ensemble in the complex chemical potential plane for a two-level system with $\Omega =100$ and $V=1.00$ for various temperatures indicated. $T_c=0.38$ for this system.

Figure 12

Same as Fig. 11, except $V=0.5$ and corresponding $T_c=0.17.$

Figure 13

Same as Fig. 11, except $V=2$ and thus $T_c=0.8.$

Figure 14

DOZ of grand canonical ensemble in the complex chemical potential plane for a 12-level picket-fence system for various temperatures indicated. Upper plot corresponds to $V=1$, in which case $T_c=2.7$; lower plot is for $V=0.2$ which is below the critical pairing strength even for zero temperature $V_c(T=0)=0.27$. The DOZ is symmetric, and only the roots with Re$(\mu )⩾0$ are shown.

Figure 15

Temperature as a function of energy in three different statistical treatments: canonical, grand canonical, and microcanonical. The ladder system with 12 levels, 12 particles, and $V=1.00$ is used for this study. The results for the microcanonical ensemble are plotted with two different choices of energy window, $\sigma =1$ and 5. The single-particle level spacing is chosen as the unit of energy.

Figure 16

Comparison of entropy as a function of excitation energy for the two-level system in three thermodynamic ensembles ($\sigma =0.5$ in microcanonical). Pairing strength $V=1.00$. (a) For $N=20$ particles; (b) $N=50$ particles; (c) $N=100$ particles. The unit of energy is determined by the single-particle level spacing.

Figure 17

Invariant correlational entropy (upper panel) and specific heat in the canonical ensemble (lower panel) as functions of excitation energy for the two-level system with $N=10$ particles and $V=1$.

Figure 18

Thermodynamics of the two-level system with $N=20$ and $V=1.00$ in the magnetic field. The (a) spin fluctuations χ, (b) specific heat, (c) magnetization (ratio to the maximum possible value), and (d) energy are shown as functions of temperature for different field strengths. For this model, $T_{\mathrm{cr}}=0.46$ at $B=0.$ The unit of energy is determined by the single-particle level spacing.

Figure 19

Same system as in Fig. 18, but concentrating on the low magnetic fields. (a) Spin fluctuations; (b) specific heat.

Figure 20

Thermodynamics within the canonical ensemble of a 12-level ladder system with $N=12$ particles in the magnetic field. The pairing strength is $V=1$. The panels include plots of magnetization, entropy, magnetic susceptibility, and specific heat as functions of temperature. The curves correspond to five different values of the magnetic field $B=0$, 5, 7, 10 and 20.

Figure 21

Same as Fig. 20, but with 11 particles in the magnetic field. The magnetic field is expressed in the units of energy with one unit taken equal to the single-particle level spacing.

Figure 22

Distribution of zeros without and with the presence of external magnetic field for two-level system $N=60$ and $G=1.00$.

Figure 23

Evolution of critical parameters α [panel (a)] and ν [panel (b)] as a function of applied magnetic field $B$ in system with $N=100$ particles and $V=1.00$.

Figure 24

Entropy in the microcanonical picture as a function of the excitation energy for several values of the magnetic field below critical. The two-level system with 20 particles and $V=1$ pairing strength was used.

Figure 25

Same system as in Fig. 24, but the temperature as a function of energy is compared in microcanonical and canonical ensembles at the magnetic field strength $B=0.05$. The difference at high energy is due to a finite model space.