THESIS
2015

xvi, 95 pages : illustrations (some color) ; 30 cm

**Abstract**
This thesis contains two parts. The first part is about superconductivity in 4-Angstrom carbon
nanotubes (CNTs) embedded in linear, parallel pores of AFI zeolite crystals. In the second part, we
study the pseudogap problem in the BEC-BCS crossover.

The 4-Angstrom CNTs can be formed in the linear pores of AFI zeolite crystals. They exhibit
quasi one-dimensional (1D) fluctuation superconductivity below a temperature of about 15 K. In
samples with improved quality, three-dimensional (3D) superconducting behaviors were observed,
which display a sharp resistance drop around the temperature 7.5 K. We build a simple model to
explain this 1D-3D crossover. The system is inhomogeneous, and contains randomly situated
bundles of small CNTs. We use the Ginzburg-Landau (GL) model, with weak J...[

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This thesis contains two parts. The first part is about superconductivity in 4-Angstrom carbon
nanotubes (CNTs) embedded in linear, parallel pores of AFI zeolite crystals. In the second part, we
study the pseudogap problem in the BEC-BCS crossover.

The 4-Angstrom CNTs can be formed in the linear pores of AFI zeolite crystals. They exhibit
quasi one-dimensional (1D) fluctuation superconductivity below a temperature of about 15 K. In
samples with improved quality, three-dimensional (3D) superconducting behaviors were observed,
which display a sharp resistance drop around the temperature 7.5 K. We build a simple model to
explain this 1D-3D crossover. The system is inhomogeneous, and contains randomly situated
bundles of small CNTs. We use the Ginzburg-Landau (GL) model, with weak Josephson-coupling
between the bundles, to model the system. Monte Carlo (MC) simulations are employed to study the
superconducting behaviors of this model. Owing to the weak Josephson-coupling, there exists a
phase transition at a temperature around 7.5 K, which displays the signatures of the Berezinskii-Kosterlitz-Thouless (BKT) transition. Below this critical temperature, the phase fluctuations along
the c-axis of the bundles are suppressed, and the whole system gradually approaches complete
coherence as the temperature is lowered. This behavior is denoted a 1D to 3D crossover
superconducting transition. The weak Josephson-coupling does not significantly contribute to the
free energy of the system, thus the specific heat still exhibits quasi 1D characteristics, with a rounded
peak between 7.5 K and 15 K. The results of numerical simulations are in good agreement with the
experimental observations.

The BEC-BCS crossover has been studied for decades. It is still controversial about whether
there exists a pseudogap state above the critical temperature around the unitary limit, where Cooper
pairs exist but do not condense. In this thesis, we study the Fermi gas with contact interaction at
finite temperatures. Feynman diagrammatic method is employed to evaluate the self-energy in the
high-temperature virial expansion. We obtain the spectral function at the lowest non-trivial order. It
contains two terms. One is of order z (the fugacity) and the other is of order z

^{2}. The z-order term
makes the main contribution at finite momentum, while the z

^{2}-order term dominates at large
momentum, and gives the value of Tan's contact. The spectral function can be interpreted easily on
the BEC side and the BCS side, where the residual interaction is weak. However, it is not trivial to
extend this interpretation to the unitary limit. A double-peak structure emerges at small momentum
and it gradually evolves into one peak at large momentum. We attribute these features to the strong
scattering between the particles in the unitary limit. The second-order spectral function is evaluated
which includes the three-body physics. It is significantly different from the lowest-order and shows
that the three-body physics plays an important role. Similar to the lowest-order, it contains a z

^{2}-order term and a z

^{3}-order term. The z

^{3}-order term gives the value of Tan's contact. Our work shows
that considering just two-body physics (T

_{2}-matrix approximation) is not sufficient to conclude about
the pseudogap problem.

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