We investigate the superconductor-insulator phase transition in the two-dimensional boson Hubbard system with short-range interactions. Fluctuations of both the phase and the amplitude of the superfluid order parameter are included in the determination of the phase diagram at zero and finite temperatures. The mean-field phase boundary is compared to quantum Monte Carlo results. We also calculate the frequency-dependent conductivity in the vicinity of the phase transition and find it universal at the multicritical point.
We study the quantum phase transition between the insulating and the globally coherent superfluid phases in the Bose-Hubbard model with
T3 structure, the ``dice lattice.'' Even in the absence of any frustration the superfluid phase is characterized by modulation of the order parameter on the different sublattices of the
T3 structure. The zero-temperature critical point as a function of magnetic field shows the characteristic ``butterfly'' form. At full frustration the superfluid region is strongly suppressed. In addition, due to the existence of the Aharonov-Bohm cages at
f=1∕2, we find some evidence for the existence of an intermediate insulating phase characterized by a zero superfluid stiffness but finite compressibility. In this intermediate phase bosons are localized due to the external frustration and the topology of the
T3 lattice. We name this new phase the Aharonov-Bohm insulator. In the presence of charge frustration the phase diagram acquires the typical lobe structure. The form and hierarchy of the Mott insulating states with fractional fillings are dictated by the particular topology of the
T3 lattice. The results presented were obtained by a variety of analytical methods: mean-field and variational techniques to approach the phase boundary from the superconducting side and a strongly coupled expansion appropriate for the Mott insulating region. In addition we performed quantum Monte Carlo simulations of the corresponding
(2+1)-dimensional
XY model to corroborate the analytical calculations with a more accurate quantitative analysis. We finally discuss experimental realization of the
T3 lattice both with optical lattices and with Josephson junction arrays.
Despite its apparent simplicity, the two-dimensional Hubbard model for locally interacting fermions on a square lattice is widely considered as a promising approach for the understanding of Cooper pair formation in the quasi two-dimensional high-Tc cuprate materials. In the present work this model is investigated by means of the functional renormalization group, based on an exact flow equation for the effective average action. In addition to the fermionic degrees of freedom of the Hubbard Hamiltonian, bosonic fields are introduced which correspond to the different possible collective orders of the system, for example magnetism and superconductivity. The interactions between bosons and fermions are determined by means of the method of “rebosonization” (or “flowing bosonization”), which can be described as a continuous, scale-dependent Hubbard-Stratonovich transformation. This method allows an efficient parameterization of the momentumdependent effective two-particle interaction between fermions (four-point vertex), and it makes it possible to follow the flow of the running couplings into the regimes exhibiting spontaneous symmetry breaking, where bosonic fluctuations determine the types of order which are present on large length scales. Numerical results for the phase diagram are presented, which include the mutual influence of different, competing types of order.
Based on the two-band free-energy functional, we have investigated phase transitions of a three-dimensional granular superconductor through the Monte Carlo method and finite-size scaling analysis. In the ordered lattices for this superconductor consisting of Josephson-coupled grains, we obtain an intermediate phase above the critical intraband coupling strength while maintaining a constant interband coupling coefficient. This phase is distinct from both the low-temperature superconducting phase and the high-temperature normal phase, characterized by the absence of long-range phase coherence solely for the first band. This numerical prediction can be confirmed through magnetic susceptibility calculations, and the phase diagram is qualitatively consistent with the mean-field theoretical analysis. Additionally, we have considered the effects of interband interaction between the nearest-neighbor grains and random disorders in diluted lattices with three different site occupancies. When the occupation probability exceeds the percolation threshold, the results are qualitatively similar to those of ordered lattices, and the intermediate phase likewise appears by adjusting the intraband Josephson coupling coefficient. In contrast, below the percolation threshold, such a phase does not exist because the long-range phase coherence for both bands disappears.
We study the phase fluctuations in the normal state of generic two-dimensional superconducting systems with
s-wave pairing. The effect of phase fluctuations of the pairing fields can be dealt with perturbatively using disorder averaging, after we treat the local superconducting order parameter as a static disordered background. It is then confirmed that the phase fluctuations above the two-dimensional Berezinskii-Kosterlitz-Thouless transition lead to a significant broadening of the single-particle spectrum, giving birth to the pseudogap phenomenon. Quantitatively, the broadening of spectral weights near the BCS gap is characterized by the ratio of the superconducting coherence length and the spatial correlation length of the superconducting pairing order parameter. Our results are tested on the fermionic attractive-
U Hubbard model on the square lattice, using the unbiased determinant quantum Monte Carlo method and stochastic analytic continuation.
The two-dimensional Hubbard model is studied for small values of the interaction strength (
U of the order of the hopping amplitude
t), using a variational ansatz well suited for this regime. The wave function, a refined Gutzwiller ansatz, has a BCS mean-field state with
d-wave symmetry as its reference state. Superconducting order is found for densities
n<1 (but not for
n=1). This resolves a discrepancy between results obtained with the functional renormalization group, which do predict superconducting order for small values of
U, and numerical simulations, which did not find superconductivity for $U\ensuremath{\lesssim}4t$. Both the gap parameter and the order parameter have a domelike shape as a function of
n with a maximum for $n\ensuremath{\approx}0.8$. Expectation values for the energy, the particle number, and the superconducting order parameter are calculated using a linked-cluster expansion up to second order in
U. In this way large systems (millions of sites) can be readily treated and well converged results are obtained. A big size is indeed required to see that the gap becomes very small at half filling and probably tends to zero in the thermodynamic limit, whereas away from half filling a finite asymptotic limit is reached. For a lattice of a given size the order parameter becomes finite only above a minimal coupling strength
Uc. This threshold value decreases steadily with increasing system size, which indicates that superconductivity exists for arbitrarily small
U for an infinite system. For moderately large systems the size dependence is quite irregular, due to variations in level spacings at the Fermi energy. The fluctuations die out if the gap parameter spans several level spacings.