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arXiv:1508.01802 · Physical Review B

Exotic superconductivity through bosons in a dynamical cluster approximation

Thomas Bilitewski, Lode Pollet

Published Nov 10, 2015• 14 citations
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Abstract

We study the instabilities towards (exotic) superconductivity of mixtures of $\text{spin-}1/2$ fermions coupled to scalar bosons on a two-dimensional square lattice with the dynamical cluster approximation (DCA) using a numerically exact continuous-time Monte-Carlo solver. The Bogoliubov bosons provide an effective phononic bath for the fermions and induce a nonlocal retarded interaction between the fermions, which can lead to (exotic) superconductivity. Because of the sign problem the biggest clusters we can study are limited to $2\ifmmode\times\else\texttimes\fi{}2$ in size, but this nevertheless allows us to study the pairing instabilities, and their possible divergence, in the $s\ensuremath{-}$ and $d\text{-wave}$ channels as well as the competition with antiferromagnetic fluctuations. At fermionic half filling we find that the $d$ wave is stable when the mediated interaction by the bosons is of the same order as the bare fermionic repulsion. Its critical temperature can be made as high as the maximum one for $s$ wave, which opens perspectives for its detection in a cold atom experiment.

DOI
10.1103/physrevb.92.184505

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The cluster size dependence of superconductivity in the conventional two-dimensional Hubbard model, commonly believed to describe high-temperature superconductors, is systematically studied using the dynamical cluster approximation and quantum Monte Carlo simulations as a cluster solver. Because of the nonlocality of the d-wave superconducting order parameter, the results on small clusters show large size and geometry effects. In large enough clusters, the results are independent of the cluster size and display a finite temperature instability to d-wave superconductivity.
We investigate the quantum phases of mixed-dimensional cold atom mixtures. In particular, we consider a mixture of a Fermi gas in a two-dimensional lattice, interacting with a bulk Fermi gas or a Bose-Einstein condensate in a three-dimensional lattice. The effective interaction of the two-dimensional system mediated by the bulk system is determined. We perform a functional renormalization group analysis, and demonstrate that by tuning the properties of the bulk system, a subtle competition of several superconducting orders can be controlled among $s\ensuremath{-}\mathrm{wave}, p\ensuremath{-}\mathrm{wave}, {d}_{{x}^{2}\ensuremath{-}{y}^{2}}\ensuremath{-}\mathrm{wave}$, and ${g}_{xy({x}^{2}\ensuremath{-}{y}^{2})}\ensuremath{-}\mathrm{wave}$ pairing symmetries. Other instabilities such as a charge-density-wave order are also demonstrated to occur. In particular, we find that the critical temperature of the $d\ensuremath{-}\mathrm{wave}$ pairing induced by the next-nearest-neighbor interactions can be an order of magnitude larger than that of the same pairing induced by doping in the simple Hubbard model. We expect that by combining the nearest-neighbor interaction with the next-nearest-neighbor hopping (known to enhance $d\ensuremath{-}\mathrm{wave}$ pairing), an even higher critical temperature may be achieved.
We present the algorithmic details of the dynamical cluster approximation (DCA), with a quantum Monte Carlo (QMC) method used to solve the effective cluster problem. The DCA is a fully causal approach which systematically restores nonlocal correlations to the dynamical mean field approximation (DMFA) while preserving the lattice symmetries. The DCA becomes exact for an infinite cluster size, while reducing to the DMFA for a cluster size of unity. We present a generalization of the Hirsch-Fye QMC algorithm for the solution of the embedded cluster problem. We use the two-dimensional Hubbard model to illustrate the performance of the DCA technique. At half filling, we show that the DCA drives the spurious finite-temperature antiferromagnetic transition found in the DMFA slowly towards zero temperature as the cluster size increases, in conformity with the Mermin-Wagner theorem. Moreover, we find that there is a finite-temperature metal to insulator transition which persists into the weak-coupling regime. This suggests that the magnetism of the model is Heisenberg-like for all nonzero interactions. Away from half filling, we find that the sign problem that arises in QMC simulations is significantly less severe in the context of DCA. Hence, we were able to obtain good statistics for small clusters. For these clusters, the DCA results show evidence of non-Fermi-liquid behavior and superconductivity near half filling.
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