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Register a new userBoundary states with elevated critical temperatures in Bardeen-Cooper-Schrieffer superconductors
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Bardeen-Cooper-Schrieffer (BCS) theory describes a superconducting transition as a single critical point where the gap function or, equivalently, the order parameter vanishes uniformly in the entire system. We demonstrate that in superconductors described by standard BCS models, the superconducting gap survives near the sample boundaries at higher temperatures than superconductivity in the bulk. Therefore, conventional superconductors have multiple critical points associated with separate phase transitions at the boundary and in the bulk. We show this by revising the Caroli-De Gennes-Matricon theory of a superconductor-vacuum boundary and finding inhomogeneous solutions of the BCS gap equation near the boundary, which asymptotically decay in the bulk. This is demonstrated for a BCS model of almost free fermions and for lattice fermions in a tight-binding approximation. The analytical results are confirmed by numerical solutions of the microscopic model. The existence of these boundary states can manifest itself as discrepancies between the critical temperatures observed in calorimetry and transport probes.
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We propose the $ThetaPhi$ (Theta-Phi) package which addresses two of the most important extensions of the essentially single-particle mean-field paradigm of the computational solid state physics: the admission of the Bardeen-Cooper-Schrieffer electronic ground state and allowance of the magnetically ordered states with an arbitrary superstructure (pitch) wave vector. Both features are implemented in the context of multi-band systems which paves the way to an interplay with the solid state quantum physics packages eventually providing access to the first-principles estimates of the relevant matrix elements of the model Hamiltonians derived from the standard DFT calculations. Several examples showing the workability of the proposed code are given.
Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a $p$-wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling factors $ u=1/2$ and $ u=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cut-off energy of the BCS theory. Our results show no evidence for pairing at $ u=1/2$ but a clear evidence for pairing at $ u=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $ u=5/2$ and the BCS state is obtained for parameters that minimize the energy of the latter, thereby providing support for the physics of composite-fermion pairing as the mechanism for the $5/2$ fractional quantum Hall effect. We discuss the issue of modular covariance of the composite-fermion BCS wave function, and calculate its Hall viscosity and pair correlation function. By similar methods, we look for but do not find an instability to $s$-wave pairing for a spin-singlet composite-fermion Fermi sea at half-filled lowest Landau level in a system where the Zeeman splitting has been set to zero.
We determine the exact dynamics of an initial Bardeen-Cooper-Schrieffer (BCS) state of ultra-cold atoms in a deep hexagonal optical lattice. The dynamical evolution is triggered by a quench of the lattice potential, such that the interaction strength $U_f$ is much larger than the hopping amplitude $J_f$. The quench initiates collective oscillations with frequency $|U_f|/(2pi)$ in the momentum occupation numbers and imprints an oscillating phase with the same frequency on the BCS order parameter $Delta$. The oscillation frequency of $Delta$ is not reproduced by treating the time evolution in mean-field theory. In our theory, the momentum noise (i.e. density-density) correlation functions oscillate at frequency $|U_f|/2pi$ as well as at its second harmonic. For a very deep lattice, with zero tunneling energy, the oscillations of momentum occupation numbers are undamped. Non-zero tunneling after the quench leads to dephasing of the different momentum modes and a subsequent damping of the oscillations. The damping occurs even for a finite-temperature initial BCS state, but not for a non-interacting Fermi gas. Furthermore, damping is stronger for larger order parameter and may therefore be used as a signature of the BCS state. Finally, our theory shows that the noise correlation functions in a honeycomb lattice will develop strong anti-correlations near the Dirac point.
Starting from H. Frohlichs second-quantized Hamiltonian for a $d$-dimensional electron gas in interaction with lattice phonons describing the quantum vibrations of a metal, we present a rigorous mathematical derivation of the superconducting state, following the principles laid out originally in 1957 by J. Bardeen, L. Cooper and J. Schrieffer. As in the series of papers written on the subject in the 90es, of which the present paper is a continuation, the representation of ions as a uniform charge background allows for a $(1+d)$-dimensional fermionic quantum-field theoretic reformulation of the model at equilibrium. For simplicity, we restrict in this article to $d=2$ dimensions and zero temperature, and disregard effects due to electromagnetic interactions. Under these assumptions, we prove transition from a Fermi liquid state to a superconducting state made up of Cooper pairs of electrons at an energy level $Gamma_{phi}sim hbaromega_D e^{-pi/mlambda}$ equal to the mass gap, expressed in terms of the Debye frequency $omega_D$, electron mass $m$ and coupling constant $lambda$. The dynamical $U(1)$-symmetry breaking produces at energies lower than the energy gap $Gamma_{phi}$ a Goldstone boson, a non-massive particle described by an effective $(2+1)$-dimensional non-linear sigma-model, whose parameters and correlations are computed. The proof relies on a mixture of general concepts and tools (multi-scale cluster expansions, Ward identities), adapted to this quantum many-body problem with its extended infra-red singularity located on the Fermi circle, and a specific $1/N$-expansion giving the leading diagrams at intermediate energies. Ladder diagrams are proved to provide the leading behavior in the infra-red limit, in agreement with mean-field theory predictions.
Shortly after the Gorkovs microscopic derivation of Ginzburg-Landau model via a small order parameter expansion in BCS theory, the derivation was carried to next-to-leading order in that parameter and its spatial derivatives. The aim was to obtain a generalized Ginzburg-Landau free energy that approximates the microscopic model better. We prove that the resulting extended Ginzburg-Landau functional does not support a superconducting state since it does not have any solutions in the form of free energy minima.
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