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arXiv:cond-mat/0409368 · Physical Review B

Lifshitz transition in d -wave superconductors

Sergio S. Botelho, C. A. R. Sá de Melo

Published Apr 13, 2005• 28 citations
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Abstract

The BCS-to-BEC evolution has been recently the focus of studies in superconductors and cold atomic gases. For a $d$-wave system, we show that a Lifshitz transition occurs at a critical particle density which separates two topologically distinct phases according to their quasiparticle excitation energies: a BCS-like gapless superconductor in the higher-density limit and a BEC-like fully gapped superconductor in the lower-density limit. This transition is second-order according to Ehrenfest's classification, but it occurs without a change in the symmetry of the order parameter and thus cannot be classified under Landau's scheme. To illustrate the nature of the transition, we compute the compressibility and the superfluid density as functions of particle density.

DOI
10.1103/physrevb.71.134507

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The BCS-to-BEC evolution has been recently the focus of studies in superconductors and cold atomic gases. For a ddd-wave system, we show that a Lifshitz transition occurs at a critical particle density which separates two topologically distinct phases according to their quasiparticle excitation energies: a BCS-like gapless superconductor in the higher-density limit and a BEC-like fully gapped superconductor in the lower-density limit. This transition is second-order according to Ehrenfest's classification, but it occurs without a change in the symmetry of the order parameter and thus cannot be classified under Landau's scheme. To illustrate the nature of the transition, we compute the compressibility and the superfluid density as functions of particle density.
The low-temperature evolution from BCS to Bose-Einstein condensation (BEC) for a two-dimensional d-wave superconductor is discussed at the saddle point (mean field) level. A systematic study of the changes of low-temperature thermodynamic properties is presented as a function of the charge carrier density and fixed interaction. It has been found that when the interaction strength is large enough, there is a critical density below which the single quasiparticle excitation spectrum develops a gap. At higher density of carriers and lower interaction strength (towards the BCS regime) the superconductor has gapless quasiparticle excitations, while at lower densities and higher interactions (towards the BEC regime) quasiparticle excitations are fully gapped. The appearence of a full gap in the quasiparticle excitation spectrum has dramatic consequences to the compressibility, specific heat, and spin susceptibility at low temperatures, as the critical density nc{n}_{c}nc​ is crossed. The change in behavior of these quantities indicates a possible quantum phase transition between a d-wave gapless phase and a d-wave fully gapped phase, as nc{n}_{c}nc​ is crossed.
The problem of the evolution from BCS theory with cooperative Cooper pairing to the formation and condensation of composite bosons has attracted considerable attention for the past several decades. It has gained renewed impetus in the mid-eighties with the discovery of the high-Tc superconductors, which have a coherence length comparable to the interparticle spacing. More recently, this subject has spurred a great deal of research activity in connection with experiments involving dilute atomic gases of fermionic atoms. The initial objective of this work will be to use functional integral techniques to analyze the low-temperature BCS-to-BEC evolution of d-wave superconductors within the saddle point (mean field) approximation for a continuum model. Then, the same mathematical formalism will be applied to the problem of the BCS-to-BEC evolution of fully spin-polarized p-wave Fermi gases in two dimensions. We find that a quantum phase transition occurs for both systems as they are driven from the BCS-like regime of weakly interacting fermionic pairs to the opposite BEC-like regime of strongly interacting bosonic molecules. This is in contrast to the smooth crossover predicted and observed in systems that exhibit s-wave pairing symmetry. We calculate several spectroscopic and thermodynamic properties that signal the occurrence of this phase transition, and suggest some possible experimental realizations. Finally, fluctuations about the saddle point solution are included in the calculations, and the effects of such correction are analyzed in the low (T~0) and high (T~Tc) temperature limits. We conclude that, at high temperatures, the bosonic degrees of freedom that arise from two-particle bound states become essential to describe the strong coupling limit,\nas the saddle point approximation alone becomes unreliable.
We review aspects of the evolution from Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensation (BEC) in two dimensions, which have now become relevant in systems with low densities, such as gated superconductors Li xZrNCl, magic-angle twisted trilayer graphene, FeSe, FeSe1− xS x, and ultracold Fermi superfluids. We emphasize the important role played by chemical potentials in determining crossovers or topological quantum phase transitions during the BCS–BEC evolution in one-band and two-band superfluids and superconductors. We highlight that crossovers from BCS to BEC occur for pairing in nonnodal s-wave channels, whereas topological quantum phase transitions, in which the order parameter symmetry does not change, arise for pairing in any nodal higher angular momentum channels, such as d-wave. We conclude by discussing a few open questions regarding the BCS-to-BEC evolution in 2D, including modulus fluctuations of the order parameter, tighter upper bounds on critical temperatures, and the exploration of lattice effects in two-band superconductors and superfluids.
We consider the evolution of superfluid properties of a three-dimensional p-wave Fermi gas from a weak coupling Bardeen-Cooper-Schrieffer (BCS) to strong coupling Bose-Einstein condensation (BEC) limit as a function of scattering volume. At zero temperature, we show that a quantum phase transition occurs for p-wave systems, unlike the s-wave case where the BCS to BEC evolution is just a crossover. Near the critical temperature, we derive a time-dependent Ginzburg-Landau (GL) theory and show that the GL coherence length is generally anisotropic due to the p-wave nature of the order parameter, and becomes isotropic only in the BEC limit.
We discuss topological phase transitions in ultracold Fermi superfluids induced by interactions and artificial spin-orbit fields. We construct the phase diagram for population-imbalanced systems at zero and finite temperatures, and analyze spectroscopic and thermodynamic properties to characterize various phase transitions. For balanced systems, the evolution from Bardeen-Cooper-Schrieffer to Bose-Einstein-condensate superfluids in the presence of spin-orbit effects is only a crossover as the system remains fully gapped, even though a triplet component of the order parameter emerges in the helicity basis. However, for imbalanced populations, spin-orbit fields induce a triplet component for the order parameter in the helicity basis, and may produce nodes in the quasiparticle excitation spectrum leading to bulk topological phase transitions of the Lifshitz type. Additionally a fully gapped phase exists, where a crossover from indirect to direct gap occurs, but a topological transition to a gapped phase possessing Majorana fermion edge states does not occur.
The evolution from BCS to BEC superconductivity in the s-wave and d-wave channels is analyzed at zero temperature for a two-dimensional superconductor. Spectroscopic quantities for s-wave and d-wave systems present fundamental differences when particle density and attraction strength are varied. A detailed analysis of single quasiparticle properties (excitation spectrum, momentum distribution, spectral function and density of states) indicates that the evolution of these spectroscopic quantities in the d-wave case is not smooth, unlike the situation encountered for the s-wave system.