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arXiv:1211.2470 · Physical Review A

BCS-BEC quantum phase transition and collective excitations in two-dimensional Fermi gases with p - and d -wave pairings

Gaoqing Cao, Lianyi He, Pengfei Zhuang

Published Jan 14, 2013• 10 citations
arXivHTMLPDF

Abstract

It is generally believed that the BCS-BEC evolution in fermionic systems with $s$-wave pairing is a smooth crossover. However, for nonzero orbital-angular-momentum pairing such as $p$- or $d$-wave pairing, the system undergoes a quantum phase transition at the point where the chemical potential $\ensuremath{\mu}$ vanishes. In this paper, we study the BCS-BEC quantum phase transition and the collective excitations associated with the order-parameter fluctuations in two-dimensional fermionic systems with $p$- and $d$-wave pairings. We show that the quantum phase transition in such systems can be generically traced back to the infrared behavior of the fermionic excitation at $\ensuremath{\mu}=0$: ${E}_{\mathbf{k}}\ensuremath{\sim}{k}^{l}$, where $l=1,2$ is the quantum number of the orbital angular momentum. The nonanalyticity of the thermodynamic quantities is due to the infrared divergence caused by the fermionic excitation at $\ensuremath{\mu}=0$. As a result, the evolution of the Anderson-Bogoliubov mode is not smooth: Its velocity is nonanalytical across the quantum phase transition.

DOI
10.1103/physreva.87.013613

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It is generally believed that the BCS-BEC evolution in fermionic systems with sss-wave pairing is a smooth crossover. However, for nonzero orbital-angular-momentum pairing such as ppp- or ddd-wave pairing, the system undergoes a quantum phase transition at the point where the chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ vanishes. In this paper, we study the BCS-BEC quantum phase transition and the collective excitations associated with the order-parameter fluctuations in two-dimensional fermionic systems with ppp- and ddd-wave pairings. We show that the quantum phase transition in such systems can be generically traced back to the infrared behavior of the fermionic excitation at \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=0: Ek\ensuremath∼kl{E}_{\mathbf{k}}\ensuremath{\sim}{k}^{l}Ek​\ensuremath∼kl, where l=1,2l=1,2l=1,2 is the quantum number of the orbital angular momentum. The nonanalyticity of the thermodynamic quantities is due to the infrared divergence caused by the fermionic excitation at \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=0. As a result, the evolution of the Anderson-Bogoliubov mode is not smooth: Its velocity is nonanalytical across the quantum phase transition.
We present a theoretical study of the finite-temperature Kosterlitz-Thouless (KT) and vortex-antivortex lattice (VAL) melting transitions in two-dimensional Fermi gases with ppp- or ddd-wave pairing. For both pairings, when the interaction is tuned from weak to strong attractions, we observe a quantum phase transition from the Bardeen-Cooper-Schrieffer (BCS) superfluidity to the Bose-Einstein condensation (BEC) of difermions. The KT and VAL transition temperatures increase during this BCS-BEC transition and approach constant values in the deep BEC region. The BCS-BEC transition is characterized by the nonanalyticities of the chemical potential, the superfluid order parameter, and the sound velocities as functions of the interaction strength at both zero and finite temperatures; however, the temperature effect tends to weaken the nonanalyticities compared to the zero-temperature case. The effect of mismatched Fermi surfaces on the ddd-wave pairing is also studied.
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.
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.
Abstract We investigate quantum fluctuation effects in a two-dimensional Fermi superfluid with dx2-y2d_{x^2\textrm{-}y^2}dx2-y2​-wave pairing through an evolution from Bardeen-Cooper-Schrieffer (BCS) limit to Bose-Einstein condensate (BEC) limit. Using the Gaussian pair fluctuation (GPF) theory beyond the mean-field approximation, we derive a modified particle number equation and solve it self-consistently with the gap equation. We find that key physical properties, including the equation of state, the compressibility, and the sound velocity, are strongly renormalized by quantum fluctuations, reflecting their crucial role near the phase transition. Both mean-field and GPF theory reveal a pronounced Lifshitz transition at zero chemical potential μ=0\mu=0μ=0, but occurring at different interaction strengths. As the system evolves from the BCS to BEC regime, the fluctuation-induced particle number grows rapidly and saturates near the transition point, further highlighting the importance of quantum fluctuations. We also analyze the quasiparticle excitation spectrum, which exhibits a transition from a nodal BCS regime with characteristic C4C_4C4​ symmetry to a fully gapped BEC regime, providing direct insight into the interplay between pairing symmetry, phase transition of spectrum topology, and quantum fluctuations.
We analyze the evolution of superfluidity for nonzero orbital angular momentum channels from the Bardeen-Cooper-Schrieffer (BCS) to the Bose-Einstein condensation (BEC) limit in three dimensions. First, we analyze the low-energy scattering properties of finite range interactions for all possible angular momentum channels. Second, we discuss ground-state (T=0)(T=0)(T=0) superfluid properties including the order parameter, chemical potential, quasiparticle excitation spectrum, momentum distribution, atomic compressibility, ground-state energy, and low-energy collective excitations. We show that a quantum phase transition occurs for nonzero angular momentum pairing, unlike the sss-wave case where the BCS to BEC evolution is just a crossover. Third, we present a Gaussian fluctuation theory near the critical temperature (T=Tc)(T={T}_{\mathrm{c}})(T=Tc​), and we analyze the number of bound, scattering, and unbound fermions as well as the chemical potential. Finally, we derive the time-dependent Ginzburg-Landau functional near Tc{T}_{\mathrm{c}}Tc​, and compare the Ginzburg-Landau coherence length with the zero-temperature average Cooper pair size.
This thesis focuses on the analysis of Bardeen-Cooper-Schrieffer (BCS) \nto Bose-Einstein condensation (BEC) evolution in ultracold superfluid \nFermi gases when the interaction between atoms is varied. The tuning of \nattractive interactions permits the ground state of the system to evolve from a weak \nfermion attraction BCS limit of loosely bound and largely overlapping Cooper pairs \nto a strong fermion attraction limit of tightly bound small bosonic molecules \nwhich undergo BEC. \nThis evolution is accompanied by anomalous behavior of many superfluid \nproperties, and reveals several quantum phase transitions. \nThis thesis has two parts: \nIn the first part, I analyze zero and nonzero orbital angular momentum \npairing effects, and show that a quantum phase transition occurs for \nnonzero angular momentum pairing, unlike the sss-wave case where the BCS \nto BEC evolution is just a crossover. \nIn the second part, I analyze two-species fermion mixtures with mass and \npopulation imbalance in continuum, trap and lattice models. In contrast with \nthe crossover physics found in the mass and population balanced mixtures, \nI demonstrate the existence of phase transitions between normal and \nsuperfluid phases, as well as phase separation between superfluid (paired) \nand normal (excess) fermions in imbalanced mixtures as a function of scattering \nparameter and mass and population imbalance.