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Full Title

Pairing and Pair Breaking in Bilayer Composite Fermion Metals

Names

Mendoza, Luis (author)

Bonesteel, N. E. (professor directing dissertation)

Aldrovandi, Ettore (university representative)

Reina, Laura (committee member)

Yang, Kun, 1967- (committee member)

Engel, Lloyd W. (committee member)

Florida State University (degree granting institution)

College of Arts and Sciences (degree granting college)

Department of Physics (degree granting department)

Format

text

doctoral thesis

Abstract

The Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity is one of the jewels of 20th century physics. In this theory, the superconducting state of a metal is understood in terms of the formation of electron pairs which condense into a macroscopic quantum state that can carry electrical current without resistance. The theory is built upon another jewel, Landau Fermi liquid theory, which implies that even strongly interacting fermions (such as electrons in a metal) can, at low energies, behave like weakly interacting ``quasiparticles". According to BCS theory, if these quasiparticles experience an attractive force, characterized by dimensionless coupling constant $\lambda$, they will form pairs with a binding energy (or energy gap) given by $\Delta \sim \omega_0 e^{-1/\lambda}$ where $\omega_0$ is a characteristic energy scale. The fact that this binding energy depends on the inverse coupling constant $1/\lambda$ in the exponent suggests that this pairing is very sensitive to the strength of the interaction, consistent with experimental observation. The quantum Hall effect, a {\it third} 20th century jewel, occurs when a two-dimensional gas of electron is placed in a magnetic field and cooled to very low temperature. For certain field strengths the electrons can form states with an energy gap. These states then exhibit a precisely quantized Hall resistance and vanishing longitudinal resistance. For other field strengths, the system is gapless, and behaves in many ways like a metal in zero magnetic field. Much of this phenomena can be understood in terms of new particles, known as composite fermions, which consist of electrons bound to an even number of flux quanta. These new particles experience an effective magnetic field that is different than the applied field and, for the case of a composite fermion metal, this effective field is zero. There is, however, a price to be paid for this effective field. The composite fermions also experience a strongly fluctuating $U(1)$ gauge field. The bilayer composite fermion metal, i.e. two parallel composite fermion metals that interact with each other via the Coulomb repulsion, is a rich system in which all three of these jewels meet, and which has been studied both experimentally and theoretically for more than 30 years. In this system the composite fermions are predicted to experience {\it two} gauge fields, one associated with each layer. These gauge fields naturally organize themselves into in-phase and out-of-phase components. A clear prediction of the theory is that the out-of-phase fluctuations are stronger and hence should dominate the low-energy physics. One effect of these out-of-phase fluctuations is that they lead to a breakdown of Fermi liquid theory. Another is that they lead to a strong attractive interaction between composite fermions in different layers leading to the formation of interlayer pairs. Because of the singular nature of this interaction and the non-Fermi liquid nature of the system, the BCS result for pairing is no longer applicable. Instead, the binding energy of the interlayer pairs is predicted to be $\Delta \sim \omega_0 (l_0/d)^2$, where $d$ is the spacing between layers, $l_0$ is the so-called magnetic length, which is a measure of the size of the wave function of individual electrons, and $\omega_0$ is again a typical energy scale. The fact that this gap depends as a power law on $l_0/d$, which plays the role of a dimensionless coupling constant here, and not an essential singularity, as in BCS theory, suggests this pairing should be significantly more robust than conventional superconductivity. But, to date, there is no experimental evidence such a paired state of composite fermions exists in bilayer composite fermion metals at large layer spacing. The mystery of why this is one of the main motivators of this Dissertation. Here we address this question by including in our study of the bilayer not just the pairing out-of-phase fluctuations, but also the {\it pair-breaking} in-phase fluctuations. While the in-phase fluctuations are significantly less singular than the the out-of-phase fluctuations, they nonetheless can have a significant effect. We study this effect using a variety of approaches. First we carry out a renormalization group analysis and find that, while the repulsive gauge field suppresses pairing, it never leads to the destruction of the paired state. We then present numerical solutions of the full $T=0$ Eliashberg equations (coupled integral equations which can describe pairing even in systems with strong enough interactions to lead to the breakdown of Fermi liquid theory) which are broadly consistent with the renormalization group results. Finally, in the central result of the Dissertation, we link these two approaches and find an analytic solution to the Eliashberg equations (within the so-called ``local approximation", which can be used to justify the renormalization group approach) which includes both the out-of-phase pairing interaction and the in-phase pair-breaking interaction. We find that the pair-breaking fluctuations neither close the energy gap, nor change the dependence of this gap on the inverse of the layer spacing squared. Rather, the pair-breaking fluctuations lead to a strongly suppressive prefactor in the expression for the energy gap. Our approach provides, for the first time, an analytic expression for this prefactor from which it can be seen that this effect is large enough to possibly account for the fact that this paired state hasn't yet been seen in experiment.

Keywords

non-Fermi liquids, pairing instability, quantum Hall effect

Date of Defense

April 9, 2020.

Submitted Note

A Dissertation submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Bibliography Note

Includes bibliographical references.

Advisory Committee

Nicholas E. Bonesteel, Professor Directing Dissertation; Ettore Aldrovandi, University Representative; Laura Reina, Committee Member; Kun Yang, Committee Member; Lloyd Engel, Committee Member.

Publisher

Florida State University

Identifier

2020_Spring_Mendoza_fsu_0071E_15842

Mendoza, L. (no date). Pairing and Pair Breaking in Bilayer Composite Fermion Metals. Retrieved from https://purl.lib.fsu.edu/diginole/2020_Spring_Mendoza_fsu_0071E_15842