Entropy and de Haas--van Alphen oscillations of a three-dimensional marginal Fermi liquid

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Entropy and de Haas–van Alphen oscillations of a three-dimensional marginal Fermi liquid

P. A. Nosov, Yi-Ming Wu, and S. Raghu

Phys. Rev. B 109, 075107 – Published 5 February 2024

Abstract

We study de Haas-van Alphen oscillations in a marginal Fermi liquid resulting from a three-dimensional metal tuned to a quantum-critical point (QCP). We show that the conventional approach based on extensions of the Lifshitz-Kosevich formula for the oscillation amplitudes becomes inapplicable when the correlation length exceeds the cyclotron radius. This breakdown is due to (i) nonanalytic finite-temperature contributions to the fermion self-energy, (ii) an enhancement of the oscillatory part of the self-energy by quantum fluctuations, and (iii) nontrivial dynamical scaling laws associated with the quantum critical point. We properly incorporate these effects within the Luttinger-Ward-Eliashberg framework for the thermodynamic potential by treating the fermionic and bosonic contributions on equal footing. As a result, we obtain the modified expressions for the oscillations of entropy and magnetization that remain valid in the non-Fermi-liquid regime.

Received 11 November 2023
Accepted 17 January 2024

DOI:https://doi.org/10.1103/PhysRevB.109.075107

Physics Subject Headings (PhySH)

Quantum criticality de Haas-van Alphen effect

Strongly correlated systems

Eliashberg theory Non-Fermi-liquid theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

P. A. Nosov ^1,2, Yi-Ming Wu¹, and S. Raghu^1,3

¹Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA
²Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA
³Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

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Vol. 109, Iss. 7 — 15 February 2024

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Images

Figure 1
The temperature dependence of the DHVA oscillation amplitude $A_{1} (T)$ in the undamped ( $z = 1$ ) and overdamped ( $z = 3$ ) cases.
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Figure 2
Numerical evaluation of the first oscillation amplitude $A_{1} (T)$ and its temperature derivative $\partial A_{1} / \partial T$ (inset), based on the extended LK formula Eq. (25). For comparison, two different expressions for the fermionic self-energy are used: The full fermion self-energy Eq. (B11), and the marginal Fermi-liquid self-energy in the absence of any thermal effects $Σ (ɛ_{m}) = {\bar{g}}^{2} ɛ_{m} (1 + ln ω_{D} / | ɛ_{m} |)$ . We note that $A_{1} (T = 0)$ is the same for both cases. The parameters used are ${\bar{g}}^{2} = 6, Λ_{D} = 10 ω_{c}, v_{F} / c = 100$ .
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Figure 3
(a) Imaginary part of the full self-energy (‘‘smooth” and ‘‘oscillatory” parts combined) evaluated at the extremal orbit $k_{z} = 0$ . The dashed line indicates the marginal FL self-energy at $B = 0$ . (b) Imaginary part of the ‘‘oscillatory” contribution to the self-energy averaged over all values of $k_{z}$ . Here ${〈 . . . 〉}_{k_{z}}$ stands for $\int d k_{z} . . . / 2 k_{F}$ . In both (a) and (b), the parameters are $ω_{c} / μ = 200$ and $ω_{D} / ω_{c} = 20$ , with two different values of the coupling constant ${\bar{g}}^{2}$ .
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Figure 4
(Left) Numerical evaluation of the imaginary part of the full self-energy at extremal orbit $k_{z} = 0$ in the overdamped case, based on Eq. (80), with $μ / ω_{c} = 200.6, Λ_{D} / ω_{c} = 10$ . (Right) The same but on a log-log scale, with emphasized asymptotic ${\sim | Ω |}^{2 / 3}$ behavior for $| Ω | ≪ ω_{c}$ [cf. Eq. (82)].
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Figure 5
(Left) In the ME theory at $B = 0$ , typical transferred momentum is predominantly transverse to the Fermi surface due to the $z = 3$ overdamped dynamics of the bosons. This leads to a $\sim | Ω |$ scaling of the imaginary part of the self-energy for all momentum states on the Fermi surface. (Right) At finite $B$ , only the states in the width $\sim \sqrt{ω_{c} / μ}$ around the extremal orbit $k_{z} = 0$ contribute to the oscillations. Thus, the oscillatory part of the self-energy at the external momenta $k_{z} = 0$ is contributed only by the scattering processes that do not take the fermion away from $k_{z} \approx 0$ in the intermediate state. This condition makes the bosonic kinematics essentially two dimensional, leading to the ${| Ω |}^{2 / 3}$ scaling.
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Figure 6
Self-energy obtained in different ways at $T = 2 \times 10^{- 4}$ . All the energy scales are measured in units of $g^{2}$ , and we have made the dimensionless coupling ${\bar{g}}^{2} = 1$ . Here the exact result (blue) is obtained by directly solving Eq. (B1) in the limit of $m_{b} \to 0$ and neglecting $Π$ . The first asymptotic result (red) is obtained from numerically solving Eq. (B14), which does not require iteration. And the second asymptotic result (brown) is from Eq. (B19).
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