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arXiv:2506.20735

Bosonized theory of de Haas-van Alphen quantum oscillation in Fermi liquids

Yuxuan Wang

Published Jun 25, 2025• 0 citations
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Abstract

The de Haas-van Alphen effect (dHvA) of a 2d Fermi liquid remains poorly understood, due to the $\sim\mathcal{O}(1)$ contribution to the oscillations of grand potential from the oscillatory part of the fermionic self energy, which has no known closed-form solution. In this work, we solve this problem via coadjoint-orbit bosonization of the Fermi surface. Compared with the fermionic formalism, the issue of the oscillatory self energy is circumvented. As an effective field theory, Landau parameters $F_{n}$ directly enter the theory. We use the bosonized theory to derive the energies of cyclotron resonance and specific heat, which are consistent with Fermi liquid theory. Via a mode expansion, we show that the problem of dHvA is reduced to 0+1D quantum mechanics. We obtain analytic expressions for the behavior of dHvA at low and high temperatures, which deviate from the well-known Lifshitz-Kosevich formula. We contrast this behavior with that of 3d Fermi liquids, for which we show such deviations are parametrically small. We discuss the effects of disorder on dHvA within the bosonized theory.

DOI
10.48550/arxiv.2506.20735

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The de Haas-van Alphen effect (dHvA) of a 2d Fermi liquid remains poorly understood, due to the ∼O(1)\sim\mathcal{O}(1)∼O(1) contribution to the oscillations of grand potential from the oscillatory part of the fermionic self energy, which has no known closed-form solution. In this work, we solve this problem via coadjoint-orbit bosonization of the Fermi surface. Compared with the fermionic formalism, the issue of the oscillatory self energy is circumvented. As an effective field theory, Landau parameters FnF_{n}Fn​ directly enter the theory. We use the bosonized theory to derive the energies of cyclotron resonance and specific heat, which are consistent with Fermi liquid theory. Via a mode expansion, we show that the problem of dHvA is reduced to 0+1D quantum mechanics. We obtain analytic expressions for the behavior of dHvA at low and high temperatures, which deviate from the well-known Lifshitz-Kosevich formula. We contrast this behavior with that of 3d Fermi liquids, for which we show such deviations are parametrically small. We discuss the effects of disorder on dHvA within the bosonized theory.
We present a bosonized effective field theory for a 2d Fermi surface in a weak magnetic field using the coadjoint orbit approach, which was recently developed as a nonlinear bosonization method in phase space for Fermi liquids and non-Fermi liquids. We show that by parametrizing the phase space with the guiding center and the mechanical momentum, and by using techniques in noncommutative field theory, the physics of Landau levels and Landau level degeneracy (NΦN_ΦNΦ​) naturally arises. For a parabolic dispersion, the resulting theory describes NΦN_ΦNΦ​ flavors of free chiral bosons propagating in \emph{momentum space}. In addition, the action contains a linear term in the bosonic field, which upon mode expansion becomes a topological θθθ-term. By properly quantizing this theory, we reproduce the well-known thermal and magnetic responses of a Fermi surface, including linear-in-TTT specific heat, Landau diamagnetism, and the de Haas-van Alphen effect. {In particular, the de Haas-van Alphen effect is shown to be a direct consequence of the topological θθθ-term.} Our theory paves the way toward understanding correlated gapless fermionic systems in a magnetic field using the powerful approach of bosonization.
Novel controlled nonperturbative techniques are a must in the study of strongly correlated systems, especially near quantum criticality. One of these techniques, bosonization, has been extensively used to understand one-dimensional, as well as higher-dimensional electronic systems at finite density. In this paper, we generalize the theory of two-dimensional bosonization of Fermi liquids, in the presence of a homogeneous weak magnetic field perpendicular to the plane. Here, we extend the formalism of bosonization to treat free spinless fermions at finite density in a uniform magnetic field. We show that particle-hole fluctuations of a Fermi surface satisfy a covariant Schwinger algebra, allowing to express a fermionic theory with forward scattering interactions as a quadratic bosonic theory representing the quantum fluctuations of the Fermi surface. By means of a coherent-state path integral formalism, we compute the fermion propagator as well as particle-hole bosonic correlations functions. We analyze the presence of de Haas-van Alphen oscillations and show how the quantum oscillations of the orbital magnetization, the Lifshitz-Kosevich theory, are obtained by means of the bosonized theory. We also study the effects of forward scattering interactions. In particular, we obtain oscillatory corrections to the Landau zero sound collective mode.
In this paper we investigate magneto-oscillations in the specific heat and the magnetization of nonrelativistic fermions coupled to a fluctuating U(1)U(1)U(1) gauge field. This model obtains as an effective model for the underdoped cuprates realizing a so-called ``algebraic charge liquid,'' which is a true non-Fermi-liquid state. Our study is driven by very recent observations of quantum oscillations in the underdoped cuprates. We calculate corrections to the standard Lifshitz-Kosevich expression due to the internal gauge degree of freedom for the oscillation amplitude. We perform this calculation in the dirty limit in a model with N\mathcal{N}N species of fermions. The $\mathcal{N}\ensuremath{\rightarrow}\ensuremath{\infty}$ result corresponds to the well-known Fermi-liquid result reproducing the Lifshitz-Kosevich result. We capture the effect of the gauge field on the oscillation amplitude to Gaussian accuracy controlled in the small parameter 1/N1/\mathcal{N}1/N. Our main finding is the presence of qualitative and quantitative differences compared to standard Fermi-liquid theory.
A simple model of spinless fermions with nested Fermi surface leading to a charge density wave is considered. A quantum critical point (QCP) is obtained by tuning the nesting mismatch of the Fermi surface so that the critical temperature is zero. The amplitudes of de Haas-van Alphen oscillations are calculated using a Lifshitz-Kosevich expression modified by the quasi-particle selfenergy. The amplitudes are considerably reduced by the interaction. The effects of the QCP extend over a large temperature interval.
Since the Wigner function (WF) is related to a Lindard-constant type linear dielectric function derived in the symmetric gauge,7it is expected to show de Haas-van Alphen (dHvA) oscillations. Starting with the symmetric eigenfunctions, we derived the pure-state WF in a magnetic field, whose plots in phase space and in term of B-1for increasing n are consistent with the dHvA effect. Furthermore the asymptotic expansion of WF at large n show periodic oscillations with a period related to the Fermi energy. The phase space plots of WF also show that dHvA and similar oscillations could be a consequence of Nature's strategy for increasing the effective spatial range without violating the uncertainty principle. Properties of the symmetric eigenfunctions were derived. The dynamics of WF can be obtained from the solution of the time-dependent Schrödinger equation (SE). A new method to solve the SE in a magnetic field in the interaction picture based on expansion in term of symmetric eigenfunctions has been developed. The matrix element for a Gaussian potential were derived explicitly, plotted against B-1, and showed oscillations. The total WF was shown to be a linear combination of the diagonal pure-state WF's by using the orthogonality for symmetric eigenfunctions. The no-special-point property for WF was confirmed, which is important for the construction of a numerical algorithm to solve the SE in a magnetic field.
We have developed a novel quantum transfer matrix method to study thermodynamic properties of the Hofstadter model on square lattices. This method overcomes the finite size effect and allows the magnetization and other thermodynamic quantities to be accurately and efficiently calculated. The interplay between the lattice potential and the applied magnetic field leads to substantial corrections to the conventional de Haas-van Alphen (dHvA) oscillation of magnetization. It yields many paramagnetic peaks or diamagnetic deeps above the background of dHvA oscillation. This correction bears the fractal feature of energy spectra and can be used to probe the hierarchical structure of Hofstadter butterfly by experimental measurements.