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arXiv:0910.4917 · Physical Review B

de Haas–van Alphen oscillations for nonrelativistic fermions coupled to an emergent U(1) gauge field

Lars Fritz, Subir Sachdev A

Published Jul 26, 2010• 4 citations
arXivHTMLPDF

Abstract

In this paper we investigate magneto-oscillations in the specific heat and the magnetization of nonrelativistic fermions coupled to a fluctuating $U(1)$ 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 $\mathcal{N}$ 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 $1/\mathcal{N}$. Our main finding is the presence of qualitative and quantitative differences compared to standard Fermi-liquid theory.

DOI
10.1103/physrevb.82.045123

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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 N\ensuremath→\ensuremath∞\mathcal{N}\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ 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.
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 study quantum oscillations for a system of fermionic and bosonic dimers and compare the results to those experimentally observed in the cuprate superconductors in their underdoped regime. We argue that the charge carriers obey the Onsager quantization condition and quantum oscillations take on a Lifshitz-Kosevich form. We obtain the effective mass and find good qualitative agreement with experiments if we tune the model to the point where the observed mass divergence at optimum doping is associated to a van Hove singularity at which four free-dimer Fermi pockets touch pairwise in the interior of the Brillouin zone. The same van Hove singularity leads to a maximum in the ddd-wave superconducting pairing amplitude when antiferromagnetic interactions are included. Our combined results therefore suggest that a quantum critical point separating the underdoped and overdoped regimes is marked by the location of the van Hove saddle point in the fermionic dimer dispersion.
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In this paper, we study the superconductivity of nonrelativistic fermions at finite-density coupled to a transverse U(1)U(1)U(1) gauge field, with the effective interaction including the Landau-damping. This model, first studied by Holstein, Norton, and Pincus [Phys. Rev. B 8, 2649 (1973)] has been known as an example of a non-Fermi liquid, i.e., a metallic state in which the decay rate of a quasiparticle is large compared to the characteristic quasiparticle energy; other examples of the non-Fermi liquid includes the two dimensional (2d) electron gas in a magnetic field at $\ensuremath{\nu}=1/2$ and the normal state of optimally doped cuprate superconductors. Our study thus addresses the question of whether or not non-Fermi liquids, like Fermi liquids, are unstable towards the formation of superconductivity. The results are (i) the non-Fermi liquid is stable against superconductivity below a critical gauge coupling, (ii) above this critical coupling, the ground state is an unconventional superconductor with angular momentum $\ensuremath{\ell}\ensuremath{\ge}2$. Our results are obtained from a solution of the Dyson-Nambu equation. Note that in this problem there is a quantum critical point between a non-Fermi liquid state and the superconducting state, as the critical coupling is nonzero. This is in contrast to a weakly coupled metal, which exhibits superconductivity for infinitesimally weak interaction regardless of its sign.
We study 2+12+12+1 dimensional massless Dirac fermions and bosons coupled to a U(1) gauge field as a model for underdoped cuprates. We find that the uniform susceptibility and the specific heat coefficient are logarithmically enhanced (compared to linear-in- TTT behavior) due to the fluctuation of a transverse gauge field which is the only massless mode at finite boson density. We analyze existing data, and find good agreement in the spin gap phase. Within our picture, the drop of the susceptibility below the superconducting Tc{T}_{c}Tc​ arises from the suppression of gauge fluctuations.