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Register a new userBosonization of Fermi liquids in a weak magnetic field
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Novel controlled non-perturbative 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 {em 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.
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This review is a summary of my work (partially in collaboration with Kurt Schoenhammer) on higher-dimensional bosonization during the years 1994-1996. It has been published as a book entitled Bosonization of interacting fermions in arbitrary dimensions by Springer Verlag (Lecture Notes in Physics m48, Springer, Berlin, 1997). I have NOT revised this review, so that there is no reference to the literature after 1996. However, the basic ideas underlying the functional bosonization approach outlined in this review are still valid today.
An introductory survey of the theoretical ideas and calculations and the experimental results which depart from Landau Fermi-liquids is presented. Common themes and possible routes to the singularities leading to the breakdown of Landau Fermi liquids are categorized following an elementary discussion of the theory. Soluble examples of Singular Fermi liquids (often called Non-Fermi liquids) include models of impurities in metals with special symmetries and one-dimensional interacting fermions. A review of these is followed by a discussion of Singular Fermi liquids in a wide variety of experimental situations and theoretical models. These include the effects of low-energy collective fluctuations, gauge fields due either to symmetries in the hamiltonian or possible dynamically generated symmetries, fluctuations around quantum critical points, the normal state of high temperature superconductors and the two-dimensional metallic state. For the last three systems, the principal experimental results are summarized and the outstanding theoretical issues highlighted.
We calculate the damping gamma_q of collective density oscillations (zero sound) in a one-dimensional Fermi gas with dimensionless forward scattering interaction F and quadratic energy dispersion k^2 / 2 m at zero temperature. For wave-vectors | q| /k_F small compared with F we find to leading order gamma_q = v_F^{-1} m^{-2} Y (F) | q |^3, where v_F is the Fermi velocity, k_F is the Fermi wave-vector, and Y (F) is proportional to F^3 for small F. We also show that zero-sound damping leads to a finite maximum proportional to |k - k_F |^{-2 + 2 eta} of the charge peak in the single-particle spectral function, where eta is the anomalous dimension. Our prediction agrees with photoemission data for the blue bronze K_{0.3}MoO_3.
We present detailed calculations of the magnetic ground state properties of Cs$_2$CuCl$_4$ in an applied magnetic field, and compare our results with recent experiments. The material is described by a spin Hamiltonian, determined with precision in high field measurements, in which the main interaction is antiferromagnetic Heisenberg exchange between neighboring spins on an anisotropic triangular lattice. An additional, weak Dzyaloshinkii-Moriya interaction introduces easy-plane anisotropy, so that behavior is different for transverse and longitudinal field directions. We determine the phase diagram as a function of field strength for both field directions at zero temperature, using a classical approximation as a first step. Building on this, we calculate the effect of quantum fluctuations on the ordering wavevector and components of the ordered moments, using both linear spinwave theory and a mapping to a Bose gas which gives exact results when the magnetization is almost saturated. Many aspects of the experimental data are well accounted for by this approach.
We study a one-dimensional lattice model of fractional statistics in which particles have next-nearest-neighbor hopping between sites which depends on the occupation number at the intermediate site and a statistical parameter $phi$. The model breaks parity and time-reversal symmetries and has four-fermion interactions if $phi e 0$. We first analyze the model using mean field theory and find that there are four Fermi points whose locations depend on $phi$ and the filling $eta$. We then study the modes near the Fermi points using the technique of bosonization. Based on the quadratic terms in the bosonized Hamiltonian, we find that the low-energy modes form two decoupled Tomonaga-Luttinger liquids with different values of the Luttinger parameters which depend on $phi$ and $eta$; further, the right and left moving modes of each system have different velocities. A study of the scaling dimensions of the cosine terms in the Hamiltonian indicates that the terms appearing in one of the Tomonaga-Luttinger liquids will flow under the renormalization group and the system may reach a non-trivial fixed point in the long distance limit. We examine the scaling dimensions of various charge density and superconducting order parameters to find which of them is the most relevant for different values of $phi$ and $eta$. Finally we look at two-particle bound states that appear in this system and discuss their possible relevance to the properties of the system in the thermodynamic limit. Our work shows that the low-energy properties of this model of fractional statistics have a rich structure as a function of $phi$ and $eta$.
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