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Peter Kopietz
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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.
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We study in this paper the properties of a gas of fermions interacting {em via} a scalar potential $v(q)=4pi{e}^2/q^2$ for $q<Lambda<<k_F$ at dimensions larger than one, where $Lambda$ is a high momentum cutoff and $k_F$ is the fermi wave vector. In particular, we shall consider the $e^2toinfty$ limit where the potential becomes confining. Within a bosonization approximation, effective Hamiltonians describing the low energy physics of the system are constructed, where we show that the system can be described as a fermi liquid formed by chargeless quasi-particles which has vanishing wavefunction overlap with the bare fermions in the system.
We construct fixed point lattice models for group supercohomology symmetry protected topological (SPT) phases of fermions in 2+1D. A key feature of our approach is to construct finite depth circuits of local unitaries that explicitly build the ground states from a tensor product state. We then recover the classification of fermionic SPT phases, including the group structure under stacking, from the algebraic composition rules of these circuits. Furthermore, we show that the circuits are symmetric, implying that the group supercohomology phases can be many body localized. Our strategy involves first building an auxiliary bosonic model, and then fermionizing it using the duality of Chen, Kapustin, and Radicevic. One benefit of this approach is that it clearly disentangles the role of the algebraic group supercohomology data, which is used to build the auxiliary bosonic model, from that of the spin structure, which is combinatorially encoded in the lattice and enters only in the fermionization step. In particular this allows us to study our models on 2d spatial manifolds of any topology and to define a lattice-level procedure for ungauging fermion parity.
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.
An unbiased zero-temperature auxiliary-field quantum Monte Carlo method is employed to analyze the nature of the semimetallic phase of the two-dimensional Hubbard model on the honeycomb lattice at half filling. It is shown that the quasiparticle weight $Z$ of the massless Dirac fermions at the Fermi level, which characterizes the coherence of zero-energy single-particle excitations, can be evaluated in terms of the long-distance equal-time single-particle Greens function. If this quantity remains finite in the thermodynamic limit, the low-energy single-particle excitations of the correlated semimetallic phase are described by a Fermi-liquid-type single-particle Greens function. Based on the unprecedentedly large-scale numerical simulations on finite-size clusters containing more than ten thousands sites, we show that the quasiparticle weight remains finite in the semimetallic phase below a critical interaction strength. This is also supported by the long-distance algebraic behavior ($sim r^{-2}$, where $r$ is distance) of the equal-time single-particle Greens function that is expected for the Fermi liquid. Our result thus provides a numerical confirmation of Fermi-liquid theory in two-dimensional correlated metals.
We investigate the 1D interacting two-component Fermi gas with arbitrary polarization. Exact results for the ground state energy, quasimomentum distribution functions, spin velocity and charge velocity reveal subtle polarization dependent quantum effects.
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