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arXiv:cond-mat/9811045

Myopic Bosonization

Girish S. Setlur

Published Nov 4, 1998• 0 citations
arXivHTMLPDF

Abstract

As the title suggests, this is an attempt at bosonizing fermions in any number of dimensions without paying attention to the fact that the Fermi surface is an extended object. One is tempted to introduce the density fluctuation and its conjugate and recast the interacting problem in terms of these canonical Bose fields. However, we find that the attempt is short-sighted figuratively as well for the same reason. But surprisingly, this flaw, which manifests itself as an inconsistency between Menikoff-Sharp's construction of the kinetic energy operator in terms of currents and densities, and our ansatz for this operator, is nevertheless able to reproduce(although reluctantly) many salient features of the free theory. Buoyed by this success, we solve the interacting problem and compute the full propagator.

DOI
10.48550/arxiv.cond-mat/9811045

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As the title suggests, this is an attempt at bosonizing fermions in any number of dimensions without paying attention to the fact that the Fermi surface is an extended object. One is tempted to introduce the density fluctuation and its conjugate and recast the interacting problem in terms of these canonical Bose fields. However, we find that the attempt is short-sighted figuratively as well for the same reason. But surprisingly, this flaw, which manifests itself as an inconsistency between Menikoff-Sharp's construction of the kinetic energy operator in terms of currents and densities, and our ansatz for this operator, is nevertheless able to reproduce(although reluctantly) many salient features of the free theory. Buoyed by this success, we solve the interacting problem and compute the full propagator.
We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (qc{\mathit{q}}_{\mathit{c}}qc​/kF{\mathit{k}}_{\mathit{F}}kF​)d{)}^{\mathit{d}})dF0{\mathit{F}}_{0}F0​ [1+F0{\mathit{F}}_{0}F0​${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here qc{\mathit{q}}_{\mathit{c}}qc​ is the cutoff of the interaction in wave-vector space, kF{\mathit{k}}_{\mathit{F}}kF​ is the Fermi wave vector, \ensuremath{\mu} is the chemical potential, F0{\mathit{F}}_{0}F0​ is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff qc{\mathit{q}}_{\mathit{c}}qc​ is small compared with kF{\mathit{k}}_{\mathit{F}}kF​, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.
Luttinger’s exactly soluble model of a one-dimensional many-fermion system is discussed. We show that he did not solve his model properly because of the paradoxical fact that the density operator commutators [p(p), p(−p′)], which always vanish for any finite number of particles, no longer vanish in the field-theoretic limit of a filled Dirac sea. In fact the operators p(p) define a boson field which is ipso facto associated with the Fermi-Dirac field. We then use this observation to solve the model, and obtain the exact (and now nontrivial) spectrum, free energy, and dielectric constant. This we also extend to more realistic interactions in an Appendix. We calculate the Fermi surface parameter , and find: (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.
Luttinger's exactly soluble model of a one-dimensional many-fermion system is discussed. We show that he did not solve his model properly because of the paradoxical fact that the density operator commutators [ρ(p), ρ(−p′)], which always vanish for any finite number of particles, no longer vanish in the field-theoretic limit of a filled Dirac sea. In fact the operators ρ(p) define a boson field which is ipso facto associated with the Fermi-Dirac field. We then use this observation to solve the model, and obtain the exact (and now nontrivial) spectrum, free energy, and dielectric constant. This we also extend to more realistic interactions in an Appendix. We calculate the Fermi surface parameter n̄k, and find: ∂n̄k/∂k|kF = ∞ (i.e., there exists a sharp Fermi surface) only in the case of a sufficiently weak interaction.
I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin, or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (nonrelativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the nonrelativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.
We study, via bosonization, the Landau fixed point for the problem of interacting spinless fermions near the Fermi surface in dimensions higher than one. We rederive the bosonic representation of the Fermi operator and use it to find the general form of the fermion propagator for the Landau fixed point. Using a generalized Bogoliubov transformation we diagonalize exactly the bosonized Hamiltonian for the fixed point and calculate the fermion propagator (and the quasiparticle residue) for isotropic interactions (independently of their strength). We reexamine two well-known problems in this context: the screening of long-range potentials and the Landau damping of gauge fields. We also discuss the origin of the Luttinger fixed point in one dimension in contrast with the Landau fixed point in higher dimensions.
An attempt is made to generalise the ideas introduced by Haldane and others regarding Bosonizing the Fermi surface. The present attempt involves introduction of Bose fields that correspond to displacements of the Fermi sea rather than just the Fermi surface. This enables the study of short wavelength fluctuations of the Fermi surface and hence the dispersion of single particle excitations with high energy. The number conserving product of two Fermi fields is represented as a simple combination of these Bose fields. It is shown that most(!) commutation rules involving these number conserving products are reproduced exactly, as are the dynamical correlation functions of the free theory. Also the work of Sharp, Menikoff and Goldin has shown that the field operator may be viewed as a unitary representation of the current algebra. An explicit realisation of this unitary representation is given in terms of canonical conjugate of the density operator.