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arXiv:1705.01280 · New Journal of Physics

Microscopic bosonization of band structures: x-ray processes beyond the Fermi edge

Izak Snyman, Serge Florens

Published Oct 23, 2017• 3 citations
arXivHTMLPDF

Abstract

Bosonization provides a powerful analytical framework to deal with one-dimensional strongly interacting fermion systems, which makes it a cornerstone in quantum many-body theory. However, this success comes at the expense of using effective infrared parameters, and restricting the description to low energy states near the Fermi level. We propose a radical extension of the bosonization technique that overcomes both limitations, allowing computations with microscopic lattice Hamiltonians, from the Fermi level down to the bottom of the band. The formalism rests on the simple idea of representating the fermion kinetic term in the energy domain, after which it can be expressed in terms of free bosonic degrees of freedom. As a result, one- and two-body fermionic scattering processes generate anharmonic boson−boson interactions, even in the forward channel. We show that up to moderate interaction strengths, these non-linearities can be treated analytically at all energy scales, using the x-ray emission problem as a showcase. In the strong interaction regime, we employ a systematic variational solution of the bosonic theory, and obtain results that agree quantitatively with an exact diagonalization of the original one-particle fermionic model. This provides a proof of the fully microscopic character of bosonization, on all energy scales, for an arbitrary band structure. Besides recovering the known x-ray edge singularity at the emission threshold, we find strong signatures of correlations even at emission frequencies beyond the band bottom.

DOI
10.1088/1367-2630/aa9558

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Bosonization provides a powerful analytical framework to deal with one-dimensional strongly interacting fermion systems, which makes it a cornerstone in quantum many-body theory. However, this success comes at the expense of using effective infrared parameters, and restricting the description to low energy states near the Fermi level. We propose a radical extension of the bosonization technique that overcomes both limitations, allowing computations with microscopic lattice Hamiltonians, from the Fermi level down to the bottom of the band. The formalism rests on the simple idea of representating the fermion kinetic term in the energy domain, after which it can be expressed in terms of free bosonic degrees of freedom. As a result, one- and two-body fermionic scattering processes generate anharmonic boson−boson interactions, even in the forward channel. We show that up to moderate interaction strengths, these non-linearities can be treated analytically at all energy scales, using the x-ray emission problem as a showcase. In the strong interaction regime, we employ a systematic variational solution of the bosonic theory, and obtain results that agree quantitatively with an exact diagonalization of the original one-particle fermionic model. This provides a proof of the fully microscopic character of bosonization, on all energy scales, for an arbitrary band structure. Besides recovering the known x-ray edge singularity at the emission threshold, we find strong signatures of correlations even at emission frequencies beyond the band bottom.
A continuum of excitations in interacting one-dimensional systems is bounded from below by a spectral edge that marks the lowest possible excitation energy for a given momentum. We analyze short-range interactions between Fermi particles and between Bose particles (with and without spin) using Bethe-ansatz techniques and find that the dispersions of the corresponding spectral edge modes are close to a parabola in all cases. Based on this emergent phenomenon we propose an empirical model of a free, nonrelativistic particle with an effective mass identified at low energies as the bare electron mass renormalized by the dimensionless Luttinger parameter KKK (or ${K}_{\ensuremath{\sigma}}$ for particles with spin). The relevance of the Luttinger parameters beyond the low-energy limit provides a more robust method for extracting them experimentally using a much wider range of data from the bottom of the one-dimensional band to the Fermi energy. The empirical model of the spectral edge mode complements the mobile impurity model to give a description of the excitations in proximity of the edge at arbitrary momenta in terms of only the low-energy parameters and the bare electron mass. Within such a framework, for example, exponents of the spectral function are expressed explicitly in terms of only a few Luttinger parameters.
Strong repulsive interactions in a one-dimensional electron system suppress the exchange coupling JJJ of electron spins to a value much smaller than the Fermi energy EF{E}_{F}EF​. The conventional theoretical description of such systems based on the bosonization approach and the concept of Tomonaga-Luttinger liquid is applicable only at energies below JJJ. In this paper, we develop a theoretical approach valid at all energies below the Fermi energy, including a broad range of energies between JJJ and EF{E}_{F}EF​. The method involves bosonization of the charge degrees of freedom, while the spin excitations are treated exactly. We use this technique to calculate the spectral functions of strongly interacting electron systems at energies in the range $J⪡\ensuremath{\epsilon}⪡{E}_{F}$. We show that in addition to the expected features at the wave vector kkk near the Fermi point kF{k}_{F}kF​, the spectral function has a strong peak centered at k=0k=0k=0. Our theory also provides analytical description of the spectral function singularities near 3kF3{k}_{F}3kF​ (the ``shadow band'' features).
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.
We develop a bosonization formalism that captures nonperturbatively the interaction effects on the Q=0\mathbf{Q}=0Q=0 continuum of excitations of nodal fermions above one dimension. Our approach is a natural extension of the classic bosonization scheme for higher dimensional Fermi surfaces to include the Q=0\mathbf{Q}=0Q=0 neutral excitations that would be absent in a single-band system. The problem is reduced to solving a boson bilinear Hamiltonian. We establish a rigorous microscopic footing for this approach by showing that the solution of such boson bilinear Hamiltonian is exactly equivalent to performing the infinite sum of Feynman diagrams associated with the Kadanoff-Baym particle-hole propagator that arises from the self-consistent Hartree-Fock approximation to the single-particle Green's function. We apply this machinery to compute the interaction corrections to the optical conductivity of two-dimensional Dirac fermions with Coulomb interactions reproducing the results of perturbative renormalization group at weak coupling and extending them to the strong-coupling regime.
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.
Bosonization technique for one-dimensional fermions out of equilibrium is developed in the framework of the Keldysh action formalism. We first demonstrate how this approach is implemented for free fermions and for the problem of nonequilibrium Fermi edge singularity. We then employ the technique to study an interacting quantum wire attached to two electrodes with arbitrary energy distributions. The nonequilibrium electron Green's functions, which can be measured via tunneling spectroscopy technique and carry the information about energy distribution, zero-bias anomaly, and dephasing, are expressed in terms of functional determinants of single-particle ``counting'' operators. The corresponding time-dependent scattering phase is found to be intrinsically related to ``fractionalization'' of electron-hole excitations in the tunneling process and at boundaries with leads. Results are generalized to the case of spinful particles as well to Green's functions at different spatial points (relevant to the problem of dephasing in Luttinger liquid interferometers). For double-step distributions, the dephasing rates are oscillatory functions of the interaction strength.