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International Journal of Modern Physics B

THE FERMION-BOSON MAPPING APPLIED TO LAGRANGIAN MODELS FOR CHARGE-DENSITY-WAVES IN ONE-DIMENSIONAL SYSTEMS

L. V. Belvedere, R. L. P. G. Amaral, A. F. DE QUEIROZ

Published Dec 10, 2002• 1 citation
Publisher

Abstract

We use the two-dimensional Fermion-Boson mapping to perform a field theory analysis of the effective Lagrangian model for incommensurate charge-density waves (ICDW) in one-dimensional systems. We consider an approach in which both the phase of the complex phonon field and the electron field are dynamical degrees of freedom contributing to the quantum dynamics and symmetry-related features of the ICDW phenomenon. We obtain the bosonized and fermionized versions of the effective electron–phonon Lagrangian. The phase of the phonon field and the phase of the bosonized chiral density of the electron field condense as a soliton order parameter, carrying neither the charge nor the chirality of the electron–phonon system, leading to a periodic sine-Gordon potential. The phonon field is fermionized in terms of a chiral fermionic condensate and the effective model is mapped into the chiral Gross–Neveu (GN) model with two Fermi field species. The linked electron–phonon symmetry of the ICDW system is mapped into the chiral symmetry of the GN model. Within the functional integral formulation, we obtain for the vacuum expectation value of the phonon field < ϕ > = 0 and < ϕ ϕ* > ≠ 0, due to the charge selection rule associated with the chiral electron–phonon symmetry. We show that the two-point correlation function of the phonon field satisfies the cluster decomposition property, as required by the chiral symmetry of the underlying GN model. The quantum description of the ICDW corresponds to charge transport through the lattice, due to the propagation of a "Goldstone mode" carrying the effective charge of the electron–phonon system, is accomplished by an electron–lattice energy redistribution. This accounts for a dynamical Peierls's energy gap generation.

DOI
10.1142/s0217979202014929

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We use the two-dimensional Fermion-Boson mapping to perform a field theory analysis of the effective Lagrangian model for incommensurate charge-density waves (ICDW) in one-dimensional systems. We consider an approach in which both the phase of the complex phonon field and the electron field are dynamical degrees of freedom contributing to the quantum dynamics and symmetry-related features of the ICDW phenomenon. We obtain the bosonized and fermionized versions of the effective electron–phonon Lagrangian. The phase of the phonon field and the phase of the bosonized chiral density of the electron field condense as a soliton order parameter, carrying neither the charge nor the chirality of the electron–phonon system, leading to a periodic sine-Gordon potential. The phonon field is fermionized in terms of a chiral fermionic condensate and the effective model is mapped into the chiral Gross–Neveu (GN) model with two Fermi field species. The linked electron–phonon symmetry of the ICDW system is mapped into the chiral symmetry of the GN model. Within the functional integral formulation, we obtain for the vacuum expectation value of the phonon field < ϕ > = 0 and < ϕ ϕ* > ≠ 0, due to the charge selection rule associated with the chiral electron–phonon symmetry. We show that the two-point correlation function of the phonon field satisfies the cluster decomposition property, as required by the chiral symmetry of the underlying GN model. The quantum description of the ICDW corresponds to charge transport through the lattice, due to the propagation of a "Goldstone mode" carrying the effective charge of the electron–phonon system, is accomplished by an electron–lattice energy redistribution. This accounts for a dynamical Peierls's energy gap generation.
We consider the Thermofield Dynamics bosonization to perform a field theory analysis of the effective Lagrangian model for incommensurate charge density waves (ICDW) in one-dimensional systems at finite temperature. The phonon degree of freedom is carryied by a dynamical phase field, contributing to the quantum dynamics and symmetry related features of the ICDW phenomenon. The electron chiral density and the phase of the phonon field condensate as a thermal soliton, carrying the symmetry under the linked electron-phonon Ue5(1)⊗Uph(1)U_e^5(1) \otimes U_{ph}(1)Ue5​(1)⊗Uph​(1) global transformations. Using the Gell'Mann-Low formula for finite temperature, the perturbative series of the phonon thermal correlation function is obtained. Due to the electron-phonon charge selection rule we obtain for the thermal vacuum expectation value for the order parameter <0(β)∣Φ∣0(β)>=0<0 (β) |Φ| 0 (β)> = 0<0(β)∣Φ∣0(β)>=0, in accordance with the cluster decomposition property of the corresponding correlation function. This reflects the fact that the quantum description of the ICDW corresponds to a local charge transport through the lattice which is accomplished by an electron-lattice energy redistribution, which accounts for a thermal dynamical mass gap generation. The electron-phonon coupling can be rewritten in terms of a mass operator for the "physical" fermion operator ΨΨΨ such that <0(β)∣Ψˉ(x)Ψ(x)∣0(β)>≠0<0 (β)| \barΨ(x)Ψ(x) | 0 (β)> \neq 0<0(β)∣Ψˉ(x)Ψ(x)∣0(β)>=0, without the breakdown of the linked electron-phonon symmetry.
The effective Lagrangian model for charge-density waves interacting with an impurity potential in one-dimensional systems is considered in the dynamical phonon phase approach. Using the fermion–boson mapping we obtain the effective bosonized version of the model. The impurity potential breaks the linked electron–phonon symmetry, the phason field turns out to be in interaction with the electron–phonon condensate and the phonon field develops a non-vanishing vacuum expectation value. The effective fermionized version of the model corresponds to the chiral Gross–Neveu model with quartic self-interaction among a massless and a massive (electron) Fermi fields. The electron–phonon system exhibits a conserved topological charge which is independent of local variations of the phases of the phonon and electron fields. The equation of state of the associated statistical-mechanical system is obtained. For the dynamical phonon phase field the Kosterlitz–Thouless phase transition is suppressed.
The spontaneous generation of charge-density-wave order in a Dirac fermion system via the natural mechanism of electron-phonon coupling is studied in the framework of the Holstein model on the honeycomb lattice. Using two independent and unbiased quantum Monte Carlo methods, the phase diagram as a function of temperature and coupling strength is determined. It features a quantum critical point as well as a line of thermal critical points. Finite-size scaling appears consistent with fermionic Gross-Neveu-Ising universality for the quantum phase transition and bosonic Ising universality for the thermal phase transition. The critical temperature has a maximum at intermediate couplings. Our findings motivate experimental efforts to identify or engineer Dirac systems with sufficiently strong and tunable electron-phonon coupling.
We discuss a new method to perform numerical simulations of one-dimensional systems with fermion and boson degrees of freedom. The method is based on a direct-space, imaginary-time representation of the fermion field. It is fast so that systems having up to 100 sites can easily be simulated. In addition, the method provides an intuitive physical "picture" of the ground state of a one-dimensional many-body system. We discuss in detail how to implement the method and how to compute various physical quantities. In particular, we show how to extend the method to study averages of off-diagonal quantities in an occupation-number representation. To assess the accuracy of our procedure, we apply it to free fermions in one dimension and compare with exact results. We then study a model of spinless interacting fermions and obtain the expected phase structure and behavior of correlation functions. We also consider the extended Hubbard model at various points in its phase diagram and study the behavior of spin-density, charge-density, and pairing correlation functions. We then study the Gross-Neveu model and show how the behavior depends on the number of fermion flavors. Finally, we consider an electron-phonon model and study its behavior both in the one-particle polaron sector and in the half-filled-band case. Along the way we show pictures of the ground-state configurations that give physical insight into the properties of the systems, like charge-density-wave, spin-density-wave, and superconducting states, "fractional charges," and solitons. We conclude by comparing our method with other methods and discuss the possibility of extending it to higher dimensions.
We study a mixture of fermionic and bosonic cold atoms on a two-dimensional optical lattice, where the fermions are prepared in two isospin states and the bosons have Bose-Einstein condensed. Number density fluctuations of the condensate form delocalized bosonic modes which couple to the fermionic atoms similarly to the electron-phonon coupling in crystals. We study the phase diagram for this system at fixed fermion density of one per site. We find that tuning of the lattice parameters and interaction strengths drives the system to undergo antiferromagnetic ordering, s-wave and d-wave pairing superconductivity, or a charge density-wave phase. We use functional renormalization group analysis where retardation effects are fully taken into account. We calculate response functions and also provide estimates of the energy gap associated with the dominant order, and how it depends on different parameters of the problem.
Semiclassical approach has been developed for the one-dimensional interacting fermion systems. Starting from the incommensurate spin density wave (SDW) mean field state for the repulsive Hubbard model in 1D, the non-Abelian bosonized Lagrangian describing the spin-charge separation is obtained. The Berry phase term is derived from the chiral anomaly, and we obtain the massless Tomonaga-Luttinger liquid in the single chain case while the spin gap opens in the double-chain system. This approach offers a new method to identify the strong-coupling fixed point, and its relation to the Abelian bosonization formalism is discussed on the spin gap state. The generalization to higher dimensions is also discussed.