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arXiv:cond-mat/0011253

Eigen-functional bosonization and Eikonal-type equations in one-dimensional strongly correlated electron system

Yu-Liang Liu

Published Nov 15, 2000• 0 citations
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Abstract

With the eigen-functional bosonization method, we study one-dimensional strongly correlated electron systems with large momentum ($2k_{F}$ and/or $4k_{F}$) transfer term(s), and demonstrate that this kind of problems ends in to solve the Eikonal-type equations, and these equations are universal, and independent of whether or not the system is integrable. In contrast to usual perturbation theory, this method is valid not only for weak electron interaction, but also for strong electron interaction. Comparing with exact solution of some integrable models, it can give correct results in one-loop approximation. This method can also be used to study electron-phonon interaction systems, and two coupled spin chain or quantum wire systems.

DOI
10.48550/arxiv.cond-mat/0011253

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With the eigen-functional bosonization method, we study one-dimensional strongly correlated electron systems with large momentum (2kF2k_{F}2kF​ and/or 4kF4k_{F}4kF​) transfer term(s), and demonstrate that this kind of problems ends in to solve the Eikonal-type equations, and these equations are universal, and independent of whether or not the system is integrable. In contrast to usual perturbation theory, this method is valid not only for weak electron interaction, but also for strong electron interaction. Comparing with exact solution of some integrable models, it can give correct results in one-loop approximation. This method can also be used to study electron-phonon interaction systems, and two coupled spin chain or quantum wire systems.
Using eigen-functional method, we study one-dimensional strongly correlated electron systems with large momentum (2k F and/or 4k F ) transfer term(s), and demonstrate that this kind of problems ends in to solve the Eikonal-type equations, and these equations are universal, and independent of whether or not the system is integrable. In contrast to usual perturbation theory, this method is valid not only for weak electron interaction, but also for strong electron interaction. Comparing with exact solution of some integrable models, it can give correct results in linearization approximation. This method can also be used to study electron-phonon interaction systems, and two coupled spin chain (or quantum wire) systems.
In the eigen-functional theoretical framework, exact/accurate treatment of high-dimensional strongly correlated electron systems may be realized. The two key steps of this method are that under the path integral formulation we first change a D-dimensional quantum many-particle system into an (D+1)-dimensional (time-dependent) effective "single-particle" problem, then by solving the eigen-equation of the propagator operator of the particles, we can obtain the ground state energy functional and action, respectively. Under the eigen-functional theory, the problems of quantum many-particle systems end in to solve the equation of phase fields that are completely determined by the electron interaction. In practice this equation of the phase fields can be numerically solved for large lattice sites. After replacing the real time by the imaginary time, this method can also be applied for finite temperature cases.
Using eigen-functional bosonization method, we study quantum many-particle systems, and show that the quantum many-particle problems end in to solve the differential equation of the phase fields which represent the particle correlation strength. Thus, the physical properties of these systems are completely determined by the differential equation of the phase fields. We mainly focus on the study of D-dimensional electron gas with/without transverse gauge fields, two-dimensional electron gas under an external magnetic field, D-dimensional boson systems, a D-dimensional Heisenberg model and a one-band Hubbard model on a square lattice, and give their exact (accurate for Heisenberg model) functional expressions of the ground state energy and action, and the eigen-functional wave functions of the fermions/bosons. With them, we can calculate a variety of correlation functions of the systems, such as single particle Green's functions and their ground state wave functions. In present theoretical framework, we can unifiably represent the Landau Fermi liquid, non-Fermi liquid (D≥2D\geq 2D≥2) and Tomonaga-Luttinger liquid.
Three different approaches to correlated electron systems are presented.(i)The single impurity Anderson model is solved by a semi-analytical approach based on the flow equation method.(ii)A numerically implementable Multi-scale Many-Body approach to strongly correlated electron systems is introduced.An extension to quantum cluster methods, it approximates correlations on any given length-scale commensurate with the strength of the correlations on the respective scale.(iii) The spectral properties of the one-dimensional Holstein and breathing polaron models using the self-consistent Born approximation are presented.4.10 a) Energy levels for zero coupling (g = 0).The solid line shows the non-interacting electron dispersion, the dashed ones are one phonon + one electron states and the dotted one two phonons + one electron, etc. k 1 is the momentum where the electron dispersion reaches ω 0 and has the most significant contribution to the scattering at the first phonon threshold.k 2 is the value of momentum with significant contribution to the scattering at the second phonon threshold.b) For the polaron with momentum k the scattering with momentum q has a significant contribution.Although the polaron momentum is close to k F , thus k a = k -k F is small, q = k a + k b is large because k b which connects two points on the Fermi surface can be large. . . . . . . . . . .
By introducing a set of auxiliary equations representing a many-body system, we have derived an extension of the Kohn-Sham scheme for the density functional theory. These equations consist of a Kohn-Sham-type equation determining single-particle orbitals and an eigen-value equation for an effective many-body problem. A variational method similar to the Kohn-Sham technique was utilized to derive effective interactions as well as effective potentials without artificial substitution of a Hubbard-type interaction and a mean-field correction in the energy functional. The second equation is described by an effective many-body Hamiltonian with both 2-body interactions and mean-field terms. Rigorous formulation of the extended Kohn-Sham equation is also given in accordance with the Hadjisavvas-Theophilou formulation. Our formulation can be interpreted as a way to define models of the strongly correlated electron systems, e.g. the Hubbard model
"Strong electron correlation phenomena are ubiquitous, such as unconventional superconductivity and the metal-insulator transitions. Having a proper understanding of strongly correlated electron systems requires a non-perturbative analysis because Fermi liquid theory inevitably breaks down due to the strong correlation. In this thesis, we will study two non-perturbative approaches to tackle strongly correlated electronic system: 1.) multi-dimensional bosonization and 2.) the AdS/CFT correspondence. Multidimensional bosonization is the generalization of bosonization to spatial dimensions larger than one. The bosonized theory is an effective field theory in terms of the bosonic particle-hole fluctuations in different patches of the Fermi surface. The bosonized theory is quadratic and hence can be solved non-perturbatively. The AdS/CFT correspondence is a conjecture that d-dimensional field theory is dual to a d+1 dimensional quantum gravitational theory. Problems in strongly coupled field theory then have an equivalent description using weakly interacting gravitational theory, allowing a non-perturbative analysis. Chapter 1 will serve as an introduction. We will review several experiments which show a breakdown of Fermi liquid theory. Formalisms for multidimensional bosonization and the AdS/CFT correspondence will then be reviewed. In chapter 2, the two-orbital Hubbard model with degenerate d{xz} and d{yz} orbitals are investigated. We apply multidimensional bosonization to solve this problem exactly and discover a z=3 overdamped collective modes that emerges at the orbital-ordering quantum critical point. These modes modify the single-particle density of states and lead to non-Fermi liquid behavior which can provide a possible explanation for the recently observed zero-bias enhancement in the point contact spectroscopy signal on iron pnictides. Chapters 3, 4 and 5 discuss the applications of the AdS/CFT correspondence to model various condensed matter systems. In chapter 3, We consider an interaction term between a bulk spinor field and a gauge field in the Reissner-Nordstr\""om AdS background. When the Pauli interaction term is large enough, a dynamical gap is generated and spectral weight transfer is observed in the spectral density, mimicking the behavior of the Hubbard model. We further consider the finite temperature case and discover that the ratio between the dynamical gap and the critical temperature has the same order as that of VO^2. The Pauli coupling is also studied in the superconducting background. Chapter 4 consider the propagation of a neutral scalar field in the geometry called the electron star. The electron star has Lifshitz scaling with finite dynamical critical exponent at the interior, which is suitable for the modeling of quantum criticality and quantum phase transition with a neutral order parameter, for example for antiferromagnetism. We find that the quantum phase transition has the Berezinski-Kosterlitz-Thouless characteristic and the dynamical critical exponent can change across the quantum critical point. In chapter 5, we construct a gravity dual of the nematic phase by studying the condensation of a spin-two field in the Schwarzschild-AdS background. The condensation of a spin-two field can distort the rotational symmetry of the Fermi surface as long as the spin-two field is coupled to a probe spinor field in the bulk gravity dual."