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International Journal of Modern Physics B

UNIVERSAL DESCRIPTION OF STRONGLY CORRELATED SYSTEMS

Yu-Liang Liu

Published Feb 20, 2002• 17 citations
Publisher

Abstract

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.

DOI
10.1142/s0217979202009949

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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.
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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.
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.
In the first part of the Thesis we mostly concentrate on spectral properties of strongly correlated systems and on their equilibrium properties. This is accomplished by the general concept of imaginary-time dynamics which we apply to a number of different problems in which different strengths of this approach emerge. In Chapter 1 we introduce the formalism that allows for a connection between the quantum and the classical worlds. The connection is established by means of the imaginary-time quantum evolution which, under certain circumstances, is shown to be equivalent to a classical stochastic process. It is further shown that exact static and spectral properties of correlated systems can be obtained when this mapping is feasible. The relationship between the imaginary-time dynamics in different frameworks such as the path-integral and the perturbative one is also underlined. In Chapter 2 we present a specific implementation of the general ideas previously presented. In particular we introduced an extension to lattice systems of the Reptation Monte Carlo algorithm [30] which benefits of a sampling scheme based on directed updates. Specific improvements over the existing methodologies consist in the unbiased evaluation of the imaginary-time path integrals for bosons and a systematic scheme to improve over the Fixed-node approximation for fermions. Applications to the Hubbard and the Heisenberg models are presented. In Chapter 3 we demonstrate the application of the imaginary-time dynamics to the exact study of spectral properties. Subject of our attention is a highly anharmonic and correlated quantum crystal such as Helium 4 at zero temperature.[33] Concerning this system, we have obtained the first ab-initio complete phonon dispersion in good agreement with neutron spectroscopy experiments. Moreover, we have also studied the density excitations of solid helium in a region of wave-vectors in between the collective (phonon) and the single-particle regimes, where the presence of residual coherence in the dynamics shows analogies between the highly anharmonic crystal and the superfluid phase. In Chapter 4 we introduce a novel method, based on the imaginary-time dynamics, to obtain unbiased estimates of fermionic properties.[34] By means of this method and of a very accurate variational state, we provide strong evidence for the stability of a saturated ferromagnetic phase in the high-density regime of the two-dimensional infinite-U Hubbard model. By decreasing the electron density, we observe a discontinuous transition to a paramagnetic phase, accompanied by a divergence of the susceptibility on the paramagnetic side. This behavior, resulting from a high degeneracy among different spin sectors, is consistent with an infinite-order phase transition scenario. 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In Chapter 6 exact diagonalization schemes and renormalization-based methods for one-dimensional systems are introduced. We identify key phenomenological traits resulting from the many-body correlation in closed systems driven sufficiently away from equilibrium.[31] We provide evidences that the dynamics of interacting lattice bosons away from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. In Chapter 7 we present an extension to dynamical properties of the Variational Quantum Monte Carlo method.[32] This is accomplished by introducing a general class of time-dependent variational states which is based on the mapping of the many-body dynamics onto an instantaneous ground-state problem. The application of the method to the experimentally relevant quantum quenches of interacting bosons reveals the accuracy and the reliability of the introduced numerical scheme. We indeed obtain for the first time a consistent variational description of the approach to the equilibrium of local observables and underline the origin of the metastability and glassy behavior previously identified. In the very last part we draw our conclusions and show some possible paths for stimulating future research.
We first introduce the basic ingredients of the eigenfunctional theory, and show that a D-dimensional quantum many-particle system is mapped into a (D+1)-dimensional time-depending single-particle problem, and in the representation of the eigenfunctionals of the particle propagator, the particles become free. Then using this method, we study five kinds of quantum many-particle systems: interacting boson system, repulsive, attractive interacting fermion systems, Hubbard model and single-impurity scattering in one-dimensional fermion system, and demonstrate that the microscopic Bogoliubov theory and the phenomenological Bijl–Feynman theory of the bosons are closely related, and apart from an anti-symmetry factor Det ‖eikj·xl‖ the ground state wave function of the repulsive interacting fermion system has a similar form to that of the interacting boson system. Moreover, we show that the attractive interacting fermion system has a sound-type excitation spectrum like that in the interacting boson system. For one-dimensional Hubbard model we calculate the electron Green's function, and charge and spin density–density correlation functions which are consistent with the exact ones obtained by the Bethe ansatz and numerical calculations, and show that the ground state energy is increasing with U, and the electrons has single-occupied constraint in the large U limit. Finally, we demonstrate clearly the evolution of the system from its ultraviolet fixed point to infrared critical fixed point as the impurity potential increases. At the infrared critical fixed point, the fermion Green's function shows that the fermions are completely reflected on the impurity site.
By introducing a phase field and solving the eigen-functional equation of particles, we obtain the exact expressions of the ground state energy as a functional of the particle density for interacting electron/boson systems, and a two-dimensional electron gas under an external magnetic field, respectively. With the eigen-functionals of the particles, we can construct the ground state wave-function of the systems. Moreover, with the expressions of the ground state energy, we can exactly determine the ground state energy and the ground state particle density of the systems by taking % δE_g[ρ]/δρ(x)=0.