[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] In this dissertation, we have presented our research on new developments in the Gutzwiller variational method as well as the applications of the method to two specific correlated electron systems of current interest: (i) the spinless fermion model with non-local Coulomb interaction in one dimension and (ii) the pressure-induced insulator-metal transition in the colossal magneto-resistive compound LaMnO3. Our results for the ground-state energy of the spinless fermion model are obtained by careful enumeration of the many-body configurations of the system and calculating the Gutzwiller reduction factor to the kinetic energy for general filling. We resolve the inconsistency in the existing literature for this problem as we find that our result for the half-filled case agrees with one of the three papers and attribute the discrepancy between them to the incorrect counting method and the assumption that the Gutzwiller approximation is equivalent to the mean-field slave-boson approach, which is although correct for the on-site Hubbard model, turns out not to be so in the present case. Compared with the exact diagonalization results, we show that the Gutzwiller method indeed offers a better solution than the slave-boson approach. As the second problem, we have studied the ground-state properties of paramagnetic LaMnO3 under hydrostatic pressure at room temperature. We apply the Gutzwiller method to an extended Hubbard model for spinless eg electrons on Mn3+ ion including all important ingredients such as cohesive energy as well as two main competing Coulomb and Jahn-Teller (JT) interactions. The plot of the energy of the paramagnetic LaMnO3 as a function of volume clearly confirms that 1) contrary to the existing accepted theoretical picture for vanishing lattice distortion as the metallic state emerges, the JT distortion survives even beyond the transition pressure into the metallic region and 2) the application of pressure leads to a structural phase separation and breaks the material into two blocks composed of domains of undistorted (metallic character) and distorted (insulating character) MnO6 octahedra; both of these findings have been already observed in experiments with no theoretical explanations. As the key unifying connection between both of these findings we suggest that the pressure-induced transition and the phase coexistence is indeed percolative. Finally, applying the idea of universality in percolation quantities near the threshold, we support our hypothesis by proposing a scaling law for the pressure-dependence of the resistance and use it as a best fit to recently-acquired experimental data. We find good agreement between our proposed model and the experimental results, confirming that the pressure-induced insulator-metal transition in LaMnO3 is indeed percolative.
[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] In this dissertation, we have presented our research on new developments in the Gutzwiller variational method as well as the applications of the method to two specific correlated electron systems of current interest: (i) the spinless fermion model with non-local Coulomb interaction in one dimension and (ii) the pressure-induced insulator-metal transition in the colossal magneto-resistive compound LaMnO3. Our results for the ground-state energy of the spinless fermion model are obtained by careful enumeration of the many-body configurations of the system and calculating the Gutzwiller reduction factor to the kinetic energy for general filling. We resolve the inconsistency in the existing literature for this problem as we find that our result for the half-filled case agrees with one of the three papers and attribute the discrepancy between them to the incorrect counting method and the assumption that the Gutzwiller approximation is equivalent to the mean-field slave-boson approach, which is although correct for the on-site Hubbard model, turns out not to be so in the present case. Compared with the exact diagonalization results, we show that the Gutzwiller method indeed offers a better solution than the slave-boson approach. As the second problem, we have studied the ground-state properties of paramagnetic LaMnO3 under hydrostatic pressure at room temperature. We apply the Gutzwiller method to an extended Hubbard model for spinless eg electrons on Mn3+ ion including all important ingredients such as cohesive energy as well as two main competing Coulomb and Jahn-Teller (JT) interactions. The plot of the energy of the paramagnetic LaMnO3 as a function of volume clearly confirms that 1) contrary to the existing accepted theoretical picture for vanishing lattice distortion as the metallic state emerges, the JT distortion survives even beyond the transition pressure into the metallic region and 2) the application of pressure leads to a structural phase separation and breaks the material into two blocks composed of domains of undistorted (metallic character) and distorted (insulating character) MnO6 octahedra; both of these findings have been already observed in experiments with no theoretical explanations. As the key unifying connection between both of these findings we suggest that the pressure-induced transition and the phase coexistence is indeed percolative. Finally, applying the idea of universality in percolation quantities near the threshold, we support our hypothesis by proposing a scaling law for the pressure-dependence of the resistance and use it as a best fit to recently-acquired experimental data. We find good agreement between our proposed model and the experimental results, confirming that the pressure-induced insulator-metal transition in LaMnO3 is indeed percolative.
We study the Gutzwiller method for the spinless fermion model in one dimension, which is one of the simplest models that incorporates the intersite Coulomb interaction. The Gutzwiller solution of this model has been studied in the literature but with differing results. We obtain the Gutzwiller solution of the problem by a careful enumeration of the many-particle configurations and explain the origin of the discrepancy in the literature to be due to the neglect of the correlation between the neighboring bond occupancies. The correct implementation of the Gutzwiller approach yields results different from the slave-boson mean-field theory, unlike for the Hubbard model with on-site interaction, where both methods are known to be equivalent. The slave-boson and the Gutzwiller results are compared to the exact solution, available for the half-filled case, and to the numerical exact diagonalization results for the case of general filling.
Introduction In this chapter we study metal–insulator transitions which occur due to correlations between the charge carriers. This quantum phase transitions results from the competition between kinetic energy and Coulomb repulsion where the latter has the tendency to localise the electrons. The simplest model which describes this competition is the one-band Hubbard model with its two energy scales, the bandwidth W and the Coulomb repulsion U between electrons of opposite spins on the same site. For a fixed density of electrons, n = 1; as the ratio U/W increases, we expect to find a zero-temperature phase transition from a metallic state to an insulator where the electrons are frozen at the lattice sites. The method employed here to investigate this transition is the Gutzwiller approach (Gutzwiller, 1965). This is a variational method which played an important role in providing a framework for the ideas of Mott on the problem of the metal–insulator transition (Mott, 1974). On the metallic side of the transition, this approach describes a metal with renormalised Fermi liquid parameters, essentially with enhanced effective mass and uniform susceptibility. This is the so-called highly correlated electron gas (Brinkman and Rice, 1970) for which the metal oxide V 2 O 3 is the prototype system. In the insulating side, Gutzwiller's approximation yields a noninteracting system of localised electrons. This insulating phase is incompressible , i.e. has zero compressibility and infinite zero temperature susceptibility. The latter is not a consequence of interactions but simply due to the divergence of the Curie susceptibility, χ ∝ 1 /T , of non-interacting moments at T = 0. Then, in Gutzwiller's approach the transition is from a paramagnetic metal to a paramagnetic insulator. We will continue using the designation Mott transition to describe this type of phase transition where the insulating phase has no long-range magnetic order. In real systems, most frequently, localisation due to interactions is accompanied by the appearance of antiferromagnetic order.
Correlated electron systems represent a wide class of materials which at equilibrium display fascinating properties. Several recent experimental breakthroughs in the field of femtosecond spectroscopy and cold atomic gases allow nowadays to investigate the real time dynamics of these many-body quantum systems. Since strongly correlated systems usually escape single particle approaches, the theoretical study of their dynamics constitutes a formidable problem which necessitates the development of novel techniques. In this Thesis we investigate the out-of-equilibrium physics of simple paradigmatic models that are believed to capture some essential physics of interacting fermions by means of the time dependent extension of the Gutzwiller Variational Approach. After an introductory Chapter on the recent results in this field, in Chapter 2 we present the Gutzwiller Approach in-and-out of equilibrium. In Chapter 3 we investigate the dynamics for the single band Hubbard model after a linear ramp of the Coulomb interaction. We will show that a dynamical transition appears for any duration of the ramp; this dynamical point is adiabatically connected to the zero temperature Metal-to-Insulator transition. We will then consider the role of quantum fluctuations beyond mean field. In Chapter 4 we consider the dynamics of an initial antiferromagnetic state under a quench of the interaction in the single band fermionic Hubbard model. We will show that non-thermal ordered states survive more than expected and that two different nonequilibrium antiferromagnets can be distinguished. Finally in Chapter 5 we will consider a two-band Hubbard model which we believe captures the main physics of the paradigmatic compound vanadium sesquioxide, V2 O3 . After an investigation of the equilibrium properties for this model, we will provide evidences that non-thermal metallic phases can emerge upon an excitation of a Mott insulator.
We review the recently proposed extension of the Gutzwiller approximation, M. Schiro' and M. Fabrizio, Phys. Rev. Lett. 105, 076401 (2010), designed to describe the out-of-equilibrium time-evolution of a Gutzwiller-type variational wave function for correlated electrons. The method, which is strictly variational in the limit of infinite lattice-coordination, is quite general and flexible, and it is applicable to generic non-equilibrium conditions, even far beyond the linear response regime. As an application, we discuss the quench dynamics of a single-band Hubbard model at half-filling, where the method predicts a dynamical phase transition above a critical quench that resembles the sharp crossover observed by time-dependent dynamical mean field theory. We next show that one can actually define in some cases a multi-configurational wave function combination of a whole set of mutually orthogonal Gutzwiller wave functions. The Hamiltonian projected in that subspace can be exactly evaluated and is equivalent to a model of auxiliary spins coupled to non-interacting electrons, closely related to the slave-spin theories for correlated electron models. The Gutzwiller approximation turns out to be nothing but the mean-field approximation applied to that spin-fermion model, which displays, for any number of bands and integer fillings, a spontaneous Z2 symmetry breaking that can be identified as the Mott insulator-to-metal transition.
Materials with strongly correlated electrons show a multitude of unconventional phenomena, such as superconductivity or magnetism, that cannot be explained in a single-particle picture and necessitate to take many-body effects into account. A route to explain the origin of these phenomena consists in constructing effective models for these materials that are then solved, often using numerical techniques. Paradigmatic models for two different classes of strongly correlated electron systems are investigated in this thesis: The Hubbard model is a paradigmatic model to describe the Mott metal-insulator transition (MIT) and the Kondo lattice model describes the low-energy physics in heavy fermion compounds. Both models are analyzed numerically using different variations of a quantum cluster technique, the variational cluster approximation (VCA). The first part focuses on the MIT, as it is investigated experimentally in quasi-two-dimensional charge-transfer salts. Its nature and associated universality class are still heavily discussed and both experiments and theoretical approaches come to different conclusions. In this thesis, the Mott transition is investigated on an anisotropic two-dimensional Hubbard model at half-filling. By using a control parameter, which induces this anisotropy, strong evidence for Mott quantum criticality is found in weakly coupled Hubbard chains. The results at zero temperature show that the second-order critical end point Tc of the interaction-driven metal-insulator transition can be tuned down to zero at strong anisotropy. Further results for the antiferromagnetic phase suggest a similar picture and motivate adding the anisotropy as a new axis in the phase diagram to account for a low-temperature critical end point of the Mott transition. The second part of this thesis focuses on unconventional phases that emerge in a paradigmatic model for heavy fermion systems, the Kondo lattice model. Using the VCA, the two-dimensional Kondo lattice model is investigated in the paramagnetic phase as well as in phases with broken symmetry. An antiferromagnetic phase at weak coupling and a phase with Kondo-singlet formation at strong coupling are analyzed. Within the antiferromagnetically ordered region, two phases with different Fermi-surface topology, which are separated by a discontinuous transition, are identified. The model is also tested for s-wave superconductivity as found recently using dynamical mean-field theory, but no indications for robust local pairing are seen within VCA. Instead, nodal d-wave superconductivity is found and analyzed in a large range of couplings and electron fillings, and its interplay with antiferromagnetic order at weak coupling is discussed. This motivates further studies on extended models that might allow for a close comparison with experiments on heavy fermion systems. Understanding the formation of unconventional phases in strongly correlated electron systems is not only of fundamental interest, but it is also essential for designing new functional materials. Although the results of approximate cluster techniques have to be handled with care, the new aspects presented in this thesis pave the way for further investigations in this direction.
The physics of strongly correlated materials poses one of the most challenging problems in condensed-matter sciences. Standard approximations applicable to wide classes of materials such as the local density approximation fail, due to the importance of the Coulomb repulsion between localized electrons. Instead, we resort to non-perturbative many-body methods. The calculations are, however, only feasible for rather small model systems. The full Hamiltonian of a real material is approximated by a model Hamiltonian comprising only the most important electronic degrees of freedom, while the effect of all other electrons is included in an average way by renormalizing the parameters. Realistic calculations of strongly correlated materials need to include sufficiently many of these electronic degrees of freedom. The new generation of massively parallel supercomputers allows for these realistic calculations. However, exploiting their computational power requires newly devised algorithms. As a solver we use the Lanczos method, which needs the full many-body state of the correlated system. It is thus limited by the available main memory. The foremost problem for a distributed-memory implementation is that the multiplication of the Hamiltonian to the many-body state leads to highly non-local memory access patterns. A simple yet important observation leads to an efficient solution: in the kinetic term of the Hamiltonian the electron-spin is conserved. Thus, writing the many-body vector as a matrix v(iup,idn), where the indices label spin-configurations, we find that the hopping term only connects vector elements that differ in one index. Hence, storing entire slices v(iup,:) on one node, the kinetic term for the spin-down electrons is local to that process. After transposing v, the same is true for the hopping of the spin-up electrons. With an efficient matrix transpose, implemented via MPI collectives, we thus obtain a version of the Lanczos method that efficiently well on JUBL, Jülich's Blue Gene/L. Using the code we treated two-orbital Hubbard Models which show interesting Mott-band insulator transitions. Kohn's criterium which determines the phase of a system by ground-state properties alone, namely the response of the ground-state energies to a vector potential, was employed. This vector potential can be equivalently regarded as a change in boundary conditions, requiring complex wave vectors. We found that half-filled systems become metallic in the region between a band and Mott insulating phase. To extrapolate to infinite system sizes and therefore getting access to an arbitrarily high resolution in reciprocal space, we use cluster perturbation theory (CPT). It works by diagonalizing a finite cluster exactly and then treating the hopping between identical clusters perturbatively in strong coupling perturbation theory. This leads to an effectively infinite chain. Due to our efficient Lanczos implementation it is possible to calculate relatively large clusters, minimizing finite-size effects. The combination of our Lanczos code and CPT allowed for a treatment of the the organic metal TTF-TCNQ. We were able to solve a long-standing problem in the interpretation of its experimental spectra. Being an interesting one-dimensional metal TTF-TCNQ has been studied thoroughly for more than 30 years. Theoretical estimates as well as experimental measurements suggest values for the parameters of the Hubbard model t and U. With these values Hubbard model calculations show qualitatively the same features as APRES experiments, namely the signatures of spin-charge separation at the Gamma-point. But in order to fit the experimental data to the numerical calculations t has to be doubled, which translates into a broadening of the spectra. This ad-hoc adjustment is not only unsatisfactory but also the temperature dependence of the spectral function is incorrectly reproduced. We resolved this problem by properly including nearest-neighbor Coulomb interactions. Recent DFT calculations show that indeed the commonly used values of t and U are correct. But Coulomb interactions between electrons on neighboring sites are also significant and must not be neglected. Thus, we studied the effect of the next-neighbor interaction parameter V being at about U/2. Including V in our Lanczos calculations for obtaining the angular-integrated spectral function shows that its effect is to broaden the spectra, comparable to increasing the value of t. Moreover, we found that doubling t indeed mimics the effect of Vapprox U/2. Surprisingly, the broadening of the spectrum as a result of V can be understood in first-order Rayleigh-Schrödinger perturbation theory.