Thesis advisor: Ziqiang . Wang / The transition between many-body localized states and the delocalized thermal states is an eigenstate phase transition at finite energy density outside the scope of conventional quantum statistical mechanics. We apply support vector machine (SVM) to study the phase transition between many-body localized and thermal phases in a disordered quantum Ising chain in a transverse external field. The many-body eigenstate energy E is bounded by a bandwidth W=Eₘₐₓ-Eₘᵢₙ. The transition takes place on a phase diagram spanned by the energy density ϵ=2(Eₘₐₓ-Eₘᵢₙ)/W and the disorder strength ẟJ of the spin interaction uniformly distributed within [-ẟJ, ẟJ], formally parallel to the mobility edge in Anderson localization. In our study we use the labeled probability density of eigenstate wavefunctions belonging to the deeply localized and thermal regimes at two different energy densities (ϵ's) as the training set, i.e., providing labeled data at four corners of the phase diagram. Then we employ the trained SVM to predict the whole phase diagram. The obtained phase boundary qualitatively agrees with previous work using entanglement entropy to characterize these two phases. We further analyze the decision function of the SVM to interpret its physical meaning and find that it is analogous to the inverse participation ratio in configuration space. Our findings demonstrate the ability of the SVM to capture potential quantities that may characterize the many-body localization phase transition. To further investigate the properties of the transition, we study the behavior of the entanglement entropy of a subsystem of size L_A in a system of size L > L_A near the critical regime of the many-body localization transition. The many-body eigenstates are obtained by exact diagonalization of a disordered quantum spin chain under twisted boundary conditions to reduce the finite-size effect. We present a scaling theory based on the assumption that the transition is continuous and use the subsystem size L_A/ξ as the scaling variable, where ξ is the correlation length. We show that this scaling theory provides an effective description of the critical behavior and that the entanglement entropy follows the thermal volume law at the transition point. We extract the critical exponent governing the divergence of ξ upon approaching the transition point. We again study the participation entropy in the spin-basis of the domain wall excitations and show that the transition point and the critical exponent agree with those obtained from finite size scaling of the entanglement entropy. Our findings suggest that the many-body localization transition in this model is continuous and describable as a localization transition in the many-body configuration space. Besides the many-body localization transition driven by disorder, We also study the Coulomb repulsion and temperature driving phase transitions. We apply a finite-temperature Gutzwiller projection to two-dimensional Hubbard model by constructing a "Gutzwiller-type" density matrix operator to approximate the real interacting density matrix, which provides the upper bound of free energy of the system. We firstly investigate half filled Hubbard model without magnetism and obtain the phase diagram. The transition line is of first order at finite temperature, ending at 2 second order points, which shares qualitative agreement with dynamic mean field results. We derive the analytic form of the free energy and therefor the equation of states, which benefits the understanding of the different phases. We later extend our approach to take anti-ferromagnetic order into account. We determine the Neel temperature and explore its interesting behavior when varying the Coulomb repulsion. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

Thesis advisor: Ziqiang Wang / Gutzwiller wave function is an important theoretical technique for treating local electron-electron correlations nonperturbatively in condensed matter and materials physics. It is concerned with calculating variationally the ground state wave function by projecting out multi-occupation configurations that are energetically costly. The projection can be carried out analytically in the Gutzwiller approximation that offers an approximate way of calculating expectation values in the Gutzwiller projected wave function. This approach has proven to be very successful in strongly correlated systems such as the high temperature cuprate superconductors, the sodium cobaltates, and the heavy fermion compounds. In recent years, it has become increasingly evident that strongly correlated systems have a strong propensity towards forming inhomogeneous electronic states with spatially periodic superstrutural modulations. A good example is the commonly observed stripes and checkerboard states in high-$T_\mathrm c$ superconductors under a variety of conditions where superconductivity is weakened. There exists currently a real challenge and demand for new theoretical ideas and approaches that treats strongly correlated inhomogeneous electronic states, which is the subject matter of this thesis. This thesis contains four parts. In the first part of the thesis, the Gutzwiller approach is formulated in the grand canonical ensemble where, for the first time, a spatially (and spin) unrestricted Gutzwiller approximation (SUGA) is developed for studying inhomogeneous (both ordered and disordered) quantum electronic states in strongly correlated electron systems. The second part of the thesis applies the SUGA to the $t$-$J$ model for doped Mott insulators which led to the discovery of checkerboard-like inhomogeneous electronic states competing with $d$-wave superconductivity, consistent with experimental observations made on several families of high-$T_{\mathrm c}$ superconductors. In the third part of the thesis, new concepts and techniques are developed to study the Mott transition in inhomogeneous electronic superstructures. The latter is termed ``SuperMottness'' which is shown to be a general framework that unifies the two paradigms in the physics of strong electronic correlation: Mott transition and Wigner crystallization. A cluster Gutzwiller approximation (CGA) approach is developed that treats the local ($U$) and extended Coulomb interactions ($V$) on equal footing. It is shown with explicit calculations that the Mott-Wigner metal-insulator transition can take place far away from half-filling. The mechanism by which a superlattice potential enhances the correlation effects and the tendency towards local moment formation is investigated and the results reveal a deeper connection among the strongly correlated inhomogeneous electronic states, the Wigner-Mott physics, and the multiorbital Mott physics that can all be united under the notion of SuperMottness. It is proposed that doping into a superMott insulator can lead to coexistence of local moment and itinerant carriers. The last part of the thesis studies the possible Kondo effect that couples the local moment and the itinerant carriers. In connection to the sodium rich phases of the cobaltates, a new Kondo lattice model is proposed where the itinerant carriers form a Stoner ferromagnet. The competition between the Kondo screening and the Stoner ferromagnetism is investigated when the conduction band is both at and away from half-filling. / Thesis (PhD) — Boston College, 2009. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.