The central problem in the modern electronic structure theory is the calculation of cor- relation energy, possibly by an approach that would account for both static and dynamic correlation in an efficient, balanced and accurate way. In this thesis, I present a collection of methods that combine the effective treatment of dynamic correlation by the coupled cluster theory with density matrix renormalization group, a well-established technique for calculations of strongly correlated systems. The connection between them is achieved via the tailored coupled clusters (TCC) ansatz, which conveniently does not impose any ad- ditional computational costs. After the successful initial assessment, we developed more efficient implementations of these methods by employing the local approaches based on pair natural orbitals. This way, we extended the range of possible applications to larger systems with thousands of basis functions. To assess the accuracy of TCC as well as its local counterparts, we performed a variety of benchmark calculations ranging from small, yet challenging systems such as the nitrogen molecule or tetramethyleneethane diradical, to larger molecules like oxo-Mn(Salen) or Fe(II)-porphyrin model. 1

<p>In this thesis we discuss in detail the density-matrix renormalization group (DMRG) for simulating low-energy properties of quantum spin models. We implement an original DMRG routine on the S=1/2 antiferromagnetic Heisenberg chain and benchmark its efficiency against exact results (energies, correlation functions, etc.) as well as conformal field-theoretical calculations due to finite-size scaling (ground-state energy and spin gap logarithmic corrections). Moreover, we apply the DMRG to a two-leg square ladder system, where in addition to bilinear exchange terms, we also consider an additional cyclic four-spin ring-exchange. The transposition of four spins gives rise to biquadratic exchange terms which are non-trivial to implement in the DMRG. Intermediate results of the ring-exchange are presented along with the difficulties presently encountered.</p> / Master of Science (MSc)

In this thesis, we investigate the entanglement and magnetic properties of frustrated spin systems and correlated electronic properties of conjugated carbon systems. In chapter 1, we present different approaches to solve the time-independent, nonrelativistic Schr¨odinger equation for a many-body system. We start with the full non-relativistic Hamiltonian of a multi nuclear system to describe the Born - Oppenheimer approximation which allows the study of electronic Hamiltonian which treats nuclear positions parametrically. We then also describe ab initio techniques such as the Hartree-Fock Method and density functional theories. We then introduce model Hamiltonians for strongly correlated systems such as the Hubbard, Pariser-Parr-Pople and Heisenberg models, and show how they result from the noninteracting one-band tight-binding model. In chapter 2, we discuss various numerical techniques like the exact diagonalization methods and density matrix renormalization group (DMRG) method. We also discuss quantum entanglement and the success of DMRG which can be attributed to the area law of entanglement entropy. In chapter 3, we study here different regions in phase diagrams of the spin-1/2, spin-1 and spin-3/2 one-dimensional antiferromagnetic Heisenberg systems with nearest-neighbor (J1) and next-nearest-neighbor (J2) interactions and dimerization (d ). Frustration arises for specific relative signs of the interactions J1 and J2. In particular, we analyze the behavior of the bipartite entanglement entropy and fidelity at the gapless to gapped phase transitions and across the lines separating different phases in the J2−d plane. All the calculations in this work are based on exact diagonalizations of finite systems. In chapter 4, we study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (JF 1 ) and antiferromagnetic (JA 1 ) nearest-neighbor interactions and a ferromagnetic next-nearest-neighbor interaction (JF 2 ). In this model frustration is present due to non-zero JF 2 . The model with site spin s behaves like a Haldane spin chain with site spin 2s in the limit of vanishing JF 2 and large JF 1 /JA 1 . We show that the exact ground state of the model can be found along a line in the parameter space. For fixed JF 1 , the phase diagram in the space of JA 1 −JF 2 is determined using numerical techniques complemented by analytical calculations. A number of quantities, including the structure factor, energy gap, entanglement entropy and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and potentially important feature of this model is that it can exhibit a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian. In chapter 5, we study correlated electronic properties of zigzag and armchair fused naphthalenes and polyperylene systems in the presence of long-range electronelectron interactions. We find that the ground state of zigzag fused naphthalene system is a higher spin state, while the ground state of armchair fused naphthalene is a singlet. The spin gap of polyperylene is unusually small and the ground state is a singlet. Our calculations of optical gap and two-photon gap suggest that polyperylene should exhibit fluorescence. From the charge gap calculation, we predict that in zigzag fused naphthalene and polyperylene systems, excitons are weakly binding. Peierls type of distortion is negligible in zigzag fused naphthalene and polyperylene systems, however, in armchair fused naphthalene system, interior bonds have tendency to distort in low-lying excited states. In chapter 6, we study the ground state spin of the Heisenberg spin-1/2 nearestneighboring antiferromagnetic exchange models of systems with fused odd member rings. In particular, we compute the ground state spin of fused three and five membered rings as well as fused five membered rings. In the thermodynamic limit, the ground state of the fused three and five membered system is a higher spin state, while fused five membered system shows a singlet ground state, for all system sizes.

This thesis deals with out-of-equilibrium transport phenomena in strongly correlated low-dimensional fermionic systems, with special emphasis on π-conjugated molecular materials. The focus of this work is to study real-time dynamics of spin and charge transport in these systems in order to investigate non-equilibrium transport in single-molecule electronic and spintronic devices. Chapter 1 describes the electronic structure and dynamics of strongly correlated fermionic systems in general, and in one-dimension, in particular. For this purpose, effective low-energy model Hamiltonians (used in this work) are discussed. Whenever applicable, approximate analytical and numerical methods commonly used in the literature to deal with these model Hamiltonians, are outlined. In the context of one-dimensional strongly correlated fermionic systems, analytical techniques like the Bethe ansatz and bosonization, and numerical procedures like exact diagonalization and DMRG, used for solving finite systems, are discussed in detail. Chapter 2 provides an overview of the different zero-temperature (T = 0) time-dependent DMRG algorithms, which have been used to study out-of-equilibrium time-dependent phenomena in low-dimensional strongly correlated systems. In Chapter 3 we employ the time-dependent DMRG algorithm proposed by Luo, Xiang and Wang [Phys. Rev. Lett. 91, 049701 (2003)], to study the role of dimerization and electronic correlations on the dynamics of spin-charge separation. We employ the H¨uckel and Hubbard models for our studies. We have modified the algorithm proposed by Luo et. al to overcome some of its limitations. Chapter 4 presents a generalized adaptive time-dependent density matrix renormalization group (DMRG) scheme developed by us, called the Double Time Window Targeting (DTWT) technique, which is capable of giving accurate results with lesser computational resources than required by the existing methods. This procedure originates from the amalgamation of the features of pace keeping DMRG algorithm, first proposed by Luo et. al, [Phys.Rev. Lett. 91, 049701 (2003)], and the time-step targeting (TST) algorithm by Feiguin and White [Phys. Rev. B 72, 020404 (2005)]. In chapter 5 we apply the Double Time Window Targeting (DTWT) technique, which was discussed in the previous chapter, for studying real-time quantum dynamics of spin-charge separation in π-conjugated polymers. We employ the Pariser-Parr-Pople (PPP) model which has long-range electron-electron interactions. For investigating real-time dynamics of spin and charge transport, we inject a hole at one end of polyene chains of different lengths and study the temporal evolution of its spin and charge degrees of freedom, using the DTWT td-DMRG algorithm. Chapter 6 we investigate the effect of terminal substituents on the dynamics of spin and charge transport in donor-acceptor substituted polyenes (D- (CH)x- A) chains, also known as push-pull polyenes. We employ long-range correlated model Hamiltonian for the D- (CH)x- A system and, real-time DMRG dynamics for time propagating the wave packet obtained by injecting a hole at a terminal site in the ground state of the system. Our studies reveal that the end groups do not affect the spin and charge velocities in any significant way, but change the amount of charge transported. We have compared these with the polymethineimine (CN)x system in which besides electron affinities, the nature of pz orbitals in conjugation also alternate from site to site. Chapter 7 presents our investigation on the effect of static electron-phonon coupling (dimerization) on the dynamics of spin-charge separation in particular, and transport in general, in π-conjugated polyene chains. The polyenes are modeled by the Pariser-Parr-Pople Hamiltonian, having long-range electron-electron correlations. Our studies reveal that spin and charge velocities depend both on the chain length and dimerization. The spin and charge velocities increase as dimerization increases, but the amount of charge and spin transported along the chain decrease with enhancement in dimerization. Furthermore, in the range 0.3≤ δ≤0.5, it is observed that the dynamics of spin-charge separation becomes complicated, and the charge degree of freedom is affected more by electron-phonon coupling compared to the spin degree of freedom.

The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.

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