Dokument

Summary:
The present dissertation concerns itself with the many body theory of quantum mechanics. In particular, the Hubbard model is examined, which has served as a testing environment for strongly correlated electron systems since the 1960s, and is not completely described despite decades of intense research. Here, the focus is on a strong, repulsive electron-electron interaction, and doping slightly below half-filling. The repulsion between electrons favors antiferromagnetic order, while the presence of holes leads to a frustrated configuration, which can usually not be characterized using perturbative approaches. The reason for examining this particular point in the phase diagram is the conjecture that it is a simplified model of the cuprate superconductors, whose pairing mechanism is not entirely understood despite their discovery in 1986. Countless analytical and numerical methods have been developed to calculate the ground state of this parameter set and other complicated models. The method of this thesis uses a tensor network representation, which can be viewed as a means of data compression for quantum mechanics. The most prominent algorithm in this area is the Density-Matrix Renormalization Group (DMRG), which is a reliable method for the ground state calculation of one-dimensional quantum systems. In this context, the present thesis introduces a prototype for the generalization of the DMRG to two dimensions. This is done by representing the electronic wavefunction as a Projected entangled Pair State (PEPS), whose quantum mechanical entanglement is tailored to the structure of a two-dimensional lattice. The ground state can then be determined through local, variational optimization, which scales linearly with system size. The thesis is structured as follows: First, the iterative diagonalization is outlined (Sec. 2.1), which is used to determine extremal eigenvalues. It is followed by a detailed description of symmetries within the Hubbard model (Sec. 2.2), since their exploitation is essential for an efficient implementation of tensor networks. Afterwards, the Wigner-Eckart theorem is derived, which is needed for non-abelian symmetries. Chapter 3 concerns itself with quantum mechanical entanglement and how it can be utilized in many body physics. Sec. 3.1 presents the AKLT model, which serves as a motivation for tensor network representations of ground states. Subsequently, the von Neumann entropy is elucidated (Sec. 3.2), which quantifies the entanglement inside of wavefunctions. Sec. 3.3 makes a connection to physical systems by describing several models and their scaling of the entropy. Chapter 4 explains elementary tensor operations that take both abelian and non-abelian symmetries into account. The emphasis is less on mathematical rigor than on intelligibility and pragmatism. Mechanisms are often explained using examples, assuming the general case is self-explaining. First, tensors are defined in general (Sec. 4.1), in particular, how their symmetries are taken advantage of and how to store them. There follows an explanation of the permutation of indices (Sec. 4.2), the pairwise contraction of tensors (Sec. 4.3), the fusion and splitting of indices (Sec. 4.4), and the factorization of a tensor into two (Sec. 4.5). Finally, we present an efficient method for contracting multiple tensors, which usually poses the main bottleneck in tensor-network algorithms. Chapter 5 delivers a compact explanation of the DMRG in the language of matrix product states. Although the DMRG itself is not the goal of this research project, it is worthwhile to describe its general principles, before moving on to PEPS. Multiple concepts can then be used as a stepping stone to treating two dimensions. Sec. 6.1 finally takes on PEPS itself. Since only open boundary conditions are considered, we have to consider finite PEPS (fPEPS), as opposed to iPEPS, which is fundamentally based on translational invariance. Subsequently, a scheme that adapts the representation of a local Hamiltonian to the topology of a PEPS is presented (Sec. 6.2). This is followed by a detailed explanation of how to determine expectation values approximately (Sec. 6.3), which is one of the central difficulties of the algorithm. Sec. 6.4 finally puts all of the pieces together to define the overarching algorithm for the variational optimization of fPEPSs as used to determine ground states of two-dimensional quantum systems. In Chapter 7, the fPEPS algorithm is applied to the two-dimensional Hubbard model. First, the influence of the approximate calculation of expectation values is investigated and the error is quantified. Afterwards, a few test simulations are conducted on 3x3 and 8x8 lattices. The algorithm yields a stable convergence of the energy and local charge and spin densities. The local observables resemble those of previous publications qualitatively. However, our version of fPEPS is not yet able to reproduce ground state energies up to more than a couple of significant figures due to some technical subtleties. Finally, Chapter 8 discusses the development status of the optimization in detail, what improvements are pending, and what physical phenomena could be analyzed in the future.

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