Tensor network methods for low-dimensional quantum systems

Li, Jheng-Wei (2022): Tensor network methods for low-dimensional quantum systems. Dissertation, LMU München: Fakultät für Physik

Abstract

This thesis contributes to developing and applying tensor network methods to simulate correlated many-body quantum systems. Numerical simulations of correlated quantum many-body systems are challenging. To describe a many-body wavefunction, the required number of parameters grows exponentially with respect to the system size. This exponential wall fundamentally limits our progress on correlated quantum systems in low dimensions. Tensor network methods in recent years have proven to be a useful framework to understand, control and possibly reduce this intrinsic complexity. The basic idea of tensor network methods is to decompose a many-body wave function as a network of small, multi-index tensors. A one-dimensional (1D) tensor network factorizes a wave function into a train of three-index tensors. This 1D tensor network ansatz is called a matrix product state (MPS) or a tensor train. A two-dimensional (2D) tensor network state is called a projected entangled-pair state (PEPS). This peculiar name PEPS comes from a quantum information perspective, where each local tensor is interpreted as a projector and correlates with the rest of the tensor network through (auxiliary) maximally entangled pairs. In the first part, we consider MPSs to study 1D and quasi-2D quantum systems. The key parameter of an MPS is its bond dimension, which controls the numerical accuracy. How large a bond dimension can be reached highly depends on the algorithms employed. The contemporary algorithms, although widely used, have to limit the bond dimension due to their high numerical costs. We develop a controlled bond expansions (CBE) scheme that allows us to grow the bond dimensions with marginal computational efforts. This CBE scheme stems from a geometric point of view to parametrize the variational space of an MPS and can be applied in various contexts. Here, we focus on applying the CBE scheme to two types of problems. The first are optimization problems, like solving the extremal eigenvalue problem. This is relevant for the ground state search, and we show that CBE can accelerate the convergence of MPS in terms of CPU time. The second is to solve ordinary differential equations, such as the time-dependant Schrödinger equation. With the help of CBE, it becomes feasible to use MPS to simulate long-time dynamics that could not be accurately computed hitherto. In the second part, we employ PEPS to simulate 2D quantum systems. PEPS is an expensive but powerful tool to simulate 2D lattices directly in the thermodynamic limit. The PEPS on infinite lattices is acronymed iPEPS. For completeness, a pedagogical review of iPEPS based on Benedikt Bruognolo’s PhD work, which I helped polsih for publication in Scipost, is included to cover the algorithmic details. Using iPEPS methods, we study the two-dimensional t-J model on square lattices at the small doping. In this work, we uncover the importance of spin rotational symmetry. Our numerics suggest that by allowing spontaneous spin-symmetry breaking or not, we can supress or permit the emergence of superconducting order in the thermodynamic limit. This finding provides useful insight to cuprate materials. Also, we use iPEPS to investigate the ground state nature of the honeycomb Kitaev-Γ model. Through a joint effort of classical and iPEPS simulations, we identify an exotic magnetic order in the parameter regime relevant to α-RuCl3 materials. In the third and final part, we study the parton construction of tensor network states. Here, we do not simulate the ground state of a given many-body Hamiltonian. Instead, we take an indirect route that first constructs a parton state in an enlarged Hilbert space, and then applies the Gutzwiller projection to return to the original physical Hilbert space. Such a parton approach has been an important theoretical technique to treat electron-electron correlations nonperturbatively in condensed matter physics. Its marriage with tensor network methods furthers its influence. Various properties of parton wave functions, which are difficult to compute previously, can now be easily accessed. We first use the parton approach to construct MPSs that harbor SU(N) chiral topological orders. The MPS representation of these Gutzwiller projected parton states allows us to compute entanglement spectra, which hold crucial information to characterize different chiral topological orders. We also develop a method to construct parton states using PEPSs. In this project, we use PEPS to approximate parton states of the π-flux models that host U(1)-Dirac spin liquids. Our approach enables us to compute the critical exponent of the spin-spin correlations for the spin-half system, whose value is still currently under debate.

• Dokumententyp: Dissertationen (Dissertation, LMU München)
• Themengebiete: 500 Naturwissenschaften und Mathematik ; 500 Naturwissenschaften und Mathematik > 530 Physik
• Fakultäten: Fakultät für Physik
• Sprache der Hochschulschrift: Englisch
• Datum der mündlichen Prüfung: 21. Oktober 2022
• 1. Bericht­erstatter:in: Delft, Jan von