Phase Transitions, Critical Phenomena, and Correlation Functions in the 2D Ising Model and its Applications to Quantum Dynamics: A Tensor Network Approach

This thesis explores several aspects of the 2D Ising Model at both real and complex temperatures utilizing tensor network algorithms. We briefly discuss the importance of tensor networks in the context of forming efficient representations of wavefunctions and partition functions for quantum and classical many-body systems respectively, followed by a brief review of the tensor network renormalization algorithms to compute the one point and two point correlation functions. We use the Tensor Renormalization Group (TRG) to study critical phenomena and examine feasibility of accurate estimations of universal critical data for three critical points for three critical points in two dimensions -- the critical points for the isotropic and the anisotropic square lattice Ising models, and the Yang-Lee critical endpoint. The latter two exhibit appreciable corrections to scaling, making a clear case for uniform convergence in bond dimension apparent in our results. We are able to reproduce exactly known values to within 1 percent with modest effort of bond dimension 28. We analytically continue into the complex temperature plane to identify and study novel phases and phase transitions of the 2D isotropic as well as the 2D anisotropic Ising model. Regions of the phase space that lie in the complex coupling plane cannot be studied using standard Quantum Monte Carlo techniques (phase problem for complex coupling). Tensor networks provides us with a powerful tool to analyze and study regions of phase space in complex coupling planes. Evidence from tensor network renormalization techniques for the infinte 2D isotropic Ising model suggest the presence of only the standard paramagnetic (PM) and the ferromagnetic (FM) phases in the complex temperature plane. On the other hand, evidence from magnetization using tensor network renormalization techniques for the 2D anisotropic Ising model, and also numerical evidence based on the exact Onsager solution for the infinte 2D anisotropic Ising model suggests the presence of novel phases in certain regions of the complex temperature $\tanh(\beta)$ plane that exhibits quasi-long range modulated correlations. Phase transitions from the paramagnetic (PM) phase to these novel phases (which we label as non-ferromagnet or NFM phase) exhibit a one-sided square root singularity akin to a commensurate-incommensurate phase transition. In this thesis, we also present a quantum circuit with measurements and post-selection that exhibits a panoply of space- and/or time-ordered phases. These phases can range from ferromagnetic order to spin-density waves to time crystals. The corresponding time crystals for the quantum circuit presented in this thesis are incommensurate with the drive frequency, a behavior that is a deviation from the time crystal that have been found in unitary circuits. The period of the incommensurate time-crystal phase can be tuned by adjusting the parameters for the quantum circuit. We demonstrate that these novel phases, including the inherently non-equilibrium dynamical phases, correspond to the complex-temperature equilibrium phases of the exactly solvable infinite square-lattice anisotropic Ising model with doubly periodic boundary conditions. We also present a quantum circuit and briefly discuss special dual unitary points in the complex temperature plane for the square-lattice isotropic Ising Model and its implications in quantum information theory