An Algorithm Based On The Graded Projected Entangled State Representation For Strongly Correlated Electronic Systems In Two Spatical Dimensions And Ground-State Phase Diagram Of The Two Dimensional T-J Model

The problem of high temperature superconductor mechanism is one of the mostimportant problems in condensed matter. Investigated for more than twenty years fromboth experimental and theoritical approaches, however, physicists can hardly extract auseful clue for the truth of the high temperature superconductivity mechanism. As thenumerical simulations methods turn to be the most important tool to study quantummany body system, theorists begin to study the problem by building theoritical modelfor high temperature superconductor with the help of numerical simulations. For severalyears, a large amount of results has been pubulished; however, different conclusions aredrawn from different methods. In the thesis, based on the original tensor networkstructures and algorithms, we build an algorithm to find the ground state wave functionswith the graded Projected Entangled-Pair State representation for translationallyinvariant strongly correlated electronic systems on infinite-size lattices in two spatialdimensions, with which, we simulate the two-dimensional t-J model suggested byAnderson at and away from half filling, with truncation dimensions up to6. Both thephase separation line and the ground-state phase diagram of the model are investigatedin the context of the tensor network algorithm in terms of the graded ProjectedEntangled-Pair State representation of the ground-state wave functions. We are able tolocate a line of phase separation between the Heisenberg anti-ferromagnetic statewithout hole and a hole-rich state, which qualitatively agrees with the results based onthe high-temperature expansions. For both J=0.4t and J=0.8t, a systematiccomputation is performed to identify all the competing ground states for variousdopings. It is found that, besides a possible Nagaoka’s ferromagnetic state, thehomogeneous regime consists of four different phases: one phase with charge and spindensity wave order coexisting with ap_x (p_y) wave superconducting state, one phasewith the symmetry mixing of d+s wave superconductivity in the spin-singlet channelandp_x (p_y) wave superconductivity in the spin-triplet channel in the presence of ananti-ferromagnetic background, one superconducting phase with extended s wavesymmetry, and one superconducting phase withp_x (p_y) wave symmetry in aferromagnetic background.The thesis is organized as follows. In Chapter1, the two dimentional t-J model isintroduced and a brief review for the tensor network reperentations and the corresponding algorithms are presented. In Chapter2, the foundmental knowledgesspecially about tensor network to develop the gPEPS algorithm are presented. First,the imaginary time evolution, the idea to generate the approximated ground states forquantum systems, is introduced and modified for the tensor network reprentationcontaining both one dimentional infinite Matrix Product States (iMPS) representationand two dimentional infinite Projected Entangled Pair State (iPEPS) representationscorresponding to one dimentional and two dimentional systems. In the second part ofthe Chapter2, the row to row tranfer matrix scheme to contract the infinitetensor-network plane is explained step by step. The main part of the thesis is presentedin Chapter3and Chapter4. In Chapter3, an interpretation of the graded ProjectedEntangled Pair State (gPEPS) representation from the Valence Bond State (VBS)approach is displayed in detail, based on which, the newly-develped gPEPS algorithm isbuilt. First, based on the concept of the VBS, a naturally generalization for2-D systemgVBS is obtained. Then considering the order and sign generating by ordering fromgVBS, a gPEPS representation of the ground state for the two dimensional fermenicsystem is built. In the end, with a sign line, we naturally develop a gPEPS algorithm fora ferminic system, which is generated from the original iPEPS algorithm for bosonicsystem with the same imaginary time evolution idea. In Chaper4, we show thesimulation results for two dimentional t-J model on a square lattice at and away halffilling for different values of parameters, especially for J/t=0.4and J/t=0.8with thegPEPS algorithm. First we compare the phase separation line with the results obtainedfrom the other algorithms. Eventually, the proposed ground-state phase diagram ofthe t-J model is discovered based on the result of J/t=0.4and J/t=0.8for the filling from0to1, obtained by performing a systematic computation identifing all the competingground states for various dopings. Chapter5is devoted to a summary.