The matrix product state (MPS) is utilized to study the ground-state properties and quantum phase transitions (QPTs) of the one-dimensional extended quantum compass model (EQCM). The MPS wave functions are argued to be very efficient descriptions of the ground states, and are numerically determined by imaginary-time projections. The ground-state energy, correlations, quantum entanglement and its spectrum, local and nonlocal order parameters, etc., are calculated and studied in detail. It is revealed that the von Neumann entanglement entropy, as well as the nearest-neighbor correlation functions, can be used to detect the second-order QPTs, but not the first-order ones, while fidelity detections can recognize both. The entanglement spectrum is extracted from the MPS wave function and found to be doubly degenerate in disordered phases, where nonzero string order parameters exist. Moreover, with the linearized tensor renormalization group method, the specific-heat curves are evaluated and their low-temperature behaviors are investigated. Compared with the exact solutions, our results verify that these MPS-based numerical methods are very accurate and powerful, and can be employed to investigate other EQCMs which do not permit exact solutions at present.
The matrix product state (MPS) is utilized to study the ground-state properties and quantum phase transitions (QPTs) of the one-dimensional extended quantum compass model (EQCM). The MPS wave functions are argued to be very efficient descriptions of the ground states, and are numerically determined by imaginary-time projections. The ground-state energy, correlations, quantum entanglement and its spectrum, local and nonlocal order parameters, etc., are calculated and studied in detail. It is revealed that the von Neumann entanglement entropy, as well as the nearest-neighbor correlation functions, can be used to detect the second-order QPTs, but not the first-order ones, while fidelity detections can recognize both. The entanglement spectrum is extracted from the MPS wave function and found to be doubly degenerate in disordered phases, where nonzero string order parameters exist. Moreover, with the linearized tensor renormalization group method, the specific-heat curves are evaluated and their low-temperature behaviors are investigated. Compared with the exact solutions, our results verify that these MPS-based numerical methods are very accurate and powerful, and can be employed to investigate other EQCMs which do not permit exact solutions at present.
The matrix product state (MPS) is utilized to study the ground state properties and quantum phase transitions (QPTs) of the one-dimensional quantum compass model (QCM). The MPS wavefunctions are argued to be very efficient descriptions of QCM ground states, and are numerically determined by imaginary time projections. The ground state energy, correlations, quantum entanglement and its spectrum, local and nonlocal order parameters, etc., are calculated and studied in details. It is revealed that the bipartite and block entanglement entropies, as well as the nearest neighbor correlation functions can be used to detect the second-order QPTs, but not the first-order ones, while fidelity detections can recognize both. The entanglement spectrum is extracted from the MPS wavefunction, and found to be doubly degenerate in disordered phases of QCM, where non-local string order parameters exist. Moreover, with linearized tensor renormalization group method, the specific heat curves are evaluated and their low temperature behaviors are investigated.
The matrix product state (MPS) is utilized to investigate the ground state properties and quantum phase transitions (QPTs) of the dimerized antiferromagnetic Heisenberg (DAH) model. The ground state MPS wavefunctions determined by the infinite time-evolving block decimation (iTEBD) algorithm are shown to be very efficient descriptions of DAH model. In the thermodynamic limit, the quantum entanglement, the bond energy, and the nearest-neighbor correlations are calculated. It is revealed that the singular behavior of the bipartite entanglement can detect the QPTs directly. The critical point J c 2 = 1.0 is determined evidently, and the quantum phase transition is argued to belong to the second-order category. At the critical point, logarithmic divergent character of the block entanglement is observed, and the system can be described by a free bosonic field theory.
In order to investigate the quantum phase transition in the one-dimensional quantum compass model, we numerically calculate non-local string correlations, entanglement entropy and fidelity per lattice site by using the infinite matrix product state representation with the infinite time evolving block decimation method. In the whole range of the interaction parameters, we find that four distinct string orders characterize the four different Haldane phases and the topological quantum phase transition occurs between the Haldane phases. The critical exponents of the string order parameters β = 1/8 and the cental charges c = 1/2 at the critical points show that the topological phase transitions between the phases belong to an Ising type of universality classes. In addition to the string order parameters, the singularities of the second derivative of the ground state energies per site, the continuous and singular behaviors of the Von Neumann entropy and the pinch points of the fidelity per lattice site manifest that the phase transitions between the phases are of the second-order, in contrast to the first-order transition suggested in previous studies.
By using the infinite time evolving block decimation in the presentation of infinite matrix product states, we study an extended quantum compass model (EQCM). This model does not only include extremely rich phase diagrams due to competitions of orbital degrees of freedom and anisotropic couplings between pseudospin-1/2 operators but also have the capacity to describe property of protected qubits for quantum computation which leads to lots of attentions paid to the phase boundaries of the EQCM. However, few attentions are paid to long-range topological string correlation order parameters of the EQCM. To study order parameters, one should understand spontaneous symmetry breaking which relates to Landau quantum phase transitions theory. Once spontaneous symmetry breaking happens, there should exist some local order which can be described by a local order parameter. This order parameter can be used to distinguish the phase from others. For continuous quantum phase transitions, in the critical regime, critical exponents can be extracted. Unfortunately, the long-range topological string correlation orders are beyond Landau quantum phase transitions theory, one can not directly use two paradigms of Landau-Ginzburg-Wilson. Usually, one can define a local order parameter by local magnetization. Naturally, one can also refer to this way to define the long-range topological string correlation order parameters by long-range topological string correlations on the following conditions, i.e. the quantum system undergoes a hidden spontaneous symmetry breaking; the long-range topological string correlation order parameter can be used to distinguish the phase from others; for continuous quantum phase transitions, the long-range topological string correlation order parameter satisfies scaling law when control parameter getting close to critical point. Based on above idea, in order to characterize the topological ordered phases and quantum phase transitions in the EQCM, even/odd long-range topological string correlations are introduced based on even/odd bonds. Hereafter, fidelity per lattice site, even/odd long-range topological string correlations, the saturation behavior of odd long-range topological string correlations and order parameters are calculated. The long-range topological string correlations show three distinguished behaviors which include decaying to zero, monotonic saturation and oscillatory saturation. By the above characterizations, oscillatory/monotonic odd long-range topological string correlation order parameter is derived. Then ground-state phase diagram of order parameters is computed which includes oscillatory/monotonic odd long-range topological string correlation order phase and antiferromagnetic phase. In the critical regime, critical exponent β=1/8 extracted from monotonic odd long-range topological string correlation order parameter and local magnetization shows the phase transition belongs to Ising universality. In addition, the phase transition points, the order of the phase transitions of fidelity show consistent with the results of order parameters.
The study of low-dimensional, strongly-interacting quantum systems has been of interest in the past few decades due to new and exotic physics that show great potential in the advancement of fundamental physics as well as technological applications. In order to investigate such systems, a vast number of techniques have been developed throughout the years, all of which have their respective strengths and weaknesses. In the world of one-dimensional (1D) quantum systems, the density-matrix renormalization group (DMRG) algorithm and matrix product states (MPS) have proven to be powerful and reliable tools that provide invaluable information while providing a wide range of data that support both theoretical and experimental studies. Coupled with the development and refinement of tensor networks - an ansatz for quantum many-body wavefunctions, in particular MPS, the DMRG algorithm has seen a spike of improvements that has pushed the boundaries of its application into more sophisticated and complex systems.This thesis broadly covers the study of several 1D many-body quantum systems represented by infinite MPS (iMPS) - a class of ansatz that represent translationally-invariant, 1D many-body quantum states. More specifically, it is divided into two main parts.The first part investigates the 1D topological Kondo insulator (TKI). This is an effective model designed to investigate how strong interactions between conduction electrons and localized moments result in a symmetry-protected topological (SPT) phase, specifically the Haldane phase. By studying the effect of the electron repulsion as well as the competition between the local and non-local couplings between the electrons and the local moments, it is found that a topological phase transition from the SPT phase to a topologically trivial phase occurs with the assistance of electron interaction. The properties of both topologically trivial and nontrivial groundstates were revealed via means of quantum information theoretic quantities such as the entanglement entropy, entanglement spectrum and non-local order parameters. These quantities unveil the nature, structure and the effect that the different interactions have on the groundstate properties. Besides the phases, the properties of the critical point was studied and classified. By obtaining the critical exponents and central charge, the universality class of this transition was found to be that of 1D free fermions/bosons. Furthermore, the presence of electron interactions was found not to alter this universality class.The second part of this thesis is a project that stems from the necessity to overcome the inherent limitation of an iMPS in representing a system at a cusp of a quantum phase transition (QPT), specifically, in determining the critical point and exponents. In the thermodynamic limit, the laws of statistical mechanics dictates that the system’s correlation length diverges at the critical point. However, when representing a critical state with an iMPS, the finite bond dimension implies that the correlation length of the iMPS remains finite, albeit large. This presents a fundamental challenge in extracting critical data since the critical exponents are defined for systems in the thermodynamic limit. To overcome this shortcoming, finite-entanglement scaling (FES) - the analog of finite-size scaling, has previously been used to locate the critical point and extract the critical exponents for systems represented by finite-bond dimensional iMPS. While FES serves its purpose sufficiently well, the computational cost is high for just about any moderately complex, real-world system. To overcome such difficulties, the second part of this thesis combines new ideas of using higher-order cumulants of the order parameter with previously-known iMPS techniques to determine the critical point and extract the exponents, thus relating statistical mechanics of QPTs and tensor networks of critical states. Some of the main methods used and developed here are FES, the application of the Binder cumulant in iMPS and the new cumulant exponent relation. Putting these tools together provides a scheme that can be used to efficiently locate the critical point and extract the critical exponents at a lower computational cost than previously-known methods. This is shown through comparisons of several exemplary models such as the 1D and quasi-1D transverse field Ising model, the 1D TKI, the S = 1 Heisenberg chain with single-ion anisotropy and the 1D Bose-Hubbard model.