Tensor networks are used to represent the wave functions of quantum many-body systems, but the standard approaches only consider two-body entanglement and do not work well for ``frustrated'' systems, where the underlying lattice geometry makes three- or many-body entanglement also important. A new tensor-network approach based on a novel tensor concept for describing such ``simplex'' entanglement shows great promise.
Tensor networks are used to represent the wave functions of quantum many-body systems, but the standard approaches only consider two-body entanglement and do not work well for frustrated'' systems, where the underlying lattice geometry makes three- or many-body entanglement also important. A new tensor-network approach based on a novel tensor concept for describing such simplex'' entanglement shows great promise.
In this thesis, we use the new numerical method multiscale entanglement
renormalization ansatz (MERA) to study the one dimensional spin system. This
method, MERA, utilizes a network of tensors to represent quantum many-body
systems on a lattice. Other examples of tensor network state are matrix product
states (MPSs) for 1D systems, tree tensor networks (TTNs) for systems with a
tree shape, and projected entangled-pair states (PEPSs) for 2D systems and be-
yond. The three structures dier in the graph that denes how the tensors are
interconnected into a network: The graphs for MPSs, TTNs, and 2D PEPSs are,
respectively, a chain, a tree, and a 2D lattice. Importantly, from these tensor
networks the expectation value of local observables can be computed eciently.
Developing algorithms to simulate quantum many-body systems is important
for understanding and studying the physical principles. And these may become
powerful tools for the analysis of quantum computation and quantum information
theory. But in general ecient simulation is dicult, due to the fundamental
quantum physical laws.
The main method we have studied and worked on is multiscale entangle-
ment renormalization ansatz (MERA) which is a variational ansatz for many-
body states. MERA is a tree tensor network for 1D, 2D systems and beyond.
We have developed code of ternary-MERA to compute the ground state energy,
expected values of local observables, and correlators for 1D lattice systems. In
a nutshell, MERA is an algorithm for entanglement renormalization which is a
numerical technique based on locally reorganizing the Hilbert space of a quantum
many-body system with the aim to reduce the amount of entanglement in its wave
function. We use this method to study 1D Ising model, XXZ model, and J1 - J2
model, and then comparing the numerical results with exact diagonalization (ED)
and density matrix renormalization group (DMRG).
Emergent phenomena of interacting quantum many-body systems arguably pose some of the greatest challenges to modern theoretical physics. Many of these systems are characterized by drastic changes in their properties as they are driven into a regime where quantum effects and interactions become relevant. Examples are the fractional quantum Hall effect and high-temperature superconductors, to name just two. New theoretical tools are necessary to study such systems, as even simple model Hamiltonians are not fully understood after decades of intense research. Numerical simulations are emerging as an indispensable tool alongside more traditional analytic approaches and have seen immense progress in recent years. In this thesis, we study tensor network states, a new class of numerical methods that have been developed in recent years drawing on insights from quantum information theory. Combining renormalization group methods with knowledge about the entanglement structure of ground states of quantum systems, tensor network states aim to efficiently describe ground states of strongly correlated quantum systems. Center stage is taken by the projected entangled-pair states. This ansatz for two-dimensional systems is discussed in detail and its capabilities are assessed by comparison to established methods and by exploring a frustrated system. We then develop a general formalism to exploit Abelian symmetries in a tensor network algorithm and demonstrate its application to projected entangled-pair states. A large part of this thesis is devoted to the discussion of a class of supersymmetric models for interacting lattice fermions. After introducing the concept of supersymmetry in quantum mechanics and an extensive overview of conformal field theory and its relation to numerical simulations, we study the critical theory of the supersymmetric model on the square ladder. We use two tensor network state algorithms, the density-matrix renormalization group and multi-scale entanglement renormalization along with exact diagonalization to explore this challenging system. In the final two chapters, we discuss two applications of tensor network states, namely simulations of the SU(3) Heisenberg model in two dimensions using two different tensor network state algorithms with the goal of illuminating the nature of its ground state, and the indistinguishability, a measure to detect phase transitions and classify wave functions which is particularly suitable for calculations based on tensor network states.
Tensor network states and parton wave functions are two pivotal methods for studying quantum many-body systems. This work connects these two subjects as we demonstrate that a variety of parton wave functions, such as projected Fermi sea and projected fermionic or bosonic paired states, can be represented exactly as tensor networks. The results can be compressed into matrix product states with moderate bond dimensions so various physical quantities can be computed efficiently. For the projected Fermi sea, we develop an excellent compression scheme with high fidelity using maximally localized Wannier orbitals. Numerical calculations on two parton wave functions demonstrate that our method exceeds commonly adopted Monte Carlo methods in some aspects. It produces energy and correlation function with very high accuracy that is difficult to achieve using Monte Carlo method. The entanglement measures that were almost impossible to compute before can also be obtained easily using our method.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
Tensor network states have proven to be powerful tools for investigating quantum many-body systems. Here, the authors propose a tensor network method, based on an adaptation of entanglement renormalization, for investigating strongly disordered systems. The method makes use of the strong disorder renormalization group to determine the order in which lattice sites are coarse grained, which sets the overall structure of the corresponding tensor network ansatz, before optimization using variational energy minimization. This approach leads to a new class of efficiently contractible tensor network ansatz for 1D systems, which may be understood as a generalization of the multiscale entanglement renormalization ansatz for disordered systems.