Interacting systems of anyons pose a unique challenge to condensed-matter simulations due to their nontrivial exchange statistics. These systems are of great interest as they have the potential for robust universal quantum computation but numerical tools for studying them are as yet limited. We show how existing tensor network algorithms may be adapted for use with systems of anyons and demonstrate this process for the one-dimensional multiscale entanglement renormalization ansatz (MERA). We apply the MERA to infinite chains of interacting Fibonacci anyons, computing their scaling dimensions and local scaling operators. The scaling dimensions obtained are seen to be in agreement with conformal field theory. The techniques developed are applicable to any tensor network algorithm, and the ability to adapt these ans\"atze for use on anyonic systems opens the door for numerical simulation of large systems of free and interacting anyons in one and two dimensions.
Interacting systems of anyons pose a unique challenge to condensed-matter simulations due to their nontrivial exchange statistics. These systems are of great interest as they have the potential for robust universal quantum computation but numerical tools for studying them are as yet limited. We show how existing tensor network algorithms may be adapted for use with systems of anyons and demonstrate this process for the one-dimensional multiscale entanglement renormalization ansatz (MERA). We apply the MERA to infinite chains of interacting Fibonacci anyons, computing their scaling dimensions and local scaling operators. The scaling dimensions obtained are seen to be in agreement with conformal field theory. The techniques developed are applicable to any tensor network algorithm, and the ability to adapt these ans"atze for use on anyonic systems opens the door for numerical simulation of large systems of free and interacting anyons in one and two dimensions.
Interacting systems of anyons pose a unique challenge to condensed matter simulations due to their non-trivial exchange statistics. These systems are of great interest as they have the potential for robust universal quantum computation, but numerical tools for studying them are as yet limited. We show how existing tensor network algorithms may be adapted for use with systems of anyons, and demonstrate this process for the 1-D Multi-scale Entanglement Renormalisation Ansatz (MERA). We apply the MERA to infinite chains of interacting Fibonacci anyons, computing their scaling dimensions and local scaling operators. The scaling dimensions obtained are seen to be in agreement with conformal field theory. The techniques developed are applicable to any tensor network algorithm, and the ability to adapt these ansaetze for use on anyonic systems opens the door for numerical simulation of large systems of free and interacting anyons in one and two dimensions.
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).
The numerical study of anyonic systems is known to be highly challenging due to their nonbosonic, nonfermionic particle-exchange statistics, and with the exception of certain models for which analytical solutions exist, very little is known about their collective behavior as a result. Meanwhile, the density matrix renormalization group (DMRG) algorithm is an exceptionally powerful numerical technique for calculating the ground state of a low-dimensional lattice Hamiltonian, and has been applied to the study of bosonic, fermionic, and group-symmetric systems. The recent development of a tensor network formulation for anyonic systems opened up the possibility of studying these systems using algorithms such as DMRG, though this has proved challenging in terms of both programming complexity and computational cost. This paper presents the implementation of DMRG for finite anyonic systems, including a detailed scheme for the implementation of anyonic tensors with optimal scaling of computational cost. The anyonic DMRG algorithm is demonstrated by calculating the ground-state energy of the ``golden chain,'' which has become the benchmark system for the numerical study of anyons and is shown to produce results comparable to those of the anyonic time-evolving block decimation algorithm and superior to the variationally optimized anyonic multiscale renormalization ansatz, at far lesser computational cost.
In a recent contribution [P. Corboz, G. Evenbly, F. Verstraete, and G. Vidal, arXiv:0904.4151 (unpublished)] entanglement renormalization was generalized to fermionic lattice systems in two spatial dimensions. Entanglement renormalization is a real-space coarse-graining transformation for lattice systems that produces a variational ansatz, the multiscale entanglement renormalization ansatz (MERA), for the ground states of local Hamiltonians. In this paper we describe in detail the fermionic version of the MERA formalism and algorithm. Starting from the bosonic MERA, which can be regarded both as a quantum circuit or in relation to a coarse-graining transformation, we indicate how the scheme needs to be modified to simulate fermions. To confirm the validity of the approach, we present benchmark results for free and interacting fermions on a square lattice with sizes between $6\ifmmode\times\else\texttimes\fi{}6$ and $162\ifmmode\times\else\texttimes\fi{}162$ and with periodic boundary conditions. The present formulation of the approach applies to generic tensor network algorithms.
Tensor networks are a class of methods for studying many-body systems. They give a geometrical description of the internal structure of many-body states, operators, and partition functions, that can be used to implement efficient algorithms to simulate them numerically. In this thesis, after providing a very brief overview of the field, we present new tensor network methods for three different use cases. \n \nFirst, we present a new real-space renormalization group algorithm for tensor networks. Compared to existing methods, its advantages are its low computational cost, simplicity of implementation, and applicability to any network. We benchmark it on the 2D classical Ising model and find accuracy comparable with the best existing tensor network methods. Due to its simplicity and generalizability, we consider our algorithm to be an excellent candidate for implementation of real-space renormalization in higher dimensions, and discuss some of the details and the remaining challenges in 3D. \n \nSecond, we show how certain topological conformal defects of critical lattice theories can be represented as tensor networks, using the 2D classical Ising model as an example. Furthermore, we show how such tensor network descriptions, combined with a renormalization group algorithm, can be used to obtain accurate estimates of the universal properties of these defects. We also show how coarse-graining of defects can be applied to any conformal defect (i.e. not just topological ones), and yields a set of associated scaling dimensions. \n \nFinally, we leave behind the focus on renormalization group methods, and present a method for spatially resolving the overlap between two tensor network states, to obtain localized information about the similarities and differences between them. For a given region, the similarity of two states in this region can be quantified by the Uhlmann fidelity of their reduced density matrices, and we show how such fidelities can be efficiently computed in many cases when the two states are represented as tensor networks. We demonstrate the usefulness of evaluating such subsystem fidelities with three example applications: studying local quenches, comparing critical and non-critical states, and quantifying convergence in tensor network simulations.
Tensor network algorithms (TNAs) represent one of the most recent developments in the field of numerical methods for the simulation of strongly-correlated manybody systems.Arising as a natural consequence of an improved understanding of the entanglement structures intrinsic to the ground-state manifolds of manybody Hilbert spaces, they correspond to algorithms exploiting entropic constraints expected to arise in various classes of many-body ground states.In this work we apply and develop TNAs for the simulation of various stronglycorrelated lattice models.Concretely, we apply tensor network renormalization (TNR) to the study of the classical Blume-Capel model (BCM).We propose to exploit the RG features specific to TNR to obtain an indicator for the vicinity of (multi-)critical points.We show that in the case of the BCM it leads to a location of its tricritical point matching the accuracy of state-of-the-art Monte Carlo approaches.This allows us to characterize the underlying c = 7/10 conformal field theory with an excellent accuracy.We then present a self-contained introduction to the most widely used techniques for the simulation of one-and two-dimensional quantum systems, where we cover matrix product states (MPS) and projected entangled-pair states (PEPS) in detail.We briefly discuss the multi-scale entanglement renormalization Ansatz (MERA).We apply infinite PEPS (iPEPS) to the simulation of the Kitaev-Heisenberg (KH) model, proposed as an effective low-energy theory for the so-called Iridate compounds of the form A 2 IrO 3 (A = Na, Li).We show the ability of iPEPS to accurately encode the complex ground-state physics of Kitaev's honeycomb model.When considering the KH model we confirm the existence of all previously found phases, locate all phase transitions in the phase diagram, finding good agreement with previous studies, and provide estimates for the survival regions of the spinliquid phases in the thermodynamic limit.We briefly discuss the nature of these transitions.We conclude this work with a study of various formulations of iPEPS on cylindrical geometries.We benchmark the proposed formulations by studying the transversefield Ising model and find good performance for a subset of the formulations studv ied.We then carry out a comparison between iPEPS and iMPS methods for the Heisenberg and Hubbard models and find a range of cylinder widths over which both methods exhibit comparable performance.We find evidence for the potential of iPEPS simulations on cylinders and argue that our findings provide support for future studies employing both MPS and PEPS methods in conjunction.vi Wir schlieSSen diese Arbeit mit einer Untersuchung verschiedener Formulierungen von iPEPS für zylindrische Geometrien ab.Wir bewerten die vorgeschlagenen Formulierungen durch die Untersuchung des Ising-Modells im transversalen Feld und finden gute Ergebnisse für eine Teilmenge der untersuchten Formulierungen.Wir führen dann einen Vergleich zwischen iPEPS und iMPS Methoden für die Heisenberg und Hubbard Modelle durch und finden einen Bereich von Zylinderbreiten, in dem beide Methoden vergleichbare Leistungen bringen.Wir finden Beweise für das Potenzial von iPEPS-Simulationen auf Zylinder und argumentieren, dass unsere Erkenntnisse Unterstützung für zukünftige Studien liefern, die sowohl MPSals auch PEPS-Methoden zusammen verwenden.viii Despite the considerable progress made towards the understanding of stronglycorrelated systems, much work remains to be done and, as is perhaps to be expected, the broad nature of the problems encountered means that no single approach is likely to be applicable in every circumstance.From a numerical perspective a wide variety of techniques have been developed over time.Of these, the most accurate is exact diagonalization (ED) which, as its name indicates, attempts to partially diagonalize the Hamiltonian operator.However, owing to the exponential growth of Hilbert space dimension as the system size is increased, this method rapidly becomes inapplicable.Indeed, even when targeting