The iPEPS tensor network, which is free of the sign problem notorious for the quantum Monte Carlo, is the method of choice to search for topological order in two-dimensional strongly correlated systems. However, finding a ground state represented by iPEPS is one problem, while extracting from this representation the desired information about topological order is quite another. The algorithm, proposed by the authors, does precisely this in a wide class of non-Abelian string net models.
The iPEPS tensor network, which is free of the sign problem notorious for the quantum Monte Carlo, is the method of choice to search for topological order in two-dimensional strongly correlated systems. However, finding a ground state represented by iPEPS is one problem, while extracting from this representation the desired information about topological order is quite another. The algorithm, proposed by the authors, does precisely this in a wide class of non-Abelian string net models.
We present a method of extracting information about the topological order from the ground state of a strongly correlated two-dimensional system computed with the infinite projected entangled pair state (iPEPS). For topologically ordered systems, the iPEPS wrapped on a torus becomes a superposition of degenerate, locally indistinguishable ground states. Projectors in the form of infinite matrix product operators onto states with well-defined anyon flux are used to compute topological S and T matrices (encoding mutual and self-statistics of emergent anyons). The algorithm is shown to be robust against a perturbation driving string-net toric code across a phase transition to a ferromagnetic phase. Our approach provides accurate results near quantum phase transition, where the correlation length is prohibitively large for other numerical methods. Moreover, we used numerically optimized iPEPS describing the ground state of the Kitaev honeycomb model in the toric code phase and obtained topological data in excellent agreement with theoretical prediction.
We present a method of extracting information about topological order from the ground state of a strongly correlated two-dimensional system represented by an infinite projected entangled pair state (iPEPS). As in previous works [A. Francuz et al., Phys. Rev. B 101, 041108(R) (2020) and A. Francuz and J. Dziarmaga, Phys. Rev. B 102, 235112 (2020)] we begin by determining symmetries of the iPEPS represented by infinite matrix product operators (iMPO) that map between the different iPEPS transfer matrix fixed points, to which we apply the fundamental theorem of matrix product states to find zipper tensors between products of iMPO's that encode fusion properties of the anyons. The zippers can be combined to extract topological F symbols of the underlying fusion category, which unequivocally identifies the topological order of the ground state. We bring the F symbols to the canonical gauge, and also compute the Drinfeld center of this unitary fusion category to extract the topological S and T matrices encoding mutual statistics and self-statistics of the emergent anyons. The algorithm is applied to Abelian toric code, Kitaev model, double semion, and twisted quantum double of Z3, as well as to non-Abelian double Fibonacci, double Ising, and quantum double of S3 and Rep(S3) string-net models.
Infinite projected entangled pair states (iPEPS) have emerged as a powerful tool for studying interacting two-dimensional fermionic systems. In this review, we discuss the iPEPS construction and some basic properties of this tensor network (TN) ansatz. Special focus is put on (i) a gentle introduction of the diagrammatic TN representations forming the basis for deriving the complex numerical algorithm, and (ii) the technical advance of fully exploiting non-abelian symmetries for fermionic iPEPS treatments of multi-band lattice models. The exploitation of non-abelian symmetries substantially increases the performance of the algorithm, enabling the treatment of fermionic systems up to a bond dimension D=24 D = 24 on a square lattice. A variety of complex two-dimensional (2D) models thus become numerically accessible. Here, we present first promising results for two types of multi-band Hubbard models, one with 2 2 bands of spinful fermions of \mathrm{SU}(2)_\mathrm{spin} \otimes \mathrm{SU}(2)_\mathrm{orb} S U ( 2 ) s p i n ⊗ S U ( 2 ) o r b symmetry, the other with 3 3 flavors of spinless fermions of \mathrm{SU}(3)_\mathrm{flavor} S U ( 3 ) f l a v o r symmetry.
Determining the properties of quantum many-body systems is a central challenge in modern physics. Being able to determine the macroscopic properties of a system from its microscopic description would hasten progress in many fields of science and technology. However, we currently lack the tools to solve such problems generally, or even to develop a theoretical intuition about how many systems might behave. From a simple Hamiltonian description of the system, one may obtain complex, highly correlated collective behaviour. Computational techniques have played a major part in the effort to determine properties of quantum many-body systems. However, as the total degrees of freedom in the system scales exponentially in the system size, numerical diagonalization of the Hamiltonian quickly becomes computationally intractable and one must develop more efficient approximate techniques to explore the system. Present numerical methods such as quantum Monte Carlo and series expansion have provided insight into many systems of interest, but are also held back by fundamental difficulties. In this thesis, we focus on tensor networks, a relatively new ansatz for representing quantum many-body states. Tensor networks are motivated by two ideas from quantum information: firstly, that quantum entanglement is the source of the immense difficulty of simulating quantum systems classically, and secondly that the ground states of certain Hamiltonians exist in a low-entanglement region of the entire Hilbert space. The strength of tensor networks is that they provide a systematic way of representing this class of low-entanglement quantum states. In particular, this thesis describes the iPEPS algorithm for computing the ground states of infinite, two-dimensional quantum lattice systems based on the Projected Entangled Pair States (PEPS) ansatz. We then benchmark the algorithm by computing the phase diagrams of several systems that have been studied with other techniques. Lastly, we apply our algorithm to problems that are not well solved by current approaches, such as frustrated spin systems.
Infinite Projected entangled-pair state (iPEPS) is a two-dimensional tensor network (TN) ansatz targeting the ground states of lattice models directly in the thermodynamic limit. It is the most successful among the true two-dimensional TN methods, challenging the more common density matrix renormalization group computations conducted on a finite-width cylinders. iPEPS, being a variational method does not suffer from the infamous sign problem, and hence could be readily applied to both fermionic problems off the half-filling and frustrated spin system. Using the imaginary-time optimization, the so called Full Update, led to the celebrated results of iPEPS such as the analysis of stripes in doped t-J and Hubbard models or the insight into the magnetization plateaus in Shastry-Sutherland model. However, the physics of the continuous phase transitions from the symmetry-broken Neel phase into the valence-bond solids or spin-liquids in frustrated magnets is still eluding iPEPS. The main difficulty lies in the optimization, since Full Update is just not precise enough. In the pursuit of the best variational iPEPS states, the gradient methods were put forward in 2016 by Corboz and Vanderstraeten. However, due to their complex nature, they have not been widely adopted especially for iPEPS ansatze with larger unit cells. We will introduce an alternative approach first applied in the context of iPEPS by Liao et al., based on the well established method of Algorithmic differentiation (AD) which allows us to evaluate gradients for iPEPS in a robust and conceptually simple way. Finally, we will present the results of our investigation into the physics of frustrated magnets putting AD to work.
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