The Tensor Network (TN) algorithm for strongly related electronic quantum lattice systems has become the uppermost algorithm for studying strongly related electronic quantum lattice systems in numerical computation. Focusing on the TN algorithm based on 2-D spin systems, this paper discusses the theoretical principles of the Projected Entangled Pair States (PEPS) algorithm and its implementation process. It also explores the infinite PEPS (iPEPS) algorithm for simulating the spin-1/2 infinite 2-D square lattice quantum XYX model. The critical phenomena of the two-dimensional XYX quantum system are analyzed from the perspectives of ground-state fidelity and order parameters, and the quantum critical point is identified. It is anticipated that with the further development of TN algorithms, the single-site fidelity theory of quantum phase transitions (QPTs) can be applied to various quantum many-body systems in condensed matter physics, thereby deepening our understanding of the extremely rich states of matter in the quantum world. Research indicates that quantum information theory provides a universal method for the study of QPTs, which can identify several ground-state fidelities and order parameters through the ground-state wave function of the system, several ground state fidelity and local order parameters can be identified to fully describe some quantum phase transition points, greatly enriching the traditional Landau theory of symmetry breaking QPTs.
The Tensor Network (TN) algorithm for strongly related electronic quantum lattice systems has become the uppermost algorithm for studying strongly related electronic quantum lattice systems in numerical computation. Focusing on the TN algorithm based on 2-D spin systems, this paper discusses the theoretical principles of the Projected Entangled Pair States (PEPS) algorithm and its implementation process. It also explores the infinite PEPS (iPEPS) algorithm for simulating the spin-1/2 infinite 2-D square lattice quantum XYX model. The critical phenomena of the two-dimensional XYX quantum system are analyzed from the perspectives of ground-state fidelity and order parameters, and the quantum critical point is identified. It is anticipated that with the further development of TN algorithms, the single-site fidelity theory of quantum phase transitions (QPTs) can be applied to various quantum many-body systems in condensed matter physics, thereby deepening our understanding of the extremely rich states of matter in the quantum world. Research indicates that quantum information theory provides a universal method for the study of QPTs, which can identify several ground-state fidelities and order parameters through the ground-state wave function of the system, several ground state fidelity and local order parameters can be identified to fully describe some quantum phase transition points, greatly enriching the traditional Landau theory of symmetry breaking QPTs.
With the deepening of people's understanding of quantum phase transitions, various efficient numerical algorithms have developed rapidly. It is necessary to develop tensor network algorithms to efficiently generate the ground state wave function of quantum many-body system and then calculate the ground state fidelity per site. This mini review article aims to discusses the tensor network algorithm for studying 2-dimensional quantum lattice systems, namely projected entangled pair state, and introduce how to characterize phase transitions in quantum many-body systems from a novel perspective-fidelity. It includes the wave function updating of projected entangled pair state and the tensor network contraction scheme, which can be used to calculate the ground-state fidelity per site of the quantum system, and the phase diagram of the ground-state can be determined without any knowledge of the order of the 2-dimensional quantum system, thus indicating that tensor network algorithms can provide a means to describe quantum phase transitions.
Tensor network (TN) algorithms for strongly correlated electron lattice systems have become one of the most important numerical methods for studying such systems. Based on the tensor network algorithm for spin ladder systems, the study of quantum phase transitions (QPTs) in spin ladder systems through string order parameters has shown that quantum information theory provides a universal method for studying quantum phase transitions: without knowing the local or non-local order parameters of the system, or whether local or non-local order parameters exist, the ground-state wave function of the system can be obtained. If the ground-state lacks the traditional Landau long-range order and thus has no local order parameters, or if several non-local order parameters are found through other means, some topological quantum phases can be completely described, thereby greatly enriching the traditional Landau symmetry-breaking phase transition theory.
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
The purpose of this paper is to describe how to describe the phase transition of quantum multibody system from the perspective of the basic concept of quantum information science - fidelity. For systems the traditional way of characterization that undergo a quantum phase transition is Landau's introduction of order and the fluctuation. In particular, for phase transitions caused by spontaneous breaking symmetry, that is, different ground state wave functions are orthogonal to each other. This suggests that, contrary to the ups and downs of traditional expressions, the concept of irrelevant information and vital information can be introduced. By introducing the basic amount of ground lattice fidelity, quantifying irrelevant information and critical information which can identify the quantum phase transition of the system. It is worth emphasizing that no matter the internal order of quantum multibody systems is a traditional symmetry or other novel quantum order, they all can be applied such as topological order, single-grid ground state fidelity. In order to efficiently calculate the single-grid ground state fidelity of quantum multibody systems, tensor network algorithm is needed. This kind of algorithm is the result of the deepening of quantum entanglement in recent years. Quantum entanglement expounds the working principle of real-space renormalization group algorithm, especially density matrix re-grouping algorithm.
In the numerical calculation, the projected entangled pair state (PEPS) algorithm is the most important tensor network algorithm for two-dimensional strongly correlated electron quantum lattice system. In this paper, the optimization of PEPS for two-dimensional quantum system is discussed. An optimization connection between how to update the PEPS tensor and how to calculate the physical observable is investigated, for the tensor network algorithm based on the PEPS representation, which can greatly improve the utilization of computing resources. In this case, optimized PEPS algorithm, as a powerful tool, can be used to study quantum phase transitions and quantum critical phenomena in the thermodynamic limit of the two-dimensional strongly correlated electron quantum lattice system. Of course, optimization of PEPS algorithm program has many other applications, such as adding U(1) and SO(2) symmetry in PEPS algorithm, etc.
We have investigated an intriguing connection between bifurcations, reduced fidelity per lattice site, local order parameter, universal order parameter, entropy and quantum phase transitions in the ground state for quantum three-state Potts model with two coupled infinite-size ladder system, in the context of the tensor network algorithm. The tensor network algorithm produces degenerate symmetry-breaking ground-state wave functions arising from the Z3 symmetry breaking, each of results from a randomly chosen initial state. We expect that our approach might provide further insights into critical phenomena in quantum many-body infinite lattice systems in condensed matter physics.