The geometric entanglement per lattice site, as a holistic measure of the multipartite entanglement, serves as a universal marker to detect quantum phase transitions in quantum many-body systems. However, it is very difficult to compute the geometric entanglement due to the fact that it involves a complicated optimization over all the possible separable states. In this paper, we propose a systematic method to efficiently compute the geometric entanglement per lattice site for quantum many-body lattice systems in two spatial dimensions in the context of a newly-developed tensor network algorithm based on an infinite projected entangled pair state representation. It is tested for quantum Ising model in a transverse magnetic field and anisotropic spin 1/2 anti-ferromagnetic XYX model in an external magnetic field on an infinite-size square lattice. In addition, the geometric entanglement per lattice site is able to detect the so-called factorizing field. Our results are in a quantitative agreement with Quantum Monte Carlo simulations.
The geometric entanglement per lattice site, as a holistic measure of the multipartite entanglement, serves as a universal marker to detect quantum phase transitions in quantum many-body systems. However, it is very difficult to compute the geometric entanglement due to the fact that it involves a complicated optimization over all the possible separable states. In this paper, we propose a systematic method to efficiently compute the geometric entanglement per lattice site for quantum many-body lattice systems in two spatial dimensions in the context of a newly-developed tensor network algorithm based on an infinite projected entangled pair state representation. It is tested for quantum Ising model in a transverse magnetic field and anisotropic spin 1/2 anti-ferromagnetic XYX model in an external magnetic field on an infinite-size square lattice. In addition, the geometric entanglement per lattice site is able to detect the so-called factorizing field. Our results are in a quantitative agreement with Quantum Monte Carlo simulations.
A systematic analysis is performed for quantum phase transitions in a two-dimensional anisotropic spin-1/2 antiferromagnetic XYX model in an external magnetic field. With the help of an innovative tensor network algorithm, we compute the fidelity per lattice site to demonstrate that the field-induced quantum phase transition is unambiguously characterized by a pinch point on the fidelity surface, marking a continuous phase transition. We also compute an entanglement estimator, defined as a ratio between the one-tangle and the sum of squared concurrences, to identify both the factorizing field and the critical point, resulting in a quantitative agreement with quantum Monte Carlo simulation. In addition, the local order parameter is "derived" from the tensor network representation of the system's ground-state wave functions.
Geometric entanglement (GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. In this paper we outline a systematic method to compute GE for two-dimensional (2D) quantum many-body lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the q-state quantum Potts model on the square lattice with $q\ensuremath{\in}{2,3,4,5}$ is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models: the antiferromagnetic spin-1/2 XXX and anisotropic XYX models in an external magnetic field, and the antiferromagnetic spin-1 XXZ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with nonmaximum value at the transition point and the phase transition is continuous. For the models under consideration, we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively.
An efficient scheme for computing the geometric entanglement (GE) per lattice site for quantum many-body systems on a periodic finite-size chain is proposed in the context of a tensor network algorithm based on matrix product state representations. It has been systematically tested for three prototypical critical quantum spin chains, which belong to the same Ising universality class. The simulation results lend strong support to the previous claim (Shi et al 2010 New J. Phys. 12 025008; Stéphan et al 2010 Phys. Rev. B 82 180406R) that the leading finite-size correction to the GE per lattice site is universal, with its remarkable connection to the celebrated Affleck–Ludwig boundary entropy corresponding to a conformally invariant boundary condition.
The global geometric entanglement (GE) is studied in the context of newly developed tensor network algorithms for finite systems. For one-dimensional quantum spin systems it is found that, at criticality, the leading finite-size correction to the global GE per site behaves as b / n , where n is the size of the system and b a given coefficient. Our conclusion is based on the computation of the GE per spin for the quantum Ising model in a transverse magnetic field and for the spin-1/2 XXZ model. We also discuss the possibility of coefficient b being universal.
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.
We present a general graph-based projected entangled-pair state (gPEPS) algorithm to approximate ground states of nearest-neighbor local Hamiltonians on any lattice or graph of infinite size. By introducing the structural matrix, which codifies the details of tensor networks on any graphs in any dimension d, we are able to produce a code that can be essentially launched to simulate any lattice. We further introduce an optimized algorithm to compute simple tensor updates as well as expectation values and correlators with a mean-field-like effective environments. Though not being variational, this strategy allows to cope with PEPS of very large bond dimension (e.g., D=100) and produces remarkably accurate results in the thermodynamic limit in many situations, and specially when the correlation length is small and the connectivity of the lattice is large. We prove the validity of our approach by benchmarking the algorithm against known results for several models, i.e., the antiferromagnetic Heisenberg model on a chain, star and cubic lattices, the hardcore Bose-Hubbard model on square lattice, the ferromagnetic Heisenberg model in a field on the pyrochlore lattice, as well as the three-state quantum Potts model in field on the kagome lattice and the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. We further demonstrate the performance of gPEPS by studying the quantum phase transition of the 2d quantum Ising model in transverse magnetic field on the square lattice, and the phase diagram of the Kitaev-Heisenberg model on the hyperhoneycomb lattice. Our results are in excellent agreement with previous studies.