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arXiv:1905.02164 · Physical review. B./Physical review. B

Optimization of infinite projected entangled pair states: The role of multiplets and their breaking

Juraj Hašík, Federico Becca

Published Aug 21, 2019• 6 citations
arXivHTMLPDF

Abstract

The infinite projected entangled pair states (iPEPS) technique [J. Jordan et al., Phys. Rev. Lett. 101, 250602 (2008)] has been widely used in the recent years to assess the properties of two-dimensional quantum systems, working directly in the thermodynamic limit. This formalism, which is based upon a tensor-network representation of the ground-state wave function, has several appealing features, e.g., encoding the so-called area law of entanglement entropy by construction; still, the method presents critical issues when dealing with the optimization of tensors in order to find the best possible approximation to the exact ground state of a given Hamiltonian. Here, we discuss the obstacles that arise in the optimization by imaginary-time evolution within the so-called simple and full updates and connect them to the emergence of a sharp multiplet structure in the ``virtual'' indices of tensors. In this case, a generic choice of the bond dimension $D$ is not compatible with the multiplets and leads to a symmetry breaking (e.g., generating a finite magnetic order). In addition, varying the initial guess, different final states may be reached, with very large deviations in the magnetization value. In order to exemplify this behavior, we show the results of the $S=1/2$ Heisenberg model on an array of coupled ladders, for which a vanishing magnetization below the critical interladder coupling is recovered only for selected values of $D$, while a blind optimization with a generic $D$ gives rise to a finite magnetization down to the limit of decoupled ladders.

DOI
10.1103/physrevb.100.054429

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The infinite projected entangled pair states (iPEPS) technique [J. Jordan et al., Phys. Rev. Lett. 101, 250602 (2008)] has been widely used in the recent years to assess the properties of two-dimensional quantum systems, working directly in the thermodynamic limit. This formalism, which is based upon a tensor-network representation of the ground-state wave function, has several appealing features, e.g., encoding the so-called area law of entanglement entropy by construction; still, the method presents critical issues when dealing with the optimization of tensors in order to find the best possible approximation to the exact ground state of a given Hamiltonian. Here, we discuss the obstacles that arise in the optimization by imaginary-time evolution within the so-called simple and full updates and connect them to the emergence of a sharp multiplet structure in the ``virtual'' indices of tensors. In this case, a generic choice of the bond dimension DDD is not compatible with the multiplets and leads to a symmetry breaking (e.g., generating a finite magnetic order). In addition, varying the initial guess, different final states may be reached, with very large deviations in the magnetization value. In order to exemplify this behavior, we show the results of the S=1/2S=1/2S=1/2 Heisenberg model on an array of coupled ladders, for which a vanishing magnetization below the critical interladder coupling is recovered only for selected values of DDD, while a blind optimization with a generic DDD gives rise to a finite magnetization down to the limit of decoupled ladders.
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.
Infinite projected entangled-pair states (iPEPS) have been introduced to accurately describe many-body wave functions on two-dimensional lattices. In this context, two aspects are crucial: the systematic improvement of the Ansatz by the optimization of its building blocks, i.e., tensors characterized by bond dimension DDD, and the extrapolation scheme to reach the ``thermodynamic'' limit $D\ensuremath{\rightarrow}\ensuremath{\infty}$. Recent advances in variational optimization and scaling based on correlation lengths demonstrated the ability of iPEPS to capture phases with spontaneously broken continuous symmetry such as the antiferromagnetic (N\'eel) one with high fidelity, in addition to valence-bond solids which are already well described by finite-DDD iPEPS. In contrast, systems in the vicinity of continuous quantum phase transitions still present a challenge for iPEPS, especially when non-Abelian symmetries are involved. Here, we consider the iPEPS Ansatz to describe the continuous transition between the (gapless) antiferromagnet and the (gapped) paramagnet that exists in the S=1/2S=1/2S=1/2 Heisenberg model on coupled two-leg ladders. In particular, we show how accurate iPEPS results can be obtained down to a narrow interval around criticality and analyze the scaling of the order parameter in the N\'eel phase in a spatially anisotropic situation.
The infinite projected entangled pair states (iPEPS) algorithm [J. Jordan et al., Phys. Rev. Lett. 101, 250602 (2008)] has become a useful tool in the calculation of ground-state properties of two-dimensional quantum lattice systems in the thermodynamic limit. Despite its many successful implementations, the method has some limitations in its present formulation which hinder its application to some highly entangled systems. The purpose of this paper is to unravel some of these issues, in turn enhancing the stability and efficiency of iPEPS methods. For this, we first introduce the fast full update scheme, where effective environment and iPEPS tensors are both simultaneously updated (or evolved) throughout time. As we shall show, this implies two crucial advantages: (i) dramatic computational savings and (ii) improved overall stability. In addition, we extend the application of the local gauge fixing, successfully implemented for finite-size PEPS [M. Lubasch et al., Phys. Rev. B 90, 064425 (2014)], to the iPEPS algorithm. We see that the gauge fixing not only further improves the stability of the method but also accelerates the convergence of the alternating least-squares sweeping in the (either ``full'' or ``fast full'') tensor update scheme. The improvement in terms of computational cost and stability of the resulting ``improved'' iPEPS algorithm is benchmarked by studying the ground-state properties of the quantum Heisenberg and transverse-field Ising models on an infinite square lattice.
An infinite projected entangled-pair state (iPEPS) is a powerful variational tensor network ansatz for two-dimensional ground states in the thermodynamic limit, and can be seen as a natural generalization of a matrix-product state to two dimensions. One of the main challenges in iPEPS simulations is the optimization of the tensors, i.e., finding the optimal variational parameters, in order to have the best representation of the ground state of a given Hamiltonian. The author presents a variational optimization scheme, in which the energy is minimized in an iterative way by sweeping over all the tensors in the ansatz, in a similar spirit as done in the density-matrix renormalization group method. Benchmark results for challenging problems are presented that show that the variational scheme yields considerably more accurate results than the previously best imaginary-time evolution algorithm, with a similar computational cost and with a faster convergence towards the ground state.
We investigate the imaginary-time relaxation critical dynamics of the two-dimensional transverse-field Ising model using infinite projected entangled pair states (iPEPS) with the full-update strategy. Simulating directly in the thermodynamic limit, we explore the relaxation process near the critical point with two types of initial states: a fully polarized state and a product state with a small magnetization. For the fully polarized state, the magnetization shows a power law scaling M∝τ−β/(νz)M\propto τ^{-β/(νz)}M∝τ−β/(νz) in the imaginary-time evolution, from which both the critical point and critical exponent can be determined with high accuracy. For the nearly paramagnetic state, the relaxation process exhibits a behavior of M∝τθM\propto τ^θM∝τθ with θ=0.1958θ=0.1958θ=0.1958 being the critical initial-slip exponent, which is in good agreement with that obtained from the dynamic scaling of the self-correlation in quantum Monte Carlo method. These universal features emerge well before the system converges to the ground state, demonstrating the efficiency of imaginary-time evolution for probing quantum criticality. Our results demonstrate that iPEPS can serve as a robust and scalable method for studying dynamical critical phenomena in two-dimensional quantum many-body systems.
An infinite projected entangled pair state (iPEPS) is a tensor network ansatz to represent a quantum state on an infinite 2D lattice whose accuracy is controlled by the bond dimension DDD. Its real, Lindbladian, or imaginary time evolution can be split into small time steps. Every time step generates a new iPEPS with an enlarged bond dimension ${D}^{\ensuremath{'}}>D$, which is approximated by an iPEPS with the original DDD. In P. Czarnik and J. Dziarmaga, Phys. Rev. B 98, 045110 (2018), an algorithm was introduced to optimize the approximate iPEPS by maximizing directly its fidelity to the one with the enlarged bond dimension ${D}^{\ensuremath{'}}$. In this paper, we implement a more efficient optimization employing a local estimator of the fidelity. For imaginary time evolution of a thermal state's purification, we also consider using unitary disentangling gates acting on ancillas to reduce the required DDD. We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field hx{h}_{x}hx​ in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy: 0.10.1%0.1 for hx=2.5{h}_{x}=2.5hx​=2.5 and 0.50.5%0.5 for the more challenging case of hx=2.9{h}_{x}=2.9hx​=2.9 close to the quantum critical point at hx=3.04438(2){h}_{x}=3.04438(2)hx​=3.04438(2).