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arXiv:2107.03399 · SciPost Physics

Automatic differentiation applied to excitations with projected entangled pair states

Boris Ponsioen, Fakher F. Assaad, Philippe Corboz

Published Jan 6, 2022• 48 citations
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Abstract

The excitation ansatz for tensor networks is a powerful tool for simulating the low-lying quasiparticle excitations above ground states of strongly correlated quantum many-body systems. Recently, the two-dimensional tensor network class of infinite projected entangled-pair states gained new ground state optimization methods based on automatic differentiation, which are at the same time highly accurate and simple to implement. Naturally, the question arises whether these new ideas can also be used to optimize the excitation ansatz, which has recently been implemented in two dimensions as well. In this paper, we describe a straightforward way to reimplement the framework for excitations using automatic differentiation, and demonstrate its performance for the Hubbard model at half filling.

DOI
10.21468/scipostphys.12.1.006

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The excitation ansatz for tensor networks is a powerful tool for simulating the low-lying quasiparticle excitations above ground states of strongly correlated quantum many-body systems. Recently, the two-dimensional tensor network class of infinite projected entangled-pair states gained new ground state optimization methods based on automatic differentiation, which are at the same time highly accurate and simple to implement. Naturally, the question arises whether these new ideas can also be used to optimize the excitation ansatz, which has recently been implemented in two dimensions as well. In this paper, we describe a straightforward way to reimplement the framework for excitations using automatic differentiation, and demonstrate its performance for the Hubbard model at half filling.
Numerical treatment of two-dimensional strongly-correlated systems is both extremely challenging and of fundamental importance. Infinite projected entangled-pair states (PEPS), a class of tensor networks, have demonstrated cutting-edge performance for ground-state calculations, working directly in the thermodynamic limit. Furthermore, in recent years the application of PEPS has been extended to also low-lying excited states, using an ansatz that targets quasiparticle states above the ground state with high accuracy. A major technical challenge for those simulations is the accurate evaluation of summations of two- and three-point correlation functions with reasonable computational cost. In this paper, we show how a reformulation of nnn-point functions in the context of PEPS leads to extra contributions to the results that prove to play an important role. Benchmarks for the frustrated ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg model illustrate the improved precision, efficiency, and stability of the simulations compared to previous approaches. Leveraging automatic differentiation to generate the most tedious and error-prone parts of the computation, the straightforward implementation presented here is a step towards broader adoption of the PEPS excitation ansatz in future applications.
Tensor networks capture large classes of ground states of phases of quantum matter faithfully and efficiently. Their manipulation and contraction has remained a challenge over the years, however. For most of the history, ground state simulations of two-dimensional quantum lattice systems using (infinite) projected entangled pair states have relied on what is called a time-evolving block decimation. In recent years, multiple proposals for the variational optimization of the quantum state have been put forward, overcoming accuracy and convergence problems of previously known methods. The incorporation of automatic differentiation in tensor networks algorithms has ultimately enabled a new, flexible way for variational simulation of ground states and excited states. In this work we review the state-of-the-art of the variational iPEPS framework, providing a detailed introduction to automatic differentiation, a description of a general foundation into which various two-dimensional lattices can be conveniently incorporated, and demonstrative benchmarking results.
Numerical treatment of two dimensional strongly-correlated systems is both extremely challenging and of fundamental importance. Infinite projected entangled-pair states (PEPS), a class of tensor networks, have demonstrated cutting-edge performance for ground state calculations, working directly in the thermodynamic limit. Furthermore, in recent years the application of PEPS has been extended to also low-lying excited states, using an ansatz that targets quasiparticle states above the ground state with high accuracy. A major technical challenge for those simulations is the accurate evaluation of summations of two- and three-point correlation functions with reasonable computational cost. In this work, we show how a reformulation of nnn-point functions in the context of PEPS leads to extra contributions to the results that prove to play an important role. Benchmarks for the frustrated J1−J2J_1-J_2J1​−J2​ Heisenberg model illustrate the improved precision, efficiency and stability of the simulations compared to previous approaches. Leveraging automatic differentiation to generate the most tedious and error-prone parts of the computation, the straightforward implementation presented here is a step towards broader adoption of the PEPS excitation ansatz in future applications.
We present an extension of a framework for simulating single quasiparticle or collective excitations on top of strongly correlated quantum many-body ground states using infinite projected entangled pair states, a tensor network ansatz for two-dimensional wave functions in the thermodynamic limit. Our approach performs a systematic summation of locally perturbed states in order to obtain excited eigenstates localized in momentum space, using the corner transfer matrix method, and generalizes the framework to arbitrary unit cell sizes, the implementation of global Abelian symmetries and fermionic systems. Results for several test cases are presented, including the transverse Ising model, the spin-1/21/21/2 Heisenberg model, and a free fermionic model, to demonstrate the capability of the method to accurately capture dispersions. We also provide insight into the nature of excitations at the $k=(\ensuremath{\pi},0)$ point of the Heisenberg model.
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.
An infinite projected entangled-pair state (iPEPS) is a variational tensor network ansatz for two-dimensional wave functions in the thermodynamic limit where the accuracy can be systematically controlled by the bond dimension DDD. We show that for the doped Hubbard model in the strongly correlated regime (U/t=8,n=0.875)(U/t=8,n=0.875)(U/t=8,n=0.875) the iPEPS yields lower variational energies than state-of-the-art variational methods in the large two-dimensional limit, which demonstrates the competitiveness of the method. In order to obtain an accurate estimate of the energy in the exact infinite-DDD limit we introduce and test an extrapolation technique based on a truncation error computed in the iPEPS imaginary time evolution algorithm. The extrapolated energies are compared with accurate quantum Monte Carlo results at half-filling and with various other methods in the doped, strongly correlated regime.