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Register a new userVariational optimization with infinite projected entangled-pair states
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Philippe Corboz
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We present a scheme to perform an iterative variational optimization with infinite projected entangled-pair states (iPEPS), a tensor network ansatz for a two-dimensional wave function in the thermodynamic limit, to compute the ground state of a local Hamiltonian. The method is based on a systematic summation of Hamiltonian contributions using the corner transfer-matrix method. Benchmark results for challenging problems are presented, including the 2D Heisenberg model, the Shastry-Sutherland model, and the t-J model, which 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.
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We analyze a criterion which guarantees that the ground states of certain many body systems are stable under perturbations. Specifically, we consider PEPS, which are believed to provide an efficient description, based on local tensors, for the low energy physics arising from local interactions. In order to assess stability in the framework of PEPS, one thus needs to understand how physically allowed perturbations of the local tensor affect the properties of the global state. In this paper, we show that a restricted version of the Local Topological Quantum Order (LTQO) condition provides a checkable criterion which allows to assess the stability of local properties of PEPS under physical perturbations. We moreover show that LTQO itself is stable under perturbations which preserve the spectral gap, leading to nontrivial examples of PEPS which possess LTQO and are thus stable under arbitrary perturbations.
The recently developed stochastic gradient method combined with Monte Carlo sampling techniques [PRB {bf 95}, 195154 (2017)] offers a low scaling and accurate method to optimize the projected entangled pair states (PEPS). We extended this method to the fermionic PEPS (fPEPS). To simplify the implementation, we introduce a fermi arrow notation to specify the order of the fermion operators in the virtual entangled EPR pairs. By defining some local operation rules associated with the fermi arrows, one can implement fPEPS algorithms very similar to that of standard PEPS. We benchmark the method for the interacting spinless fermion models, and the t-J models. The numerical calculations show that the gradient optimization greatly improves the results of simple update method. Furthermore, much larger virtual bond dimensions ($D$) and truncation dimensions ($D_c$) than those of boson and spin systems are necessary to converge the results. The method therefore offer a powerful tool to simulate fermion systems because it has much lower scaling than the direct contraction methods.
Variational Monte Carlo studies employing projected entangled-pair states (PEPS) have recently shown that they can provide answers on long-standing questions such as the nature of the phases in the two-dimensional $J_1 - J_2$ model. The sampling in these Monte Carlo algorithms is typically performed with Markov Chain Monte Carlo algorithms employing local update rules, which often suffer from long autocorrelation times and interdependent samples. We propose a sampling algorithm that generates independent samples from a PEPS, bypassing all problems related to finite autocorrelation times. This algorithm is a generalization of an existing direct sampling algorithm for unitary tensor networks. We introduce an auxiliary probability distribution from which independent samples can be drawn, and combine it with importance sampling in order to evaluate expectation values accurately. We benchmark our algorithm on the classical Ising model and on variational optimization of two-dimensional quantum spin models.
The infinite Projected Entangled-Pair State (iPEPS) algorithm is one of the most efficient techniques for studying the ground-state properties of two-dimensional quantum lattice Hamiltonians in the thermodynamic limit. Here, we show how the algorithm can be adapted to explore nearest-neighbor local Hamiltonians on the ruby and triangle-honeycomb lattices, using the Corner Transfer Matrix (CTM) renormalization group for 2D tensor network contraction. Additionally, we show how the CTM method can be used to calculate the ground state fidelity per lattice site and the boundary density operator and entanglement entropy (EE) on an infinite cylinder. As a benchmark, we apply the iPEPS method to the ruby model with anisotropic interactions and explore the ground-state properties of the system. We further extract the phase diagram of the model in different regimes of the couplings by measuring two-point correlators, ground state fidelity and EE on an infinite cylinder. Our phase diagram is in agreement with previous studies of the model by exact diagonalization.
Based on the scheme of variational Monte Carlo sampling, we develop an accurate and efficient two-dimensional tensor-network algorithm to simulate quantum lattice models. We find that Monte Carlo sampling shows huge advantages in dealing with finite projected entangled pair states, which allows significantly enlarged system size and improves the accuracy of tensor network simulation. We demonstrate our method on the square-lattice antiferromagnetic Heisenberg model up to $32 times 32$ sites, as well as a highly frustrated $J_1-J_2$ model up to $24times 24$ sites. The results, including ground state energy and spin correlations, are in excellent agreement with those of the available quantum Monte Carlo or density matrix renormalization group methods. Therefore, our method substantially advances the calculation of 2D tensor networks for finite systems, and potentially opens a new door towards resolving many challenging strongly correlated quantum many-body problems.
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