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arXiv:1508.04003 · Physical review. B./Physical review. B

Improved energy extrapolation with infinite projected entangled-pair states applied to the two-dimensional Hubbard model

Philippe Corboz

Published Jan 14, 2016• 110 citations
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Abstract

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 $D$. We show that for the doped Hubbard model in the strongly correlated regime $(U/t=8,n=0.875)$ 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-$D$ 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.

DOI
10.1103/physrevb.93.045116

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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.
Accurate computation of the energy variance and ⟨⟨L†L⟩⟩\langle\langle \mathcal{L}^\dagger \mathcal{L} \rangle\rangle⟨⟨L†L⟩⟩ using iPEPS
Infinite projected entangled-pair states (iPEPS) provide a powerful tensor network ansatz for two-dimensional quantum many-body systems in the thermodynamic limit. In this paper we introduce an approach to accurately compute the energy variance of an iPEPS, enabling systematic extrapolations of the ground-state energy to the exact zero-variance limit. It is based on the contraction of a large cell of tensors using the corner transfer matrix renormalization group (CTRMG) method, to evaluate the correlator between pairs of local Hamiltonian terms. We show that the accuracy of this approach is substantially higher than that of previous methods, and we demonstrate the usefulness of variance extrapolation for the Heisenberg model, for a free fermionic model, and for the Shastry-Sutherland model. Finally, we apply the approach to compute ⟨⟨L†L⟩⟩\langle \langle \mathcal{L}^\dagger \mathcal{L} \rangle \rangle⟨⟨L†L⟩⟩ for an open quantum system described by the Liouvillian L\mathcal{L}L, in order to assess the quality of the steady-state solution and to locate first-order phase transitions, using the dissipative quantum Ising model as an example.
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.
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 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).
A typical quantum state obeying the area law for entanglement on an infinite two-dimensional (2D) lattice can be represented by a tensor network Ansatz, known as an infinite projected entangled pair state (iPEPS), with a finite bond dimension DDD. Its real-imaginary time evolution can be split into small time steps. An application of a time step generates a new iPEPS with a bond dimension kkk times the original one. The new iPEPS does not make optimal use of its enlarged bond dimension kDkDkD; hence, in principle, it can be represented accurately by a more compact Ansatz, preferably with the original DDD. In this work we show how the more compact iPEPS can be optimized variationally to maximize its overlap with the new iPEPS. To compute the overlap we use the corner-transfer-matrix renormalization group. By simulating sudden quench of the transverse field in the 2D quantum Ising model with the proposed algorithm, we provide a proof of principle that real-time evolution can be simulated with iPEPS. A similar proof is provided with the same model for imaginary-time evolution of purification of its thermal states.
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.