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arXiv:0907.5520

Graded Projected Entangled-Pair State Representations and An Algorithm for Translationally Invariant Strongly Correlated Electronic Systems on Infinite-Size Lattices in Two Spatial Dimensions

Qian‐Qian Shi, Sheng-Hao Li, Jianhui Zhao, Huan-Qiang Zhou

Published Jul 31, 2009• 14 citations
arXivHTMLPDF

Abstract

An algorithm to find a graded Projected Entangled-Pair State representation of the ground state wave functions is developed for translationally invariant strongly correlated electronic systems on infinite-size lattices in two spatial dimensions. It is tested for the two-dimensional t-J model at and away from half filling, with truncation dimensions up to 6. We are able to locate a line of phase separation, which qualitatively agrees with the results based on the high-temperature expansions. We find that the model exhibits an extended s-wave superconductivity for J=0.4t at quarter filling. However, we emphasize that the currently accessible truncation dimensions are not large enough, so it is necessary to incorporate the symmetry of the system into the algorithm, in order to achieve results with higher precision.

DOI
10.48550/arxiv.0907.5520

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An algorithm to find a graded Projected Entangled-Pair State representation of the ground state wave functions is developed for translationally invariant strongly correlated electronic systems on infinite-size lattices in two spatial dimensions. It is tested for the two-dimensional t-J model at and away from half filling, with truncation dimensions up to 6. We are able to locate a line of phase separation, which qualitatively agrees with the results based on the high-temperature expansions. We find that the model exhibits an extended s-wave superconductivity for J=0.4t at quarter filling. However, we emphasize that the currently accessible truncation dimensions are not large enough, so it is necessary to incorporate the symmetry of the system into the algorithm, in order to achieve results with higher precision.
We present an algorithm to simulate two-dimensional quantum lattice systems in the thermodynamic limit. Our approach builds on the projected entangled-pair state algorithm for finite lattice systems [F. Verstraete and J. I. Cirac, arxiv:cond-mat/0407066] and the infinite time-evolving block decimation algorithm for infinite one-dimensional lattice systems [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)10.1103/PhysRevLett.98.070201]. The present algorithm allows for the computation of the ground state and the simulation of time evolution in infinite two-dimensional systems that are invariant under translations. We demonstrate its performance by obtaining the ground state of the quantum Ising model and analyzing its second order quantum phase transition.
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.
The ground-state phase diagram of the two-dimensional t-J model is investigated in the context of the tensor network algorithm in terms of the graded Projected Entangled-Pair State representation of the ground-state wave functions. There is a line of phase separation between the Heisenberg anti-ferromagnetic state without hole and a hole-rich state. For both J=0.4t and J=0.8t, a systematic computation is performed to identify all the competing ground states for various dopings. It is found that, besides a possible Nagaoka's ferromagnetic state, the homogeneous regime consists of four different phases: one phase with charge and spin density wave order coexisting with a p_x (p_y)-wave superconducting state, one phase with the symmetry mixing of d+s-wave superconductivity in the spin-singlet channel and p_x (p_y)-wave superconductivity in the spin-triplet channel in the presence of an anti-ferromagnetic background, one superconducting phase with extended s-wave symmetry, and one superconducting phase with p_x (p_y)-wave symmetry in a ferromagnetic background.
By introducing the concepts and methods of machine learning, the traditional tensor network algorithm is upgraded to the machine learning tensor network algorithm in order to obtain more accurate results in complex systems. Under the context of the virtual time evolution tensor network algorithm established in the graded projection entangled pair state (gPEPS) representation of the two-dimensional strongly correlated electron infinite fermion system based on machine learning, this paper mainly investigates the quantum phase diagram, quantum phase transition and quantum entanglement of the two-dimensional t-J model related to high temperature superconductivity through the ground state energy. The research results showed that the t-J model has a phase separation line between the Heisenberg antiferromagnetic state with no vacancies and the state rich in vacancies, with the upper critical point at Jc=0.95t and the lower critical point at Jc=3.45t. It was found that at J=0.4t, in addition to the potential long-range ferromagnetic state, four different phases were composed in the different doping uniform regions: one wasis the superconducting phase with coexistence of charge and spin density waves and px(py) wave symmetry, one wasis the mixed superconducting phase with d+s wave superconducting pairing and spin triplet px(py) wave superconducting symmetry in the antiferromagnetic background, one wasis the superconducting phase with extended s-wave symmetry, and the other one iswas the superconducting phase with px(py) wave symmetry in the ferromagnetic background.
Tensor network methods have become a powerful class of tools to capture strongly correlated matter, but methods to capture the experimentally ubiquitous family of models at finite temperature beyond one spatial dimension are largely lacking. We introduce a tensor network algorithm able to simulate thermal states of two-dimensional quantum lattice systems in the thermodynamic limit. The method develops instances of projected entangled pair states and projected entangled pair operators for this purpose. It is the key feature of this algorithm to resemble the cooling down of the system from an infinite temperature state until it reaches the desired finite-temperature regime. As a benchmark, we study the finite-temperature phase transition of the Ising model on an infinite square lattice, for which we obtain remarkable agreement with the exact solution. We then turn to study the finite-temperature Bose-Hubbard model in the limits of two (hard-core) and three bosonic modes per site. Our technique can be used to support the experimental study of actual effectively two-dimensional materials in the laboratory, as well as to benchmark optical lattice quantum simulators with ultracold atoms.
Being able to accurately describe the dynamics steady states of driven and/or dissipative but quantum correlated lattice models is of fundamental importance in many areas of science: from quantum information to biology.An efficient numerical simulation of large open systems in two spatial dimensions is a challenge.In this work, we develop a tensor network method, based on an infinite projected entangled pair operator ansatz, applicable directly in the thermodynamic limit.We incorporate techniques of finding optimal truncations of enlarged network bonds by optimizing an objective function appropriate for open systems.Comparisons with numerically exact calculations, both for the dynamics and the steady state, demonstrate the power of the method.In particular, we consider dissipative transverse quantum Ising, driven-dissipative hard-core boson, and dissipative anisotropic XY models in non-mean-field limits, proving able to capture substantial entanglement in the presence of dissipation.Our method enables us to study regimes that are accessible to current experiments but lie well beyond the applicability of existing techniques.