Time evolution of an infinite projected entangled pair state: An algorithm from first principles

Czarnik, Piotr; Dziarmaga, Jacek

doi:10.1103/physrevb.98.045110

Cited by 21 publications

(20 citation statements)

References 77 publications

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“…It is clear that repeated application of time steps would result in an exponential growth of the bond dimension. Therefore, after each time step it is necessary to approximate the exact new iPEPS |ψ by an approximate iPEPS -representing a state |ψwith all bonds having the original bond dimension D. A straightforward optimization of the fidelity between the approximate |ψ and the exact |ψ is feasible 69 and, in principle, it should give the most accurate |ψ , but it is not the most efficient one.…”

Section: Outlinementioning

confidence: 99%

Time evolution of an infinite projected entangled pair state: An efficient algorithm

2019

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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 D. 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 > D, which is approximated by an iPEPS with the original D. In 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 . In this work 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 D. We test the algorithm simulating Lindbladian evolution and unitary evolution after a sudden quench of transverse field hx in the 2D quantum Ising model. Furthermore, we simulate thermal states of this model and estimate the critical temperature with good accuracy: 0.1% for hx = 2.5 and 0.5% for the more challenging case of hx = 2.9 close to the quantum critical point at hx = 3.04438(2). arXiv:1811.05497v2 [cond-mat.str-el]

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Section: Outlinementioning

confidence: 99%

Time evolution of an infinite projected entangled pair state: An efficient algorithm

2019

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“…While the rank of the tensors used in the representation is determined by the physical problem, the amount of information they contain is determined by their bond dimension, D, which is used to control the accuracy of the ansatz. Although iPEPS were introduced originally for representing the ground states of local Hamiltonians, more recently several iPEPS methods have been developed for the representation of thermal states [36,[52][53][54][55][56][57][58][59][60][61][62]. Here we focus on the approaches discussed in Ref.…”

Section: Thermodynamics From Ipepsmentioning

confidence: 99%

Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase

Wietek

Corboz

Weßel

et al. 2019

Phys. Rev. Research

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The thermodynamic properties of the Shastry-Sutherland model have posed one of the longestlasting conundrums in frustrated quantum magnetism. Over a wide range on both sides of the quantum phase transition (QPT) from the dimer-product to the plaquette-based ground state, neither analytical nor any available numerical methods have come close to reproducing the physics of the excited states and thermal response. We solve this problem in the dimer-product phase by introducing two qualitative advances in computational physics. One is the use of thermal pure quantum (TPQ) states to augment dramatically the size of clusters amenable to exact diagonalization. The second is the use of tensor-network methods, in the form of infinite projected entangled pair states (iPEPS), for the calculation of finite-temperature quantities. We demonstrate convergence as a function of system size in TPQ calculations and of bond dimension in our iPEPS results, with complete mutual agreement even extremely close to the QPT. Our methods reveal a remarkably sharp and low-lying feature in the magnetic specific heat around the QPT, whose origin appears to lie in a proliferation of excitations composed of two-triplon bound states. The surprisingly low energy scale and apparently extended spatial nature of these states explain the failure of less refined numerical approaches to capture their physics. Both of our methods will have broad and immediate application in addressing the thermodynamic response of a wide range of highly frustrated magnetic models and materials.

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“…The implementation of finite-temperature tensor network algorithms in two-dimensions has been steadily developed and improved over the last ten years [19][20][21][22], and benchmarked on simple models as the quantum Ising model [20,21,23,24], the quantum compass model [25] or the Shastry-Sutherland Heisenberg model [26,27]. These methods are based on a parametrization of the thermal density operator (TDO) in terms of a Projected Entangled Pair Operator (PEPO).…”

Section: Introductionmentioning

confidence: 99%

Finite-temperature symmetric tensor network for spin-1/2 Heisenberg antiferromagnets on the square lattice

2021

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Within the tensor network framework, the (positive) thermal density operator can be approximated by a double layer of infinite Projected Entangled Pair Operators (iPEPO) coupled via ancilla degrees of freedom. To investigate the thermal properties of the spin-1/2 Heisenberg model on the square lattice, we introduce a family of fully spin-SU(2) and lattice-C 4v symmetric on-site tensors (of bond dimensions D = 4 or D = 7) and a plaquette-based Trotter-Suzuki decomposition of the imaginary-time evolution operator. A variational optimization is performed on the plaquettes, using a full (for D = 4) or simple (for D = 7) environment obtained from the single-site Corner Transfer Matrix Renormalization Group fixed point. The method is benchmarked by a comparison to quantum Monte Carlo in the thermodynamic limit. Although the iPEPO spin correlation length starts to deviate from the exact exponential growth for inverse-temperature β 2, the behavior of various observables turns out to be quite accurate once plotted w.r.t the inverse correlation length. We also find that a direct T = 0 variational energy optimization provides results in full agreement with the β → ∞ limit of finite-temperature data, hence validating the imaginary-time evolution procedure. Extension of the method to frustrated models is described and preliminary results are shown.

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Time evolution of an infinite projected entangled pair state: An algorithm from first principles

Cited by 21 publications

References 77 publications

Time evolution of an infinite projected entangled pair state: An efficient algorithm

Time evolution of an infinite projected entangled pair state: An efficient algorithm

Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase

Finite-temperature symmetric tensor network for spin-1/2 Heisenberg antiferromagnets on the square lattice

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