Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell

Nietner, A.; Vanhecke, Bram; Verstraete, Frank; Eisert, Jens; Vanderstraeten, Laurens

doi:10.22331/q-2020-09-21-328

Cited by 14 publications

(15 citation statements)

References 58 publications

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“…Our techniques have the added value of directly tackling the thermodynamic limit: Residual finite-size effects are encompassed by the so-called bond dimension of the ansatz and could be accounted for in a rather systematic way [40][41][42][43][44]. These features have made two-dimensional tensor networks a very suitable tool for studying intricate condensed matter problems, not only via their ground states [45][46][47][48] but even beyond [49][50][51][52][53][54], as well as finite temperature properties of both classical and quantum models in two spatial dimensions [33,34,[55][56][57][58][59][60][61][62]. The present work builds upon and develops this substantial technical machinery.…”

Section: Introductionmentioning

confidence: 99%

The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

et al. 2021

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The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the O(3)O(3) non-linear sigma model in 1+11+1 dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in 3+13+1 dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an SU(2)SU(2) symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to \chi_E^\text{eff} \sim 1500χEeff∼1500, a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite-TT transition and asymptotic freedom, though with a slight preference for the second.

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Section: Introductionmentioning

confidence: 99%

The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

et al. 2021

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“…2(b). However, we find the standard contraction algorithms such as variational uniform matrix product state (VUMPS) [42][43][44] and corner transfer matrix renormalization group (CTMRG) [52][53][54] fail to converge in such a construction of local tensors.…”

Section: A Representations Of Partition Functionmentioning

confidence: 99%

“…where |Ψ(A, B) is the leading eigenvector represented by matrix product states (MPS) made up of two-site unit cell of local A and B tensors 43 . This fixed-point equation can be accurately solved by the multiple lattice-site VUMPS algorithm 42 , which provides an efficient variational scheme to approximate the largest eigenvector |Ψ(A, B) . The precision of this approximation is controlled by the auxiliary bond dimension D of local A and B tensors.…”

Section: B Multisite Vumps Algorithmmentioning

confidence: 99%

“…It is demonstrated that the extension of the applicability of tensor networks to the fully frustrated systems with continuous U (1) degrees of freedom is nontrivial, because the standard formulation of the tensor network fails to converge. Here we propose a new construction strategy based on the splitting of U (1) spins, and then the partition function of the FFXY model is transformed into an infinite 2D tensor network with an enlarged unit cell, which can be efficiently contracted by a recently proposed tensor network algorithm 42 under optimal variational principles 43,44 .…”

Section: Introductionmentioning

confidence: 99%

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Tensor network approach to the two-dimensional fully frustrated XY model and a chiral ordered phase

Song,

Zhang

2021

Preprint

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A general framework is proposed to solve the two-dimensional fully frustrated XY model for the Josephson junction arrays in a perpendicular magnetic field. The essential idea is to encode the ground-state local rules induced by frustrations in the local tensors of the partition function. The partition function is then expressed in terms of a product of one-dimensional transfer matrix operator, whose eigen-equation can be solved by an algorithm of matrix product states rigorously. The singularity of the entanglement entropy for the one-dimensional quantum analogue provides a stringent criterion to distinguish various phase transitions without identifying any order parameter a prior. Two very close phase transitions are determined at Tc1 ≈ 0.4459 and Tc2 ≈ 0.4532, respectively. The former corresponding to a Berezinskii-Kosterlitz-Thouless phase transition describing the phase coherence of Cooper pairs, and the latter is an Ising-like continuous phase transition below which a chirality order with spontaneously broken time-reversal symmetry is established. This intermediate temperature phase represents a bosonic metallic phase and a local order parameter is given by the phase differences of the Cooper pairs on four lattice sites of each plaquette.

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mentioning

confidence: 99%

The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

Schmoll,

Kshetrimayum,

Eisert

et al. 2021

Preprint

View full text Add to dashboard Cite

The classical Heisenberg model in two spatial dimensions constitutes one of the most paradigmatic spin models, taking an important role in statistical and condensed matter physics to understand magnetism. Still, despite its paradigmatic character and the widely accepted ban of a (continuous) spontaneous symmetry breaking, controversies remain whether the model exhibits a phase transition at finite temperature. Importantly, the model can be interpreted as a lattice discretization of the O(3) non-linear sigma model in 1 + 1 dimensions, one of the simplest quantum field theories encompassing crucial features of celebrated higher-dimensional ones (like quantum chromodynamics in 3 + 1 dimensions), namely the phenomenon of asymptotic freedom. This should also exclude finite-temperature transitions, but lattice effects might play a significant role in correcting the mainstream picture. In this work, we make use of state-of-the-art tensor network approaches, representing the classical partition function in the thermodynamic limit over a large range of temperatures, to comprehensively explore the correlation structure for Gibbs states. By implementing an SU (2) symmetry in our two-dimensional tensor network contraction scheme, we are able to handle very large effective bond dimensions of the environment up to χ eff E ∼ 1500, a feature that is crucial in detecting phase transitions. With decreasing temperatures, we find a rapidly diverging correlation length, whose behaviour is apparently compatible with the two main contradictory hypotheses known in the literature, namely a finite-T transition and asymptotic freedom, though with a slight preference for the second.

show abstract

Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell

Cited by 14 publications

References 58 publications

The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

Tensor network approach to the two-dimensional fully frustrated XY model and a chiral ordered phase

The classical two-dimensional Heisenberg model revisited: An $SU(2)$-symmetric tensor network study

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