Excitations with projected entangled pair states using the corner transfer matrix method

Ponsioen, Boris; Corboz, Philippe

doi:10.1103/physrevb.101.195109

Cited by 39 publications

(30 citation statements)

References 61 publications

(122 reference statements)

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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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“…In that respect, it would be extremely interesting to look at fractionalized excitations such as spinons or holons: In principle, the excitation ansatz can be straightforwardly generalized to also capture these topological excitations [20,52]. Another interesting question is how this MPS excitation ansatz on the cylinder compares to the excitation ansatz for projected entangled-pair states [53][54][55], which is formulated directly on the infinite plane.…”

Section: Discussionmentioning

confidence: 99%

Efficient matrix product state methods for extracting spectral information on rings and cylinders

et al. 2021

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Based on the MPS formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing DMRG or MPS ground-state code. Although this approach is built on open-boundary MPS, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number, and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2-D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2-D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.

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“…The excitation ansatz for iPEPS, based on its one-dimensional MPS equivalent [20][21][22][23], offers the possibility to accurately simulate quasiparticle excitations on top of a strongly correlated ground state. After its original development [12] it has been extended to more general spin models [24] and fermionic models [25], the latter using a new implementation based on the corner transfer matrix (CTM) [26][27][28][29] contraction method.…”

Section: Conclusion 15mentioning

confidence: 99%

“…The different dimensions d and D refer to the local Hilbert space on a single site and the bond dimension, respectively, the latter of which controls the accuracy of the ansatz. For simplicity, in this paper we limit the discussion to translationally invariant systems, with a one-site unit cell with a single tensor A that is repeated on the lattice, though the implementation can be easily extended to arbitrary unit cell sizes [25].…”

Section: Ipepsmentioning

confidence: 99%

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Automatic differentiation applied to excitations with projected entangled pair states

2022

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The excitation ansatz for tensor networks is a powerful tool for simulating the low-lying quasiparticle excitations above ground states of strongly correlated quantum many-body systems. Recently, the two-dimensional tensor network class of infinite projected entangled-pair states gained new ground state optimization methods based on automatic differentiation, which are at the same time highly accurate and simple to implement. Naturally, the question arises whether these new ideas can also be used to optimize the excitation ansatz, which has recently been implemented in two dimensions as well. In this paper, we describe a straightforward way to reimplement the framework for excitations using automatic differentiation, and demonstrate its performance for the Hubbard model at half filling.

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Excitations with projected entangled pair states using the corner transfer matrix method

Cited by 39 publications

References 61 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

Efficient matrix product state methods for extracting spectral information on rings and cylinders

Automatic differentiation applied to excitations with projected entangled pair states

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