Approaching the Kosterlitz-Thouless transition for the classical 
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 model with tensor networks

Vanderstraeten, Laurens; Vanhecke, Bram; Läuchli, Andreas M.; Verstraete, Frank

doi:10.1103/physreve.100.062136

Cited by 27 publications

(33 citation statements)

References 44 publications

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“…The same behaviour is found for the O(2)-case (i.e., XY) with the leading eigenvalue λ(β) = I 0 (β)[33,34].…”

supporting

confidence: 74%

“…Note that the spirit of the mapping in Eq. ( 8) has also been used in previous works on the O(2) and O(3) non-linear sigma model [33,34,71]. The same decomposition can also be used in the anti-ferromagnetic version of the Heisenberg model, in which additional factors of (i) 2 appear in the plane-wave expansion of Eq.…”

Section: The Partition Function As a Tensor Networkmentioning

confidence: 99%

“…We achieve such a formally exact rewriting via an expansion over the fundamental characters of the (non-Abelian) group, in a similar fashion to what was recently performed for the Abelian O(2) case in Refs. [33,34]. We then compute a number of relevant quantities such as spin-spin correlators, the thermodynamic entropy S and geometric entanglement ε, among others.…”

Section: Introductionmentioning

confidence: 99%

“…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%

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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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“…The same behaviour is found for the O(2)-case (i.e., XY) with the leading eigenvalue λ(β) = I 0 (β)[33,34].…”

supporting

confidence: 74%

Section: The Partition Function As a Tensor Networkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

et al. 2021

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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

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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) 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.

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Ordering kinetics and steady states of XY-model with ferromagnetic and nematic interaction

Mondal,

Mishra,

Mishra

2024

J. Phys.: Condens. Matter

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Previous studies on the generalized XY model have concentrated on the equilibrium phase diagram and the equilibrium nature of distinct phases under varying parameter conditions. We direct our attention towards examining the system's evolution towards equilibrium states across different parameter values, specifically by varying the relative strengths of ferromagnetic and nematic interactions. We study the kinetics of the system, using the temporal annihilation dynamics of defects at varying temperatures and its impact on the coarsening behavior of the system. For both pure polar and pure nematic systems, we observe temperature-dependent decay of the exponent, leading to a decelerated growth of domains within the system. At parameter values where both ferromagnetic and nematic interactions are simultaneously present, we show a phase diagram highlighting three low-temperature phases—polar, nematic, and coexistence—alongside a high-temperature disordered phase. Our study provides valuable insights into the complex interplay of interactions, offering a comprehensive understanding of the system's behavior during its evolution towards equilibrium.

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Approaching the Kosterlitz-Thouless transition for the classical XY model with tensor networks

Cited by 27 publications

References 44 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

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

Ordering kinetics and steady states of XY-model with ferromagnetic and nematic interaction

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