Lee-Yang zeros of the antiferromagnetic Ising model

“…Our work falls into the broader context of showing how zeros in the complex plane actually relate to the existence and design of approximation algorithms. This connection has been well studied for general graphs, see, for example, [20,18,15]; for bounded-degree graphs, the picture is less clear, but the key seems to lie in understanding the underlying complex dynamical systems [6,9,10,42,4,7]. A general theory is so far elusive, but it seems that the chaotic behavior of the underlying complex dynamical system is linked to the presence of zeros of the partition function and to the #P-hardness of approximation.…”

Section: Introductionmentioning

confidence: 99%

“…4 ) and hence| 𝛼 − 𝛼 |, |𝛽 − 𝛽 | ≤ 𝜖 + 1/(8𝑀 4 ) ≤ 1/(4𝑀 4 ). Now, for distinct 𝛾, 𝛿 ∈ Q 2𝑀 2 , we have that |𝛾 − 𝛿 | ≥ 1/(2𝑀 2), so it must be that 𝛼 = 𝛼 and 𝛽 = 𝛽 , completing the computation of 𝑅 goal .…”

mentioning

confidence: 96%

Lee–Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs

Buys

Galanis

Patel

et al. 2022

Forum of Mathematics, Sigma

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We study the computational complexity of approximating the partition function of the ferromagnetic Ising model with the external field parameter $\lambda $ on the unit circle in the complex plane. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics but have also been key in the recent deterministic approximation scheme for all $|\lambda |\neq 1$ by Liu, Sinclair and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle and on the tantalising question of what happens around $\lambda =1$ , where, on one hand, the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability and, on the other hand, the presence of Lee–Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point $\lambda =1$ and, more generally, on the entire unit circle. For an integer $\Delta \geq 3$ and edge interaction parameter $b\in (0,1)$ , we show $\mathsf {\#P}$ -hardness for approximating the partition function on graphs of maximum degree $\Delta $ on the arc of the unit circle where the Lee–Yang zeros are dense. This result contrasts with known approximation algorithms when $|\lambda |\neq 1$ or when $\lambda $ is in the complementary arc around $1$ of the unit circle. Our work thus gives a direct connection between the presence/absence of Lee–Yang zeros and the tractability of efficiently approximating the partition function on bounded-degree graphs.

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“…It should be noted that the existence of zeros does not imply hardness in a straightforward manner. 3 2 Notably, the correlation decay approach, which also yields deterministic approximation algorithms and was key in the full classification of antiferromagnetic 2-spin systems [25,37,38,12], somewhat surprisingly does not perform as well for ferromagnetic systems, see [19] for the state-of-the-art on this front.…”

Section: Introductionmentioning

confidence: 99%

“…We obtain the connection between the Lee-Yang circle of zeros and computational complexity via tools from complex dynamical systems; our work falls into the broader context of showing how zeros in the complex plane actually relate to the existence and design of approximation algorithms. This connection has been well studied for general graphs, see, e.g., [15,14,11]; for bounded-degree graphs, the picture is less clear, but the key seems to lie in understanding the underlying complex dynamical systems [5,8,9,35,3,6]. A general theory is so far elusive, but it seems that the chaotic behaviour of the underlying complex dynamical system is linked to the presence of zeros of the partition function and to the #P-hardness of approximating the partition function, see Section 3 for a more detailed explanation.…”

Section: Introductionmentioning

confidence: 99%

Lee-Yang zeros and the complexity of the ferromagnetic Ising Model on bounded-degree graphs

Buys¹,

Galanis

Patel

et al. 2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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We study the computational complexity of approximating the partition function of the ferromagnetic Ising model in the Lee-Yang circle of zeros given by |λ| = 1, where λ is the external field of the model.Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic approximation scheme for all |λ| = 1 by Liu, Sinclair, and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle, and on the tantalising question of what happens in the circular arc around λ = 1, where on one hand the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability, and on the other hand the presence of Lee-Yang zeros alludes to computational hardness.Our main result establishes a sharp computational transition at the point λ = 1; in fact, our techniques apply more generally to the whole unit circle |λ| = 1. We show #P-hardness for approximating the partition function on graphs of maximum degree ∆ when b, the edge-interaction parameter, is in the interval (0, ∆−2 ∆ ] and λ is a nonreal on the unit circle. This result contrasts with known approximation algorithms when |λ| = 1 or b ∈ ( ∆−2 ∆ , 1), and shows that the Lee-Yang circle of zeros is computationally intractable, even on bounded-degree graphs.Our inapproximability result is based on constructing rooted tree gadgets via a detailed understanding of the underlying dynamical systems, which are further parameterised by the degree of the root. The ferromagnetic Ising model has radically different behaviour than previously considered antiferromagnetic models, and showing our #P-hardness results in the whole Lee-Yang circle requires a new high-level strategy to construct the gadgets. To this end, we devise an elaborate inductive procedure to construct the required gadgets by taking into account the dependence between the degree of the root of the tree and the magnitude of the derivative at the fixpoint of the corresponding dynamical system.The full version (with all proofs) is available on arXiv at arxiv.org/abs/2006.14828.

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Lee-Yang zeros of the antiferromagnetic Ising model

Cited by 5 publications

References 17 publications

Generic family displaying robustly a fast growth of the number of periodic points

Generic family displaying robustly a fast growth of the number of periodic points

Lee–Yang zeros and the complexity of the ferromagnetic Ising model on bounded-degree graphs

Lee-Yang zeros and the complexity of the ferromagnetic Ising Model on bounded-degree graphs

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