Location of zeros for the partition function of the Ising model on bounded degree graphs

Peters, Hans; Regts, Guus

doi:10.1112/jlms.12286

Cited by 32 publications

(65 citation statements)

References 37 publications

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“…For this to be done, we need an algorithm that given a local Hamiltonian H = O (n) ℓ=1 H ℓ , computes tr[H k ] in time O(n • 2 O (k ) ) instead of the current n O (k ) running time. This has been achieved for the classical Ising model [34,37] by relating the derivatives of the partition function to combinatorial objects that can be efficiently counted. Another approach for an improved running time is recently introduced in [29], where the authors apply a multivariate version of the cluster expansion that we use to compute the derivatives of log Z β (H ) efficiently.…”

Section: Discussion and Open Questionsmentioning

confidence: 99%

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Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems

Harrow

Mehraban

Soleimanifar

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We present a quasi-polynomial time classical algorithm that estimates the partition function of quantum many-body systems at temperatures above the thermal phase transition point. It is known that in the worst case, the same problem is NP-hard below this point. Together with our work, this shows that the transition in the phase of a quantum system is also accompanied by a transition in the hardness of approximation. We also show that in a system of n particles above the phase transition point, the correlation between two observables whose distance is at least Ω(log n) decays exponentially. We can improve the factor of log n to a constant when the Hamiltonian has commuting terms or is on a 1D chain. The key to our results is a characterization of the phase transition and the critical behavior of the system in terms of the complex zeros of the partition function. Our work extends a seminal work of Dobrushin and Shlosman on the equivalence between the decay of correlations and the analyticity of the free energy in classical spin models. On the algorithmic side, our result extends the scope of a recent approach due to Barvinok for solving classical counting problems to quantum many-body systems.

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Section: Discussion and Open Questionsmentioning

confidence: 99%

“…Since its introduction [5], this method has been used to obtain deterministic algorithms for various interesting problems such as the ferromagnetic and antiferromagnetic Ising models [34,37] on bounded graphs.…”

Section: Previous Workmentioning

confidence: 99%

Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems

Harrow

Mehraban

Soleimanifar

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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“…In this paper, we look at the computational complexity and complex zeros of the partition function in the Ising model. This is a classical and also currently very active area of research; see [6], [9], [10], [13], [16], [14], [15], [18], [19] and [22] for some recent results.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We consider the partition function S as a function of a complex parameter . It is known that for a fixed Δ, as grows and ranges over all graphs with maximum degree Δ of a vertex, the zeros of the univariate function ↦ −→ S with either choice of can get arbitrarily close to = 0; see [2], [1], [18], [6]. We have…”

Section: The Bounds For the Zero-free Region Are Asymptotically Optimalmentioning

confidence: 99%

“…A novelty of our approach with respect to locating zero-free regions of S , compared to those of [9], [14], [6] and [18], is that we allow all parameters and to vary: this concerns both Lee-Yang zeros [12], [6], [18] of S as a function of with fixed and Fisher zeros [14] of S as a function of with fixed. It appears that Theorem 1 is the first result establishing an asymptotically optimal zero-free region when the interactions are allowed to differ for different pairs { , } and even to be of different signs, so that we have a mixture of ferromagnetic and antiferromagnetic interactions.…”

Section: What's Newmentioning

confidence: 99%

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More on zeros and approximation of the Ising partition function

Barvinok

2021

Forum of Mathematics, Sigma

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We consider the problem of computing the partition function $\sum _x e^{f(x)}$ , where $f: \{-1, 1\}^n \longrightarrow {\mathbb R}$ is a quadratic or cubic polynomial on the Boolean cube $\{-1, 1\}^n$ . In the case of a quadratic polynomial f, we show that the partition function can be approximated within relative error $0 < \epsilon < 1$ in quasi-polynomial $n^{O(\ln n - \ln \epsilon )}$ time if the Lipschitz constant of the non-linear part of f with respect to the $\ell ^1$ metric on the Boolean cube does not exceed $1-\delta $ , for any $\delta>0$ , fixed in advance. For a cubic polynomial f, we get the same result under a somewhat stronger condition. We apply the method of polynomial interpolation, for which we prove that $\sum _x e^{\tilde {f}(x)} \ne 0$ for complex-valued polynomials $\tilde {f}$ in a neighborhood of a real-valued f satisfying the above mentioned conditions. The bounds are asymptotically optimal. Results on the zero-free region are interpreted as the absence of a phase transition in the Lee–Yang sense in the corresponding Ising model. The novel feature of the bounds is that they control the total interaction of each vertex but not every single interaction of sets of vertices.

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Thermodynamic limit of the first Lee‐Yang zero

Jiang,

Newman

2023

Comm Pure Appl Math

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We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in of finite‐volume singularities in . For the Ising model defined on a finite at inverse temperature and external field h, let be the modulus of the first zero (that closest to the origin) of its partition function (in the variable h). We prove that decreases to as Λ increases to where is the radius of the largest disk centered at the origin in which the free energy in the thermodynamic limit is analytic. We also note that is strictly positive if and only if β is strictly less than the critical inverse temperature.

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Location of zeros for the partition function of the Ising model on bounded degree graphs

Cited by 32 publications

References 37 publications

Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems

Classical algorithms, correlation decay, and complex zeros of partition functions of Quantum many-body systems

More on zeros and approximation of the Ising partition function

Thermodynamic limit of the first Lee‐Yang zero

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