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Analytic properties of the two-dimensional Ising model

Pedro D. Fonseca

Published Sep 1, 2002• 0 citations
OpenAlex

Abstract

After a brief presentation of the two-dimensional Ising model, we study the analytic properties of the scaling function associated with the free energy in the critical domain T → Tc, H → 0. The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach. We determine the discontinuities across the Yang-Lee and Langer branch cuts. We confirm the standard analyticity assumptions and propose “extended analyticity”; roughly speaking, the latter states that the Yang-Lee branching point is the nearest singularity under Langer's branch cut. We support the extended analyticity by evaluating numerically the associated “extended dispersion relation”.

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The Ising model in two dimensions: summary of results

Barry M. McCoy

Abstract This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.

•Dec 1, 2009•0 citations
After a brief presentation of the two-dimensional Ising model, we study the analytic properties of the scaling function associated with the free energy in the critical domain T → Tc, H → 0. The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach. We determine the discontinuities across the Yang-Lee and Langer branch cuts. We confirm the standard analyticity assumptions and propose “extended analyticity”; roughly speaking, the latter states that the Yang-Lee branching point is the nearest singularity under Langer's branch cut. We support the extended analyticity by evaluating numerically the associated “extended dispersion relation”.
We study the scaling functions associated with the lightest particle mass M1M_1M1​ in 2d Ising field theory in external magnetic field. The scaling functions depend on the scaling parameter ξ=h/∣m∣158ξ= h/|m|^{\frac{15}{8}}ξ=h/∣m∣815​, or related parameter η=m/h815η= m / h^{\frac{8}{15}}η=m/h158​. Analytic properties of M1M_1M1​ in the high-T domain were discussed in arXiv:2203.11262. In this work, we study analyticity of M1M_1M1​ in the low-T domain. Important feature of this analytic structure is represented by the Fisher-Langer's branch cut. The discontinuity across this branch cut determines the behavior of M1M_1M1​ at all complex ξξξ via associated low-T dispersion relation. Also, we put forward the "extended analyticity" conjecture for M1M_1M1​ in the complex ηηη-plane, similar to the analyticity of the free energy density previously proposed in arXiv:hep-th/0112167. The extended analyticity implies the "extended dispersion relation", which we verify against the numerics from the Truncated Free Fermion Approach (TFFSA), giving strong support to the conjecture.
Abstract. We investigate the tricritical Ising model in complex magnetic field in order to characterize the analytic structure of its free energy. By supplementing analytic methods with the truncation of conformal space technique we obtain nonperturbative data even if the field theories we consider are not integrable. The existence of edge singularities analogous to the Yang–Lee points in the Ising field theory is confirmed. A surprising result, due to the conformal dimensions of the operators involved, is the appearance of two branching points which seems appealing to identify with a pair of complex conjugate spinodal singularities.
We investigate the tricritical Ising model in a complex magnetic field in order to characterize the analytic structure of its free energy. By supplementing analytic methods with the truncation of conformal space technique we obtain non-perturbative data even if the field theories that we consider are not integrable. The existence of edge singularities analogous to the Yang–Lee points in the Ising field theory is confirmed. A surprising result, due to the conformal dimensions of the operators involved, is the appearance of two branching points which it seems appealing to identify with a pair of complex conjugate spinodal singularities.
The analytic properties of the Onsager solution of the two-dimensional Ising lattice in zero external magnetic field are investigated. The free energy above the critical temperature Tc{T}_{c}Tc​ can be regarded as an analytic function in the uuu plane, where $u=\mathrm{exp}(\ensuremath{-}\frac{4J}{{k}_{B}T})$, with singularities at $u=0, \ensuremath{\infty}, \mathrm{and} \mathrm{exp}(\ifmmode\pm\else\textpm\fi{}\frac{4J}{{k}_{B}{T}_{c}})$. But its analytic continuation below the critical temperature does not describe a stable thermodynamic phase. Similar statements are true for the free energy below the critical temperature. The famous logarithmic singularity is briefly discussed. Finally, the knowledge of the analytic properties is shown, by elementary considerations, to explain the nonanalytic behavior at vanishing external field below the critical temperature---a result first enunciated by Yang and Lee for the free energy in the presence of an external magnetic field.
Renormalization group ideas and results from critical droplet theory are used to construct a scaling ansatz for the imaginary component of the free energy of an Ising model in its metastable state close to the critical point. The analytic properties of the free energy are used to determine scaling functions for the free energy in the vicinity of the critical point and the abrupt transition line. These functions have essential singularities at the abrupt transition. Analogous forms for the magnetization and susceptibility in two dimensions are fit to numeric data and show good agreement, especially when their nonsingular behavior is modified to match existing numeric results.
Abstract This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.