Field theoretical approach to spin models

Abstract

We developed a systematic non-perturbative method base on Dyson–Schwinger theory and the Φ-derivable theory for Ising model at broken phase. Based on these methods, we obtain critical temperature and spin spin correlation beyond mean field theory. The spectrum of Green function obtained from our methods become gapless at critical point, so the susceptibility become divergent at T_c. The critical temperature of Ising model obtained from this method is fairly good in comparison with other non-cluster methods. It is straightforward to extend this method to more complicate spin models for example with continue symmetry.

Graphical abstract

Keywords

Ising model / mean field theory / Dyson–Schwinger equations

1 Introduction

The Ising model is the simplest spin model, and it has been studied for many years. Rigorous solutions have been given by Ising for the one-dimensional case [1] and by Onsager in the case of the two-dimensional square lattice [2], which provide a benchmark for other approximate method. There are several techniques for solving such kind statistical models, such as Monte Carlo simulations, mean-field-type methods, cluster mean-field methods, and renormalization-group methods (RG). Although RG methods have a good description of system in the vicinity of the critical point, it neither predicts the behavior of the system in the area far away from critical point nor give the phase transition temperature. Many attempts to calculate quantities in the statistical systems beyond the mean field theory have been made in recent years [3].

The mean-field (MF) approach, base on one-site approximation, began from Pierre Weiss [4], which gives the well-know solution for the critical temperature of the transition from symmetry phase to broken phase T_c/J = z, where z is the number of nearest neighbors and J is the coupling strength. Wysin and Kaplan [5] made a significant improvement to the MF in a simple way. Their “self-consistent correlated molecular-field theory” (SCCF) takes into account the impact of the spin state of the central spin on the effective field of neighboring spins. They obtained more accurate critical temperature compared with some other methods, such as MF or Bethe–Peierls–Weiss (BPW) approximation (often called Bethe approximation in short) [6–8]. Zhuravlev [9] introduced “screened magnetic field” approximation which further improves the result of the SCCF method and allows one to obtain critical temperature with better accuracy. Beyond on-site approximation, BPW approach can be considered as the simplest case of cluster approach. Based on cluster idea, many new approximations have been proposed, such as “correlated cluster mean-field” (CCMF) theory introduced by Yamamoto [10], and “effective correlated mean-field approach” (ECMF) developed by Viana [11]. For a large enough cluster, this approach can give a good estimate for critical temperature.

In this work, we use two kind field theoretical methods to treat Ising model, which is based on Schwinger–Dyson equations (1PI) approach and two-particle irreducible (2PI) Φ-derivable theory [12–14], respectively. Within the approximations, both methods do not necessarily guarantee the identity which respects the fluctuation-dissipation theorem, thus the susceptibility of Ising model obtained from above methods do not diverge at critical temperature. Fortunately, for 1PI approach, a general method to preserve the identity in an approximation scheme was developed long time ago [15] in the context of field theory as covariant Gaussian approximation (CGA) to solve unrelated problems in quantum field theory and superfluidity. For 2PI method, Van Hees and Knoll developed an improved Φ-derivable theory which preserve the identity by approximating the 1PI functional with the 2PI functional [16]. After modified procedure mentioned above, the susceptibility of Ising model diverges at critical temperature for both case. With a relatively low cost, the critical temperature T_c obtained from them is quite accurately comparing with other non-cluster method. More importantly, since our methods base on Hubbard–Stratonovich transformation, it is straightforward to extend these methods to more complicate models, like XY model, Heisenberg model, with preserving the identity for the fluctuation-dissipation theorem and Ward–Takahashi identity (WTI) for models with continue symmetries, which are crucial for the description of such systems.

The paper is organized as following. In Section 2 and Section 3, we derive the equations for Ising model base on 1PI approach and the Φ-derivable theory respectively. Numerical results including the critical temperature, susceptibility and Green’s function are given in Section 4. Finally, we give a summary in Section 5.

2 1PI formalism

The Hamiltonian of the Ising model in a two-dimension square lattice can be expressed as

\begin{array}{}\text{(1)}& H=-\frac{1}{2}{\displaystyle \sum _{i,j}{J}_{i\mathrm{,}j}{\sigma }_i{\sigma }_j-}{\displaystyle \sum _i{\sigma }_i{h}_i\mathrm{,}}\end{array}

where J_i,j is the coupling strength between i and j, which equal to J for any two nearest neighboring sites, otherwise equal to zero. The spin σ_i takes either +1 or−1.

After　Hubbard–Stratonovich transformation, the grand-canonical partition function of this system can be written as path integral over continue variable parameter ϕ [17]:

\begin{array}{}\text{(2)}& \begin{array}{ll}Z(h)\hfill & ={\displaystyle \sum _{\left\{{\sigma }_i\right\}}}\exp (-\beta H)\hfill \\ \hfill & ={\displaystyle \int }D(\varphi )\exp (-\frac{\beta }{2}{\displaystyle \sum _{i\mathrm{,}j}}{J}_{i\mathrm{,}j}^{-1}({\varphi }_i{\varphi }_j-h_i{\varphi }_j\hfill \\ \hfill & -{\varphi }_i{h}_j+h_i{h}_j)+{\displaystyle \sum _n}\ln [\cosh (\beta {\varphi }_n)])\mathrm{.}\hfill \end{array}\end{array}

Based on the above formula, we can get the relationship between σ and ϕ for zero-external field case, i.e., h_i = 0 for each i,

\begin{array}{}\text{(3)}& \langle {\sigma }_m\rangle ={\displaystyle \sum _i{J}_{m\mathrm{,}i}^{-1}\langle {\varphi }_i\rangle \mathrm{,}}\end{array}

\begin{array}{}\text{(4)}& {\langle {\sigma }_m{\sigma }_n\rangle }_c={\displaystyle \sum _{i\mathrm{,}j}{J}_{m\mathrm{,}i}^{-1}{J}_{n\mathrm{,}j}^{-1}{\langle {\varphi }_i{\varphi }_j\rangle }_c-{\beta }^{-1}{J}_{m\mathrm{,}n}^{-1}\mathrm{.}}\end{array}

Here we have used the property J_i,j = J_j,i, thus J_{i\mathrm{,}j}^{-1}= J_{j\mathrm{,}i}^{-1} .

For zero-external field case, we add a new auxiliary source H_i to generate Green function. This auxiliary source has to be set to zero at the end of the calculation. The partition function can be rewritten as

\begin{array}{}\text{(5)}& \begin{array}{ll}Z\left(H\right)=\hfill & {\displaystyle \int }D(\varphi )\exp (-\beta \frac{1}{2}{\displaystyle \sum _{i\mathrm{,}j}}{J}_{ij}^{-1}{\varphi }_i{\varphi }_j\hfill \\ \hfill & +{\displaystyle \sum _n}\ln [\cosh \left(\beta {\varphi }_n\right)]-{\displaystyle \sum _i}{H}_i{\varphi }_i)\mathrm{.}\hfill \end{array}\end{array}

The generating functional W for connected diagrams reads W (H) =−ln Z (H), From this we can define the mean field and the connected Green’s function:

\begin{array}{}\text{(6)}& {\phi }_i=\frac{\delta W\left(H\right)}{\delta H_i}=\langle {\varphi }_i\rangle \mathrm{,}\end{array}

\begin{array}{}\text{(7)}& G_{ij}=-\frac{{\delta }^{2}W(H)}{\delta H_i\delta H_j}=\langle {\varphi }_i{\varphi }_j\rangle -\langle {\varphi }_i\rangle \langle {\varphi }_j\rangle \mathrm{.}\end{array}

By a functional Legendre transformation on φ one obtains the effective action:

\begin{array}{}\text{(8)}& \text{Γ}(\phi )=W(H)-{\displaystyle \sum _i{H}_i{\phi }_i\mathrm{.}}\end{array}

The first equation in the series of the DS equations, i.e., the off-shell (H≠ 0) “shift” equation is

\begin{array}{}\text{(9)}& 0=H_m+\beta {\displaystyle \sum _i{J}_{im}^{-1}{\phi }_i-\beta \langle \tanh \left(\beta {\varphi }_m\right)\rangle \mathrm{.}}\end{array}

Higher-order DS equations in the cumulant form are obtained by differentiating the equation above. The second DS equation is

\begin{array}{}\text{(10)}& {\text{Γ}}_{ij}=-\frac{\delta H_i}{\delta {\phi }_j}=\beta J_{ij}^{-1}-\beta \frac{\delta \langle \tanh \left(\beta {\varphi }_i\right)\rangle }{\delta {\phi }_j}\mathrm{.}\end{array}

Γ_ij is the inverse of G_ij since

\begin{array}{}\text{(11)}& {\displaystyle \sum _n{G}_{in}{\text{Γ}}_{nj}=}{\displaystyle \sum _n\frac{{\delta }^{2}W(H)}{\delta H_i\delta H_n}\frac{\delta H_n}{\delta {\phi }_j}=}{\displaystyle \sum _n\frac{\delta H_n}{\delta {\phi }_j}\frac{\delta {\phi }_i}{\delta H_n}={\delta }_{ij}\mathrm{.}}\end{array}

Considering leading correction to mean field theory, the 〈tanh (βϕ_i)〉 can be expanded as

\begin{array}{}\text{(12)}& \langle \tanh (\beta {\varphi }_i)\rangle =\tanh (\beta {\phi }_i)-{\beta }^2{\text{sech}}^2\left(\beta {\phi }_i\right)\tanh \left(\beta {\phi }_i\right)G_{ii}\mathrm{.}\end{array}

Substitute Eqs. (12) into Eq. (9) and (10), and neglect the derivative of G_ii with respect to φ_j according to leading order approximation. Now we could set H = 0, for homogeneous system we have φ_i = φ for any site i. Thus, we could express above equations in momentum space using Fourier transformation G_ij = {\displaystyle {\sum }_{\alpha =x\mathrm{,}y}}{\displaystyle {\int }_{-\pi }^{\pi }}\frac{{\text{d}}^{2}k}{{(2\pi )}^2}\exp (-\text{i}{k}_{\alpha }(i_{\alpha }-j_{\alpha }))G(k):

\begin{array}{}\text{(13)}& 0=\frac{\beta \phi }{4J}-\beta \tanh \left(\beta \phi \right)+{\beta }^3{G}_{ii}{\text{sech}}^2\left(\beta \phi \right)\text{tanh}\left(\beta \phi \right)\mathrm{,}\end{array}

\begin{array}{}\text{(14)}& \begin{array}{ll}\text{Γ}(k)=\hfill & \beta J^{-1}(k)-{\beta }^2{\text{sech}}^2\left(\beta \phi \right)\hfill \\ \hfill & +{\beta }^4{G}_{ii}\left[{\text{sech}}^4\left(\beta \phi \right)-2{\text{sech}}^2\left(\beta \phi \right){\text{tanh}}^2\left(\beta \phi \right)\right]\mathrm{,}\hfill \end{array}\end{array}

where J⁻¹(k) = [2J (cos k_x + cos k_y)]⁻¹, G_ii = \begin{array}{l}{\displaystyle {\int }_{-\pi }^{\pi }}\frac{{\text{d}}^{2}k}{{(2\pi )}^2}G(k)={\displaystyle {\int }_{-\pi }^{\pi }}\frac{{\text{d}}^{2}k}{{(2\pi )}^2}{\text{Γ}}^{-1}(k)\hfill \end{array} . Notice G_ii is not a function of i due to translation invariance of system. Then we can get φ and G at fixed β and J with Eq. (13) and Eq. (14).

The susceptibility obtained from above calculation do not diverge at phase transition temperature since the truncation applied to the formula (12) and (14) will break the fluctuation–dissipation theorem. It is not surprising since such a method does not respect also WTI for systems with continue symmetry and there is not Goldstone mode for broken phase [18].

A general method to preserve both identities in an approximation scheme was developed long time ago [15, 19], in the context of field theory as the covariant Gaussian approximation(CGA). In this improved method, the full covariant correlator is defined by functional derivative:

\begin{array}{}\text{(15)}& \begin{array}{ll}{(G_{\text{full}})}_{ij}^{-1}\hfill & =-\frac{\delta H_i}{\delta {\phi }_j}\hfill \\ \hfill & ={\text{Γ}}_{ij}+{\beta }^3{\text{Λ}}_{iij}{\text{sech}}^2\left(\beta {\phi }_i\right)\text{tanh}\left(\beta {\phi }_i\right)\mathrm{,}\hfill \end{array}\end{array}

where

\begin{array}{}\text{(16)}& {\text{Λ}}_{iij}=\frac{\delta G_{ii}}{\delta {\phi }_j}\mathrm{,}\end{array}

which can be obtained by taking the derivative of

\begin{array}{}\text{(17)}& {\delta }_{ij}={\displaystyle \sum _n{G}_{in}{\text{Γ}}_{nj}\mathrm{.}}\end{array}

We get

\begin{array}{}\text{(18)}& {\text{Λ}}_{iim}=-{\displaystyle \sum _{k\mathrm{,}j}{G}_{ik}\frac{\delta {\text{Γ}}_{kj}}{\delta {\phi }_m}{G}_{ji}\mathrm{,}}\end{array}

here

\begin{array}{}\text{(19)}& \begin{array}{ll}\hfill & \frac{\delta \text{Γ}(i\mathrm{,}j)}{\delta {\phi }_m}=-{\beta }^2\frac{\delta [{\text{sech}}^2(\beta {\phi }_i)]}{\delta {\phi }_m}{\delta }_{ij}\hfill \\ \hfill & +{\beta }^3{G}_{ii}\frac{\delta \left[\beta {\text{sech}}^4\left(\beta {\phi }_i\right)-2\beta {\text{sech}}^2\left(\beta {\phi }_i\right){\text{tanh}}^2\left(\beta {\phi }_i\right)\right]}{\delta {\phi }_m}{\delta }_{ij}\hfill \\ \hfill & +{\beta }^3{\text{Λ}}_{iim}\left[\beta {\text{sech}}^4\left(\beta {\phi }_i\right)-2\beta {\text{sech}}^2\left(\beta {\phi }_i\right){\text{tanh}}^2\left(\beta {\phi }_i\right)\right]{\delta }_{ij}\mathrm{.}\hfill \end{array}\end{array}

These equations are actually the Bethe–Salpeter equation. After Fourier transformation, We can get full covariant correlator:

\begin{array}{}\text{(20)}& G_{\text{full}}^{-1}(k)=\text{Γ}(k)+{\beta }^3\text{Λ}(k){\text{sech}}^2\left(\beta \phi \right)\text{tanh}\left(\beta \phi \right)\mathrm{,}\end{array}

where Λ(k) is the Fourier transform of Λ_iim,

\begin{array}{}\text{(21)}& {\text{Λ}}_{iim}={\displaystyle \sum _{\alpha =x\mathrm{,}y}{\displaystyle {\int }_{-\pi }^{\pi }}\frac{{\text{d}}^{2}k}{{(2\pi )}^2}\exp (-\text{i}{k}_{\alpha }(i_{\alpha }-m_{\alpha }))\text{Λ}(k)\mathrm{,}}\end{array}

which can be solved by the Bethe–Salpeter equation (18),

\begin{array}{}\text{(22)}& \text{Λ}(k)=-\frac{I(k)\left\{\left[2{\beta }^3{\text{sech}}^2(\beta \phi )\tanh (\beta \phi )\right]+{\beta }^3{G}_{ii}\left[-8{\beta }^2{\text{sech}}^4(\beta \phi )\tanh (\beta \phi )+4{\beta }^2{\text{sech}}^2(\beta \phi ){{\displaystyle \tanh }}^3(\beta \phi )\right]\right\}}{1+I(k)\left[{\beta }^4{\text{sech}}^4(\beta \phi )-2{\beta }^4{\text{sech}}^2(\beta \phi ){{\displaystyle \tanh }}^2(\beta \phi )\right]}\mathrm{,}\end{array}

where I(k) is defined as

\begin{array}{}\text{(23)}& I(k)=\frac{1}{{(2\pi )}^2}{\displaystyle {\int }_{-\pi }^{\pi }}{\text{d}}^{2}pG(k+p)G(p)\mathrm{.}\end{array}

Substituting Eq. (22) back to Eq. (20), we can get full covariant Green’s function. And the susceptibility obtained by this method will diverge at phase transition point.

3 2PI formalism

The Φ-derivable approximation possesses several intriguing features. Approximations of this kind are the so-called conserving approximations [13, 16], which means it is consistent with the conservation laws that follow from the Noether’s theorem (current conservation, total momentum, total energy, etc). The usual thermodynamic relations between pressure, energy density and entropy hold exactly within this approximation.

In addition to the usually introduced one-point auxiliary external source a two-point auxiliary external source is also included in 2PI method. The corresponding grand-canonical partition function is defined within the path integral formalism as

\begin{array}{}\text{(24)}& \begin{array}{ll}Z[H\mathrm{,}B]=\hfill & {\displaystyle \int }D(\varphi )\exp (-S(\varphi )\hfill \\ \hfill & -{\displaystyle \sum _i}{H}_i{\varphi }_i-\frac{1}{2}{\displaystyle \sum _{i\mathrm{,}j}}{B}_{ij}{\varphi }_i{\varphi }_j)\mathrm{,}\hfill \end{array}\end{array}

where

\begin{array}{}\text{(25)}& S(\varphi )=\beta \frac{1}{2}{\displaystyle \sum _{i\mathrm{,}j}{J}_{ij}^{-1}{\varphi }_i{\varphi }_j-}{\displaystyle \sum _n\ln \left[\cosh \left(\beta {\varphi }_n\right)\right]\mathrm{.}}\end{array}

The generating functional of connected Green function is defined as

\begin{array}{}\text{(26)}& W[H\mathrm{,}B]=-\ln Z[H\mathrm{,}B]\mathrm{.}\end{array}

The 2PI functional Γ[φ, G] is defined by the double Legendre transformation and can be written in the form

\begin{array}{}\text{(27)}& \begin{array}{ll}\text{Γ}[\phi \mathrm{,}G]=\hfill & S[\phi ]+\frac{1}{2}\text{Tr}\left[D^{-1}(G-D)\right]\hfill \\ \hfill & +\frac{1}{2}\text{Tr}\ln \left(G^{-1}\right)+\Phi [\phi \mathrm{,}G]\mathrm{,}\hfill \end{array}\end{array}

where {\left(\begin{array}{c}{D}^{-1}\end{array}\right)}_{ij}=\frac{{\delta }^{2}S(\phi )}{\delta {\phi }_i\delta {\phi }_j} and {\phi }_i=\frac{\delta W[H\mathrm{,}B]}{\delta H_i}=\langle {\varphi }_i\rangle , and G_ij = 〈ϕ_iϕ_j 〉−〈ϕ_i〉 〈ϕ_j 〉. Φ[φ, G] can be calculated approximately with well-known standard techniques [16]. We generalize the work [16] by Van Hees and Knoll to arbitrary interaction form, and we find the lowest order approximation of Φ[φ, G] can be demonstrated to be equal to

\begin{array}{}\text{(28)}& \Phi [\phi \mathrm{,}G]=\frac{1}{8}{\displaystyle \sum _i{S}^{(4)}({\phi }_i)G_{ii}{G}_{ii}\mathrm{.}}\end{array}

Here S⁽⁴⁾[φ] stands for the fourth derivative of S⁽⁴⁾(φ). The above expression will allow us obtaining the results with O(N ) Linear–Sigma model in Ref. [16], but for our case:

\begin{array}{}\text{(29)}& \begin{array}{ll}\Phi [\phi \mathrm{,}G]=\hfill & \frac{{\beta }^4}{4}{\displaystyle \sum _i}{(G_{ii})}^2[{\text{sech}}^4(\beta {\phi }_i)-2{\text{sech}}^2(\beta {\phi }_i)\hfill \\ \hfill & \cdot {{\displaystyle \tanh }}^2(\beta {\phi }_i)]\mathrm{.}\hfill \end{array}\end{array}

Then the equations are now given by the fact that we wish to study the theory with vanishing auxiliary sources H and B,

\begin{array}{}\text{(30)}& \frac{\delta \text{Γ}[\phi \mathrm{,}G]}{\delta {\phi }_i}=-H_i-\frac{1}{2}{\displaystyle \sum _m{B}_{im}{\phi }_m-\frac{1}{2}}{\displaystyle \sum _m{B}_{mi}{\phi }_m\stackrel{!}{=}\mathrm{0,}}\end{array}

\begin{array}{}\text{(31)}& \frac{\delta \text{Γ}[\phi \mathrm{,}G]}{\delta G_{ij}}=-\frac{1}{2}{B}_{ij}\stackrel{!}{=}0.\end{array}

Then from Eq. (30) and Eq. (31) we get the “shift” equation and gap equation:

\begin{array}{}\text{(32)}& \begin{array}{ll}\text{0}=\hfill & \beta {\displaystyle \sum _j}{J}_{ij}^{-1}{\phi }_j-\beta \tanh (\beta {\phi }_i)\hfill \\ \hfill & +G_{ii}{\beta }^3{\text{sech}}^2(\beta {\phi }_i)\tanh (\beta {\phi }_i)\hfill \\ \hfill & -{\beta }^5{G}_{ii}{G}_{ii}[2{\text{sech}}^4(\beta {\phi }_i)\tanh (\beta {\phi }_i)\hfill \\ \hfill & -sec{h}^2(\beta {\phi }_i){{\displaystyle \tanh }}^3(\beta {\phi }_i)]\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(33)}& \begin{array}{ll}{G}_{ij}^{-1}=\hfill & \beta J_{ij}^{-1}-{\delta }_{ij}[{\beta }^2{\text{sech}}^2(\beta {\phi }_i)]\hfill \\ \hfill & +{\delta }_{ij}{\beta }^4{G}_{ii}[{\text{sech}}^4(\beta {\phi }_i)\hfill \\ \hfill & -2{\text{sech}}^2(\beta {\phi }_i){{\displaystyle \tanh }}^2(\beta {\phi }_i)]\mathrm{,}\hfill \end{array}\end{array}

and in Fourier space the equations reads

\begin{array}{}\text{(34)}& \begin{array}{ll}0=\hfill & \frac{\beta \phi }{4J}-\beta \tanh (\beta \phi )+{\beta }^3{G}_{ii}{\text{sech}}^2(\beta \phi )\tanh (\beta \phi )\hfill \\ \hfill & -{\beta }^5{G}_{ii}{G}_{ii}[2{\text{sech}}^4(\beta \phi )\tanh (\beta \phi )\hfill \\ \hfill & -sec{h}^2(\beta \phi )tan{h}^3(\beta \phi )]\mathrm{,}\hfill \end{array}\end{array}

\begin{array}{}\text{(35)}& \begin{array}{ll}{G}^{-1}(k)=\hfill & \beta J^{-1}(k)-{\beta }^2{\text{sech}}^2(\beta \phi )\hfill \\ \hfill & +{\beta }^4{G}_{ii}[{\text{sech}}^4(\beta \phi )\hfill \\ \hfill & -2{\text{sech}}^2(\beta \phi ){\text{tanh}}^2(\beta \phi )]\mathrm{.}\hfill \end{array}\end{array}

We can get φ and G from Eq. (34) and Eq. (35) at fixed J and β. In general the solution of Eq. (34) and Eq. (35) do not respect symmetry of system for truncated Φ[φ, G]. In order to cure this problem we supplement the 2PI approximation scheme by an additional effective action defined with respect to the self-consistent solution as [16]

\begin{array}{}\text{(36)}& \text{Γ}(\phi )=\text{Γ}[\phi \mathrm{,}\tilde{G}(\phi )]\mathrm{,}\end{array}

where \tilde{G}[\phi ] is defined by

\begin{array}{}\text{(36)}& {\frac{\delta \text{Γ}[\phi \mathrm{,}G]}{\delta G}|}_{G=\tilde{G}[\phi ]}=0.\end{array}

We can define external Green’s function by the usual definition as double derivatives of Γ(φ) as

\begin{array}{}\text{(37)}& \begin{array}{ll}{(G_{\text{ext}})}_{ij}^{-1}\hfill & =\frac{{\delta }^2\text{Γ}(\phi )}{\delta {\phi }_i\delta {\phi }_j}\hfill \\ \hfill & =G_{ij}^{-1}+\frac{\delta \Phi [\phi \mathrm{,}G]}{\delta {\phi }_i\delta {\phi }_j}+{\displaystyle \sum _{m\mathrm{,}n}}\frac{{\delta }^2\text{Γ}[\phi \mathrm{,}G]}{\delta {\phi }_i\delta G_{mn}}{\text{Λ}}_{mnj}\mathrm{,}\hfill \end{array}\end{array}

where {\text{Λ}}_{mnj}=\frac{\delta G_{mn}}{\delta {\phi }_j} . And \frac{{\delta }^2\text{Γ}[\phi \mathrm{,}G]}{\delta {\phi }_i\delta G_{mn}}={\delta }_{in}{\delta }_{mn}\{{\beta }^3{\text{sech}}^2(\beta {\phi }_i)\text{tanh}(\beta {\phi }_i)

\begin{array}{}\text{(38)}& \begin{array}{ll}\hfill & -G_{mm}[4{\beta }^5{\text{sech}}^4(\beta {\phi }_i)\tanh (\beta {\phi }_i)\hfill \\ \hfill & -2{\beta }^5{\text{sech}}^2(\beta {\phi }_i){{\displaystyle \tanh }}^3(\beta {\phi }_i)]\}\hfill \end{array}\end{array}

due to the property of Kronecker delta only Λ’s whose first and second indices are coincident contributes to Eq. (37). Λ_mmj can be obtained by solving Bathe–Salpeter equation:

\begin{array}{}\text{(39)}& {\text{Λ}}_{iim}=-{\displaystyle \sum _{k\mathrm{,}j}{G}_{ik}\frac{\delta {\text{Γ}}_{kj}}{\delta {\phi }_m}{G}_{ji}\mathrm{.}}\end{array}

Here

\begin{array}{}\text{(40)}& \begin{array}{ll}\frac{\delta {\text{Γ}}_{ij}}{\delta {\phi }_m}=\hfill & \frac{\delta }{\delta {\phi }_m}{\left(\frac{{\delta }^2\text{Γ}[\phi \mathrm{,}G]}{\delta {\phi }_i\delta {\phi }_j}\right)}_{G=\tilde{G}(\phi )}=-{\beta }^2\frac{\delta {\text{sech}}^2\left(\beta {\phi }_i\right)}{\delta {\phi }_m}{\delta }_{ij}+{\beta }^3{G}_{ii}\frac{\delta \left[\beta {\text{sech}}^4\left(\beta {\phi }_i\right)-2\beta {\text{sech}}^2\left(\beta {\phi }_i\right){\text{tanh}}^2\left(\beta {\phi }_i\right)\right]}{\delta {\phi }_m}{\delta }_{ij}\hfill \\ \hfill & +{\beta }^3{\text{Λ}}_{iim}\left[\beta {\text{sech}}^4\left(\beta {\phi }_i\right)-2\beta {\text{sech}}^2\left(\beta {\phi }_i\right){\text{tanh}}^2\left(\beta {\phi }_i\right)\right]{\delta }_{ij}\mathrm{.}\hfill \end{array}\end{array}

In Fourier space the external Green’s function can be written as

\begin{array}{}\text{(41)}& \begin{array}{ll}{G}_{\text{ext}}^{-1}(k)=\hfill & G^{-1}(k)+\{{\beta }^3{\text{sech}}^2(\beta \phi )\tanh (\beta \phi )\hfill \\ \hfill & -G_{ii}[4{\beta }^5{\text{sech}}^4(\beta \phi )\tanh (\beta \phi )\hfill \\ \hfill & -2{\beta }^5{\text{sech}}^2(\beta \phi ){{\displaystyle \tanh }}^3(\beta \phi )]\}\text{Λ}(k)\hfill \\ \hfill & -{\beta }^6{G}_{ii}{G}_{ii}[2{\text{sech}}^6(\beta \phi )\hfill \\ \hfill & -11{\text{sech}}^4(\beta \phi ){\text{tanh}}^2(\beta \phi )\hfill \\ \hfill & +2{\text{sech}}^2(\beta \phi ){\text{tanh}}^4(\beta \phi )]\mathrm{,}\hfill \end{array}\end{array}

where Λ(k) of 2PI method has the same expression as Eq. (22) obtained in 1PI approach.

4 Numerical results

We solve 1PI equations (13, 14), and 2PI equations (34, 35), respectively. The results are shown below in k_B =J = 1 unit. φ as a function of temperature is presented in Fig. 1. The equation ceases to have a solution at T_c, which is the end point of the broken phase and is actually the critical point of a second-order phase transition. For a given φ, we can get 〈σ〉 from Eq. (3), however it is not exactly equal to the spontaneous magnetization and needs corrections to get the exact 〈σ〉 just like G needs corrections to get the exact Green function.

In Table 1 we display T_c from 1PI, 2PI, as well as the SCCF and SMF results [20, 21], together with either exact or approximate values from series estimates [5]. For the 2D square lattice Ising model, 1PI gives T_c = 2.4606, and 2PI gives T_c = 2.4390, both closer to the exact result than the BPW approximation and self-consistent correlated field method(SCCF).

According to fluctuation–dissipation theorem, we can get the susceptibility χ_ij with following relation:

\begin{array}{}\text{(42)}& {\text{χ}}_{ij}=\frac{\delta \langle {\sigma }_i\rangle }{\delta h_j}=\beta \langle {\sigma }_i{\sigma }_j\rangle -\beta \langle {\sigma }_i\rangle \langle {\sigma }_j\rangle \mathrm{.}\end{array}

The Fourier transform of the susceptibility χ_ij is

\begin{array}{}\text{(43)}& {\chi }_{ij}={\displaystyle \sum _{\alpha =x\mathrm{,}y}{\displaystyle {\int }_{-\pi }^{\pi }}\frac{{\text{d}}^{2}k}{{(2\pi )}^2}\exp (-\text{i}{k}_{\alpha }(i_{\alpha }-j_{\alpha }))\chi (k)\mathrm{.}}\end{array}

The zero momentum susceptibility, which we denote as χ, can be obtained from Eq. (20, 41) with the following expression:

\begin{array}{}\text{(44)}& \begin{array}{ll}\text{χ}\hfill & =\beta {\displaystyle \sum _m}(\langle {\sigma }_m{\sigma }_0\rangle -{\langle {\sigma }_0\rangle }^2)\hfill \\ \hfill & =\beta {\displaystyle \sum _m}(J_{m\mathrm{,}i}^{-1}{J}_{\mathrm{0,}j}^{-1}{G}_{ij}-{\beta }^{-1}{J}_{m\mathrm{,0}}^{-1})\hfill \\ \hfill & =\beta \left[J^{-2}(0)G(0)-{\beta }^{-1}{J}^{-1}(0)\right]\mathrm{.}\hfill \end{array}\end{array}

Here G(0) refers to G_full(0) or G_ext(0), under 1PI or 2PI approximation, respectively. The numerical results are plotted in Fig. 2, both susceptibility will diverge at its corresponding T_c.

We also compare results for finite size lattices (subject to periodic boundary conditions) with Monte Carlo results. And the results are illustrated in Fig. 3, under two different temperatures. For the finite size lattice of N ×N with periodic boundary conditions, the formula in the integration shall be substituted as

\begin{array}{}\text{(45)}& \frac{1}{{\left(2\text{π}\right)}^2}{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}}\text{d}{k}_x{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}}\text{d}{k}_{y}f(k_x\mathrm{,}{k}_y)\to \frac{1}{N^2}{\displaystyle \sum _{k_x\mathrm{,}{k}_y}f\left(k_x\mathrm{,}{k}_y\right)\mathrm{,}}\end{array}

where f (k_x, k_y ) is a periodic function of k_x and k_y (periodicity is 2π), and inside the summation, k_x=\frac{2\pi }{N}i , i =0, N−1,k_y=\frac{2\pi }{N}j , j = 0, N−1 due to periodic boundary conditions.

It can be seen that our results show a significant improvement compared to the mean-field approach, especially for a temperature closer to T_c. We also include the correlation function obtained by BPW method (or Bethe approximation) in Fig. 3 for comparison. The BPW calculation of the correlation is highly non trivial and complex. It was only studied quite recently, see [24] and references therein. BPW method was particular useful for studying Ising model (also useful for Random Ising model), however the generalizations to other models are too complex. BPW result is better than the field theoretical result for the correlation function below the real critical temperature (T_c = 2.26918), however the critical temperature obtained by BPW method is T_c = 2.885, worse than the critical temperature obtained by the field theoretical method (for 1PI, T_c = 2.4606, and for 2PI, T_c = 2.4390).

Although the deviation of our approach for the correlation function with respect to MC is slightly larger than BPW method below the critical temperature, however, the field theoretical approach can easily generalize to quantum many body theory, and can be also applied to complicated spin models with continue symmetry, for example XY model and Heisenberg model, etc.

5 Conclusion

In conclusion, we calculate the critical temperature, susceptibility and Green function nonperturbatively with two kind field theories developed by the Dyson–Schwinger theory and the Φ-derivable theory at leading order fluctuation correction. With relative low cost, both method are able to give fairly good predictions of T_c for the Ising model. In the area far away from critical point, the susceptibility and Green function obtained from our method is quite accurate comparing with exact solution. This is a systemic approach which can be used to treat more complex spin models. The methods will preserve fundamental identities, like the fluctuation dissipation relation and WTI identities for systems with continue symmetries, which are very crucial for giving consistent descriptions of such systems.

References

[1] E. Ising, Beitrag zur theorie des ferromagnetismus, Z.Phys. 31(1), 253 (1925)

[2] L. Onsager, Crystal statistics (i): A two-dimensional model with an order-disorder transition, Phys. Rev. 65(3–4), 117 (1944)

[3] A. Kuzemsky, Statistical mechanics and the physics of many-particle model systems, Phys. Part. Nucl. 40(7), 949(2009)

[4] P. Weiss and E. Stoner, Magnetism and atomic structure, J. Phys. 6, 667 (1907)

[5] G. Wysin and J. Kaplan, Correlated molecular-field theory for Ising models, Phys. Rev. E 61(6), 6399 (2000)

[6] H. A. Bethe, Statistical theory of superlattices, Proc. R.Soc. Lond. A 150(871), 552 (1935)

[7] R. Peierls, On Ising’s model of ferromagnetism, in: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, Cambridge University Press, 1936, pp 477–481

[8] P. R. Weiss, The application of the Bethe–Peierls method to ferromagnetism, Phys. Rev. 74(10), 1493 (1948)

[9] K. K. Zhuravlev, Molecular-field theory method for evaluating critical points of the ising model, Phys. Rev. E 72(5), 056104 (2005)

[10] D. Yamamoto, Correlated cluster mean-field theory for spin systems, Phys. Rev. B 79(14), 144427 (2009)

[11] J. R. Viana, O. R. Salmon, J. R. de Sousa, M. A. Neto, and I. T. Padilha, An effective correlated mean-field theory applied in the spin-1/2 Ising ferromagnetic model, J.Magn. Magn. Mater. 369, 101 (2014)

[12] J. M. Luttinger and J. C. Ward, Ground-state energy of a many-fermion system (ii), Phys. Rev. 118(5), 1417 (1960)

[13] G. Baym and L. P. Kadanoff, Conservation laws and correlation functions, Phys. Rev. 124(2), 287 (1961)

[14] J. M. Cornwall, R. Jackiw, and E. Tomboulis, Effective action for composite operators, Phys. Rev. D 10(8), 2428 (1974)

[15] A. Kovner and B. Rosenstein, Covariant Gaussian approximation (i): Formalism, Phys. Rev. D 39(8), 2332 (1989)

[16] H. Van Hees and J. Knoll, Renormalization in self-consistent approximation schemes at finite temperature (iii): Global symmetries, Phys. Rev. D 66(2), 025028 (2002)

[17] D. J. Amit and V. Martin-Mayor, Field Theory, the Renormalization Group, and Critical Phenomena: Graphs to Computers, World Scientific Publishing Company, 2005

[18] J. Wang, D. Li, H. Kao, and B. Rosenstein, Covariant Gaussian approximation in Ginzburg–Landau model, Ann. Phys. 380, 228 (2017)

[19] B. Rosenstein and A. Kovner, Covariant Gaussian approximation (ii): Scalar theories, Phys. Rev. D 40(2), 504 (1989)

[20] M. E. Fisher, The theory of equilibrium critical phenomena, Rep. Prog. Phys. 30(2), 615 (1967)

[21] N. W. Ashcroft, N. D. Mermin, et al., Solid state physics, Vol. 2005, Holt, Rinehart And Winston, New York, London, 1976

[22] H. Au-Yang and J. H. Perk, Correlation functions and susceptibility in the z-invariant Ising model, in: MathPhys Odyssey 2001, Springer, 2002, pp 23–48

[23] W. Orrick, B. Nickel, A. Guttmann, and J. Perk, The susceptibility of the square lattice Ising model: New developments, J. Stat. Phys. 102(3/4), 795 (2001) (for the complete set of series coefficients see https://blogs.unimelb.edu.au/tony-guttmann/)

[24] F. Ricci-Tersenghi, The Bethe approximation for solving the inverse Ising problem: A comparison with other inference methods, J. Stat. Mech. 2012(08), P08015 (2012)

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