A reduction relation is noted which allows one to relate exactly, at any temperature T, two quantities for the Ising model on the diamond lattice: (i) the ratio, essentially, of the average neighbour-neighbour correlation function and the energy per spin, P=T 2 /U 2 and (ii) a ratio of a certain pair of four-spin correlation functions Q=T4/U4. This relation is applied to Mouritsen's Monte Carlo determination of these quantities P c and Q c , at the critical point T c . It is shown that the Monte Carlo values are severely inconsistent with the exact relation. A more reliable value of the neighbour-neighbour correlation function, ( sigma 1 sigma 2 ) approximately=0.283, is calculated from the Monte Carlo data.A reduction relation is noted which allows one to relate exactly, at any temperature T, two quantities for the Ising model on the diamond lattice: (i) the ratio, essentially, of the average neighbour-neighbour correlation function and the energy per spin, P=T 2 /U 2 and (ii) a ratio of a certain pair of four-spin correlation functions Q=T4/U4. This relation is applied to Mouritsen's Monte Carlo determination of these quantities P c and Q c , at the critical point T c . It is shown that the Monte Carlo values are severely inconsistent with the exact relation. A more reliable value of the neighbour-neighbour correlation function, ( sigma 1 sigma 2 ) approximately=0.283, is calculated from the Monte Carlo data.

Journal of Physics Condensed MatterMay 10, 19931 citation

Spin-Spin Correlation Functions in Magnetic Systems.

Philip C. Clapp

Jan 1, 19631 citation

Exact Relations for Quantum Many-Body Correlation Functions

Tadashi Toyoda

AIP conference proceedingsJan 1, 20040 citations

An exact alternative solution method of 1D Ising model with Block-spin transformation at $H=0$ An exact alternative solution method of 1D Ising model with Block-spin transformation at $H=0$

Tuncer Kaya

An alternative exact explicit solution of 1D Ising chain is presented without using any boundary conditions (or free boundary condition) by the help of applying successively block-spin transformation. Exact relation are obtained between spin-spin correlation functions in the absence of external field. To evaluate average magnetization (or the order parameter), it is assumed that the average magnetization can be related to infinitely apart two spin correlation function as $^{2}=$ . A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that $=\tanh{K}$ , which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of $\{σ_{i}σ_{i+1}\}=\{s_{i}\}$ and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that $\{σ_{0}σ_{j}\}=(\tanh{K})^{j}$ , here $j$ assumes the values of $j=2^n$ , here $n$ is an integer numbers. We believe that this result is really important in that it is the only exact unique treatment of the 1D Ising chain beside with the transfer matrix method. It is also pointed out the irrelevances of some of the alternative derivation appearing in graduate level text books. The obtained unique correlation relation in this this study leads in the limit of $N\rightarrow\infty$ to $=(\tanh{K})^{N/2}$ , indicating that the second order phase transition is only possible in the limit of $K\rightarrow\infty$ . The obtained results in this work are exactly the same as those of obtained by the transfer matrix method which considers periodic boundary condition.An alternative exact explicit solution of 1D Ising chain is presented without using any boundary conditions (or free boundary condition) by the help of applying successively block-spin transformation. Exact relation are obtained between spin-spin correlation functions in the absence of external field. To evaluate average magnetization (or the order parameter), it is assumed that the average magnetization can be related to infinitely apart two spin correlation function as $^{2}=$. A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that $=\tanh{K}$, which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of $\{σ_{i}σ_{i+1}\}=\{s_{i}\}$ and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that $\{σ_{0}σ_{j}\}=(\tanh{K})^{j}$, here $j$ assumes the values of $j=2^n$, here $n$ is an integer numbers. We believe that this result is really important in that it is the only exact unique treatment of the 1D Ising chain beside with the transfer matrix method. It is also pointed out the irrelevances of some of the alternative derivation appearing in graduate level text books. The obtained unique correlation relation in this this study leads in the limit of $N\rightarrow\infty$ to $=(\tanh{K})^{N/2}$, indicating that the second order phase transition is only possible in the limit of $K\rightarrow\infty$. The obtained results in this work are exactly the same as those of obtained by the transfer matrix method which considers periodic boundary condition.

arXiv:1801.04452Jan 13, 20180 citations

Green's-Function Analysis of the Ising Ferromagnet

F. Burr Anderson

The method of the two-time temperature-dependent Green's function has been used to analyze the Ising model of a ferromagnet in an external magnetic field. The selection of a particular Green's function enables us to write an exact expression for the equation of motion. We are then led to a differential difference equation for the correlation function corresponding to the Green's function. No decoupling assumptions have been made, so the equation is exact for both arbitrary spin and range of interaction. It is shown how various approximate theories may be extracted from our formalism. The exact differential difference equation may be reduced to a partial differential equation. The latter form allows us to generate relations among the magnetization and spin-spin correlation functions. These relations are given in detail for the case of spin \textonehalf{} and $z$ nearest neighbors.The method of the two-time temperature-dependent Green's function has been used to analyze the Ising model of a ferromagnet in an external magnetic field. The selection of a particular Green's function enables us to write an exact expression for the equation of motion. We are then led to a differential difference equation for the correlation function corresponding to the Green's function. No decoupling assumptions have been made, so the equation is exact for both arbitrary spin and range of interaction. It is shown how various approximate theories may be extracted from our formalism. The exact differential difference equation may be reduced to a partial differential equation. The latter form allows us to generate relations among the magnetization and spin-spin correlation functions. These relations are given in detail for the case of spin \textonehalf{} and $z$ nearest neighbors.

Physical review. B, Solid stateJun 1, 19719 citations

An exact alternative solution method of 1D Ising model with Block-spin transformation at

H=0

^{2}=

. A discussion to justify this consideration is given in the introduction with a relevant manner. It is obtained that

=\tanh{K}

, which is exactly the same relation as the previously derived relation by considering the configurational space equivalence of

\{σ_{i}σ_{i+1}\}=\{s_{i}\}

and the result of transfer matrix method in the absence of external field. By applying further block-spin transformation, it is obtained that

\{σ_{0}σ_{j}\}=(\tanh{K})^{j}

, here

j

assumes the values of

j=2^n

, here

n

is an integer numbers. We believe that this result is really important in that it is the only exact unique treatment of the 1D Ising chain beside with the transfer matrix method. It is also pointed out the irrelevances of some of the alternative derivation appearing in graduate level text books. The obtained unique correlation relation in this this study leads in the limit of

N\rightarrow\infty

=(\tanh{K})^{N/2}

, indicating that the second order phase transition is only possible in the limit of

K\rightarrow\infty

. The obtained results in this work are exactly the same as those of obtained by the transfer matrix method which considers periodic boundary condition.

z

nearest neighbors.