An argument is presented for a certain asymptotic form for the correlation functions of two local dynamical variables when the system is near its critical point and in the one phase region. If $A(\mathbf{r})$ and $B(\mathbf{r})$ are local dynamical variables and $\ensuremath{\omega}$ stands for the variables specifying the thermodynamic state, it is found that for $\ensuremath{\omega}$ near the critical point ${\ensuremath{\omega}}_{c}$ and for $r$ large enough, $〈A(\mathbf{r})B(0)〉\ensuremath{-}〈A(\mathbf{r})〉〈B(0)〉\ensuremath{\simeq}\frac{{d}_{A}(\ensuremath{\omega}){d}_{B}(\ensuremath{\omega}){e}^{\ensuremath{-}\ensuremath{\kappa}(\ensuremath{\omega})}}{{r}^{b(\ensuremath{\omega})}}$, where $\ensuremath{\kappa}(\ensuremath{\omega})$ and $b(\ensuremath{\omega})$ are the same for a large class of $A$ and $B$ and $\ensuremath{\kappa}(\ensuremath{\omega})\ensuremath{\rightarrow}0$, $b(\ensuremath{\omega})\ensuremath{\rightarrow}b\ensuremath{\ge}0 \mathrm{as} \ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}$. For the Ising model of any dimension in zero field and below the critical temperature, the additional assumption of a single correlation length implies the relationship ${\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta}=2\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}$, which is often conjectured for the critical exponents.An argument is presented for a certain asymptotic form for the correlation functions of two local dynamical variables when the system is near its critical point and in the one phase region. If $A(\mathbf{r})$ and $B(\mathbf{r})$ are local dynamical variables and $\ensuremath{\omega}$ stands for the variables specifying the thermodynamic state, it is found that for $\ensuremath{\omega}$ near the critical point ${\ensuremath{\omega}}_{c}$ and for $r$ large enough, $〈A(\mathbf{r})B(0)〉\ensuremath{-}〈A(\mathbf{r})〉〈B(0)〉\ensuremath{\simeq}\frac{{d}_{A}(\ensuremath{\omega}){d}_{B}(\ensuremath{\omega}){e}^{\ensuremath{-}\ensuremath{\kappa}(\ensuremath{\omega})}}{{r}^{b(\ensuremath{\omega})}}$, where $\ensuremath{\kappa}(\ensuremath{\omega})$ and $b(\ensuremath{\omega})$ are the same for a large class of $A$ and $B$ and $\ensuremath{\kappa}(\ensuremath{\omega})\ensuremath{\rightarrow}0$, $b(\ensuremath{\omega})\ensuremath{\rightarrow}b\ensuremath{\ge}0 \mathrm{as} \ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}$. For the Ising model of any dimension in zero field and below the critical temperature, the additional assumption of a single correlation length implies the relationship ${\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta}=2\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}$, which is often conjectured for the critical exponents.

Physical ReviewJul 5, 196818 citations

Structure of the correlation function of a fluctuating system at the critical point

Yu. M. Ivanchenko, Alexander A. Lisyansky, Alexander E. Filippov

Theoretical and Mathematical PhysicsAug 1, 19903 citations

Static and dynamic structure factors in three-dimensional randomly diluted Ising models

Pasquale Calabrese, Andrea Pelissetto C, Ettore Vicari

We consider the three-dimensional randomly diluted Ising model and study the critical behavior of the static and dynamic spin-spin correlation functions (static and dynamic structure factors) at the paramagnetic-ferromagnetic transition in the high-temperature phase. We consider a purely relaxational dynamics without conservation laws, the so-called model A. We present Monte Carlo simulations and perturbative field-theoretical calculations. While the critical behavior of the static structure factor is quite similar to that occurring in pure Ising systems, the dynamic structure factor shows a substantially different critical behavior. In particular, the dynamic correlation function shows a large-time decay rate which is momentum independent. This effect is not related to the presence of the Griffiths tail, which is expected to be irrelevant in the critical limit, but rather to the breaking of translational invariance, which occurs for any sample and which, at the critical point, is not recovered even after the disorder average.We consider the three-dimensional randomly diluted Ising model and study the critical behavior of the static and dynamic spin-spin correlation functions (static and dynamic structure factors) at the paramagnetic-ferromagnetic transition in the high-temperature phase. We consider a purely relaxational dynamics without conservation laws, the so-called model A. We present Monte Carlo simulations and perturbative field-theoretical calculations. While the critical behavior of the static structure factor is quite similar to that occurring in pure Ising systems, the dynamic structure factor shows a substantially different critical behavior. In particular, the dynamic correlation function shows a large-time decay rate which is momentum independent. This effect is not related to the presence of the Griffiths tail, which is expected to be irrelevant in the critical limit, but rather to the breaking of translational invariance, which occurs for any sample and which, at the critical point, is not recovered even after the disorder average.

arXiv:0711.3673Physical Review EFeb 27, 20085 citations

Experimental investigation of the correlation function near the critical point

Mikhail A. Anisimov, A. M. Evtyushenkov, Yu. F. Kiyachenko +1 more

Sep 1, 19741 citation

Thermodynamic Structure of Correlation Functions near a Critical Point in Multicomponent Systems

Luciano Mistura

A simple generalization of the Griffiths and Wheeler postulates for the thermodynamic behavior near a critical point is introduced to describe fluctuations in local dynamical variables. It is proposed that in the classical Hilbert space of the dynamical variables there are only two linearly independent vectors, related to the shape of the coexistence surface and the critical surface, which contain critical fluctuations. It is shown that, as a consequence of the affine structure of the field space, the above assumption leads to the following asymptotic property for the correlation function ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})$ of any two local operators $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{a}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{b}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$: ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})={a}_{1}(q, \ensuremath{\omega}){b}_{1}(q, \ensuremath{\omega}){f}_{11}(q, \ensuremath{\omega})+{a}_{2}(q, \ensuremath{\omega}){b}_{2}(q, \ensuremath{\omega}){f}_{22}(q, \ensuremath{\omega})+{a}_{3}(q, \ensuremath{\omega}){b}_{3}(q, \ensuremath{\omega}){f}_{33}(q, \ensuremath{\omega})+\dots{}$Here $q$ is the wave number, $\ensuremath{\omega}$ is a point in the field space, the coefficients ${a}_{i}$ and ${b}_{i}$ are "generally" well-behaved thermodynamic quantities and the ${f}_{\mathrm{ii}}$ , which are universal functions, have the following properties: ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is nondivergent if $i\ensuremath{\ge}3$, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{22}(q, \ensuremath{\omega})$ is weakly divergent, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is strongly divergent. A few typical examples are worked out to show that if this property is postulated, the Griffiths and Wheeler thermodynamic behavior is reproduced.A simple generalization of the Griffiths and Wheeler postulates for the thermodynamic behavior near a critical point is introduced to describe fluctuations in local dynamical variables. It is proposed that in the classical Hilbert space of the dynamical variables there are only two linearly independent vectors, related to the shape of the coexistence surface and the critical surface, which contain critical fluctuations. It is shown that, as a consequence of the affine structure of the field space, the above assumption leads to the following asymptotic property for the correlation function ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})$ of any two local operators $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{a}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{b}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$: ${f}_{\mathrm{ab}}(q, \ensuremath{\omega})={a}_{1}(q, \ensuremath{\omega}){b}_{1}(q, \ensuremath{\omega}){f}_{11}(q, \ensuremath{\omega})+{a}_{2}(q, \ensuremath{\omega}){b}_{2}(q, \ensuremath{\omega}){f}_{22}(q, \ensuremath{\omega})+{a}_{3}(q, \ensuremath{\omega}){b}_{3}(q, \ensuremath{\omega}){f}_{33}(q, \ensuremath{\omega})+\dots{}$Here $q$ is the wave number, $\ensuremath{\omega}$ is a point in the field space, the coefficients ${a}_{i}$ and ${b}_{i}$ are "generally" well-behaved thermodynamic quantities and the ${f}_{\mathrm{ii}}$, which are universal functions, have the following properties: ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is nondivergent if $i\ensuremath{\ge}3$, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{22}(q, \ensuremath{\omega})$ is weakly divergent, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is strongly divergent. A few typical examples are worked out to show that if this property is postulated, the Griffiths and Wheeler thermodynamic behavior is reproduced.

Physical Review AJul 1, 19725 citations

Correlation Function near the Critical Point

P. Gluck

physica status solidi (b)Aug 1, 19730 citations

An argument is presented for a certain asymptotic form for the correlation functions of two local dynamical variables when the system is near its critical point and in the one phase region. If

A(\mathbf{r})

and

B(\mathbf{r})

are local dynamical variables and $\ensuremath{\omega}$ stands for the variables specifying the thermodynamic state, it is found that for $\ensuremath{\omega}$ near the critical point ${\ensuremath{\omega}}_{c}$ and for

r

large enough, $〈A(\mathbf{r})B(0)〉\ensuremath{-}〈A(\mathbf{r})〉〈B(0)〉\ensuremath{\simeq}\frac{{d}_{A}(\ensuremath{\omega}){d}_{B}(\ensuremath{\omega}){e}^{\ensuremath{-}\ensuremath{\kappa}(\ensuremath{\omega})}}{{r}^{b(\ensuremath{\omega})}}$, where $\ensuremath{\kappa}(\ensuremath{\omega})$ and $b(\ensuremath{\omega})$ are the same for a large class of

A

and

B

and $\ensuremath{\kappa}(\ensuremath{\omega})\ensuremath{\rightarrow}0$, $b(\ensuremath{\omega})\ensuremath{\rightarrow}b\ensuremath{\ge}0 \mathrm{as} \ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}$. For the Ising model of any dimension in zero field and below the critical temperature, the additional assumption of a single correlation length implies the relationship ${\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta}=2\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}}$, which is often conjectured for the critical exponents.

q

is the wave number, $\ensuremath{\omega}$ is a point in the field space, the coefficients

{a}_{i}

and

{b}_{i}

are "generally" well-behaved thermodynamic quantities and the

{f}_{\mathrm{ii}}

, which are universal functions, have the following properties: ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is nondivergent if $i\ensuremath{\ge}3$, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{22}(q, \ensuremath{\omega})$ is weakly divergent, ${\mathrm{lim}}_{q\ensuremath{\rightarrow}0,\ensuremath{\omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{c}}{f}_{11}(q, \ensuremath{\omega})$ is strongly divergent. A few typical examples are worked out to show that if this property is postulated, the Griffiths and Wheeler thermodynamic behavior is reproduced.