Critical behavior of the correlation function of three-dimensional O(N) models in the symmetric phase

TL;DR

This study investigates the critical behavior of the correlation function in three-dimensional O(N) models, revealing Gaussian behavior at low momentum across all N, supported by strong-coupling series and large-N analysis.

Contribution

The paper provides new strong-coupling series data and demonstrates Gaussian behavior of the correlation function in the symmetric phase for all N.

Findings

Correlation function is essentially Gaussian at low momentum for all N

Strong-coupling series analysis supports Gaussian behavior

Large-N analysis confirms the results

Abstract

We present new strong-coupling series for O(N) spin models in three dimensions, on the cubic and diamond lattices. We analyze these series to investigate the two-point Green's function G(x) in the critical region of the symmetric phase. This analysis shows that the low-momentum behavior of G(x) is essentially Gaussian for all N from zero to infinity. This result is also supported by a large-N analysis.

Tables

Table 1. Table 1: Comparison of strong-coupling expansion on cubic and diamond lattices with 1 / N 1 𝑁 1/N and ε 𝜀 \varepsilon -expansion

$N$	lattice	${10}^4{c}_2$		${10}^5{c}_3$		${10}^4\left(s^{\ast }-1\right)$
$0$	cubic	$\left|{10}^4{c}_2\right|\lesssim 2$		$1.$	$2\left(1\right)$	$1.$	$2\left(3\right)$
	diamond	$\left|{10}^4{c}_2\right|\lesssim 1$		$1.$	$0\left(1\right)$	$1.$	$0\left(5\right)$
	$\epsilon$-expansion	$-2.$	$35$	$0.$	$60$
$1$	cubic	$-2.$	$9\left(2\right)$	$1.$	$1\left(1\right)$	$-2.$	$3\left(5\right)$
	diamond	$-3.$	$1\left(2\right)$	$1.$	$0\left(2\right)$	$-2.$	$2\left(3\right)$
	$\epsilon$-expansion	$-2.$	$78$	$0.$	$71$
$2$	cubic	$-3.$	$8\left(3\right)$	$1.$	$1\left(1\right)$	$-3.$	$5\left(5\right)$
	diamond	$-4.$	$2\left(3\right)$	$1.$	$1\left(3\right)$	$-3.$	$5\left(2\right)$
	$\epsilon$-expansion	$-3.$	$01$	$0.$	$77$
$3$	cubic	$-4.$	$0\left(2\right)$	$1.$	$1\left(2\right)$	$-4.$	$0\left(4\right)$
	diamond	$-4.$	$2\left(3\right)$	$1.$	$1\left(3\right)$	$-3.$	$5\left(2\right)$
	$\epsilon$-expansion	$-3.$	$11$	$0.$	$79$
$4$	cubic	$-4.$	$1\left(2\right)$	$1.$	$2\left(1\right)$	$-4.$	$0\left(4\right)$
	diamond	$-4.$	$7\left(2\right)$	$1.$	$0\left(2\right)$	$-4.$	$0\left(2\right)$
	$\epsilon$-expansion	$-3.$	$13$	$0.$	$80$
	$1/N$	$-11.$	$12$	$3.$	$36$	$-11.$	$48$
$8$	cubic	$-3.$	$5\left(2\right)$	$1.$	$0\left(2\right)$	$-3.$	$7\left(3\right)$
	diamond	$-4.$	$0\left(1\right)$	$0.$	$7\left(5\right)$	$-4.$	$0\left(4\right)$
	$\epsilon$-expansion	$-2.$	$94$	$0.$	$75$
	$1/N$	$-5.$	$56$	$1.$	$18$	$-5.$	$74$
$16$	cubic	$-2.$	$4\left(2\right)$	$0.$	$70\left(5\right)$	$-2.$	$7\left(2\right)$
	diamond	$-2.$	$65\left(5\right)$	$0.$	$5\left(5\right)$	$-2.$	$9\left(2\right)$
	$\epsilon$-expansion	$-2.$	$35$	$0.$	$60$
	$1/N$	$-2.$	$78$	$0.$	$84$	$-2.$	$87$

Equations

$$|k|\lesssim 1/\xi ,0<T/T_c-1\ll 1.$$

$$S=-N\beta \sum _{\mathrm{links}}{\vec{s}}_{x_l}\cdot {\vec{s}}_{x_r},$$

$$G(x)=⟨{\vec{s}}_x\cdot {\vec{s}}_0⟩.$$

$$L(k;\beta )\equiv \frac{\tilde{G}(0;\beta )}{\tilde{G}(k;\beta )}.$$

$${\xi }_G^2\equiv \frac{1}{6}\frac{{\sum }_x{x}^{2}G(x)}{{\sum }_{x}G(x)}.$$

$$L(y)=1+y+l(y),l(y)=\sum _{i=2}^{\infty }{c}_i{y}^i.$$

$$m_{2j}=\sum _x{x}^{2j}G(x).$$

$$c_2\simeq -\frac{0.0044486}{N},c_3\simeq \frac{0.0001344}{N},$$

$$c_4\simeq -\frac{0.00000658}{N},c_5\simeq \frac{0.00000040}{N}...$$

$$c_i\ll c_2\ll 1,i\ge 3$$

$$s^{\ast }-1\simeq -\frac{0.0045900}{N}.$$

$$c_i\simeq {\epsilon }^2\frac{N+2}{(N+8{)}^2}{e}_i,$$

$$e_2\simeq -0.007520,e_3\simeq \mathrm{0.0001919.}$$

$$l_4=\sum _{x,y,z}[f_4(x,y)+f_4(y,z)+f_4(z,x)]G(x,y,z),$$

$$f_4(x,y)=(x^2+y^2{)}^2-8x^2{y}^2;$$

$$l_{6,1}=\sum _{x,y,z}[f_6(x,y)+f_6(y,z)+f_6(z,x)]$$

$$\times G(x,y,z),$$

$$f_6(x,y)=(x^2+y^2{)}^3-8(x^4{y}^2+x^2{y}^4);$$

$$l_{6,2}=\sum _{x,y,z}[x^6+y^6+z^6-45x^2{y}^2{z}^2]G(x,y,z).$$

$$A_{2j,i}\equiv \frac{l_{2j,i}}{m_{2j}}\sim {\xi }_G^{-2}.$$

$$B_{2j,i}\equiv \frac{l_{2j,i}}{m_{2j-2}};$$

Full text

1 INTRODUCTION

Three-dimensional ${\rm O}(N)$-symmetric spin models describe many important critical phenomena in nature: the case $N=3$ describes ferromagnetic materials, where the order parameter is the magnetization; the case $N=2$ describes the helium superfluid transition, where the order parameter is the quantum amplitude; the case $N=1$ (Ising model) describes liquid-vapor transitions, where the order parameter is the density.

The critical behavior of the two-point correlation function $G(x)$ is related to critical scattering, which is observed in many experiments, e.g., neutron scattering in ferromagnetic materials, light and X-rays scattering in liquid-gas systems.

In the following we will focus on the low-momentum behavior of the Fourier-transformed correlation function $\widetilde{G}(k)$ in the critical region of the symmetric phase, i.e., for

[TABLE]

2 LATTICE MODELS

Let us consider an ${\rm O}(N)$-symmetric lattice spin models described by the nearest-neighbor action

[TABLE]

where $\beta=1/T,$ $\vec{s}$ is an $N$-component real vector, and $x_{l}$, $x_{r}$ are the endpoints of the link. The two-point correlation function is defined by

[TABLE]

In order to simplify the study the critical behavior of $G(x)$, we introduce the dimensionless RG-invariant function

[TABLE]

In the critical region of the symmetric phase, $L(k,\beta)$ is a function only of the ratio $y\equiv k^{2}/M_{G}^{2}$, where $M_{G}\equiv 1/\xi_{G}$; the second-moment correlation length $\xi_{G}$ is defined by

[TABLE]

$M_{G}$ is the mass-scale which can be directly observed in scattering experiments. $L(y)$ can be expanded in powers of $y$ around $y=0$:

[TABLE]

$l(y)$ parameterizes the difference from a generalized Gaussian propagator. The coefficients $c_{i}$ can be expressed as the critical limit of appropriate dimensionless RG-invariant ratios of the spherical moments

[TABLE]

Another interesting quantity related to the low-momentum behavior of $G$ is the ratio $s=M^{2}/M_{G}^{2}$, where $M$ is the mass-gap of the theory. Its critical value is $s^{*}=-y_{0}$, where $y_{0}$ is the zero of $L(y)$ closest to the origin.

In the large-$N$ limit, $l(y)$ is depressed by a factor of $1/N$. The coefficients $c_{i}$ can be obtained from a $1/N$ expansion in the continuum [1]:

[TABLE]

We are presently computing the order $1/N^{2}$ of the expansion. We expect that the pattern established by the $1/N$ expansion

[TABLE]

will be followed by all models with sufficiently large $N$. This implies $s^{*}-1\simeq c_{2}$: indeed, in the large-$N$ limit,

[TABLE]

The coefficients $c_{i}$ can also be computed from an $\varepsilon$-expansion of the corresponding $\phi^{4}$ theory around $d=4$ [2]:

[TABLE]

where $\varepsilon=4-d$ and

[TABLE]

3 STRONG-COUPLING EXPANSION

We computed the strong-coupling expansion of $G(x)$ up to 15th order on the cubic lattice, and up to 21st order on the diamond lattice. Our technique for the strong-coupling expansion of ${\rm O}(N)$ spin models was presented in Ref. [3].

We took special care in the choice of estimators for the “physical” quantities $c_{i}$ and $s^{*}$. This step is very important from a practical point of view: better estimators can greatly improve the stability of the extrapolation to the critical point. Our search for optimal estimators was guided by the requirement of a regular strong-coupling expansion (e.g., no $\ln\beta$ terms) and by the knowledge of the large-$N$ limit (we chose estimators which are “perfect” for $N=\infty$).

The strong-coupling series of the estimators were analyzed by Padé approximants, Dlog-Padé approximants and first-order integral approximants (see Ref. [4] for a review of the resummation techniques; see also Ref. [5]). For diamond lattice models with $N\neq 0$, $\beta_{c}$ was not known, and we estimated it from the strong coupling series of the magnetic susceptibility.

Our strong-coupling results on cubic and diamond lattices are compared with the results of the $1/N$ expansion and of the $\varepsilon$-expansion in Table 1. One may notice that universality between cubic and diamond lattice is always confirmed; furthermore, the agreement with the $\varepsilon$-expansion and with the $1/N$ expansion is satisfactory.

The predicted pattern $c_{3}\ll c_{2}\ll 1$ is verified for all $N$. We can conclude that the two-point Green’s function is essentially Gaussian for all momenta with $|k^{2}|\lesssim M_{G}^{2}$, and that the small corrections are dominated by the $(k^{2})^{2}$ term.

4 APPROACH TO CRITICALITY

We investigated the approach to criticality, with special attention devoted to anisotropy (violation of rotational invariance). Let us introduce the anisotropy estimators

[TABLE]

In the critical limit, $l_{2j}$ are depressed with respect to the spherical moments $m_{2j}$. In the large-$N$ limit one can show that

[TABLE]

We analyzed the strong-coupling series of

[TABLE]

for all values of $N$, we found that $B_{2j,i}$ have a finite (but non-universal) $T\to T_{c}$ limit. This supports the validity of Eq. (15) for all $N$.

Ratios of $A_{2j,i}$ are universal quantities; we found that at criticality $A_{6,1}/A_{4}\simeq 0.95$ and $A_{6,2}/A_{6,1}\simeq 0.75$ (within one per mill) for all $N$.