The non-analytic momentum dependence of spin susceptibility of Heisenberg magnets in paramagnetic phase and its effect on critical exponents

TL;DR

This paper investigates the momentum dependence of spin susceptibility in Heisenberg magnets within the nonlinear sigma model, revealing non-analytic behavior that influences critical exponents near the upper critical dimension.

Contribution

It introduces a universal scaling function describing non-analytic momentum dependence of susceptibility and analyzes its impact on critical exponents in arbitrary dimensions.

Findings

The susceptibility exhibits non-analytic momentum dependence characterized by a universal function.

The non-analytic term dominates the correction to the critical exponent  in dimensions close to 4.

Critical exponents  and  are consistent with previous studies, with new insights into their corrections.

Abstract

We study momentum dependence of static magnetic susceptibility $\chi(q)$ in paramagnetic phase of Heisenberg magnets and its relation to critical behavior within nonlinear sigma model (NLSM) at arbitrary dimension $2<d<4$. In the first order of $1/N$ expansion, where $N$ is the number of spin components, we find $\chi(q)\propto[q^{2}+\xi^{-2}(1+f(q\xi ))]^{-1+\eta /2}$, where $\xi $ is the correlation length, $q$ is the momentum, measured from magnetic wave vector, the universal scaling function $f(x)$ describes deviation from the standard Landau-Ginzburg momentum dependence. In agreement with previous studies at large $x$ we find $f(x\gg 1)\simeq (2B_{4}/N)x^{4-d}$; the absolute value of the coefficient $B_4$ increases with $d$ at $d>5/2$. Using NLSM, we obtain the contribution of the"anomalous" term $\xi^{-2}f(q\xi )$ to the critical exponent $\nu $, comparing it to the contribution of the non-analytical dependence, originating from the critical exponent $\eta $ (the obtained critical exponents $\nu $ and $\eta $ agree with previous studies). In the range $3\leq d<4$ we find that the former contribution dominates, and fully determines $1/N$ correction to the critical exponent $\nu $ in the limit $d\rightarrow 4.$