Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gr\"{u}neisen Parameter and Thermal Expansion by the Self-Consistent
  Renormalization Theory of Spin Fluctuations   The thermal expansion coefficient $\alpha$ and the Gr\"{u}neisen parameter
$\Gamma$ near the magnetic quantum critical point (QCP) are derived on the
basis of the self-consistent renormalization (SCR) theory of spin fluctuation.
From the SCR entropy, the specific heat $C_{V}$, $\alpha$, and $\Gamma$ are
shown to be expressed in a simple form as $C_{V}=C_{\rm a}-C_{\rm b}$,
$\alpha=\alpha_{\rm a}+\alpha_{\rm b}$, and $\Gamma=\Gamma_{\rm a}+\Gamma_{\rm
b}$, respectively, where $C_{\rm i}$, $\alpha_{\rm i}$, and $\Gamma_{\rm i}$
$({\rm i}={\rm a}, {\rm b})$ are related with each other. As the temperature
$T$ decreases, $C_{\rm a}$, $\alpha_{\rm b}$, and $\Gamma_{\rm b}$ become
dominant in $C_{V}$, $\alpha$, and $\Gamma$, respectively. The inverse
susceptibility of spin fluctuation coupled to the volume $V$ in $\Gamma_{\rm
b}$ is found to give rise to the divergence of $\Gamma$ at the QCP for each
class of ferromagnetism and antiferromagnetism (AFM) in spatial dimensions
$d=3$ and $2$. This $V$-dependent inverse susceptibility in $\alpha_{b}$ and
$\Gamma_{\rm b}$ contributes to the $T$ dependences of $\alpha$ and $\Gamma$,
and even affects their criticality in the case of the AFM QCP in $d=2$.
$\Gamma_{\rm a}$ is expressed as $\Gamma_{\rm
a}(T=0)=-\frac{V}{T_0}\left(\frac{\partial T_0}{\partial V}\right)_{T=0}$ with
$T_0$ being the characteristic temperature of spin fluctuation, which has an
enhanced value in heavy electron systems.

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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: hep-lat
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The categories of this paper are: supr-con
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: nlin
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: physics