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Physical review. B, Condensed matter

Nonlinear scaling fields and corrections to scaling near criticality

Amnon Aharony A, Michael Ellis Fisher

Published Apr 1, 1983• 165 citations
Publisher

Abstract

Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and the ordering field, $h$, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, ${g}_{t}=t+{b}_{t}{h}^{2}+{c}_{t}{t}^{2}+\ensuremath{\cdots}$ and ${g}_{h}=h(1+{c}_{h}t+\ensuremath{\cdots})$. The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities, $L$, of the form ($1+{a}_{L}t+{b}_{L}{t}^{2}+\ensuremath{\cdots}$), etc., where the correction amplitudes ${a}_{L},{b}_{L},\dots{}$, are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term ${b}_{t}{h}^{2}$ also generates nonanalytic contributions such as an additive, energylike term, varying as ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ in the zero-field susceptibility, and factors like ($1+{c}_{L}{|h|}^{2\ensuremath{-}\frac{1}{\ensuremath{\Delta}}}$) on the critical isotherm, $t=0$. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in ${T}_{c}$ and various amplitudes although "universal" ratios remain constant.

DOI
10.1103/physrevb.27.4394

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It is argued that the leading corrections to scaling for planar Ising models, which occur as analytic factors, arise from the quadratic terms of the nonlinear thermal and ordering fields (rather than from irrelevant variables). This yields ${{\ensuremath{\alpha}}_{\mathrm{eff}}}^{\ensuremath{'}}+2{\ensuremath{\beta}}_{\mathrm{eff}}+{{\ensuremath{\gamma}}_{\mathrm{eff}}}^{\ensuremath{'}}=2$ for the effective critical exponents, with no\mathrm{no}no leading corrections, and is confirmed by exact square-lattice results for arbitrary anisotropy J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​. For isotropic lattices the ratios of correction amplitudes and quadratic nonlinear-field terms appear universal.
Writing thermodynamic quantities near a critical point as ${f}_{i}\ensuremath{\sim}{A}_{i}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i}}(1+{a}_{i}{|t|}^{\ensuremath{\Delta}})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, we show that to leading order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ one has $\frac{{a}_{i}}{{a}_{j}}=\frac{({\ensuremath{\lambda}}_{i}\ensuremath{-}{{\ensuremath{\lambda}}_{i}}^{0})}{({\ensuremath{\lambda}}_{j}\ensuremath{-}{{\ensuremath{\lambda}}_{j}}^{0})}$, where ${{\ensuremath{\lambda}}_{i}}^{0}$ is the mean-field value of ${\ensuremath{\lambda}}_{i}$. This ratio is also equal to $\frac{({\ensuremath{\lambda}}_{i, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{i})}{({\ensuremath{\lambda}}_{j, \mathrm{eff}}\ensuremath{-}{\ensuremath{\lambda}}_{j})}$, where ${\ensuremath{\lambda}}_{i, \mathrm{eff}}$ is derived from a fit of data with ${f}_{i}\ensuremath{\sim}{|t|}^{\ensuremath{-}{\ensuremath{\lambda}}_{i, \mathrm{eff}}}$. Various experiments are analyzed and compared to these predictions.
We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse field Ising model near its quantum critical point (QCP). Our study has been motivated by the question about the thermodynamical signatures of this paradigmatic quantum critical system and, more generally, by the issue of how quantum criticality accumulates entropy. We find that the crossovers in the phase diagram of temperature and (the nonthermal control parameter) transverse field obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Gr\"uneisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature, which follows the prediction of quantum critical scaling. However, at the critical field, upon decreasing the temperature, the Gr\"uneisen ratio approaches a constant instead of showing the expected divergence. We are able to understand this unusual result in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Gr\"uneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. In spite of this special form of the scaling function, we show that the entropy is still maximized near the QCP, as expected from the general scaling argument. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.
A simple systematic rule, inspired by high-temperature series expansion (HTSE) results, is proposed for optimizing the expression for thermodynamic observables of ferromagnets exhibiting critical behavior at Tc{T}_{c}Tc​. This ``extended scaling'' scheme leads to a protocol for the choice of scaling variables, $\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})∕T$ or $({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}$ depending on the observable instead of $(T\ensuremath{-}{T}_{c})∕{T}_{c}$, and more importantly to temperature dependent noncritical prefactors for each observable. The rule corresponds to scaling of the leading term of the reduced susceptibility above Tc{T}_{c}Tc​ as ${\ensuremath{\chi}}_{c}^{*}(T)\ensuremath{\sim}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\gamma}}$ in agreement with standard practice with scaling variable $\ensuremath{\tau}$ and for the leading term of the second-moment correlation length as ${\ensuremath{\xi}}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}1∕2}{\ensuremath{\tau}}^{\ensuremath{-}\ensuremath{\nu}}$. For the specific heat in bipartite lattices, the rule gives ${C}_{c}^{*}(T)\ensuremath{\sim}{T}^{\ensuremath{-}2}{[({T}^{2}\ensuremath{-}{T}_{c}^{2})∕{T}^{2}]}^{\ensuremath{-}\ensuremath{\alpha}}$. The latter two expressions are not standard. The scheme can allow for confluent and noncritical correction terms. A stringent test of the extended scaling is made through analyses of high-precision numerical and HTSE data, or real data, on the three-dimensional canonical Ising, XYXYXY, and Heisenberg ferromagnets. For the susceptibility $\ensuremath{\chi}(T)$ and the correlation length $\ensuremath{\xi}(T)$ of the three ferromagnets, their optimized expression, which consists of the leading terms [respectively, ${\ensuremath{\chi}}_{c}^{*}(T)$ and ${\ensuremath{\xi}}_{c}^{*}(T)$] and a quite limited number of confluent and noncritical correction terms, represents real data to surprisingly good approximations over the entire temperature range from Tc{T}_{c}Tc​ to infinity. The temperature dependent prefactors introduced are of crucial importance not only in fixing the optimized expression at relatively high temperatures but also in determining appropriately the small amplitude correction terms. For the specific heat of the Ising ferromagnet, Cc∗(T){C}_{c}^{*}(T)Cc∗​(T) combined with two noncritical correction terms which are calculated with no free parameters once the correlation length critical parameters are known reproduces real data nicely also over the whole temperature range.
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