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Physical Review

Susceptibility and Fluctuation

Harold Falk, Ludwig W. Bruch

Published Apr 10, 1969• 92 citations
Publisher

Abstract

Bounds are presented relating zero-field static isothermal magnetic susceptibilities to the mean-square fluctuations of corresponding magnetization variables. The lower bounds contain the first frequency moment of a spectral density. When this moment $\overline{\ensuremath{\omega}}$ approaches zero, the upper and lower bounds merge, and the susceptibility is determined by the mean-square fluctuation. In particular, if the susceptibility diverges at a temperature ${T}_{c}$, and if the expectation of the double commutator appearing in $\overline{\ensuremath{\omega}}$ is finite at and near ${T}_{c}$, then the fluctuation and the susceptibility diverge in the same manner, and their critical exponents will be identical.

DOI
10.1103/physrev.180.442

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Bounds are presented relating zero-field static isothermal magnetic susceptibilities to the mean-square fluctuations of corresponding magnetization variables. The lower bounds contain the first frequency moment of a spectral density. When this moment \ensuremathω‾\overline{\ensuremath{\omega}}\ensuremathω​ approaches zero, the upper and lower bounds merge, and the susceptibility is determined by the mean-square fluctuation. In particular, if the susceptibility diverges at a temperature Tc{T}_{c}Tc​, and if the expectation of the double commutator appearing in \ensuremathω‾\overline{\ensuremath{\omega}}\ensuremathω​ is finite at and near Tc{T}_{c}Tc​, then the fluctuation and the susceptibility diverge in the same manner, and their critical exponents will be identical.
For the S=12S=\frac{1}{2}S=21​, XY\mathrm{XY}XY model of a quantum lattice fluid or a ferromagnet the conventional order parameter does not commute with the Hamiltonian. As a result, the mean-square fluctuation of the order parameter and the isothermal susceptibility are not related in the usual way the general fluctuation theorem. For the above model, arguments are here presented to support the idea that as $T\ensuremath{\rightarrow}{T}_{c}$ the quantum effect due to the noncommutation becomes masked and the two quantities have the same critical behavior. This work is consistent with the exact results of Falk and Bruch who defined a certain moment of the spectral density and used inequalities to establish that if the moment \ensuremath{\rightarrow} 0 as $T\ensuremath{\rightarrow}{T}_{c}$, then the susceptibility-fluctuation ratio becomes unity thus ensuring coinciding critical behavior. The latter result applies to a large class of models including the one considered here.
An analysis is made of the fluctuations in a finite part of an infinite, ddd-dimensional ferromagnetic Ising lattice at and near the critical point. This leads to a formal proof that $\frac{2d}{(\ensuremath{\delta}+1)}$ and $\frac{2d\ensuremath{\beta}}{({\ensuremath{\gamma}}^{\ensuremath{'}}+2\ensuremath{\beta})}$ are lower bounds on the exponent characterizing the asymptotic decrease with distance of the two-body correlation function at the critical point, where $\ensuremath{\beta}$, $\ensuremath{\delta}$, and ${\ensuremath{\gamma}}^{\ensuremath{'}}$ are exponents representing the asymptotic forms of the coexistence curve, critical isotherm, and isothermal susceptibility.
The derivation of series expansions for the mean size of finite clusters in the Ising model is described briefly. From the analysis of low temperature series it is concluded that for a two-dimensional lattice in zero magnetic field the mean size probably diverges at the Ising critical temperature, T c , as (T c -T) - theta , with theta =1.91+or-0.01. It appears therefore that theta < gamma '=1.75 the corresponding Ising susceptibility exponent. For a three-dimensional lattice it is tentatively concluded that the mean size diverges at some temperature T*<T c .
The transverse components of the susceptibility and fluctuation for the spin van der Waals model are obtained for high- and low-temperature XY\mathrm{XY}XY- and Ising-like regimes. The difference between the susceptibility and fluctuation, which arises because of noncommutativity, is related to certain correlation functions of Sz{S}_{z}Sz​. In the XY\mathrm{XY}XY-like regime, the difference vanishes since the principal long-range order is 〈Sx〉〈{S}_{x}〉〈Sx​〉. But in the Ising-like regime, the difference persists since the principal long-range order now is 〈Sz〉〈{S}_{z}〉〈Sz​〉. Our results are compared with the bounds of Falk and Bruch on the ratio of the susceptibility to the fluctuation. In the critical region as well as in the high-temperature region the upper and lower bounds merge. In these regions, the two correlation functions become identical in the manner indicated by Falk and Bruch. In the low-temperature region, the bounds do not merge. Here we find that in the Ising-like regime of the model the susceptibility and fluctuation are properly different, but in the XY\mathrm{XY}XY-like regime the two functions nevertheless become identical. Thus, the susceptibility and fluctuation can evidently be the same without the bounds necessarily merging.
If scaling variables and scaling expressions are chosen judiciously, ``critical'' scaling analyses do not need to be restricted to a narrow range of temperatures near the critical point. The standard scaling variable $(T\ensuremath{-}{T}_{c})/{T}_{c}$ is inadapted to wide ranges of temperature because it diverges at high temperatures. With the variable $\ensuremath{\tau}=(T\ensuremath{-}{T}_{c})/T$, in the high-dimension (mean-field) limit, the reduced susceptibility and the second-moment correlation length of the Ising ferromagnet depend on temperature as $\ensuremath{\chi}(T)={\ensuremath{\tau}}^{\ensuremath{-}1}$ and $\ensuremath{\xi}(T)={T}^{\ensuremath{-}1/2}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$ exactly over the entire temperature range above the critical temperature Tc{T}_{c}Tc​. For the canonical two-dimensional square lattice near-neighbor Ising ferromagnet, it is shown that compact ``extended scaling'' expressions analogous to the high-dimensional limit form but with the appropriate exponents give accurate approximations to the true temperature dependencies, again over the entire temperature range from Tc{T}_{c}Tc​ to infinity. Within this approach, for near-neighbor interaction systems, there is no crossover temperature above which mean-field-like behavior sets in.
We derive upper and lower bounds on the fidelity susceptibility in terms of macroscopic thermodynamical quantities, such as the susceptibilities and thermal average values. The quality of the bounds is checked against the exact expressions for a single spin in an external magnetic field. Their usefulness is illustrated by two examples of many-particle models which are exactly solved in the thermodynamic limit: the Dicke superradiance model and the single-impurity Kondo model. It is shown that, when divergent behavior is considered, the fidelity susceptibility and the thermodynamic susceptibility are equivalent for a large class of models exhibiting critical behavior.