This paper contains a unified presentation of Gibbs' inequalities and some of their quantum-statistical generalizations.
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 w 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 cb 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. INTRODUCTIONW E present upper and lower bounds for the zerofield isothermal magnetic susceptibility and for a generalized wave-number-dependent susceptibility. These are a generalization of equations between the susceptibility and fluctuation beyond the case where magnetization operators commute with the zero-field Hamiltonian. The bounds are expressed in terms of the mean-square fluctuations in the magnetization variables and the first frequency moment of a spectral density. When this moment cb 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 cb is finite 1 at and near T c , then the fluctuation and the susceptibility diverge in the same manner and their critical exponents 2 will be identical. DEFINITIONS AND PRINCIPAL RESULTSThe zero-field isothermal magnetic susceptibility is defined by the limit equation(1) where the ensemble average of the magnetization is defined by (M) h =TrZMe-^H°~h M )2/Trle-fi (H°-hM) l,(2) and Ho-hM is the Hamiltonian for the system in an external magnetic field h.These definitions yield 3 the following formula for XT :x The work of N. D. Mermin and H. Wagner, [Phys. Rev. Letters 17, 1133 (1966)], establishes that the expectation of the relevant double commutator is bounded for the class of one-, two-, and three-dimensional, isotropic, spin-S Heisenberg models with finite-range exchange interaction [XRi? 2 |/(R) | finite]. 2 We thank Professor M. E. Fisher for this statement of our results following our summary of this work at the Yeshiva The zero-field expectation value is defined for a general operator A byIf M commutes with H 0 , Eq. (3) shows that $~lX T = {(AM) 2 )o, the mean-square fluctuation of the magnetization. An extension of this result to the static susceptibility for the spatial Fourier components A k of an inhomogeneous disturbance is 4 X T Uk*Af)= ( d\(e™»AA k e-™°AAj;)o =X T (A k \A k ),where the notation of Eqs. (4) and (5) has been used and A^ is the Hermitian conjugate of A k . Since Eq. (6) clearly reduces to Eq. (3) for the special case where A k is the Hermitian operator M, we present our bounds for the more general quantity X T {A k \A k ).
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