Susceptibility and Fluctuation

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 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 ).

Magnetic Properties of Layered Transition Metal Compounds

“…and also a lower bound (65) where A is a function of the temperature that may be found elsewhere [28,180].…”

Moments of the auto-correlation function and the KMS-condition

“…3, [] Remark that the inequality (5) has been obtained before by Falk and Bruch [7], the special case of inequality (7) with n = 1 was first derived by Martens and Verbeure [8]. The other inequalities relating higher moments are new as far as we know.…”

Dynamic response function and bounds of the susceptibility of a semiclassical gas and Kramers-Kronig relations in optic-data inversion

“…Introducing t =&u /2, u =Pficoo, as before, we have from (21) Finally we note that the WLB approaches the upper bound when the recoil frequency co0 vanishes, e. g., when the particle becomes very massive. The merging of the upper and lower bounds, first noted in the context of critical phenomena [6,16,17], occurs whenever co= vs/2Sk~0. Recall in our problem co=coo.…”

“…(22)(23)These expansions are useful for observing the closeness of the bounds to the susceptibility. The static susceptibility bounds, first given by Falk and Bruch[6], later by Dyson, Lieb, and Simon[7], and others[13 -15], have been rarely tested on models because exact solutions of the susceptibility are difficult to obtain[8,16]. Our solution affords an opportunity to test the bounds rather easily.The upper bound gk /Sk =gk ( 1, where for our model For our model the WLB is thus the susceptibility sum rule evaluated at the recoil frequency.…”

“…Also the zero-frequency limit of the dynamic susceptibility, being the static susceptibility, is a bounded function. The bounds, especially the lower bounds, are obtained by convexity theory via Jensen s inequality [6,7].…”