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Physical review. B, Solid state

S=12 , XY Model on Cubic Lattices. I. Susceptibility and Fluctuation near Critical Temperature

M. Howard Lee

Published Aug 1, 1973• 10 citations
Publisher

Abstract

For the $S=\frac{1}{2}$, $\mathrm{XY}$ 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.

DOI
10.1103/physrevb.8.1203

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Physical review. B, Condensed matter•Apr 1, 1986•84 citations
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→TcT\ensuremath{\rightarrow}{T}_{c}T\ensuremath→Tc​ 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→TcT\ensuremath{\rightarrow}{T}_{c}T\ensuremath→Tc​, 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.
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.
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 Tc{T}_{c}Tc​, and if the expectation of the double commutator appearing in $\overline{\ensuremath{\omega}}$ 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.
The time behavior of the spin operator ${S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}^{x}(t)$ in the XY\mathrm{XY}XY model is studied by a perturbative method. The zeroth order, which truncates a certain zero-time commutator, is shown to be equivalent to a molecular-field approximation. An analytic expression for the dynamic form factor is then obtained which satisfies basic sum rules. Through the fluctuation-dissipation theorem the dynamic susceptibility and other related physical quantities are also given. Within this zeroth-order approximation there exists a high-frequency collective mode at q12J{q}^{\frac{1}{2}}Jq21​J This mode is undamped and is similar to the plasmon.
The XY\mathrm{XY}XY model of a quantum lattice fluid or a ferromagnet is studied by the method of exact high-temperature series expansion. Nine terms are obtained in the free-energy series and seven in the series for the square of the fluctuation in the long-range order. Analysis of these series yields the critical values $\frac{k{T}_{c}}{J}=4.84\ifmmode\pm\else\textpm\fi{}0.06$, $\ensuremath{\gamma}=1.00\ifmmode\pm\else\textpm\fi{}0.07$, and $\ensuremath{\alpha}=\ensuremath{-}0.20\ifmmode\pm\else\textpm\fi{}0.20$ for the fcc lattice.
The spin one half XY model of ferromagnetism is defined by the interaction Hamiltonian [MATH] where the σ's are Pauli matrices defined at the sites i and j of a regular lattice. This model is appropriate for magnetic insulators for which g⊥ >> g|| temperature series expansions for the fluctuation in the long range order yield estimates of the critical temperatures for the f.c.c., b.c.c. and s.c. lattices and a susceptibility index γ ≈ 4/3. The fourth order fluctuation series yields the estimate Δ ≈ 5/3 from which by the scaling hypothesis all other indices are obtained. A further linearity hypothesis leads to predictions for the Heisenberg model indices from the Ising and XY indices.
The crossover from ideal-Bose-gas behavior to nonclassical, XY-like criticality in a dilute, interacting Bose fluid is considered in detail with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}$(T), below the transition. A general discussion of crossover scaling in constrained systems shows that ${\ensuremath{\rho}}_{s}$ and other temperature-dependent thermodynamic functions at controlled overall density can be expressed in terms of universal scaling functions. The finite-temperature Green's function formalism is used to calculate the associated linear scaling fields and the crossover exponent exactly for all dimensionalities d>2, scaling predictions, including logarithmic singularities just at d=3, being checked via second-order computations. A mapping of the Bose problem onto the classical s4{s}^{4}s4 spin model is achieved which is asymptotically exact in the critical region. In this way, previously known renormalization-group results are used to calculate the scaling functions for ${\ensuremath{\rho}}_{s}$, for the off-diagonal susceptibility, etc., to first order in \ensuremath{\epsilon}=4-d. Good agreement is found with the experimental measurements by Reppy and co-workers of ${\ensuremath{\rho}}_{s}$(T) for helium absorbed at low coverages in Vycor glass (of pore size 50--80 A\r{}). An effective-mass ratio m∗{m}^{\mathrm{*}}m∗/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio a∗{a}^{\mathrm{*}}a∗/a\ensuremath{\simeq}1.3 are indicated for mobile helium in the Vycor (although the irregular character of Vycor has not, at this point, been fully allowed for in the theory). Other approaches in the literature to computing the critical behavior of quantum-mechanical systems are reviewed and related to the present analysis.