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arXiv:cond-mat/0305434 · Physical review. B, Condensed matter

Critical equation of state of randomly dilute Ising systems

Pasquale Calabrese, Martino De Prato, Andrea Pelissetto C, Ettore Vicari

Published Oct 13, 2003• 8 citations
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Abstract

We determine the critical equation of state of three-dimensional randomly dilute Ising systems--- i.e., of the random-exchange Ising universality class. We first consider the small-magnetization expansion of the Helmholtz free energy in the high-temperature phase. Then, we apply a systematic approximation scheme of the equation of state in the whole critical regime, which is based on polynomial parametric representations matching the small-magnetization expansion of the Helmholtz free energy and satisfying a global stationarity condition. These results allow us to estimate several universal amplitude ratios, such as the ratio ${A}^{+}{/A}^{\ensuremath{-}}$ of the specific-heat amplitudes. Our best estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.6(3)$ is in good agreement with experimental results on dilute uniaxial antiferromagnets.

DOI
10.1103/physrevb.68.134418

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We determine the critical equation of state of three-dimensional randomly dilute Ising systems--- i.e., of the random-exchange Ising universality class. We first consider the small-magnetization expansion of the Helmholtz free energy in the high-temperature phase. Then, we apply a systematic approximation scheme of the equation of state in the whole critical regime, which is based on polynomial parametric representations matching the small-magnetization expansion of the Helmholtz free energy and satisfying a global stationarity condition. These results allow us to estimate several universal amplitude ratios, such as the ratio A+/A\ensuremath−{A}^{+}{/A}^{\ensuremath{-}}A+/A\ensuremath− of the specific-heat amplitudes. Our best estimate A+/A\ensuremath−=1.6(3){A}^{+}{/A}^{\ensuremath{-}}=1.6(3)A+/A\ensuremath−=1.6(3) is in good agreement with experimental results on dilute uniaxial antiferromagnets.
We address the problem of determining the critical equation of state of three-dimensional XY\mathrm{XY}XY systems. For this purpose we first consider the small-field expansion of the effective potential (Helmholtz free energy) in the high-temperature phase. We compute the first few nontrivial zero-momentum n-point renormalized couplings, which parametrize such expansion, by analyzing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections. These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime, satisfy the correct analytic properties (Griffith's analyticity), and take into account the Goldstone singularities at the coexistence curve. A systematic approximation scheme is introduced, which is limited essentially by the number of known terms in the small-field expansion of the effective potential. From our approximate representations of the equation of state, we derive estimates of universal ratios of amplitudes. For the specific-heat amplitude ratio we obtain ${A}^{+}{/A}^{\ensuremath{-}}=1.055(3),$ to be compared with the best experimental estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.054(1).$
High-temperature series are computed for a generalized three-dimensional Ising model with arbitrary potential. Three specific "improved" potentials (suppressing leading scaling corrections) are selected by Monte Carlo computation. Critical exponents are extracted from high-temperature series specialized to improved potentials, achieving high accuracy; our best estimates are gamma=1.2371(4), nu=0.630 02(23), alpha=0.1099(7), eta=0.0364(4), beta=0.326 48(18). By the same technique, the coefficients of the small-field expansion for the effective potential (Helmholtz free energy) are computed. These results are applied to the construction of parametric representations of the critical equation of state. A systematic approximation scheme, based on a global stationarity condition, is introduced (the lowest-order approximation reproduces the linear parametric model). This scheme is used for an accurate determination of universal ratios of amplitudes. A comparison with other theoretical and experimental determinations of universal quantities is presented.
We perform a high-statistics simulation of the three-dimensional randomly dilute Ising model on cubic lattices L3 with L< or =256. We choose a particular value of the density, x=0.8, for which the leading scaling corrections are suppressed. We determine the critical exponents, obtaining nu=0.683(3), eta=0.035(2), beta=0.3535(17), and alpha=-0.049(9), in agreement with previous numerical simulations. We also estimate numerically the fixed-point values of the four-point zero-momentum couplings that are used in field-theoretical fixed-dimension studies. Although these results somewhat differ from those obtained using perturbative field theory, the field-theoretical estimates of the critical exponents do not change significantly if the Monte Carlo result for the fixed point is used. Finally, we determine the six-point zero-momentum couplings, relevant for the small-magnetization expansion of the equation of state, and the invariant amplitude ratio R(+)(xi) that expresses the universality of the free-energy density per correlation volume. We find R(+)(xi)=0.2885(15).
The critical thermodynamics of the random quenched dilute Ising system is studied using a field-theoretical approach. A method of calculation of the universal critical amplitude combinations directly in three space dimensions is given. It is applied to both the disordered (T>Tc{\mathit{T}}_{\mathit{c}}Tc​) and the ordered (TTc{\mathit{T}}_{\mathit{c}}Tc​) phases. The detailed derivation of the relevant perturbative series expansions is presented up to the third order in powers of the renormalized coupling constants. Because of the shortness of the series, only indicative values for the amplitude combinations are proposed. Those estimates seem to be reliable except for the specific-heat amplitude ratio for which, however, the result suggests a positive sign.
We compute the 2 n -point coupling constants in the high-temperature phase of the two-dimensional Ising model by using transfer-matrix techniques. This provides the first few terms of the expansion of the effective potential (Helmholtz free energy) and of the equation of state in terms of the renormalized magnetization. By means of a suitable parametric representation, we determine an analytic extension of these expansions, providing the equation of state in the whole critical region in the t , h plane.
We improve the theoretical estimates of the critical exponents for the three-dimensional Heisenberg universality class. We find $\ensuremath{\gamma}=1.3960(9),$ $\ensuremath{\nu}=0.7112(5),$ $\ensuremath{\eta}=0.0375(5),$ $\ensuremath{\alpha}=\ensuremath{-}0.1336(15),$ $\ensuremath{\beta}=0.3689(3),$ and $\ensuremath{\delta}=4.783(3).$ We consider an improved lattice ${\ensuremath{\varphi}}^{4}$ Hamiltonian with suppressed leading scaling corrections. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods and high-temperature expansions. The critical exponents are computed from high-temperature expansions specialized to the ${\ensuremath{\varphi}}^{4}$ improved model. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine a number of universal amplitude ratios.