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Register a new userIncommensurate phases in statistical theory of the crystalline state
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Authors
V. A. Golovko
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The paper continues a series of papers devoted to treatment of the crystalline state on the basis of the approach in equilibrium statistical mechanics proposed earlier by the author. This paper is concerned with elaboration of a mathematical apparatus in the approach for studying second-order phase transitions, both commensurate and incommensurate, and properties of emerging phases. It is shown that the preliminary symmetry analysis for a concrete crystal can be performed analogously with the one in the Landau theory of phase transitions. After the analysis one is able to deduce a set of equations that describe the emerging phases and corresponding phase transitions. The treatment of an incommensurate phase is substantially complicated because the symmetry of the phase cannot be described in terms of customary space groups. For this reason, a strategy of representing the incommensurate phase as the limit of a sequence of long-period commensurate phases whose period tends to infinity is worked out. The strategy enables one to obviate difficulties due to the devils staircase that occurs in this situation.
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We have investigated the proof of the $H$ theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [Phy. Rev. E {bf 66}, 056125, 2002; {it ibid.} {bf 72}, 036108 2005]. In our analysis, however, we have not considered the so-called deformed mathematics as did in the above reference. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the $kappa$-formalism, and the second law of thermodynamics implies that the $kappa$ parameter lies on the interval [-1,1]. It is shown that the collisional equilibrium states (null entropy source term) are described by a $kappa$ power law generalization of the exponential Juttner distribution, e.g., $f(x,p)propto (sqrt{1+ kappa^2theta^2}+kappatheta)^{1/kappa}equivexp_kappatheta$, with $theta=alpha(x)+beta_mu p^mu$, where $alpha(x)$ is a scalar, $beta_mu$ is a four-vector, and $p^mu$ is the four-momentum. As a simple example, we calculate the relativistic $kappa$ power law for a dilute charged gas under the action of an electromagnetic field $F^{mu u}$. All standard results are readly recovered in the particular limit $kappato 0$.
We establish the qualitative behavior of the incommensurability $epsilon$, optimal domain wall filling $ u$ and chemical potential $mu$ for increasing doping by a systematic slave-boson study of an array of vertical stripes separated by up to $d=11$ lattice constants. Our findings obtained in the Hubbard model with the next-nearest neighbor hopping $t=-0.15t$ agree qualitatively with the experimental data for the cuprates in the doping regime $xlesssim 1/8$. It is found that $t$ modifies the optimal filling $ u$ and triggers the crossover to the diagonal (1,1) spiral phase at increasing doping, stabilized already at $xsimeq 0.09$ for $t=-0.3t$.
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