On tensor phase transitions
- PDF / 231,368 Bytes
- 8 Pages / 612 x 792 pts (letter) Page_size
- 87 Downloads / 290 Views
SORDER, AND PHASE TRANSITION IN CONDENSED SYSTEMS
On Tensor Phase Transitions A. M. Farutin Kapitsa Institute for Physical Problems, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 119334 Russia Laboratoire de Spectrométrie Physique, Université Joseph Fourier, CNRS, 38402 Saint Martin d’Hères, France e-mail: [email protected] Received October 16, 2008
Abstract—Within the Landau theory, phase diagrams are obtained for second-order phase transitions with order parameter in the form of a symmetric traceless tensor of rank at most six that is transformed in accordance with a one-dimensional representation of a crystal group. The case of two-dimensional representation is analyzed for rank equal to three. PACS numbers: 75.10.-b DOI: 10.1134/S1063776109030108
1. INTRODUCTION Spin nematics [1] and tensor magnetic materials [2] are characterized by a tensor order parameter S α1 α2 …α N , which is a completely symmetric tensor of rank N vanishing under contraction on a pair of indices. The phase diagram for N = 2 was constructed in [3]. The problem of phase transitions with a tensor order parameter also arises in other fields of physics; for example, the phase diagram for N = 3 was constructed in [4] as applied to liquid crystals. An arbitrary symmetric traceless tensor of rank N is defined by 2N + 1 independent parameters. In the general case, only three of these parameters can be annulled by an appropriate choice of the coordinate system in the spin space. Such a large number of independent variables lead to rather complicated expressions for contractions, so that the calculation of these contractions in terms of their Cartesian coordinates becomes ineffective as N increases. In this paper, we present phase diagrams for 3 < N ≤ 6 and describe a simple method for calculating contractions. In some cases solutions are obtained analytically. The methods proposed allow generalizations to the case of larger N. Exchange invariance requires that the expansion of energy in powers of the order parameter should be a linear combination of its contractions. A fourth-order expansion is expressed as 1 2 1 E = --- τS α1 α2 …α N + --2 4
∑ β Q (S i
i
α 1 α 2 …α N ),
(1)
necessary that the order parameter be transformed in accordance with a nonunitary representation of the crystal group; otherwise, contractions of degree three in S arise in expansion (1). 2. CALCULATION OF CONTRACTIONS To calculate contractions, we expand a tensor with respect to a basis of 2N + 1 spherical harmonics of momentum N and different projections of the momentum onto a chosen axis. Denote by a, b, c a basis in the spin space and introduce the complex vectors i m = --- ( b + ic ), 2
i n = --- ( b – ic ). 2
Define the kth element of the basis of rank N symmetric k traceless tensors S N with indices α1, α2, …, αN by the recurrence formula k
k
S N, α1 …α N = a α N S N – 1, α1 …α N – 1 + µ α N S N – 1, α1 …α N – 1 + ν α N S N – 1, α1 …α N – 1 , k–1
k+1
(2)
where S 0 = 1, and S N = 0 for |k | > N. Applying formula (2) and induction on N, we can eas
Data Loading...
