N-vector Field Research Articles

Critical behavior of a general O(n)-symmetric model of two n-vector fields in D = 4 − 2ϵ

The critical behavior of the O(n)-symmetric model with two n-vector fields is studied within the field-theoretical renormalization group approach in a D = 4 − 2ϵ expansion. Depending on the coupling constants, the β-functions, fixed points and critical exponents are calculated up to the one- and two-loop order, respectively (η in two- and three-loop order). Both continuous lines of fixed points and invariant discrete solutions were found. Apart from already known fixed points two new ones arise. One agrees in one-loop order with a known fixed point, but differs from it in two-loop order.

A representation theorem for volume-preserving transformations

A general solution is presented for the partial differential equation ∂u/ ∂x= k( x), where u and x are n-vector fields, ∂u/ ∂x denotes the Jacobian of the transformation x→ u and k( x) is a scalar-valued function. The solution for the case k( x)=1 is of special interest because it furnishes a representation theorem for volume-preserving transformations in an n-dimensional space. Such a representation for the case n=2 was obtained by Gauss. The solution for n=3, presented here, furnishes a representation for isochoric (volume-preserving) finite deformations, which are important in the mechanics of highly deformable incompressible solid materials.

Geometry of Hamiltonian n-vector fields in multisymplectic field theory

Multisymplectic geometry—which originates from the well known De Donder–Weyl (DW) theory—is a natural framework for the study of classical field theories. Recently, two algebraic structures have been put forward to encode a given theory algebraically. Those structures are formulated on finite dimensional spaces, which seems to be surprising at first. In this paper, we investigate the correspondence of Hamiltonian functions and certain antisymmetric tensor products of vector fields. The latter turn out to be the proper generalisation of the Hamiltonian vector fields of classical mechanics. Thus we clarify the algebraic description of solutions of the field equations.

Modified Borel summation of divergent series and critical-exponent estimates for anN-vector cubic model in three dimensions from five-loop ε expansions

A new approach to summation of divergent field-theoretical series is suggested. It is based on the Borel transformation combined with a conformal mapping and does not imply the exact asymptotic parameters to be known. The method is tested on functions expanded in their asymptotic power series. It is applied to estimating the critical exponent values for an N-vector field model, describing magnetic and structural phase transitions in cubic and tetragonal crystals, from five-loop \epsilon expansions.

RENORMALIZATION IN TRUNCATED PARISI-SOURLAS MODEL

The Parisi-Sourlas model is truncated by ignoring irrelevant terms and introducing an N-vector field. The β-functions for the model of 4−ε dimensions are calculated up to two-loop order in the dimensional regularization scheme. By using the β-functions, the critical exponents are evaluated up to ε2-order. The critical exponents of the truncated Parisi-Sourlas model are compared with those of the ordinary N-vector model. It is found that this model roughly corresponds to the (N−2)-vector model. In fact, the ghost-like fields contribute to the critical exponents in the opposite way, being contrasted with the ordinary N-vector field.

Scaling function near a fluctuation-induced tricritical point

A free energy functional containing a critical n-vector field coupled to a non-critical scalar one, which commonly appears in the spin-phonon problem and near the ferromagnetic transition of the Hubbard model, is used within a field theoretical and renormalization group framework to obtain, to first order in ϵ=4− d, the universal scaling function of the equation of state near a fluctuation-induced tricritical point.

Interpolating between Ising, XY-, and non-linear σ-models

The β-functions of O( N)-symmetric non-linear σ-models on the lattice were recently discovered to be non-monotonic for N ⩾ 3. We explain the non-monotonic behaviour as a non-perturbative lattice effect by relating it to the Kosterlitz-Thouless transition of the XY-model. We also relate the latter transition to the phase transition of the Ising model. These relationships are established by interpolating between the O( N)- and the O( N − 1)-symmetric non-linear σ-models by suppression of the Nth component of the N-vector field with a mass term. A critical line in the coupling-mass plane connects the critical point of the Ising model ( N = 1) with the critical point of the XY-model ( N = 2). This line extends towards the region of non-monotonic behaviour of the β-function of the O(3)-symmetric model. The nature of the transition lines is also investigated.

Interpolating between O( N)-symmetric σ-models with N = 1,2,3

We interpolate between the O( N)- and the O( N − 1)-symmetric non-linear σ-models with standard action by suppressing the Nth component of the N-vector field with a mass term. We find a critical line in the coupling-mass plane connecting the critical point of the Ising model ( N = 1) with the critical point of the XY-model ( N = 2). This line extends towards the region of non-monotonous behaviour of the ß-function of the O(3)-symmetric model. The massless phase of the XY-model is separated from the ordered phase of the Ising model by a transition at zero mass, while it is separated from the only phase of the O(3)-model by a transition that seems to occur at exponentially small mass.

Interpolating between O( N)-symmetric σ-models with N = 1, 2, 3

We interpolate between the O( N)- and the O( N−1)-symmetric non-linear σ-models with standard action by suppressing the Nth component of the N-vector field with a mass term. We find a critical line in the coupling-mass plane connecting the critical point of the Ising model ( N=1) with the critical point of the XY-model ( N=2). This line extends towards the region of non-monotonic behaviour of the β-function of the O(3)-symmetric model. The massless phase of the XY-model is separated from the ordered phase of the Ising model by a transition at zero mass, while it is separated from the only phase of the O(3)-model by a transition that seems to occur at exponentially small mass.

Fluctuations and renormalization of a field on a boundary

The renormalization of the ( ϕ 2) 2 n-vector field theoretical model in semi-infinite geometry, with a quadratic surface coupling, implies a derivative surface counterterm. This counterterm destroys the positive definiteness of the action, and introduces ambiguities and unacceptable infinities. We reexamine the mechanism which makes it indeed not needed, and show the role played by the fluctuations of the surface field, which build up the extra renormalization Z 1 1 2 of the surface field.

Partially ordered states in Ginzburg-Landau-Wilson models with cubic-type anisotropy.

We study the Ginzburg-Landau-Wilson M N-vector model with cubic-type anisotropy and obtain exact results in the large-N limit. We focus on the region of the parameter space in which cubic anisotropy favors, at low temperatures, long-range magnetic order of one of M N-component fields. This region is known to be, for sufficiently large MN, a runaway region of perturbative renormalization-group (RG) analysis, for the spatial dimensionality d less than four, what is usually interpreted to correspond to a fluctuation-induced first-order transition. The large-N analysis presented in this paper enables us to check the validity of this conjecture and to establish the structure of the phase diagram in any d and for all values of quartic couplings of the model. The most interesting outcome of this analysis is that, in a range of values of quartic couplings, the direct transition from paramagnetic to ferromagnetic phase splits into two consecutive phase transitions with a partially ordered phase occurring in the temperature range between these transitions. Discrete permutational symmetry of the M N-vector model (being restored in the paramagnetic phase) is spontaneously broken in the partially ordered phase; one of M N-vector fields becomes soft in this phase (as if there is a quadratic anisotropy, breaking permutational symmetry, present in the system). However, this partially ordered fluctuation induced phase has no long-range magnetic order.We find that such a phase is present in the phase diagram in any d if quartic couplings of the model are sufficiently strong. In the weak-coupling regime (which is usually believed to be tractable by RG analysis when d\ensuremath{\approxeq}4), we find (in agreement with RG) that, for 4\ensuremath{\ge}d\ensuremath{\ge}${d}_{c}$(M), the system exhibits a direct fluctuation-induced first-order transition and the partially ordered state is absent. However, for ${d}_{c}$(M)>d>2, the partially ordered state is universally present even in parts of the phase diagram having arbitrarily small quartic couplings. Possible situations occurring for finite N are discussed qualitatively. Systems favoring incommensurate modulated order on one of M equivalent pairs of wave vectors in q space can be modeled by M two-vector models. For these systems the partially ordered phase is a nematiclike orientationally ordered phase. We discuss the feasibility of observing such a phase in these systems.

Perturbative expansions for the n-vector field model of phase transition

Correlation corrections to the phenomenological expressions for the order parameter and specific heat are calculated for the n-vector field model. At n=3 the numerical coefficients for the correction terms obtained are found to coincide with those for the Heisenberg model with a large interaction range.