Marginal dimensions for multicritical phase transitions

The field-theoretical model describing multicritical phenomena with two coupled order parameters with n_{||} and n_{\perp} components and of O(n_{||}) \oplus O(n_{\perp}) symmetry is considered. Conditions for realization of different types of multicritical behaviour are studied within the field-theoretical renormalization group approach. Surfaces separating stability regions for certain types of multicritical behaviour in parametric space of order parameter dimensions and space dimension d are calculated using the two-loop renormalization group functions. Series for the order parameter marginal dimensions that control the crossover between different universality classes are extracted up to the fourth order in \varepsilon=4-d and to the fifth order in a pseudo-\varepsilon parameter using the known high-order perturbative expansions for isotropic and cubic models. Special attention is paid to a particular case of O(1) \oplus O(2) symmetric model relevant for description of anisotropic antiferromagnets in an external magnetic field.


Introduction
The concept of universality plays a paradigmatic role in the modern statistical physics. Accordingly, continuous phase transitions can be grouped into universality classes (see, e. g. [1]). Systems within the same universality class are characterized by the same set of critical exponents governing the scaling behaviour of their thermodynamical functions. Therefore, one of the aims of a theoretical description of a system is to establish its universality class.
In the theory of critical phenomena it is standard now to use methods of field theoretical renormalization group (RG) [2][3][4][5]. Within these methods, a stable fixed point (FP) corresponds to the universality class.
For systems with complex internal symmetries described by φ 4 theories with several couplings, several different nontrivial FPs may exist. Depending on global parameters of a system, these FPs can interchange their stability causing the system to trigger from one universality class to another. The lack of a stable FP can even mean that a continuous phase order transition is transformed into a discontinuous. These global parameters (that effect the FP stability) are spatial dimension d and the dimension n of the order parameter (OP). In the n-d-space, the regions of stable FPs are separated by borders and the n(d) curves define the OP marginal dimensions that control the crossover between different universality classes.
In this paper we are interested in the stability borders and marginal dimensions for a model with two coupled OP fields, namely, the model with O(n ) ⊕ O(n ⊥ ) symmetry [6][7][8]. Such a model describes,

The model and RG picture of its multicritical phenomena
The model with O(n ) ⊕ O(n ⊥ ) symmetry can be obtained from the well-known O(n)-symmetrical model [21], splitting its n-component OP φ 0 into two: φ ⊥0 and φ 0 that act in orthogonal subspaces with dimensions n and n ⊥ , respectively (n + n ⊥ = n): where three couplingsů ,ů ⊥ andů × should be introduced instead of the only one in the O(n) symmetric model, andr ⊥ andr are connected with the temperature distance to the critical line for φ ⊥0 and φ 0 , correspondingly. The first mean-field analysis of the model with two coupled OPs was performed in order to describe the supersolids [22] (see also [17]). It shows that the character of the multicritical point in such a phase diagram depends on the sign ofů ⊥ů −ů 2 × . For a positive sign, a tetracritical point is realized, while for a negative sign, it is a bicritical point. Going beyond the mean field theory, fluctuations should be taken into account. This is achieved by the field-theoretical RG approach [2], in which the large-scale behaviour of the system is connected with the stable FP of the RG transformations. The transformation of the fourth order couplings {ů} in (2.2) under renormalization is described by β-functions.
The β-functions for O(n )⊕O(n ⊥ ) model were known in a one-loop approximation [8]. The next order approximation has been found in the massive [19] as well as in the minimal subtraction RG schemes [16].
The FPs {u * } of the RG transformation are found from the zeros of the β-functions with i =⊥, , ×. A stable FP possesses positive eigenvalues ω 1 , ω 2 , ω 3 (or their real parts) of stability matrix ∂β i /∂u j . The stable FPs for O(n ) ⊕ O(n ⊥ ) are already known from the one-loop studies [8]. For d < 4 and for sufficiently low OP dimensions satisfying n ⊥ + n < 4, only the isotropic Heisenberg FP H of O(n ⊥ + n ) symmetry with {u * ⊥ = u * × = u * } is stable. When n ⊥ (or n ) increases breaking (2.7), still with n ⊥ n + 2(n ⊥ + n ) < 32,  [16,19]. In particular, in the case n = 1, n ⊥ = 2 FP B (connected with tetracriticality) appears to be stable in a two loop order [16]. Resummation of higher orders ε-expansion [20] does not change this result.

Stability border-surfaces within a two-loop order approximation
As noted above, the stability of FPs D, B, H is dependent on three parameters n , n ⊥ and d. Therefore, the borders between regions for which one or another FP is stable, form surfaces in the parametric space n − n ⊥ − d: f (n , n ⊥ , d) = 0. We call them border-surfaces (BSs).
Two alternative ways are used in practice to analyze RG functions and to get universal quantities, in particular, marginal dimensions. In one approach, i.e., the ε-expansion, the solutions are obtained as a series in ε and then they are evaluated at the value of interest (for instance, at ε = 1 for d = 3 theories). Alternatively, one may fix the space dimension d to a certain value and directly solve a system of non-linear equations obtaining the FP coordinates numerically [26]. In the next two subsections we use these approaches to obtain marginal dimensions of the O(n ) ⊕ O(n ⊥ ) model within a two-loop RG approximation.

BSs from ε-expansion
We start our analysis with establishing the border between the stability regions of the decoupled FP D and the biconical FP B. As it was noted in [16], two of the FP D stability exponents correspond to the stability exponent of , while the remaining one is defined by Since ω H (n) is always positive, only ω D 2 governs the stability of the FP D, changing its sign depending on n , n ⊥ , d. Therefore, the surface between stability regions of FPs D and B can be extracted from the condition of (3.1) vanishing. Substituting the ε-expansion for the FP D coordinates into (3.1) one collects terms up to ε 2 and sets the result equal to zero: This is analytically solved for ε = ε(n || , n ⊥ ). The result is shown as the right hand surface in figure 1 (a).
The BS between the regions of stability of the FPs B and H can be derived from the condition that FP H changes its stability. Only one of the three eigenvalues of the stability matrix changes its sign in the region considered. Calculating this eigenvalue up to the ε 2 order we get the equation for the surface: The surface is also shown in figure 1 (a) (the lower left hand surface).
The limiting borderlines in the plane ε = 0 (d = 4), are equivalent to the case when the one loop order inequalities (2.7), (2.8) are transformed into equalities, from which one obtains The vertical line in figure 1 (a) presents a system with n = 2, n ⊥ = 1, indicating which FP governs the multicritical behavior of this system with the change of ε. Note that the FP B is stable in the region 0.51 ε 1.04. We are interested in this case, since it describes anisotropic ferro-and antiferromagnets in space dimension d = 3.

BSs from resummed β-functions
Another way to obtain the BSs, is to calculate them from the β-functions (2.3)-(2.5) fixing d at certain values. Since the RG expansions have divergent [2] nature, the special resummation techniques are needed to get convergent results [29]. The two-loop β-functions (2.3)-(2.5) β = β u i have a form of polynomials in renormalized couplings: We first represent (3.5) in the form of a resolvent series [30] in one auxiliary variable t : where the expansion coefficients a α in (3.6) explicitly depend on the couplings and on the coefficients c i j k (3.5). Obviously, F ({u}, 1) = β({u}). We resume the function (3.6) as a single variable function using

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the Padé-Borel technique [31] and writing its Borel image as: Analytical continuation of the function (3.7) is achieved by representing it in the form of a Padé approximant [32]. In our case, we use the diagonal Padé approximant [1/1]: (3.8) Finally, the resummed function is obtained via an inverse Borel transform: It is technically difficult to extract the data from the resummed function in the limit ε → 0. Therefore, we present BSs for 0.002 ε 1.2, and n , n ⊥ in the range from -0.56 to 5. Limiting borderlines in the plane ε → 0 described by (3.4) give us the one-loop (thin) borderlines of figure 1 of [16], while the intersections of the surfaces with the plane ε = 1 give the two-loop (thick) borderlines of figure 1 of [16]. Similarly to what we did it in figure 1 (a), we present in figure 1

High loop order results for marginal dimensions
The marginal dimensions of the O(n ) ⊕ O(n ⊥ ) models can be defined based on the high order RG results for simpler isotropic and cubic models. In particular, exact scaling arguments [18] connect the FP D stability with the critical exponents of the O(n ⊥ ) and O(n ) models: where α(n) and ν(n) are the heat capacity and correlation length critical exponents of the O(n) model.
As it was indicated in [20], the stability of the FP H is defined by the marginal dimension n c of the cubic model. Since in the FP H , the RG functions depend only on the combination n = n ⊥ + n , the resulting marginal dimension can be presented in the form n H ⊥ (n , ε) = n c (ε) − n . In the following two subsections we present an analysis of the marginal dimensions n D ⊥ (n , ε), n H ⊥ (n , ε) based on the five-loop minimal subtraction series for the RG functions of isotropic [34] and cubic models [35], as well as for the case d = 3 based on the six-loop series for these models [36,37] obtained within the massive scheme [38,39].
Here and below, symbol o denotes the approximants which can not be constructed within the order of perturbation theory considered here. Usually, the best convergence of the results is observed along the main diagonal and the closest sub-diagonals of the Padé table [32]. However, it appears that the value of n ⊥ (1, 1) given by the Padé-aproximant [2/2] differs from those given by [1/2] and [2/1] by an order of one, leading to an uncertainty of the numerical estimate.
To obtain a reliable estimate of n ⊥ (1, ε) we rely on the Padé-Borel resummation described in subsection 3.2. We obtain a resolvent series by a substitution ε → εt . For the obtained expression we build get n H ⊥ (n , ε) using the available five-loop ε-expansion for the marginal dimension of the cubic model [35]:

Conclusion
In the present paper we have studied the conditions under which different types of multicritical be- and their stability defines the regions in the space of the dimensions of the OPs as well as in the spatial dimension where the corresponding multicritical behavior manifests itself. Using the ε-expansion for the two-loop β-functions obtained in the minimal subtraction scheme we derived the BSs separating these regions. We obtained similar BSs applying the resummation procedure. In the particular case of O(1)⊕O (2) symmetry, we confirm the previous studies finding that the biconical FP associated with a tetracritical behaviour is stable for the case d = 3. In higher space dimensions, the O(n +n ⊥ ) symmetrical FP associated with the bicritical behaviour is stable. Our analysis also made use of the results of higher order approximations within the field-theoretical RG approach. At this stage, there were used the scaling arguments connecting the stability of the FPs of O(n ) ⊕ O(n ⊥ ) model with the universal quantities of the O(n) and the cubic models. Exploiting fiveloop expressions for the O(n) model, we derived an ε-expansion for the marginal dimension n D (n , ε) separating the regions of stability for the FPs D and B. Applying the resummation procedure to this result, we have analyzed the dependence of n D (1, ε) on ε. Exploiting the five-loop expressions for the cubic model we obtained the value of n H (1, ε) separating the regions of stability for the FPs H and B. Finally, we complete our results by three-dimensional estimates of n D (1) and n H (1) based on the pseudo-ε expansions derived within a six-loop RG approximation.
These results are also important for the critical dynamics [43][44][45][46][47]. The type of a dynamical FP in such systems depend, of course, on the static FP values. In order to extend our results to the dynamics of antiferromagnets in an external field, further work is necessary. One has to extend this analysis to the statics of the corresponding model C [48][49][50].