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arXiv:1606.07449 · Physical review. E

Universal behavior of coupled order parameters below three dimensions

Julia Borchardt, Astrid Eichhorn

Published Oct 5, 2016• 18 citations
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Abstract

We explore universal critical behavior in models with two competing order parameters, and an O(N) ⊕O(M) symmetry for dimensions d≤3. In d=3, there is always exactly one stable renormalization group fixed point, corresponding to bicritical or tetracritical behavior. Employing pseudospectral techniques to solve functional renormalization group equations in a two-dimensional field space, we uncover a more intricate structure of fixed points in d<3, where two additional bicritical fixed points play a role. Towards d=2, we discover ranges of N=M with several simultaneously stable fixed points, indicating the coexistence of several universality classes.

DOI
10.1103/physreve.94.042105

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We explore universal critical behavior in models with two competing order parameters, and an O(N) ⊕O(M) symmetry for dimensions d≤3. In d=3, there is always exactly one stable renormalization group fixed point, corresponding to bicritical or tetracritical behavior. Employing pseudospectral techniques to solve functional renormalization group equations in a two-dimensional field space, we uncover a more intricate structure of fixed points in d<3, where two additional bicritical fixed points play a role. Towards d=2, we discover ranges of N=M with several simultaneously stable fixed points, indicating the coexistence of several universality classes.
We employ the nonperturbative functional renormalization group to study models with an O(N(1) ⊕O(N)(2)) symmetry. Here different fixed points exist in three dimensions, corresponding to bicritical and tetracritical behavior induced by the competition of two order parameters. We discuss the critical behavior of the symmetry-enhanced isotropic, the decoupled and the biconical fixed point, and analyze their stability in the N(1),N(2) plane. We study the fate of nontrivial fixed points during the transition from three to four dimensions, finding evidence for a triviality problem for coupled two-scalar models in high-energy physics. We also point out the possibility of noncanonical critical exponents at semi-Gaussian fixed points and show the emergence of Goldstone modes from discrete symmetries.
The critical behavior of many physical systems involves two competing n1- and n2-component order-parameters, S1 and S2, respectively, with n = n1 + n2. Varying an external control parameter g, one encounters ordering of S1 below a critical (second-order) line for g < 0 and of S2 below another critical line for g > 0. These two ordered phases are separated by a first-order line, which meets the above critical lines at a bicritical point, or by an intermediate (mixed) phase, bounded by two critical lines, which meet the above critical lines at a tetracritical point. For n = 1 + 2 = 3, the critical behavior around the (bi- or tetra-) multicritical point either belongs to the universality class of a non-rotationally invariant (cubic or biconical) fixed point, or it has a fluctuation driven first-order transition. These asymptotic behaviors arise only very close to the transitions. We present accurate renormalization-group flow trajectories yielding the effective crossover exponents near multicriticality.
We calculate the static critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by the renormalization group method within the minimal subtraction scheme in two-loop order. Summation methods lead to fixed points describing multicritical behavior. Their stability border lines in the space of the order parameter components n_|| and n_perpendicular and spatial dimension d are calculated. The essential features obtained already in two-loop order for the interesting case of an antiferromagnet in a magnetic field ( n_|| =1, n_perpendicular =2 ) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points, leading to very small transient exponents. We are also able to calculate the flow of static couplings, which allows us to consider the attraction region. Depending on the nonuniversal background parameters, the existence of different multicritical behavior (bicritical or tetracritical) is possible, including a triple point.
We calculate the relaxational dynamical critical behavior of systems of O(n_||)(plus sign in circle)O(n_perpendicular) symmetry by renormalization group method within the minimal subtraction scheme in two-loop order. The three different bicritical static universality classes previously found for such systems correspond to three different dynamical universality classes within the static borderlines. The Heisenberg and the biconical fixed point lead to strong dynamic scaling whereas in the region of stability of the decoupled fixed point weak dynamic scaling holds. Due to the neighborhood of the stability border between the strong and the weak scaling dynamic fixed point to the dynamical stable fixed point a very small dynamic transient exponent of omega(Beta)_(v) =0.0044 is present in the dynamics for the physically important case n_|| =1 and n_perpendicular =2 in d=3 .
We discuss the static and dynamic multicritical behavior of three-dimensional systems of O(n∥)⊕O(n⊥)O(n_\|)\oplus O(n_\perp)O(n∥​)⊕O(n⊥​) symmetry as it is explained by the field theoretical renormalization group method. Whereas the static renormalization group functions are currently know within high order expansions, we show that an account of two loop contributions refined by an appropriate resummation technique gives an accurate quantitative description of the multicritical behavior. One of the essential features of the static multicritical behavior obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field (n∥=1n_\|=1n∥​=1, n⊥=2n_\perp=2n⊥​=2) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents. We further pursue an analysis of dynamical multicritical behavior choosing different forms of critical dynamics and calculating asymptotic and effective dynamical exponents within the minimal subtraction scheme.
A field-theoretic description of phase transformations in complex systems with two interacting order parameters is given. For three-dimensional systems in the two-loop approximation a renormalization-group analysis of the scaling functions is carried out directly, and the fixed points corresponding to stability of the bicritical and tetracritical behavior are identified. The critical exponents at the multicritical points in the two-loop approximation are calculated with the use of the Padé-Borel summation technique.