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arXiv:cond-mat/0209580 · Physical review. B, Condensed matter

Multicritical phenomena in O(n1)⊕O(n2) -symmetric theories

Pasquale Calabrese, Andrea Pelissetto C, Ettore Vicari

Published Feb 18, 2003• 177 citations
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Abstract

We study the multicritical behavior arising from the competition of two distinct types of ordering characterized by $\mathrm{O}(n)$ symmetries. For this purpose, we consider the renormalization-group flow for the most general $\mathrm{O}{(n}_{1})\ensuremath{\bigoplus}\mathrm{O}{(n}_{2})$-symmetric Landau-Ginzburg-Wilson Hamiltonian involving two fields ${\ensuremath{\varphi}}_{1}$ and ${\ensuremath{\varphi}}_{2}$ with ${n}_{1}$ and ${n}_{2}$ components, respectively. In particular, we determine in which cases, approaching the multicritical point, one may observe the asymptotic enlargement of the symmetry to $\mathrm{O}(N)$ with ${N=n}_{1}{+n}_{2}.$ By performing a five-loop $\ensuremath{\epsilon}$-expansion computation we determine the fixed points and their stability. It turns out that for ${N=n}_{1}{+n}_{2}>~3$ the $\mathrm{O}(N)$-symmetric fixed point is unstable. For $N=3,$ the multicritical behavior is described by the biconal fixed point with critical exponents that are very close to the Heisenberg ones. For $N>~4$ and any ${n}_{1}{,n}_{2}$ the critical behavior is controlled by the tetracritical decoupled fixed point. We discuss the relevance of these results for some physically interesting systems, in particular for anisotropic antiferromagnets in the presence of a magnetic field and for high-${T}_{c}$ superconductors. Concerning the SO(5) theory of superconductivity, we show that the bicritical O(5) fixed point is unstable with a significant crossover exponent ${\ensuremath{\varphi}}_{4,4}\ensuremath{\approx}0.15;$ this implies that the O(5) symmetry is not effectively realized at the point where the antiferromagnetic and superconducting transition lines meet. The multicritical behavior is either governed by the tetracritical decoupled fixed point or is of first-order type if the system is outside its attraction domain.

DOI
10.1103/physrevb.67.054505

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We study the multicritical behavior arising from the competition of two distinct types of ordering characterized by O(n)\mathrm{O}(n)O(n) symmetries. For this purpose, we consider the renormalization-group flow for the most general O(n1)\ensuremath⨁O(n2)\mathrm{O}{(n}_{1})\ensuremath{\bigoplus}\mathrm{O}{(n}_{2})O(n1​)\ensuremath⨁O(n2​)-symmetric Landau-Ginzburg-Wilson Hamiltonian involving two fields \ensuremathφ1{\ensuremath{\varphi}}_{1}\ensuremathφ1​ and \ensuremathφ2{\ensuremath{\varphi}}_{2}\ensuremathφ2​ with n1{n}_{1}n1​ and n2{n}_{2}n2​ components, respectively. In particular, we determine in which cases, approaching the multicritical point, one may observe the asymptotic enlargement of the symmetry to O(N)\mathrm{O}(N)O(N) with N=n1+n2.{N=n}_{1}{+n}_{2}.N=n1​+n2​. By performing a five-loop \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ-expansion computation we determine the fixed points and their stability. It turns out that for N=n1+n2> 3{N=n}_{1}{+n}_{2}>~3N=n1​+n2​> 3 the O(N)\mathrm{O}(N)O(N)-symmetric fixed point is unstable. For N=3,N=3,N=3, the multicritical behavior is described by the biconal fixed point with critical exponents that are very close to the Heisenberg ones. For N> 4N>~4N> 4 and any n1,n2{n}_{1}{,n}_{2}n1​,n2​ the critical behavior is controlled by the tetracritical decoupled fixed point. We discuss the relevance of these results for some physically interesting systems, in particular for anisotropic antiferromagnets in the presence of a magnetic field and for high-Tc{T}_{c}Tc​ superconductors. Concerning the SO(5) theory of superconductivity, we show that the bicritical O(5) fixed point is unstable with a significant crossover exponent \ensuremathφ4,4\ensuremath≈0.15;{\ensuremath{\varphi}}_{4,4}\ensuremath{\approx}0.15;\ensuremathφ4,4​\ensuremath≈0.15; this implies that the O(5) symmetry is not effectively realized at the point where the antiferromagnetic and superconducting transition lines meet. The multicritical behavior is either governed by the tetracritical decoupled fixed point or is of first-order type if the system is outside its attraction domain.
We study the nature of the multicritical point in the three-dimensional $\mathrm{O}(3)\ensuremath{\bigoplus}\mathrm{O}(2)$ symmetric Landau-Ginzburg-Wilson theory, which describes the competition of two order parameters that are O(3) and O(2) symmetric, respectively. This study is relevant for SO(5) theory of high-Tc{T}_{c}Tc​ superconductors, which predicts the existence of a multicritical point in the temperature-doping phase diagram, where the antiferromagnetic and superconducting transition lines meet. We investigate whether $\mathrm{O}(3)\ensuremath{\bigoplus}\mathrm{O}(2)$ symmetry gets effectively enlarged to O(5) approaching the multicritical point. For this purpose, we study the stability of the O(5) fixed point. By means of a Monte Carlo simulation, we show that the O(5) fixed point is unstable with respect to the spin-4 quartic perturbation with the crossover exponent ${\ensuremath{\phi}}_{4,4}=0.180(15)$, in substantial agreement with recent field-theoretical results. This estimate is much larger than the one-loop ϵϵϵ-expansion estimate ${\ensuremath{\phi}}_{4,4}=1∕26$, which has often been used in the literature to discuss the multicritical behavior within SO(5) theory. Therefore, no symmetry enlargement is generically expected at the multicritical transition. We also perform a five-loop field-theoretical analysis of the renormalization-group flow. It shows that bicritical systems are not in the attraction domain of the stable decoupled fixed point. Thus, in these systems---high-Tc{T}_{c}Tc​ cuprates should belong to this class---the multicritical point corresponds to a first-order transition.
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.
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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 discuss the critical behavior of several three-dimensional magnetic systems, such as pure and randomly dilute (anti)ferromagnets and stacked triangular antiferromagnets. We also discuss the nature of the multicritical points that arise in the presence of two distinct O(n)-symmetric order parameters and, in particular, the nature of the multicritical point in the phase diagram of high-T_c superconductors that has been predicted by the SO(5) theory. For each system, we consider the corresponding Landau-Ginzburg-Wilson field theory and review the field-theoretical results obtained from the analysis of high-order perturbative series in the frameworks of the epsilon and of the fixed-dimension d=3 expansions.