Classifying perturbative fixed points near upper critical dimensions plays an important role in understanding the space of conformal field theories and critical phases of matter. In this work, we consider perturbative fixed points of N=5 scalar bosons coupled with quartic interactions preserving an arbitrary subgroup G \subset ⊂ O(5). We perform an exhaustive algorithmic search over the symmetry groups G which are irreducible and satisfy the Landau condition, so that the fixed point can be reached by fine-tuning a single mass term and there is no need to tune the cubic couplings. We also impose stability of the RG flow in the space of quartic couplings, and reality. We thus prove that there exist no new stable fixed points in d=4- \epsilon ϵ dimensions beyond the two known ones: namely the O(5) nvariant fixed point and the Cubic(5) fixed point. This work is a continuation of the classification of such fixed points with N=4 scalars by Toledano, Michel, Toledano and Brézin in 1985 [Phys. Rev. B 31, 7171 (1985)].
The renormalization-group recursion relations are solved for the effective Hamiltonians relative to phase transitions with four-component order parameters. For this value of n there are 22 types of Hamiltonians which can be classified into two categories according to the action of their normalizer
GN on the corresponding parameter space (
GN is the symmetry group leaving globally invariant this space). In the first case,
GN generates a finite number of isolated fixed points whose characteristics can be deduced from the detailed investigation of five Hamiltonians only.In the second category, for which
GN is a continuous group, there are, in addition to isolated fixed points, continuous manifolds of physically equivalent fixed points (the dimension of the manifolds is either one or three). In the search for a stable fixed point, the continuous manifolds can be ignored, while the isolated points are related to the five former Hamiltonians. For n=4, it is necessary to solve the recursion relations to two-loop order. The only possible stable ones among the fixed points then arise from a splitting of points which coincide, to one-loop order, with the isotropic fixed point. Extending, to two-loop order, a result recently established to the preceding order, we show that if a stable fixed point exists, it is unique. For n=4, the stable fixed point has one of three possible symmetries : di-icosahedral, hypercubic, or dicylindrical.Despite this anisotropy of the critical fluctuations, the exponents associated with any of the stable fixed points are identical to order ${\ensuremath{\epsilon}}^{2}$ to the ``isotropic'' exponents corresponding to n=4. The cubic point is destabilized by any operator of symmetry lower than cubic. The dicylindrical one remains stable with respect to certain anisotropies of lower symmetry. We examine the available experimental data in light of the preceding theoretical results concerning the critical behavior and the thermodynamic order of the transitions. On the other hand, we establish two general symmetry conditions relative to the stable fixed points determined by the renormalization-group equations. The first one specifies group theoretically, for each value of n, the possible symmetries
Gi∗ of the stable fixed points. The second one formulates a necessary condition for the occurrence of an anisotropic stable fixed point: The normalizer
GN of the considered parameter space must fulfill the condition
GN\ensuremath{\subset}@;
BGi∗ for one, at least, of the former
Gi∗ groups. These rules are shown to be very restrictive for n=4: On the basis of symmetry the lack of a stable fixed point can be asserted for 10 Hamiltonians out of 22, without solving the fixed-point equations.
We saw in Chapter 3 that the hamiltonian for a System at a critical point flows under the renormalization group into a critical fixed point. Under a renormalization group transformation, the microscopic length scale is rescaled by a constant factor b , and so the coordinates of a given point, as measured in units of this length scale, transform according to r → b −1 r. This is called a scale transformation . Once the flows reach such a fixed point, the parameters of the hamiltonian no longer change, and it is said to be scale invariant . As well as being scale invariant, the fixed point hamiltonian usually possesses other spatial symmetries. For example, if the underlying model is defined on a lattice, so that its hamiltonian is invariant under lattice translations, the corresponding critical fixed point hamiltonian is generally invariant under arbitrary uniform translations. This is because terms which might be added to the hamiltonian which break the symmetry under continuous translations down to its subgroup of lattice translations are irrelevant at such a fixed point. Similarly, if the lattice model is invariant under a sufficiently large subgroup of the rotation group (for example, if the interactions in the x and y directions on a Square lattice are equal), then the fixed point hamiltonian enjoys full rotational invariance. As discussed on p.58, even if the interactions are anisotropic, rotational invariance may often be recovered by a suitable finite relative rescaling of the coordinates.
For a Landau-Ginzburg-Wilson Hamiltonian of any given symmetry we show how one can find a group
GT of orthogonal transformations in parameter space, which commute with renormalization-group transformations. Then a renormalization-group transformation may be expanded into covariants of
GT. We also present a systematic procedure for finding fixed points; they are most likely to decouple the Hamiltonian or to increase its symmetry. The merit of the conclusions obtained is illustrated using an example of a system with
C4 symmetry. Agreement with the results of $\ensuremath{\epsilon}$-expansion calculations has been found.
The canonical Gross-Neveu model for N two-component Dirac fermions in 2+1 dimensions suffers a continuous phase transition at a critical interaction gc1∼1/N at large N , at which its continuous symmetry SO(2N) is preserved and a discrete (Ising) symmetry becomes spontaneously broken. A recent mean-field calculation, however, points to an additional transition at a different critical gc2∼−Ngc1 , at which SO(2N)→SO(N)×SO(N) . To study the latter phase transition we rewrite the Gross-Neveu interaction g(ψ¯ψ)2 in terms of three different quartic terms for the single ( L=1 ) 4N -component real (Majorana) fermion, and then extend the theory to L>1 . This allows us to track the evolution of the fixed points of the renormalization group transformation starting from L≫1 , where one can discern three distinct critical points which correspond to continuous phase transitions into (1) SO(2N) -singlet mass-order-parameter, (2) SO(2N) -symmetric-tensor mass-order-parameters, and (3) SO(2N) -adjoint nematic-order-parameters, down to L=1 value that is relevant to the standard Gross-Neveu model. Below the critical value of Lc(N)≈0.35N for N≫1 only the Gross-Neveu critical point (1) still implies a diverging susceptibility for its corresponding ( SO(2N) -singlet) order parameter, whereas the two new critical points that existed at large L ultimately become equivalent to the Gaussian fixed point at L=1 . We interpret this metamorphosis of the SO(2N) -symmetric-tensor fixed point from critical to spurious as an indication that the transition at gc2 in the original Gross-Neveu model is turned first-order by fluctuations. Published by the American Physical Society 2024
We investigate the relativistic SO(2)- and SO(3)-invariant Gross-Neveu-Yukawa field theories for real, rank-two, symmetric, traceless tensor order parameters coupled to
Nf flavors of two-component Dirac fermions. These field theories arise as an effective description of fractionalized spin-orbital liquids. The two theories are the simplest and special cases of the more general class of field theories with
SO(N) symmetric tensor order parameter coupled to Dirac fermions, in which the symmetry is low enough to allow only one and not the usual two quartic self-interaction terms. Using a two-loop renormalization group near the upper critical dimension, we demonstrate that the theory exhibits an exotic critical fixed point for
N=3 and the concomitant continuous phase transition for any value of
Nf. For
N=2 the theory is equivalent to the chiral XY model. We discuss the crucial role of the symmetry-allowed sextic self-interactions in the selection of the ground state configuration in the case of SO(3). The universal quantities such as the anomalous dimensions of order parameters and fermions, the correlation length exponent, and the mass gap ratio between order parameter and fermion masses are computed up to ${\ensuremath{\epsilon}}^{2}$ order.