Beyond the Isotropic Lifshitz Endpoint

The puzzle of the disappearance of isotropic Lifshitz points in condensed matter physics is explained from the point of view of the Wilsonian renormalization group. In analogy to the commensurate ideas of metamagnetic phase transitions, we describe the physics of thermodynamic states beyond an isotropic Lifshitz endpoint. Such phenomenon may be understood in terms of a statistically isotropic environment of coexisting multi-incommensurate helicoidal states. In addition to the magnetic and condensed matter discussions, we consider also an interesting example in the context of dynamical evolution.


I. Introduction
Helical ordering in magnetic systems was initially discovered by Yoshimori [1], Kaplan [2], and Villain [3] in terms of the mean field theory. The tricritical or Lifshitz behavior of the helical, ferromagnetic, and disordered phases was first examined in terms of the renormalization group (RG) by Hornreich et al. [4] by considering the nonquadratic effect of the critical propagator of the effective action for such a system. Later, this idea was generalized by Nicoll et al. [5] to arbitrary order and also with anisotropy [6]. In these studies, it was discovered that for isotropic systems, critical points may not exist beyond certain value of the isotropic Lifshitz order L which needs not to be an integer. In this letter, we study in detail when and why the non-existence of this criticality set in as well as the interesting thermodynamic behavior beyond this isotropic Lifshitz endpoint.

II. Renormalization-Group Analysis
Exact RG formulations with UV cutoffs at momentum scales much smaller than Planckian [7][8][9] are sufficient for our discussion. We search for infrared fixed points (IRFPs) from such a formulation perturbatively using the method of Gaussian eigenfunction expansions as first described by Nicoll et al. [10] and Chang et al. [11] and extended later to systems with generalized critical propagators [5,6] with []  being the integer part of the enclosed quantity.
These results lead to physically realizable attractive IRFPs when the spatial dimension lies within the upper d  (above which mean field theory applies) and lower d  (below which catastrophic divergences occur) borderline dimensions given below [12]: Thus, for realizable isotropic IRFPs for 3 d  , we must have 3 For anisotropic generalized Lifshitz ordering, the physically realizable dimension lies within the upper and lower dimensions that satisfy [12] / respectively. Unlike the isotropic situation, anisotropic generalized Lifshitz ordering of the above type are generally realizable for 3 d  . For example, the upper and lower borderline dimensions for a uniaxial generalized critical propagator with only one of the 3-dimensional wave vector 1 k raised to a power larger than 2 are:

III. Beyond the Isotropic Lifshitz Endpoint
To understand the thermodynamics beyond the physically realizable isotropic Lifshitz state, let us consider first a somewhat similar phenomenon: the metamagnetic phase transition in condensed matter physics. Generally for an antiferromagnet, its critical (Néel) point decreases with the application of an applied magnetic field. At some critical field, however, the criticality for a metamagnet terminates at a tricritical point [13].
Beyond this field, the associated symmetry breaking leads to the first order transition to the tri-coexistence of two opposite antiferromagnetic-sublattice states and the paramagnetic state. In other words, it is the critical point of the tricoexistence line.
Returning to the discussion of the appearance of the isotropic Lifshitz endpoint, the physics for the point of demarcation along the isotropy line in the Affine space of critical propagator exponents (Fig. 1)

IV. An Example
We consider the example of gravitational evolution at large cosmological spatial (or small k) scales due to classical fluctuations. Our attention will be restricted to the matterdominated era of non-relativistic dynamic motion. We therefore assume: (i) the gravitational field is weak, (ii) the variations of fields are slow in time, and (iii) the particles are nonrelativistic. Thus, the Einstein-Hilbert effective action reduces to: Lifshitz) exponents,  and i  , will be considered real and positive but not necessarily integers [5,6]. For our interests here, we shall consider the situations for physically realizable gravitational dynamics at 3 d  dimensions.
At small spatial scales (such as stellar distances), we do not expect much appreciable scale-running effects of G (and  ). Beyond the intermediate (larger than galactic) scales, we expect the scale-running effects of G and  begin to come into play. Initially, the evolution at such scales can be anisotropic and this may be one of the reasons for the continuing development of large-scale structures in the Universe.
Although the dynamic interactions during the cosmological evolution will include other orderings in addition to the gravitational potential (such as those related to the fields m and v in (1)), we do not expect these effects to influence the helicoidal orderings due to the scale-running of the propagator coupling constant G . In particular, we notice that the lower borderline dimension generally depends on the critical propagator exponents only.
We may therefore use the RG results of Eqs. (2) and (4b) for d  to understand the gravitational phenomenon by setting 1 n  , with the scalar field s representing the gravitational potential () x  . Such complex effects (CILOMAS) involving helicoidal orderings have been treated in detail by Chang [15] in a Physics Letter.
Beyond the intermediate scales, we expect the evolutional dynamics to become more and more isotropic. When full isotropy is statistically achieved at very large spatial scales, we may discuss the gravitational evolution in terms of the Friedmann-Robertson-Walker (FRW) paradigm. For a late matter-dominated ( 0 p  ) flat Universe, the Friedmann equation and the conservation equation of the energy-momentum tensor are: where () at is the scale factor of the expanding Universe, () m t  is the matter density, and the over dot indicates differentiation with respect to time t . We shall follow the arguments given by Bonanno and Reuter [14] and relate the momentum and time in leading order in the infrared as /  (Fig. 1).

V. Summary
We considered the thermodynamics beyond the admissible range of isotropic Lifshitz fixed points in RG calculations. Such phenomenon is akin to the well-known result of tricritical-tricoexistence effect of metamagnetic phase transitions in condensed matter physics. The isotropic Lifshitz endpoint is the critical point of the statistically isotropic multiphase coexisting thermodynamic state of subdomains of anisotropic helicoidal neighborhoods. An example of cosmic evolution based on the effective action with scale-running gravitational and cosmological constants at large spatial scales due to classical fluctuations is considered. The gravitational symmetry-breaking effects at intermediate (e.g., galaxy) scales are related to the development and formation of cosmic structures with multifractal characteristics [15,20,21]. At large cosmological scales, the FRW formulism leads to the understanding of statistically isotropic multi-phase coexisting states. We presently live in a matter-dominated statistically isotropic fractal Universe with a gravitational singularity index of 4.230

 
. The result provides a natural explanation to the cosmic accelerated expansion.