Indications of isotropic Lifshitz points in four dimensions

Abstract

The presence of isotropic Lifshitz points for a U(1) symmetric scalar theory is investigated with the help of the Functional Renormalization Group at the conjectured lower critical dimension d=4. To this aim, a suitable truncation in the expansion of the effective action in powers of the field is considered and, consequently, the Renormalization Group flow is reduced to a set of ordinary differential equations for the parameters that define the effective action. Within this approximation, indications of a line of Lifshitz points are found, that present evident similarities with the properties shown by the line of fixed points observed in the two dimensional Berezinsky-Kosterlitz-Thouless phase. In particular, this line of Lifshitz points exhibits the vanishing of the expectation value of the field, together with a finite stiffness and, for specific combinations of the parameters that define the effective action, also the algebraic decay at large distance of the order parameter correlations.

Isotropic Lifshitz scaling in four dimensions

“…where four field derivative as well as two field derivative terms, with field independent parameters, In this scheme one can compute the flow equation for the various k-dependent parameters [17]. By starting the flow at an initial scale k = Λ, with large values of α Λ , one immediately observes for this parameter a power law when k → 0, with exponent η, according to its scaling dimension : α 2 k ∝ k η .…”

“…In this paper, after reviewing some general properties of the Lifshitz points in Section 2, we discuss in Section 3 the isotropic points for O(N ) theories, which show up in the dimensional range 4 < d < 8, by means of a non-perturbative approach, namely the Functional Renormalization Group (FRG) flow equations [12,13,14], which is especially useful in cases that are out of reach of other approaches such as the -expansion [1,2,15]. Then, in Section 4, we concentrate on the case d = 4 [16,17], where we observe very interesting properties associated to the Lifshitz scaling, such as the appearance of a continuous line of fixed points, that present clear similarities with those observed in the d = 2 Berezinskii -Kosterlitz -Thouless phase [18,19]. Our conclusions are reported in Section 5.…”

Fluctuations in cool quark matter and the phase diagram of quantum chromodynamics

“…In condensed matter it is well known that a third phase can arise, in which spatially inhomogeneous structures form. If so, the three phases meet at a Lifshitz point [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

“…As demonstrated first by Brazovskii [89,[107][108][109], the line of transitions between the symmetric phase and that with chiral spirals becomes a line of first order transitions. Most importantly, the infrared fluctuations about a Lifshitz point are so strong that there is, in fact, no true Lifshitz point [15][16][17][23][24][25]. This is known to occur in inhomogenous polymers, both from experiment and numerical simulations [18][19][20][21][22].…”

Quantum Lifshitz points and fluctuation-induced first-order phase transitions in imbalanced Fermi mixtures

“…As was indicated by mean-field studies, there is however another, quite distinct multicritical phenomenon present at T > 0, related to the so-called Lifshitz point [36,37], where two ordered phases (uniform and nonuniform superfluids) coexist with the normal phase. The universal critical singularities at the classical (T > 0) Lifshitz point [38][39][40] are completely different from those controlling the critical or tricritical point. Notably, while the upper critical dimension d u for the tricritical transition is 3, it is much higher (at least 9/2) for the Lifshitz transition [38,41,42], making the conventional approaches to critical phenomena in spatial dimensionality d = 3 based on the expansion ( = d u − d) problematic [41].…”