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arXiv:1806.00043 · Physical review. D/Physical review. D.

Indications of isotropic Lifshitz points in four dimensions

Dario Zappalà

Published Oct 8, 2018• 14 citations
arXivHTMLPDF

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 Berezinskii-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 a large distance of the order parameter correlations.

DOI
10.1103/physrevd.98.085005

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The presence of isotropic Lifshitz points for a U(1)U(1)U(1) symmetric scalar theory is investigated with the help of the functional renormalization group at the conjectured lower critical dimension d=4d=4d=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 Berezinskii-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 a large distance of the order parameter correlations.
We employ the derivative expansion of the nonperturbative renormalization group to address the phenomenon of anisotropic scale invariance and the associated functional fixed points, also known as Lifshitz points, in systems characterized by a scalar order parameter. We demonstrate the existence of the Lifshitz fixed point featuring a nonclassical value of the anisotropy exponent θ<1/2 and provide estimates for values of a set of critical exponents in the physically most relevant case of the three-dimensional uniaxial Lifshitz point (d,m)=(3,1), m denoting the anisotropy index. We compare our predictions with existing estimates from perturbative expansions around dimensionality d=4+1/2 as well as those from the 1/N expansion.
In the framework of the renormalization-group (RG) theory of critical phenomena, a quantitative description of many continuous phase transitions can be obtained by considering an effective Φ 4 theories, having an N -component fundamental field Φ i and containing up to fourth-order powers of the field components. Their RG flow is usually characterized by several fixed points (FPs). We give here strong arguments in favour of the following conjecture: the stable FP corresponds to the fastest decay of correlations, that is, is the one with the largest values of the critical exponent η describing the power-law decay of the two-point function at criticality. We prove this conjecture in the framework of the ε-expansion. Then, we discuss its validity beyond the ε-expansion. We present several lower-dimensional cases, mostly three-dimensional, which support the conjecture. We have been unable to find a counterexample.
We analyze the interplay of longitudinal and transverse thermal fluctuations in a U(1)\text{U}(1)U(1) symmetric two-dimensional ${\ensuremath{\phi}}^{4}$ theory. Toward this end, we derive coupled renormalization-group equations for both types of fluctuations obtained from a linear (Cartesian) decomposition of the order-parameter field. Discarding the longitudinal fluctuations, the expected Berezinskii-Kosterlitz-Thouless phase characterized by a finite stiffness and an algebraic decay of order-parameter correlations is recovered. Renormalized by transverse fluctuations, the longitudinal mass scales to zero so that longitudinal fluctuations become increasingly important for small momenta. Within our expansion of the effective action, they generate a logarithmic decrease of the stiffness, in agreement with previous functional renormalization-group calculations. The logarithmic terms imply a deviation from the vanishing $\ensuremath{\beta}$ function for the stiffness in the nonlinear $\ensuremath{\sigma}$ model describing the phase fluctuations at three-loop order. To gain further insight, we also compute the flow of the parameters characterizing longitudinal and transverse fluctuations from a density-phase representation of the order-parameter field, with a cutoff on phase fluctuations. The power-law flow of the longitudinal mass and other quantities is thereby confirmed, but the stiffness remains finite in this approach. We conclude that the marginal flow of the stiffness obtained in the Cartesian representation is an artifact of the truncated expansion of momentum dependences.
By writing the flow equations for the continuum Legendre effective action (a.k.a. Helmholtz free energy) with respect to a particular form of smooth cutoff, and performing a derivative expansion up to some maximum order, a set of differential equations are obtained which at FPs (Fixed Points) reduce to non-linear eigenvalue equations for the anomalous scaling dimension \\eta. Illustrating this by expanding (single component) scalar field theory, in two, three and four dimensions, up to second order in derivatives, we show that the method is a powerful and robust means of discovering and quantifying non-perturbative continuum limits (continuous phase transitions).