SciPost Phys. 7, 010 (2019) ·
published 17 July 2019
Conformal field theories (CFTs) with MN and tetragonal global symmetry in
$d=2+1$ dimensions are relevant for structural, antiferromagnetic and
helimagnetic phase transitions. As a result, they have been studied in great
detail with the $\varepsilon=4-d$ expansion and other field theory methods. The
study of these theories with the nonperturbative numerical conformal bootstrap
is initiated in this work. Bounds for operator dimensions are obtained and they
are found to possess sharp kinks in the MN case, suggesting the existence of
full-fledged CFTs. Based on the existence of a certain large-$N$ expansion in
theories with MN symmetry, these are argued to be the CFTs predicted by the
$\varepsilon$ expansion. In the tetragonal case no new kinks are found,
consistently with the absence of such CFTs in the $\varepsilon$ expansion.
Estimates for critical exponents are provided for a few cases describing phase
transitions in actual physical systems. In two particular MN cases,
corresponding to theories with global symmetry groups $O(2)^2\rtimes S_2$ and
$O(2)^3\rtimes S_3$, a second kink is found. In the $O(2)^2\rtimes S_2$ case it
is argued to be saturated by a CFT that belongs to a new universality class
relevant for the structural phase transition of NbO$_2$ and
paramagnetic-helimagnetic transitions of the rare-earth metals Ho and Dy. In
the $O(2)^3\rtimes S_3$ case it is suggested that the CFT that saturates the
second kink belongs to a new universality class relevant for the
paramagnetic-antiferromagnetic phase transition of the rare-earth metal Nd.
SciPost Phys. 6, 035 (2019) ·
published 21 March 2019
Three-dimensional theories with cubic symmetry are studied using the
machinery of the numerical conformal bootstrap. Crossing symmetry and unitarity
are imposed on a set of mixed correlators, and various aspects of the parameter
space are probed for consistency. An isolated allowed region in parameter space
is found under certain assumptions involving pushing operator dimensions above
marginality, indicating the existence of a conformal field theory in this
region. The obtained results have possible applications for ferromagnetic phase
transitions as well as structural phase transitions in crystals. They are in
tension with previous $\varepsilon$ expansion results, as noticed already in
SciPost Phys. 6, 008 (2019) ·
published 17 January 2019
Fixed points of scalar field theories with quartic interactions in
$d=4-\varepsilon$ dimensions are considered in full generality. For such
theories it is known that there exists a scalar function $A$ of the couplings
through which the leading-order beta-function can be expressed as a gradient.
It is here proved that the fixed-point value of $A$ is bounded from below by a
simple expression linear in the dimension of the vector order parameter, $N$.
Saturation of the bound requires a marginal deformation, and is shown to arise
when fixed points with the same global symmetry coincide in coupling space.
Several general results about scalar CFTs are discussed, and a review of known
fixed points is given.