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We study how universality classes of O(N)-symmetric models depend continuously on the dimension d and the number of field components N. We observe, from a renormalization group perspective, how the implications of the Mermin-Wagner-Hohenberg theorem set in as we gradually deform theory space towards d=2. For fractal dimension in the range 2<d<3 we observe, for any N bigger than or equal to 1, a finite family of multi-critical effective potentials of increasing order. Apart for the N=1 case, these disappear in d=2 consistently with the Mermin-Wagner-Hohenberg theorem. Finally, we study O(N=0)-universality classes and find an infinite family of these in two dimensions.
Just over fifty years ago, Pierre Hohenberg developed a rigorous proof of the non-existence of long-range order in a two-dimensional superfluid or superconductor at finite temperatures. The proof was immediately extended by N. D. Mermin and H. Wagner
We study the conformal bootstrap for 3D CFTs with O(N) global symmetry. We obtain rigorous upper bounds on the scaling dimensions of the first O(N) singlet and symmetric tensor operators appearing in the $phi_i times phi_j$ OPE, where $phi_i$ is a fu
We use numerical bootstrap techniques to study correlation functions of a traceless symmetric tensors of $O(N)$ with two indexes $t_{ij}$. We obtain upper bounds on operator dimensions for all the relevant representations and several values of $N$. W
Motivated by applications to critical phenomena and open theoretical questions, we study conformal field theories with $O(m)times O(n)$ global symmetry in $d=3$ spacetime dimensions. We use both analytic and numerical bootstrap techniques. Using the
We apply the methods of modern analytic bootstrap to the critical $O(N)$ model in a $1/N$ expansion. At infinite $N$ the model possesses higher spin symmetry which is weakly broken as we turn on $1/N$. By studying consistency conditions for the corre