The abelian Higgs model is the textbook example for the superconducting transition and the Anderson-Higgs mechanism, and has become pivotal in the description of deconfined quantum criticality. We study the abelian Higgs model with $n$ complex scalar
fields at unprecedented four-loop order in the $4-epsilon$ expansion and find that the annihilation of the critical and bicritical points occurs at a critical number of $n_c approx 182.95left(1 - 1.752epsilon + 0.798 epsilon^2 + 0.362epsilon^3right) + mathcal{O}left(epsilon^4right) onumber$. Consequently, below $n_c$, the transition turns from second to first order. Resummation of the series to extract the result in three-dimensions provides strong evidence for a critical $n_c(d=3)$ which is significantly below the leading-order value, but the estimates for $n_c$ are widely spread. Conjecturing the topology of the renormalization group flow between two and four dimensions, we obtain a smooth interpolation function for $n_c(d)$ and find $n_c(3)approx 12.2pm 3.9$ as our best estimate in three dimensions. Finally, we discuss Miransky scaling occurring below $n_c$ and comment on implications for weakly first-order behavior of deconfined quantum transitions. We predict an emergent hierarchy of length scales between deconfined quantum transitions corresponding to different $n$.
The QED$_3$-Gross-Neveu model is a (2+1)-dimensional U(1) gauge theory involving Dirac fermions and a critical real scalar field. This theory has recently been argued to represent a dual description of the deconfined quantum critical point between Ne
el and valence bond solid orders in frustrated quantum magnets. We study the critical behavior of the QED$_3$-Gross-Neveu model by means of an epsilon expansion around the upper critical space-time dimension of $D_c^+=4$ up to the three-loop order. Estimates for critical exponents in 2+1 dimensions are obtained by evaluating the different Pade approximants of their series expansion in epsilon. We find that these estimates, within the spread of the Pade approximants, satisfy a nontrivial scaling relation which follows from the emergent SO(5) symmetry implied by the duality conjecture. We also construct explicit evidence for the equivalence between the QED$_3$-Gross-Neveu model and a corresponding critical four-fermion gauge theory that was previously studied within the 1/N expansion in space-time dimensions 2<D<4.
Gapless Dirac fermions appear as quasiparticle excitations in various condensed-matter systems. They feature quantum critical points with critical behavior in the 2+1 dimensional Gross-Neveu universality class. The precise determination of their crit
ical exponents defines a prime benchmark for complementary theoretical approaches, such as lattice simulations, the renormalization group and the conformal bootstrap. Despite promising recent developments in each of these methods, however, no satisfactory consensus on the fermionic critical exponents has been achieved, so far. Here, we perform a comprehensive analysis of the Ising Gross-Neveu universality classes based on the recently achieved four-loop perturbative calculations. We combine the perturbative series in $4-epsilon$ spacetime dimensions with the one for the purely fermionic Gross-Neveu model in $2+epsilon$ dimensions by employing polynomial interpolation as well as two-sided Pade approximants. Further, we provide predictions for the critical exponents exploring various resummation techniques following the strategies developed for the three-dimensional scalar $O(n)$ universality classes. We give an exhaustive appraisal of the current situation of Gross-Neveu universality by comparison to other methods. For large enough number of spinor components $Ngeq 8$ as well as for the case of emergent supersymmetry $N=1$, we find our renormalization group estimates to be in excellent agreement with the conformal bootstrap, building a strong case for the validity of these values. For intermediate $N$ as well as in comparison with recent Monte Carlo results, deviations are found and critically discussed.
Dirac and Weyl fermions appear as quasi-particle excitations in many different condensed-matter systems. They display various quantum transitions which represent unconventional universality classes related to the variants of the Gross-Neveu model. In
this work we study the bosonized version of the standard Gross-Neveu model -- the Gross-Neveu-Yukawa theory -- at three-loop order, and compute critical exponents in $4-epsilon$ dimensions for general number of fermion flavors. Our results fully encompass the previously known two-loop calculations, and agree with the known three-loop results in the purely bosonic limit of the theory. We also find the exponents to satisfy the emergent super-scaling relations in the limit of a single-component fermion, order by order up to three loops. Finally, we apply the computed series for the exponents and their Pade approximants to several phase transitions of current interest: metal-insulator transitions of spin-1/2 and spinless fermions on the honeycomb lattice, emergent supersymmetric surface field theory in topological phases, as well as the disorder-induced quantum transition in Weyl semimetals. Comparison with the results of other analytical and numerical methods is discussed.
