Operator mixing in the ε -expansion: Scheme and evanescent-operator independence

Abstract

We consider theories with fermionic degrees of freedom that have a fixed point of Wilson-Fisher type in noninteger dimension d ¼ 4 − 2ϵ. Due to the presence of evanescent operators, i.e., operators that vanish in integer dimensions, these theories contain families of infinitely many operators that can mix with each other under renormalization. We clarify the dependence of the corresponding anomalous-dimension matrix on the choice of renormalization scheme beyond leading order in ϵ-expansion. In standard choices of scheme, we find that eigenvalues at the fixed point cannot be extracted from a finite-dimensional block. We illustrate in examples a truncation approach to compute the eigenvalues. These are observable scaling dimensions, and, indeed, we find that the dependence on the choice of scheme cancels. As an application, we obtain the IR scaling dimension of four-fermion operators in QED in d ¼ 4 − 2ϵ at order Oðϵ 2 Þ.

Critical properties of the valence-bond-solid transition in lattice quantum electrodynamics

“…Finally, in the limit e 2 ¼ 0 the model reduces to the ungauged NJL or chiral Oð2Þ=XY GNY model, and the results in Eqs. (27)- (29) and Ref. [54] agree with the four-loop result in Ref.…”

“…In contrast to the Z 2 case, however, gapless Uð1Þ gauge fluctuations drive the ground state away from the free-Dirac fixed point. For small values of the gauge coupling, the numerical results are consistent with a gapless phase described by the deconfined, conformal QED 3 fixed point, which can be accessed either in the large-N f expansion [22][23][24][25][26] or in the ϵ expansion below four spacetime dimensions [27][28][29][30]. When the gauge coupling becomes strong, a quantum phase transition from the deconfined QED 3 phase to a confining phase occurs, accompanied by chiral symmetry breaking and dynamical mass generation for the fermions, and is found to be continuous [19][20][21].…”

Critical properties of the Néel–algebraic-spin-liquid transition

“…These matrices generate an SU (2) nodal subgroup of the enlarged SU (4) flavor symmetry of the pure ASL state, i.e., the internal symmetry of the first term in Eq. (18), but in the four-dimensional theory elements of this subgroup act as Lorentz group elements on four-component Dirac spinors. In other words, the Yukawa coupling breaks the SU (2) nodal symmetry in the (2+1)D theory, which means it breaks Lorentz symmetry in the four-dimensional theory.…”

“…The low-energy effective theory of the model in this regime -(2+1)D quantum electrodynamics (QED 3 ) with N f = 1 flavor of spinful Dirac fermions, where N f is defined precisely here as the number of spin SU (2) doublets of four-component Dirac spinors -is identical to that of the ASL as defined above, and for simplicity we will refer to the phase observed numerically as the ASL. At low energies the ASL is believed to be described by a strongly coupled (2+1)D conformal field theory whose universal properties, notably the critical exponents controlling the power-law correlations mentioned above, can be systematically computed using field-theoretic approaches such as the -expansion [16][17][18][19] and the large-N f expansion [13,15,[20][21][22].…”

Noncommutativity between the low-energy limit and integer dimension limits in the ε expansion: A case study of the antiferromagnetic quantum critical metal

“…For recent discussions on subtle issues in the -expansion for relativistic QFTs [55,56], see Refs. [57][58][59]. T denotes temperature and ρ denotes a tunning parameter that drives the transition from a paramagnetic Fermi liquid (FL) to an antiferromagnetically ordered Fermi liquid (AFM).…”