We use the epsilon expansion to explore a new universality class of second order quantum phase transitions associated with a four-dimensional Yukawa field theory coupled to a traceless Hermitian matrix scalar field. We argue that this class includes four-fermion modes in $2<D<4$ dimensions with $\mathrm{SU}({N}_{F})\ifmmode\times\else\texttimes\fi{}\mathrm{U}(N)$ symmetry and a $\mathrm{U}(N)$ scalar, $\mathrm{SU}({N}_{F})$ isovector four-fermion coupling. The epsilon expansion indicates that there is a second order phase transition for $N\ensuremath{\ge}{N}^{*}({N}_{F})$, where ${N}^{*}({N}_{F})\ensuremath{\simeq}{0.27N}_{F}\mathrm{if}{N}_{F}\ensuremath{\rightarrow}\ensuremath{\infty}$.
We use the epsilon expansion to explore a new universality class of second order quantum phase transitions associated with a four-dimensional Yukawa field theory coupled to a traceless Hermitian matrix scalar field. We argue that this class includes four-fermion modes in 2<D<4 dimensions with SU(NF)\ifmmode×\else\texttimes\fiU(N) symmetry and a U(N) scalar, SU(NF) isovector four-fermion coupling. The epsilon expansion indicates that there is a second order phase transition for N\ensuremath≥N∗(NF), where N∗(NF)\ensuremath≃0.27NFifNF\ensuremath→\ensuremath∞.
In this set of lectures, I shall review some recent work on quantum critical phenomena which involve the changes in phase of the ground state of a quantum antiferromagnetic system when external parameters are varied through some critical values. I will also draw analogies between antiferromagnetic phase transitions and chiral phase transitions in lower (particularly 2 + l)-dimensional gauge theories and also some 2+1-dimensional four-fermi theories where chiral symmetry breaking patterns depend on the coupling constants and also the matter and gauge field content of the theory. I will begin by reviewing some arguments that, in the infrared limit, the dynamics of Goldstone bosons in the confining and chiral symmetry breaking phase of 2 + 1 dimensional QCD is identical to the effective spin wave dynamics in 2 dimensional generalized quantum antiferromagnets. I also discuss the possibility that certain quantum critical behaviors in generalized antiferromagnets can be analyzed using the principle of universality and are in the universality class of a particular type of relativistic quantum field theory with gauge and chiral symmetry. I show how the epsilon expansion can be used and argue that, for a large range of symmetry groups and other parameters, the phase transitions are expected to be fluctuation induced first order transitions. For the small range of parameters where they are second order, the epsilon expansion and universality give information about the critical behavior.
We study the universal critical properties of the QED3-Gross-Neveu-Yukawa model with N flavors of four-component Dirac fermions coupled to a real scalar order parameter at four-loop order in the $\ensuremath{\epsilon}$ expansion below four dimensions. For N=1, the model is conjectured to be infrared dual to the SU(2)-symmetric noncompact CP1 model, which describes the deconfined quantum critical point of the N\'eel-valence-bond-solid transition of spin-1/2 quantum antiferromagnets on the two-dimensional square lattice. For N=2, the model describes a quantum phase transition between an algebraic spin liquid and a chiral spin liquid in the spin-1/2 kagome antiferromagnet. For general N we determine the order-parameter anomalous dimension, the correlation length exponent, the stability critical exponent, as well as the scaling dimensions of SU(N) singlet and adjoint fermion mass bilinears at the critical point. We use Pad\'e approximants to obtain estimates of critical properties in 2+1 dimensions.
We use the Wilson-Fisher ǫ expansion to study quantum critical behavior in gauged Yukawa matrix field theories with weak quenched disorder. We find that the resulting quantum critical behavior is in the universality class of the pure system. As in the pure system, the phase transition is typically first order, except for a limited range of parameters where it can be second order with computable critical exponents. Our results apply to the study of two-dimensional quantum antiferromagnets with weak quenched disorder and provide an example for fluctuation-induced first order phase transitions in circumstances where naively none is expected. One of the most remarkable features of quantum spin systems is their relationship to gauge theories. This connection, which was originally used to study chiral symmetry breaking in QCD in the strong coupling limit [1], remains a powerful tool. Recently this analogy has been used to prove that certain gauge theories break chiral symmetry in the strong coupling limit [2-4]. Also, it has been further suggested [5] that the underlying analogy can be used in a much broader sense to account for the quasi-particle spectrum and other infrared features of two-dimensional (2D) quantum antiferromagnets and three-dimensional
Wilson–Fisher expansion near upper critical dimension has proven to be an invaluable conceptual and computational tool in our understanding of the universal critical behavior in the [Formula: see text] field theories that describe low-energy physics of the canonical models, such as Ising, XY, and Heisenberg. Here, I review its application to a class of the Gross–Neveu–Yukawa (GNY) field theories, which emerge as possible universal description of a number of quantum phase transitions in electronic two-dimensional systems such as graphene and d-wave superconductors. GNY field theories may be viewed as minimal modifications of the [Formula: see text] field theories in which the order parameter is coupled to relativistic Dirac fermions through Yukawa term and which still exhibit critical fixed points in the suitably formulated Wilson–Fisher [Formula: see text]-expansion. I discuss the unified GNY field theory for a set of different symmetry-breaking patterns, with focus on the semimetal-Néel-ordered-Mott insulator quantum phase transition in the half-filled Hubbard model on the honeycomb lattice, for which a comparison between the state-of-the-art [Formula: see text]-expansion, quantum Monte Carlo, large N, and functional renormalization-group calculations can be made.
A second-order phase transition for the three-dimensional Gross-Neveu model is established for one fermion species, N=1. This transition breaks a paritylike discrete symmetry. It constitutes its peculiar universality class with critical exponent $\ensuremath{\nu}=0.63$ and scalar and fermionic anomalous dimensions ${\ensuremath{\eta}}_{\ensuremath{\sigma}}=0.31$ and ${\ensuremath{\eta}}_{\ensuremath{\psi}}=0.11,$ respectively. We also compute critical exponents for other N. Our results are based on exact renormalization-group equations.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.