Quantum Critical Behavior in Gauged Yukawa Matrix Field Theories with Quenched Disorder

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 non-Abelian gauge theories. Using the quantum chromodynamics (QCD) terminology, the quantum phase transition with a chiral symmetry breaking pattern is then characterized by the number of flavors (NF ) and colors (NC) of quarks in the gauge theory. In the antiferromagnet the rank of the spin algebra and the size of its representation play the same role as the number of flavors and colors respectively. The study of 2D quantum antiferromagnets, particulary in connection with high-TC superconductivity, is a rapidly developing subject of great current interest [6]. An important question that naturally arises is the stability of the antiferromagnetic long-range order (LRO) in the presence of quenched disorder (QD). In fact, since the Néel temperature of planar highTC superconductors, such as La2CuO4, is extremely sensitive to impurities and defects, it is probably not correct to ignore the impurity effects even in high-quality samples used in laboratories. The critical behavior of low-dimensional systems with quenched disorder and an O(N)-vector magnetic order parameter has been addressed by a number of authors [7, 8] by using a Landau-Ginzburg-Wilson (LGW) Hamiltonian and applying the Wilson-Fisher (WF) ǫ expansion [9]. In particular, Boyanovsky and Cardy [8] have carried out a two-loop (double) ǫ expansion in systems in which the impurities are correlated in ǫd dimensions and randomly distributed in d − ǫd dimensions. They take the full structure of the theory into account which, as a result of anisotropies, leads to highly nonlocal interactions. In this paper we shall study the stability of LRO in 2D quantum antiferromagnets with QD by using the WF ǫ expansion and renormalization group (RG) techniques. We shall approach this problem by examining the critical behavior of the related zero temperature gauged Yukawa matrix field theories in three (Euclidean) dimensions in the presence of quenched disorder. This requires that the anisotropies be properly taken into account, as is done (see Ref. [8], e.g.) for the O(N)-vector model in the presence of QD. However we shall go one step further and also examine the stability of the theory against quantum fluctuations which can arise through the Coleman-Weinberg mechanism [10]. Our results enable us to argue that below a certain critical doping (impurity concentration) ∆crit., and in a narrow region in parameter space, 2D doped quantum antiferromagnets belong to the universality class of gauged Yukawa matrix field theories which describe the pure system. We check this result through an explicit calculation of the critical exponent ν and find that the Harris criterion [11] is indeed satisfied. However, we find that a careful examination of the RG flows in the space of couplings indicates that LRO can be destroyed due to quantum fluctuations. This means that there are cases where there is an infrared (IR) stable fixed point and the Harris criterion is satisfied, but the phase transition is fluctuation-induced first order rather than second order, contrary to what is naively expected. We shall begin with a brief review of some recent work on pure quantum antiferromagnets and gauged Yukawa matrix theories.