This work deals with quantum phase transitions in Dirac systems where the symmetries involved play a key role in the nature of the transition and are enlarged at criticality. By using functional renormalization group (FRG) methods, we show that an emergent relativistic symmetry not present at the bare level is a common feature of several different kinds of phase transitions. In three different projects, the interplay of emergent symmetries and universality in critical phenomena are explored through FRG techniques. In the first project, we studied the relation between discrete symmetry breaking and the emergence of two diverging length scales with their respective critical exponents. A second project explored the relation between multicritical points where two adjacent symmetry-breaking phases are compatible and the possibility of a continuous, order-to-order transition. The third and final project was concerned with the persistence of chiral symmetry in three-dimensional quantum electrodynamics - here seen as an effective description of different condensed matter systems - for the lowest possible number of fermion flavours, as well as with a dual description that has the potential to provide exact results for this strongly correlated theory.
The recently discovered fermion-induced quantum critical points of Dirac materials present a new opportunity to revisit and extend the standard paradigms of quantum criticality. The appearance of a second diverging length scale at a continuous quantum phase transition is one such extension. The authors present a functional renormalization group study of fermion-induced quantum critical points in the broken-symmetry phase, in which the presence of gapless fermions and discrete symmetry breaking provide the natural setup for this phenomenon to occur.
Supersymmetry, originally proposed in high-energy physics, can emerge as a remarkable low-energy structure in condensed matter systems. While emergent supersymmetry at quantum critical points is widely discussed in models with continuous symmetries, real materials are constrained by microscopic discrete symmetries. To address this, we investigate (2+1)-dimensional Gross-Neveu-Yukawa theories coupling Dirac fermions to a complex order parameter with discrete
Zn anisotropy. Using the functional renormalization group, we find that for
n>3, the anisotropic perturbations are irrelevant at the fixed point, yielding a
N=2 Wess-Zumino supersymmetric critical point. In the ordered phase, this dangerously irrelevant anisotropy gives rise to a second characteristic length scale,
ξ′, alongside the usual correlation length,
ξ. By tracking mass thresholds along symmetry-broken renormalization group trajectories, we extract the exponents
ν′ and
ν without imposing prior scaling assumptions. For the
Z4,
Z5, and
Z6 models, our results support the scaling relation
ν′/ν=1+∣yn∣/p with
p=2 in the isotropic framework used here.
First-order phase transitions in many-fermion systems are not detected in the susceptibility analysis of common renormalization-group (RG) approaches. Here, we introduce a counterterm technique within the functional renormalization-group (fRG) formalism which allows access to all stable and metastable configurations. It becomes possible to study symmetry-broken states which occur through first-order transitions as well as hysteresis phenomena. For continuous transitions, the standard results are reproduced. As an example, we study discrete-symmetry breaking in a mean-field model for a commensurate charge-density wave. An additional benefit of the approach is that away from the critical temperature for the breaking of discrete symmetries large interactions can be avoided at all RG scales.
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 quantum multicritical point in a (2+1)-dimensional Dirac system between the semimetallic phase and two ordered phases that are characterized by anticommuting mass terms with
O(N1) and
O(N2) symmetries, respectively. Using $\ensuremath{\epsilon}$ expansion around the upper critical space-time dimension of four, we demonstrate the existence of a stable renormalization-group fixed point, enabling a direct and continuous transition between the two ordered phases directly at the multicritical point. This point is found to be characterized by an emergent
O(N1+N2) symmetry for arbitrary values of
N1 and
N2 and fermion flavor numbers
Nf as long as the corresponding representation of the Clifford algebra exists. Small
O(N)-breaking perturbations near the chiral
O(N) fixed point are therefore irrelevant. This result can be traced back to the presence of gapless Dirac degrees of freedom at criticality, and it is in clear contrast to the purely bosonic
O(N) fixed point, which is stable only when
N<3. As a by-product, we obtain predictions for the critical behavior of the chiral
O(N) universality classes for arbitrary
N and fermion flavor number
Nf. Implications for critical Weyl and Dirac systems in 3+1 dimensions are also briefly discussed.
First-order phase transitions in many-fermion systems are not detected\n in the susceptibility analysis of common renormalization-group (RG)\n approaches. Here, we introduce a counterterm technique within the\n functional renormalization-group (fRG) formalism which allows access to\n all stable and metastable configurations. It becomes possible to study\n symmetry-broken states which occur through first-order transitions as\n well as hysteresis phenomena. For continuous transitions, the standard\n results are reproduced. As an example, we study discrete-symmetry\n breaking in a mean-field model for a commensurate charge-density wave.\n An additional benefit of the approach is that away from the critical\n temperature for the breaking of discrete symmetries large interactions\n can be avoided at all RG scales.