Compatible orders and fermion-induced emergent symmetry in Dirac systems

Abstract

We study the quantum multicritical point in a (2+1)D Dirac system between the semimetallic phase and two ordered phases that are characterized by anticommuting mass terms with O(N1) and O(N2) symmetry, respectively. Using 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 N f , 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 N f . Implications for critical Weyl and Dirac systems in (3+1)D are also briefly discussed.Introduction. The interplay and competition of different ordering tendencies in many-body systems are the source of various exciting phenomena, including unconventional superconductivity, the nature of quantum spin liquids, and the physics of deconfined criticality. These notoriously challenging problems sometimes become theoretically accessible when an emergent higher symmetry can be found. The complex phase diagram of the high-T c superconductors, for instance, has been argued to be deducible from an emergent symmetry in which the O(3) Néel and U(1) superconducting order parameters are combined into a five-tuplet which turns out to be a vector under O(5).1 Numerical simulations of the deconfined critical point between the Néel and valence bond solid orders on the square lattice also find evidence for an emergent O(5) symmetry.2 The emergence of this symmetry can be made natural by postulating duality relations between the bosonic gauge theory describing the deconfined critical point and certain fermionic theories.3,4 Similarly, recent quantum Monte Carlo simulations of Dirac fermions in 2 + 1 dimensions find a direct and continuous transition between O(3) and Z 2 ordered phases with an emergent O(4) symmetry at criticality.

Fermion-bag inspired Hamiltonian lattice field theory for fermionic quantum criticality

“…Study of quantum critical points in these four-fermion models has reemerged as an exciting area of research [15], especially due to the recent discovery that many materials can be described by Dirac fermions in the low energy limit and such materials can have new phases and quantum critical points that separate them [16,17]. Massless fermions can even help induce new quantum critical points and multi-critical points that do not exist in purely bosonic models [18][19][20][21]. New analytical studies of the Gross-Neveu transitions using -expansions [22][23][24], large-N expansions [25,26], functional renormalization group techniques [27] and the bootstrap approach [28] have been performed recently.…”

Emergent symmetries and coexisting orders in Dirac fermion systems

“…30. The non-perturbative FRG can be performed directly in 2 + 1 dimensions and allows us to assess the multicritical behavior of the model more precisely than leading-order expansions [33][34][35]. We firmly establish the emergence of O(N 1 + N 2 ) symmetry at the multicritical point for all consistent values of N 1 and N 2 .…”

“…Further evidence of coexistence for N 1 = 3, N 2 = 1 is provided by our refined QMC analysis.Effective field theory. For the FRG analysis, we consider the low-energy effective Gross-Neveu-Yukawa (GNY) model with two OP fields [33][34][35][36][37][38][39][40], describing interacting spin-1/2 fermions on the honeycomb lattice in the vicinity of a multicritical point. The Euclidean Lagrangian is L = L F + L B , wherewhere the mass matrices β a φ , β b χ anticommute among each other as well as with the Hamiltonian.…”

“…(1) includes a symmetry-enlarged O(N 1 + N 2 )invariant subspace, when u 1 = u 2 = u 3 , m 2 1 = m 2 2 , and g 2 1 = g 2 2 . The leading-order -expansion analysis [33,34] studied the RG fixed point with the enlarged O(N ) symmetry, referred to as the isotropic fixed point (IFP). Within that approach, the IFP is found to be stable for all consistent values of N .…”

“…For the symmetry-enhanced case, ∆ = 0, the above classification is valid if the submanifold in coupling space determined by ∆ = 0 is closed under the RG flow. This is not generally the case in theories with massless fermions as can be shown using the -expansion [33]: the submanifold in theory space defined by u 1 = u 2 = u 3 =: u = 0 satisfies ∆ = 0, but its dependence on the logarithmic RG scale t is given by ∂ t ∆ = 2u(g 2 1 − g 2 2 ) 2 . Therefore, in the presence of Dirac fermions, the sign of ∆ may be subject to change.…”