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arXiv:1610.07603 · Physical review. B./Physical review. B

Fermion-induced quantum critical points in two-dimensional Dirac semimetals

Shao-Kai Jian, Hong Yao

Published Nov 28, 2017• 35 citations
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Abstract

In this paper we investigate the nature of quantum phase transitions between two-dimensional Dirac semimetals and ${Z}_{3}$-ordered phases (e.g., Kekule valence-bond solid), where cubic terms of the order parameter are allowed in the quantum Landau-Ginzberg theory and the transitions are putatively first order. From large-$N$ renormalization-group (RG) analysis, we find that fermion-induced quantum critical points (FIQCPs) [Z.-X. Li et al., Nat. Commun. 8, 314 (2017)] occur when $N$ (the number of flavors of four-component Dirac fermions) is larger than a critical value ${N}_{c}$. Remarkably, from the knowledge of space-time supersymmetry, we obtain an exact lower bound for ${N}_{c}$, i.e., ${N}_{c}>1/2$. (Here the ``1/2'' flavor of four-component Dirac fermions is equivalent to one flavor of four-component Majorana fermions). Moreover, we show that the emergence of two length scales is a typical phenomenon of FIQCPs and obtain two different critical exponents, i.e., $\ensuremath{\nu}\ensuremath{\ne}{\ensuremath{\nu}}^{\ensuremath{'}}$, by large-$N$ RG calculations. We further give a brief discussion of possible experimental realizations of FIQCPs.

DOI
10.1103/physrevb.96.195162

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A unified theory of quantum critical points beyond the conventional Landau-Ginzburg-Wilson paradigm remains unknown. According to Landau cubic criterion, phase transitions should be first-order when cubic terms of order parameters are allowed by symmetry in the Landau-Ginzburg free energy. Here, from renormalization group analysis, we show that second-order quantum phase transitions can occur at such putatively first-order transitions in interacting two-dimensional Dirac semimetals. As such type of Landau-forbidden quantum critical points are induced by gapless fermions, we call them fermion-induced quantum critical points. We further introduce a microscopic model of SU(N) fermions on the honeycomb lattice featuring a transition between Dirac semimetals and Kekule valence bond solids. Remarkably, our large-scale sign-problem-free Majorana quantum Monte Carlo simulations show convincing evidences of a fermion-induced quantum critical points for N = 2, 3, 4, 5 and 6, consistent with the renormalization group analysis. We finally discuss possible experimental realizations of the fermion-induced quantum critical points in graphene and graphene-like materials.Quantum phase transitions are governed by Landau-Ginzburg theory and the exceptions are rare. Here, Li et al. propose a type of Landau-forbidden quantum critical points induced by gapless fermions in two-dimensional Dirac semimetals.
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In this paper we discuss the Neˊ\acute{e}eˊel and Kekuleˊ\acute{e}eˊ valence bond solids quantum criticality in graphene Dirac semimetal. Considering the quartic four-fermion interaction g(ψˉiΓijψj)2g(\barψ_iΓ_{ij}ψ_j)^2g(ψˉ​i​Γij​ψj​)2 that contains spin,valley, and sublattice degrees of freedom in the continuum field theory, we find the microscopic symmetry is spontaneously broken when the coupling ggg is greater than a critical value gcg_cgc​. The symmetry breaking gaps out the fermion and leads to semimetal-insulator transition. All possible quartic fermion-bilinear interactions give rise to the uniform critical coupling, which exhibits the multicritical point for various orders and the Landau-forbidden quantum critical point. We also investigate the typical critical point between Neˊ\acute{e}eˊel and Kekuleˊ\acute{e}eˊ valence bond solid transition when the symmetry is broken. The quantum criticality is captured by the Wess-Zumino-Witten term and there exist a mutual-duality for Neˊ\acute{e}eˊel-Kekuleˊ\acute{e}eˊ VBS order. We show the emergent spinon in the Neˊ\acute{e}eˊel-Kekuleˊ\acute{e}eˊ VBS transition , from which we conclude the phase transition is a deconfined quantum critical point. Additionally, the connection between the index theorem and zero energy mode bounded by the topological defect in the Kekuleˊ\acute{e}eˊ VBS phase is studied to reveal the Neˊ\acute{e}eˊel-Kekuleˊ\acute{e}eˊ VBS duality.
Quantum criticality is closely related to the existence of two phases with unrelated symmetry breaking. In this paper, we study Néel and Kekulé valence bond state (VBS) quantum criticality in Dirac semimetals with four-fermion interactions. Our results show that all possible dynamical masses yield the same critical coupling, which exhibits the phenomenon that all possible phases meet at a multicritical point (e.g., a tricritical point for the Néel, Kekulé-VBS and semimetallic phases). In terms of the well-established Wess–Zumino–Witten field theory, we investigate the typical criticality for the transition between Néel and Kekulé-VBS phases, and the compatible Néel–Kekulé-VBS mass matrices imply the existence of a non-Landau transition between the Néel and Kekulé-VBS phases. We show the existence of mutual duality in the defect-driven Néel–Kekulé-VBS transition near the non-Landau critical point and find that this mutual duality results from the presence of a mutual Chern–Simons term. We also study the mutual duality based on dual topological excitations.
We analyze emergent quantum multicriticality for strongly interacting, massless Dirac fermions in two spatial dimensions (d=2d=2d=2) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break O(S1)\text{O}({S}_{1})O(S1​) and O(S2)\text{O}({S}_{2})O(S2​) symmetries in the ordered phase. Performing a renormalization-group analysis by using the $\ensuremath{\epsilon}=(3\ensuremath{-}d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an enlarged O(S1+S2)\text{O}({S}_{1}+{S}_{2})O(S1​+S2​) chiral symmetry. Such a fixed point acts as an exotic quantum multicritical point (MCP), governing the continuous semimetal-insulator as well as insulator-insulator (for example, antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either O(S1)\text{O}({S}_{1})O(S1​) or O(S2)\text{O}({S}_{2})O(S2​) chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power-law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher-dimensional Dirac fermions is also outlined.