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arXiv:1705.08973 · Physical review. B./Physical review. B

Fluctuation-induced continuous transition and quantum criticality in Dirac semimetals

Laura Classen, Igor F. Herbut, Michael M. Scherer

Published Sep 20, 2017• 60 citations
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Abstract

Thermal phase transitions are successfully described by fluctuations of an appropriate order parameter, which further allows classification of the transition into first or second order. However, some quantum analogs of thermal phase transitions defy this description and identifying the mechanisms behind these unconventional processes will fundamentally improve our understanding of matter. Here, the authors provide a thorough study of such a transition, which has been experimentally accessed in graphene: the emergence of a Kekul\'e valence bond solid. Additional gapless quantum fluctuations of Dirac electrons change the nature of this transition from first to second order. Using a nonperturbative functional renormalization group approach, the authors advance the state-of-the-art estimates for the corresponding critical exponents and predict measurable corrections to scaling.

DOI
10.1103/physrevb.96.115132

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Thermal phase transitions are successfully described by fluctuations of an appropriate order parameter, which further allows classification of the transition into first or second order. However, some quantum analogs of thermal phase transitions defy this description and identifying the mechanisms behind these unconventional processes will fundamentally improve our understanding of matter. Here, the authors provide a thorough study of such a transition, which has been experimentally accessed in graphene: the emergence of a Kekul'e valence bond solid. Additional gapless quantum fluctuations of Dirac electrons change the nature of this transition from first to second order. Using a nonperturbative functional renormalization group approach, the authors advance the state-of-the-art estimates for the corresponding critical exponents and predict measurable corrections to scaling.
The fluctuations-driven continuous quantum criticality has sparked tremendous interest in condensed matter physics. It has been verified that the fluctuations of gapless fermions can change the nature of phase transition at criticality. In this paper, we study the fermionic quantum criticality with enlarged Ising×\times×Ising fluctuations in honeycomb lattice materials. The perturbative renormalization group approach is employed to investigated the flow equations and stable fixed-point structure. By including cubic terms of order parameter, we found that the phase transition is always continuous for any finite value of the number of Dirac fermions NfN_fNf​. Further, we also compute the critical exponents and predict the critical scaling behavior for Nf=2N_f=2Nf​=2 case which corresponds to spin-1/2 fermions on graphene-like materials.
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.
In this paper we investigate the nature of quantum phase transitions between two-dimensional Dirac semimetals and Z3{Z}_{3}Z3​-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-NNN 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 NNN (the number of flavors of four-component Dirac fermions) is larger than a critical value Nc{N}_{c}Nc​. Remarkably, from the knowledge of space-time supersymmetry, we obtain an exact lower bound for Nc{N}_{c}Nc​, i.e., Nc>1/2{N}_{c}>1/2Nc​>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-NNN RG calculations. We further give a brief discussion of possible experimental realizations of FIQCPs.
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.
According to the Landau criterion, a phase transition should be first order when cubic terms of order parameters are allowed in its effective Ginzburg-Landau free energy. Recently, it was shown by renormalization-group (RG) analysis that continuous transition can happen at putatively first-order Z3{Z}_{3}Z3​ transitions in two-dimensional (2D) Dirac semimetals and such non-Landau phase transitions were dubbed ``fermion-induced quantum critical points'' (FIQCPs) [Z.-X. Li, Y.-F. Jiang, S.-K. Jian, and H. Yao, Nat. Commun. 8, 314 (2017)]. The RG analysis, controlled by the 1/N1/N1/N expansion with NNN the number of flavors of four-component Dirac fermions, shows that the FIQCP occurs for $N\ensuremath{\ge}{N}_{c}$. Previous quantum Monte Carlo simulations of a microscopic model of SU(N\mathrm{SU}(NSU(N) fermions on the honeycomb lattice showed that the FIQCP occurs at the transition between Dirac semimetals and Kekule valence bond solids for $N\ensuremath{\ge}2$. However, the precise value of the lower bound Nc{N}_{c}Nc​ has not been established. Especially, the case of N=1N=1N=1 has not been explored by studying microscopic models so far. Here, by introducing a generalized SU(N\mathrm{SU}(NSU(N) fermion model with N=1N=1N=1 (namely, spinless fermions on the honeycomb lattice), we perform large-scale sign-problem-free Majorana quantum Monte Carlo simulations and find convincing evidence of the FIQCP for N=1N=1N=1. Consequently, our results suggest that the FIQCP can occur in 2D Dirac semimetals for all positive integers $N\ensuremath{\ge}1$.
Near the critical point of a phase transition, physical observables are characterized by a set of pure numbers -- the critical exponents -- which are identical for a whole class of transitions defining a ``universality class''. Novel Dirac materials exhibit quantum critical points, which are not captured by conventional universality classes and their description defines new challenges for theory. The authors present here a comprehensive renormalization group study and give quantitative predictions for universality classes of Dirac fermions, as relevant to semimetal-insulator transitions in graphene and emergent supersymmetry in topological insulators. The authors also compare to independent conformal bootstrap results and quantum Monte Carlo estimates.