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arXiv:1910.14287 · Physical review. B./Physical review. B

Fermion-induced quantum critical point in Dirac semimetals: A sign-problem-free quantum Monte Carlo study

Bo-Hai Li, Zixiang Li, Hong Yao

Published Feb 5, 2020• 13 citations
arXivHTMLPDF

Abstract

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 ${Z}_{3}$ 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 $1/N$ expansion with $N$ 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 $\mathrm{SU}(N$) 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 ${N}_{c}$ has not been established. Especially, the case of $N=1$ has not been explored by studying microscopic models so far. Here, by introducing a generalized $\mathrm{SU}(N$) fermion model with $N=1$ (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 $N=1$. Consequently, our results suggest that the FIQCP can occur in 2D Dirac semimetals for all positive integers $N\ensuremath{\ge}1$.

DOI
10.1103/physrevb.101.085105

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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≥NcN\ensuremath{\ge}{N}_{c}N\ensuremath≥Nc​. 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≥2N\ensuremath{\ge}2N\ensuremath≥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≥1N\ensuremath{\ge}1N\ensuremath≥1.
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.
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.
Fluctuations can change the phase transition properties drastically. An example is the fermion-induced quantum critical point (FIQCP), in which fluctuations of the massless Dirac fermions turn a putative Landau-de Gennes first-order phase transition (FOPT) with a cubic boson interaction into a continuous one. However, for the Landau-Devonshire theory, which characterizes another very large class of FOPTs, its fate under the coupling with extra fluctuations has not been explored. Here, we discover a new type of FIQCP, in which the Dirac fermion fluctuations round the boson Landau-Devonshire FOPT into a continuous phase transition. By using the functional renormalization group analyses, we determine the condition for the appearance of this FIQCP. Moreover, we point out that the present FIQCP can be a supersymmetric critical point. We finally show that the low-temperature phase diagram can provide distinct experimental evidences to detect this FIQCP.
Fluctuations can change the phase transition properties drastically. An example is the fermion-induced quantum critical point (FIQCP), in which fluctuations of the massless Dirac fermions turn a putative Landau--de Gennes first-order phase transition (FOPT) with a cubic boson interaction into a continuous one. However, for the Landau-Devonshire theory, which characterizes another very large class of FOPTs, its fate under the coupling with extra fluctuations has not been explored. Here, we discover another type of FIQCP, in which the Dirac fermion fluctuations round the boson Landau-Devonshire FOPT into a continuous phase transition. By using the functional renormalization group analyses, we determine the condition for the appearance of this FIQCP. Moreover, we point out that the present FIQCP can be a supersymmetric critical point. We finally show that the low-temperature phase diagram can provide distinct experimental evidences to detect this FIQCP.
Spinless fermions on a honeycomb lattice provide a minimal realization of lattice Dirac fermions. Repulsive interactions between nearest neighbors drive a quantum phase transition from a Dirac semimetal to a charge-density-wave state through a fermionic quantum critical point, where the coupling of the Ising order parameter to the Dirac fermions at low energy drastically affects the quantum critical behavior. Encouraged by a recent discovery (Huffman and Chandrasekharan 2014 Phys. Rev. B 89 111101 ) of the absence of the fermion sign problem in this model, we study the fermionic quantum critical point using the continuous-time quantum Monte Carlo method with a worm-sampling technique. We estimate the transition point with the critical exponents and . Compatible results for the transition point are also obtained with infinite projected entangled-pair states.
Quantum critical phenomena may be qualitatively different when massless Dirac fermions are present at criticality. Using our recently-discovered fermion-sign-free Majorana quantum Monte Carlo method introduced by us in (Li et al 2015 Phys. Rev. B 91 241117 ), we investigate the quantum critical phenomena of spinless Dirac fermions at their charge-density-wave phase transitions on the honeycomb lattice having sites with largest L = 24. By finite-size scaling, we accurately obtain critical exponents of this so-called Gross–Neveu chiral-Ising universality class of two (two-component) Dirac fermions in 2+1D: , , and , which are qualitatively different from the mean-field results but are reasonably close to the ones obtained from renormalization group calculations.