In this paper, we discover a new quantum Monte Carlo (QMC) method to solve the fermion sign problem in interacting fermion models by employing Majorana representation of complex fermions. We call it "Majorana QMC" (MQMC). MQMC simulations can be performed efficiently both at finite and zero temperatures. Especially, MQMC is fermion sign free in simulating a class of spinless fermion models on bipartite lattices at half filling and with arbitrary range of (unfrustrated) interactions. Moreover, we find a class of SU (N ) fermionic models with odd N , which are sign-free in MQMC but whose sign problem cannot be in solved in other QMC methods such as continuous-time QMC. To the best of our knowledge, MQMC is the first auxiliary field QMC method to solve fermion sign problem in spinless (more generally, odd number of species) fermion models. We conjecture that MQMC could be applied to solve fermion sign problem in more generic fermionic models.Introduction: Interacting fermionic quantum systems with strong correlations and/or topological properties have attracted increasing attentions [1,2]. Nonetheless, in two and higher spatial dimensions, strongly interacting quantum systems are generically beyond the reach of analytical methods in the sense of solving those quantum models in an unbiased way. As an intrinsicallyunbiased numerical method, quantum Monte Carlo simulation plays a key role in understanding physics of strongly correlated many-body systems [3][4][5][6][7]. Unfortunately, in simulating fermionic many-body systems, QMC often encounters the notorious fermion minus-sign problem [8,9], which arises as a consequence of Fermi statistics [10]. Undoubtedly, generic solutions of fermion sign problems would lead to a great leap forward in understanding correlated electronic systems [9].Many QMC algorithms are based on converting an interacting fermion model into a problem of free fermions interacting with background auxiliary classical fields; the Boltzmann weight is the determinant of free fermion matrix which is a function of auxiliary fields and which can be positive, negative, or even complex. In such determinant QMC (DQMC), when the determinants are rendered to be positive definite, we say a solution to the fermion sign problem is found. For spinful electrons, conventional strategy of solving fermion sign problem is to find a symmetric treatment of both spin components of electrons such that the Boltzmann weight can be written as the product of two real determinants with the same sign and is then positive definite [11][12][13][14][15][16]. For spinless or spin-polarized fermion models, it is usually much more difficult to solve fermion sign problem because the Boltzmann weight contains only a single determinant and the usual strategy used for even species of fermions cannot be directly applied here.In this paper, based on Majorana representation of fermions, we propose a genuinely new auxiliary field QMC approach to solve fermion sign problem in spinless fermion models. We observe that each complex fermion

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 N L 2 s 2 = 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 (twocomponent) Dirac fermions in 2+1D: 0.45(2) η = , 0.77(3) ν = , and 0.60(3) β =, which are qualitatively different from the mean-field results but are reasonably close to the ones obtained from renormalization group calculations.

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.