Solving the fermion sign problem in quantum Monte Carlo simulations by Majorana representation

Abstract

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

Majorana-Hubbard model on the square lattice

“…Unlike the model of Refs. [32,33], our model does not appear to be amenable to Quantum Monte Carlo simulations. As in the 1D works [22,23], we study the model for both signs of the coupling constant.…”

Simple Fermionic Model of Deconfined Phases and Phase Transitions

“…(11), the Oð2NÞ symmetry of our model is manifest. As a consequence, sign-free quantum Monte Carlo (QMC) simulations can be carried out for all values of N [42,43]. We have used a standard implementation of the finite-temperature auxiliary-field approach [15,16,44] to evaluate…”

Split Orthogonal Group: A Guiding Principle for Sign-Problem-Free Fermionic Simulations

“…In particular, we choose the following auxiliary field decomposition of the interaction term to reveal the connections of various solutions to the sign problem [28,29],…”

“…(5), e.g., attractive interaction between sites belonging to the same sublattice [28,29]. We decouple the interactions as…”