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

Abstract

We present a guiding principle for designing fermionic Hamiltonians and quantum Monte Carlo (QMC) methods that are free from the infamous sign problem by exploiting the Lie groups and Lie algebras that appear naturally in the Monte Carlo weight of fermionic QMC simulations. Specifically, rigorous mathematical constraints on the determinants involving matrices that lie in the split orthogonal group provide a guideline for sign-free simulations of fermionic models on bipartite lattices. This guiding principle not only unifies the recent solutions of the sign problem based on the continuous-time quantum Monte Carlo methods and the Majorana representation, but also suggests new efficient algorithms to simulate physical systems that were previously prohibitive because of the sign problem.PACS numbers: 02.70. Ss, 71.10.Fd, 02.20.Tw One of the biggest challenges to classical simulation of quantum systems is the infamous fermion sign problem of quantum Monte Carlo (QMC) simulations. It appears when the weights of configurations in a QMC simulation may become negative and therefore cannot be directly interpreted as probabilities [1]. In the presence of a sign problem, the simulation effort typically grows exponentially with system size and inverse temperature.While the sign problem is nondeterministic polynomial (NP) hard [2], implying that there is little hope of finding a generic solution, this does not exclude ad hoc solutions to the sign problem for specific models. For example, one can sometimes exploit symmetries to design appropriate sign-problem-free QMC algorithms for a restricted class of models [3]. However, it is unclear how broad these classes are and it is in general hard to foresee whether a given physical model would have a sign problem in any QMC simulations. The situation is not dissimilar to the study of many intriguing problems in the NP complexity class, where a seemingly infeasible problem might turn out to have a polynomial-time solution surprisingly [4].A fruitful approach in pursuing such specific solutions is to design Hamiltonians that capture the right low energy physics and allow sign-problem-free QMC simulations at the same time, called "designer" Hamiltonians [5]. This naturally calls for design principles. For bosonic and quantum spin systems a valuable guiding principle is the Marshall sign rule [6,7] which ensures nonnegative weight for all configurations. The design of the sign-problem-free fermionic Hamiltonians is harder. The method of choice for fermionic QMC simulations are the determinantal QMC approaches, including traditional discrete-time [8] and new continuous-time approaches [9][10][11][12][13]. Both approaches map the original interacting system to free fermions with an imaginary-time dependent Hamiltonian. The partition function is then written as a weighted sum of matrix determinants after tracing out the fermions [8,9,12]:where f C is a c-number and H C (τ ) is an imaginary-time dependent single-particle Hamiltonian matrix (whose matrix elements denote hopping amplitude...

Thermal Ising transitions in the vicinity of two-dimensional quantum critical points

“…We also denote by g = V /t the dimensionless tuning parameter, and g c refers to its critical value. This model has been examined in several recent works, in particular since it was realized, that it can be studied by unbiased, sign problem-free QMC methods, based either on the fermion bag approach 18,21,29 or an appropriate decoupling of the interactions 20,23,24 , e.g., after expressing it in terms of Majorana fermions 19,22 . In the following, we use the continuous-time interaction expansion (CT-INT) approach presented in Refs.…”

“…18 and 21, also a QMC algorithm based on a Majorana formulation 19,22 (MQMC) and a projective continuous-time approach 20,23 (LCT-INT) have been applied to this model, yielding consistent findings. More recently, close connections among these algorithmic approaches have furthermore been identified [24][25][26] . From these recent QMC simulations, which concentrated on ground state properties, the value of the quantum critical interaction strength V c ≈ 1.355t in units of the hopping strength has been estimated, and approximate values of the critical exponent ν and the anomalous exponent η for the order parameter fluctuations have been obtained, which we review in more detail below.…”

Irreducible projective representations and their physical applications

“…Sign problems in Quantum Monte Carlo QMC simulations. The generalized Schur's lemma corresponding to projective Reps of anti-unitary groups can also be applied to search for models which are free of sign problems in quantum Monte Carlo (QMC) simulations [43,[63][64][65][66]. By introducing the HubbardStratonovich field ξ(x, τ ), interacting fermion models can be mapped to free fermion Hamiltonians.…”

Optimized contraction scheme for tensor-network states

“…The quantum Monte Carlo is a powerful method for studying interacting bosons, quantum spin models without frustrations, and some special interacting fermion models. However, it suffers from the so-called minussign problem [4,5] in dealing with interacting fermions or frustrated quantum spin models in which the error increases exponentially with the system size and with decreasing temperature. The DMRG is the most accurate method for studying one-dimensional systems, but in two or higher dimensions, the lattice size that can be reliably handled by the DMRG [6,7] is relatively small, limited by the entanglement area law [8] which implies that the number of states retained must increase exponentially with the boundary size.…”