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...

We present the ground state extension of the efficient quantum Monte Carlo algorithm for lattice fermions of arXiv:1411.0683. Based on continuous-time expansion of imaginary-time projection operator, the algorithm is free of systematic error and scales linearly with projection time and interaction strength. Compared to the conventional quantum Monte Carlo methods for lattice fermions, this approach has greater flexibility and is easier to combine with powerful machinery such as histogram reweighting and extended ensemble simulation techniques. We discuss the implementation of the continuous-time projection in detail using the spinless t−V model as an example and compare the numerical results with exact diagonalization, density-matrix-renormalization-group and infinite projected entangled-pair states calculations. Finally we use the method to study the fermionic quantum critical point of spinless fermions on a honeycomb lattice and confirm previous results concerning its critical exponents.

Efficient continuous time quantum Monte Carlo (CT-QMC) algorithms that do not suffer from time discretization errors have become the state-of-the-art for most discrete quantum models. They have not been widely used yet for fermionic quantum lattice models, such as the Hubbard model, nor other fermionic lattice systems due to a suboptimal scaling of O(β 3 ) with inverse temperature β, compared to the linear scaling of discrete time algorithms. Here we present a CT-QMC algorithms for fermionic lattice systems that matches the scaling of discrete-time methods but is more efficient and free of time discretization errors. This provides an efficient simulation scheme that is free from the systematic errors opening an avenue to more precise studies of large systems at low and zero temperature.

We study the anisotropic 3D Hubbard model with increased nearest-neighbor tunneling amplitudes along one direction using the dynamical cluster approximation and compare the results to a quantum simulation experiment of ultracold fermions in an optical lattice. We find that the short-range spin correlations are significantly enhanced in the direction with stronger tunneling amplitudes. Our results agree with the experimental observations and show that the experimental temperature is lower than the strong tunneling amplitude. We characterize the system by examining the spin correlations beyond neighboring sites and determine the distribution of density, entropy, and spin correlation in the trapped system. We furthermore investigate the dependence of the critical entropy at the Néel transition on anisotropy.

Monte Carlo simulations are a powerful tool for elucidating the properties of complex systems across many disciplines. Not requiring any a priori knowledge, they are particularly well suited for exploring new phenomena. However, when applied to fermionic quantum systems, quantum Monte Carlo (QMC) algorithms suffer from the so-called "negative sign problem", which causes the computational effort to grow exponentially with problem size. Here we demonstrate that the fermion sign problem originates in topological properties of the configurations. In particular, we show that in the widely used auxiliary field approaches the negative sign of a configuration is a geometric phase that is the imaginary time counterpart of the Aharonov-Anandan phase, and reduces to a Berry phase in the adiabatic limit. This provides an intriguing connection between QMC simulations and classification of topological states. Our results shed clarify the controversially debated origin of the sign problem in fermionic lattice models.Interesting phenomena can emerge from simple interactions in many-body systems, but analytic solutions are rare making numerical simulations essential for the investigation of their properties. The Monte Carlo method [1] has had great success due to its benign scaling with the system size. By randomly sampling representative configurations of the system, for example using the Metropolis algorithm [2], the properties of interacting many-body systems can be determined in an unbiased way up to statistical sampling errors that are typically small. In many cases the computational effort only increases polynomially, often linearly or with a small power of the system size.While Monte Carlo algorithms were originally developed for classical systems, Fermi early on suggested that they could be used to simulate quantum systems by using an imaginary time formulation of the Schrödinger equation, as reported in Ref. [1]. This equation describes classical particles performing a random walk in an external potential. In the infinite time limit the distribution of the particles converges to that of the ground state of the quantum system. Identifying finite imaginary times with the inverse temperature β = 1/k B T has led to the path-integral formulation of quantum mechanics, where the partition function Z = Tr exp(−βH) of the quantum system is mapped into a sum Z = c w c over the paths c with statistical weights w c . The stochastic sampling of these paths forms the basis of finite temperature quantum Monte Carlo (QMC) algorithms, which have been widely applied to simulate quantum lattice models [3][4][5][6], the electronic structure of materials [7,8], ultracold atoms [9, 10], nuclear matter [11], and lattice quantum chromodynamics [12].While in classical systems the Boltzmann weights are always positive, in QMC the weight of a path configuration can be negative due to particle statistics or gauge fields [13]. Negative weight configurations cancel contributions of positive ones, resulting in an exponential increase of statistical e...