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arXiv:1501.00986 · Physical Review B

Efficient continuous-time quantum Monte Carlo method for the ground state of correlated fermions

Lei Wang, Mauro Iazzi, Philippe Corboz, Matthias Troyer

Published Jun 30, 2015• 33 citations
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Abstract

While efficient algorithms exist for the simulation of bosons simulations of fermions are more challenging because of the infamous fermion sign problem. In this paper the authors take a step toward solution and develop continuous-time quantum Monte Carlo algorithm for lattice fermions that scales linearly with projection time and interaction strength. This method is benchmarked on the problem of the fermionic quantum critical point of spinless fermions on a honeycomb lattice.

DOI
10.1103/physrevb.91.235151

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While efficient algorithms exist for the simulation of bosons simulations of fermions are more challenging because of the infamous fermion sign problem. In this paper the authors take a step toward solution and develop continuous-time quantum Monte Carlo algorithm for lattice fermions that scales linearly with projection time and interaction strength. This method is benchmarked on the problem of the fermionic quantum critical point of spinless fermions on a honeycomb lattice.
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 $O({\ensuremath{\beta}}^{3})$ with inverse temperature $\ensuremath{\beta}$, compared to the linear scaling of discrete-time algorithms. Here we present a CT-QMC algorithm 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.
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.
We extend the continuous-time interaction-expansion quantum Monte Carlo method with respect to measuring observables for fermion-boson lattice models. Using generating functionals, we express expectation values involving boson operators, which are not directly accessible because simulations are done in terms of a purely fermionic action, as integrals over fermionic correlation functions. We also demonstrate that certain observables can be inferred directly from the vertex distribution, and present efficient estimators for the total energy and the phonon propagator of the Holstein model. Furthermore, we generalize the covariance estimator of the fidelity susceptibility, an unbiased diagnostic for phase transitions, to the case of retarded interactions. The new estimators are applied to half-filled spinless and spinful Holstein models in one dimension. The observed renormalization of the phonon mode across the Peierls transition in the spinless model suggests a soft-mode transition in the adiabatic regime. The critical point is associated with a minimum in the phonon kinetic energy and a maximum in the fidelity susceptibility.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.
Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem" when applied to fermions--causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show that such a solution is almost certainly unattainable by proving that the sign problem is nondeterministic polynomial (NP) hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP in polynomial time.