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.

See http://mathoverflow.net/questions/204460/how-to-prove-this-determinant-is-positive and https://terrytao.wordpress.com/2015/05/03/the-standard-branch-of-the-matrix-logarithm/ for discussions on mathematical aspect of the paper; v3 added footnotes [44, 59] and new supplemental material

References in corpus (17)

• Classification of topological insulators and superconductors in three spatial dimensions
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Continuous-time auxiliary field Monte Carlo for quantum impurity models
• Sign-problem-free quantum Monte Carlo of the onset of antiferromagnetism in metals
• Orbital ordering and frustration of $p$-band Mott-insulators
• Bridging lattice-scale physics and continuum field theory with quantum Monte Carlo simulations
• Fermionic Quantum Critical Point of Spinless Fermions on a Honeycomb Lattice
• Creating State-Dependent Lattices for Ultracold Fermions by Magnetic Gradient Modulation
• Orbital order in Mott insulators of spinless p-band fermions
• Fidelity susceptibility made simple: A unified quantum Monte Carlo approach
• Renyi Entanglement Entropy of Interacting Fermions Calculated Using Continuous-Time Quantum Monte Carlo Method
• Efficient continuous-time quantum Monte Carlo algorithm for fermionic lattice models
• Efficient Continuous-time Quantum Monte Carlo Method for the Ground State of Correlated Fermions
• p-wave Superfluidity by Spin-Nematic Fermi Surface Deformation
• Quantum Monte Carlo study of mass-imbalanced Hubbard models
• Marshall-positive SU($N$) quantum spin systems and classical loop models: A practical strategy to design sign-problem-free spin Hamiltonians
• The relationship between Hirsch-Fye and weak coupling diagrammatic Quantum Monte Carlo methods

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