Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1605.07420 · Physical review. E

Solution to sign problems in models of interacting fermions and quantum spins

Emilie Huffman, Shailesh Chandrasekharan A

Published Oct 19, 2016• 17 citations
arXivHTMLPDF

Abstract

We show that solutions to fermion sign problems that are found in the formulation where the path integral is expanded in powers of the interaction in continuous time can be extended to systems involving fermions interacting with dynamical quantum spins. While these sign problems seem unsolvable in the auxiliary field approach, solutions emerge in the world-line representation of quantum spins. Combining this idea with meron-cluster methods, we are able to further extend the class of models that are solvable. We demonstrate these solutions to sign problems by considering several examples of strongly correlated systems that contain the physics of semimetals, insulators, superfluidity, and antiferromagnetism.

DOI
10.1103/physreve.94.043311

Related papers

Meron-Cluster Solution of Fermion Sign Problems

Shailesh Chandrasekharan A, U.‐J. Wiese

We present a general strategy to solve the notorious fermion sign problem using cluster algorithms. The method applies to various systems in the Hubbard model family as well as to relativistic fermions. Here it is illustrated for nonrelativistic lattice fermions. A configuration of fermion world lines is decomposed into clusters that contribute independently to the fermion permutation sign. A cluster whose flip changes the sign is referred to as a meron. Configurations containing meron clusters contribute $0$ to the path integral, while all other configurations contribute $1$. The cluster representation describes the partition function as a gas of clusters in the zero-meron sector.

arXiv:cond-mat/9902128•Physical Review Letters•Oct 18, 1999•223 citations

Fermion Bag Approach to Fermion Sign Problems

Shailesh Chandrasekharan A

The fermion bag approach is a new method to tackle fermion sign problems in lattice field theories. Using this approach it is possible to solve a class of sign problems that seem unsolvable by traditional methods. The new solutions emerge when partition functions are written in terms of fermion bags and bosonic worldlines. In these new variables it is possible to identify hidden pairing mechanisms which lead to the solutions. The new solutions allow us for the first time to use Monte Carlo methods to solve a variety of interesting lattice field theories, thus creating new opportunities for understanding strongly correlated fermion systems.

arXiv:1304.4900•Apr 17, 2013•5 citations

Pacifying the Fermi-liquid: battling the devious fermion signs

Jan Zaanen, Frank Krüger, Jian-Huang She +2 more

The fermion sign problem is studied in the path integral formalism. The standard picture of Fermi liquids is first critically analyzed, pointing out some of its rather peculiar properties. The insightful work of Ceperley in constructing fermionic path integrals in terms of constrained world-lines is then reviewed. In this representation, the minus signs associated with Fermi-Dirac statistics are self consistently translated into a geometrical constraint structure (the {\em nodal hypersurface}) acting on an effective bosonic dynamics. As an illustrative example we use this formalism to study 1+1-dimensional systems, where statistics are irrelevant, and hence the sign problem can be circumvented. In this low-dimensional example, the structure of the nodal constraints leads to a lucid picture of the entropic interaction essential to one-dimensional physics. Working with the path integral in momentum space, we then show that the Fermi gas can be understood by analogy to a Mott insulator in a harmonic trap. Going back to real space, we discuss the topological properties of the nodal cells, and suggest a new holographic conjecture relating Fermi liquids in higher dimensions to soft-core bosons in one dimension. We also discuss some possible connections between mixed Bose/Fermi systems and supersymmetry.

arXiv:0802.2455•Iranian Journal of Physics Research•Feb 18, 2008•6 citations

Meron-Cluster Simulation of Quantum Spin Ladders in a Magnetic Field

Shailesh Chandrasekharan A, B. Scarlet, U.‐J. Wiese

Numerical simulations of numerous quantum systems suffer from the notorious sign problem. Meron-cluster algorithms lead to an efficient solution of sign problems for both fermionic and bosonic models. Here we apply the meron concept to quantum spin systems in an arbitrary external magnetic field, in which case standard cluster algorithms fail. As an example, we simulate antiferromagnetic quantum spin ladders in a uniform external magnetic field that competes with the spin-spin interaction. The numerical results are in agreement with analytic predictions for the magnetization as a function of the external field.

arXiv:cond-mat/9909451•Sep 30, 1999•2 citations

Sign problem in Monte Carlo simulations of frustrated quantum spin systems

Patrik Henelius, Anders W. Sandvik

We discuss the sign problem arising in Monte Carlo simulations of frustrated quantum spin systems. We show that for a class of ``semifrustrated'' systems [Heisenberg models with ferromagnetic couplings ${J}_{z}(r)<0$ along the z axis and antiferromagnetic couplings ${J}_{\mathrm{xy}}(r)=\ensuremath{-}{J}_{z}(r)$ in the $\mathrm{xy}$ plane, for arbitrary distances $r],$ the sign problem present for algorithms operating in the z basis can be solved within a recent ``operator-loop'' formulation of the stochastic series expansion method [a cluster algorithm for sampling the diagonal matrix elements of the power series expansion of $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)$ to all orders]. The solution relies on the identification of operator loops which change the configuration sign when updated (``merons'') and is similar to the meron-cluster algorithm recently proposed by Chandrasekharan and Wiese for solving the sign problem for a class of fermion models [Phys. Rev. Lett. 83, 3116 (1999)]. Some important expectation values, e.g., the internal energy, can be evaluated in the subspace with no merons, where the weight function is positive definite. Calculations of other expectation values require sampling of configurations with only a small number of merons (typically zero or two), with an accompanying sign problem which is not serious. We also discuss problems which arise in applying the meron concept to more general quantum spin models with frustrated interactions.

arXiv:cond-mat/0001351•Physical review. B, Condensed matter•Jul 1, 2000•134 citations

Solution to sign problems in half-filled spin-polarized electronic systems

Emilie Huffman, Shailesh Chandrasekharan A

We solve the sign problem in a particle-hole symmetric spin-polarized fermion model on bipartite lattices using the idea of fermion bags. The solution can be extended to a class of models at half filling but without particle-hole symmetry. Attractive Hubbard models with an odd number of fermion species can also be solved. Our solutions should allow us to study quantum phase transitions that have remained unexplored so far due to sign problems.

arXiv:1311.0034•Physical Review B•Mar 12, 2014•84 citations
We show that solutions to fermion sign problems that are found in the formulation where the path integral is expanded in powers of the interaction in continuous time can be extended to systems involving fermions interacting with dynamical quantum spins. While these sign problems seem unsolvable in the auxiliary field approach, solutions emerge in the world-line representation of quantum spins. Combining this idea with meron-cluster methods, we are able to further extend the class of models that are solvable. We demonstrate these solutions to sign problems by considering several examples of strongly correlated systems that contain the physics of semimetals, insulators, superfluidity, and antiferromagnetism.
We present a general strategy to solve the notorious fermion sign problem using cluster algorithms. The method applies to various systems in the Hubbard model family as well as to relativistic fermions. Here it is illustrated for nonrelativistic lattice fermions. A configuration of fermion world lines is decomposed into clusters that contribute independently to the fermion permutation sign. A cluster whose flip changes the sign is referred to as a meron. Configurations containing meron clusters contribute 000 to the path integral, while all other configurations contribute 111. The cluster representation describes the partition function as a gas of clusters in the zero-meron sector.
The fermion bag approach is a new method to tackle fermion sign problems in lattice field theories. Using this approach it is possible to solve a class of sign problems that seem unsolvable by traditional methods. The new solutions emerge when partition functions are written in terms of fermion bags and bosonic worldlines. In these new variables it is possible to identify hidden pairing mechanisms which lead to the solutions. The new solutions allow us for the first time to use Monte Carlo methods to solve a variety of interesting lattice field theories, thus creating new opportunities for understanding strongly correlated fermion systems.
The fermion sign problem is studied in the path integral formalism. The standard picture of Fermi liquids is first critically analyzed, pointing out some of its rather peculiar properties. The insightful work of Ceperley in constructing fermionic path integrals in terms of constrained world-lines is then reviewed. In this representation, the minus signs associated with Fermi-Dirac statistics are self consistently translated into a geometrical constraint structure (the {\em nodal hypersurface}) acting on an effective bosonic dynamics. As an illustrative example we use this formalism to study 1+1-dimensional systems, where statistics are irrelevant, and hence the sign problem can be circumvented. In this low-dimensional example, the structure of the nodal constraints leads to a lucid picture of the entropic interaction essential to one-dimensional physics. Working with the path integral in momentum space, we then show that the Fermi gas can be understood by analogy to a Mott insulator in a harmonic trap. Going back to real space, we discuss the topological properties of the nodal cells, and suggest a new holographic conjecture relating Fermi liquids in higher dimensions to soft-core bosons in one dimension. We also discuss some possible connections between mixed Bose/Fermi systems and supersymmetry.
Numerical simulations of numerous quantum systems suffer from the notorious sign problem. Meron-cluster algorithms lead to an efficient solution of sign problems for both fermionic and bosonic models. Here we apply the meron concept to quantum spin systems in an arbitrary external magnetic field, in which case standard cluster algorithms fail. As an example, we simulate antiferromagnetic quantum spin ladders in a uniform external magnetic field that competes with the spin-spin interaction. The numerical results are in agreement with analytic predictions for the magnetization as a function of the external field.
We discuss the sign problem arising in Monte Carlo simulations of frustrated quantum spin systems. We show that for a class of ``semifrustrated'' systems [Heisenberg models with ferromagnetic couplings Jz(r)<0{J}_{z}(r)<0Jz​(r)<0 along the z axis and antiferromagnetic couplings ${J}_{\mathrm{xy}}(r)=\ensuremath{-}{J}_{z}(r)$ in the xy\mathrm{xy}xy plane, for arbitrary distances r],r],r], the sign problem present for algorithms operating in the z basis can be solved within a recent ``operator-loop'' formulation of the stochastic series expansion method [a cluster algorithm for sampling the diagonal matrix elements of the power series expansion of $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}H)$ to all orders]. The solution relies on the identification of operator loops which change the configuration sign when updated (``merons'') and is similar to the meron-cluster algorithm recently proposed by Chandrasekharan and Wiese for solving the sign problem for a class of fermion models [Phys. Rev. Lett. 83, 3116 (1999)]. Some important expectation values, e.g., the internal energy, can be evaluated in the subspace with no merons, where the weight function is positive definite. Calculations of other expectation values require sampling of configurations with only a small number of merons (typically zero or two), with an accompanying sign problem which is not serious. We also discuss problems which arise in applying the meron concept to more general quantum spin models with frustrated interactions.