Hubbard-Stratonovich transformation
Encyclopedia
The Hubbard–Stratonovich (HS) transformation is an exact mathematical
transformation invented by Russian physicist Ruslan L. Stratonovich
and popularized by British physicist John Hubbard
. It is used to convert a particle theory
into its respective field theory
by linearizing the density operator
in the many-body interaction term of the Hamiltonian
and introducing a scalar
auxiliary field. It is defined via the integral identity (Baeurle 2002, Baeurle 2003)
where the real constant
. The basic idea of the HS transformation is to reformulate a system of particles interacting through two-body potentials into a system of independent particles interacting with a fluctuating field. The procedure is widely used in polymer physics
(Baeurle 2007, Schmid 1998, Matsen 2002, Fredrickson 2002), classical particle physics (Baeurle 2002a, Baeurle 2004), spin glass theory
, and electronic structure theory (Rom 1997, Baer 1998).
approximation techniques, like the mean field approximation (Matsen 2002, Fredrickson 2002) or beyond mean field approximation procedures (Baeurle 2007, Baeurle 2006, Baeurle 2007a). A major difficulty arising in the simulation with such field theories is their highly oscillatory nature in case of strong interactions, which leads to the well-known numerical sign problem
(Baeurle 2002, Baeurle 2003). The problem originates from the repulsive part of the interaction potential, which implicates the introduction of the complex factor via the HS transformation. Several analytical and numerical techniques have been developed recently to
alleviate the sign problem in Monte Carlo simulation in an efficient
way (Baeurle 2002, Baeurle 2002a, Baeurle 2003a).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
transformation invented by Russian physicist Ruslan L. Stratonovich
Ruslan L. Stratonovich
Ruslan Leont'evich Stratonovich was an outstanding physicist, engineer, and probabilist. Professor Stratonovich was born on May 31, 1930 in Moscow, Russia...
and popularized by British physicist John Hubbard
John Hubbard (physicist)
John Hubbard was a British physicist, best known for the Hubbard model for interacting electrons, the Hubbard-Stratonovich transformation, and the Hubbard approximations. He graduated from Imperial College London, receiving a B.Sc. and a Ph.D...
. It is used to convert a particle theory
Particle physics
Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
into its respective field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
by linearizing the density operator
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
in the many-body interaction term of the Hamiltonian
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...
and introducing a scalar
Scalar
Scalar may refer to:*Scalar , a quantity used to multiply vectors in the context of vector spaces*Scalar , a quantity which is independent of specific classes of coordinate systems...
auxiliary field. It is defined via the integral identity (Baeurle 2002, Baeurle 2003)
where the real constant
. The basic idea of the HS transformation is to reformulate a system of particles interacting through two-body potentials into a system of independent particles interacting with a fluctuating field. The procedure is widely used in polymer physicsPolymer physics
Polymer physics is the field of physics that studies polymers, their fluctuations, mechanical properties, as well as the kinetics of reactions involving degradation and polymerisation of polymers and monomers respectively....
(Baeurle 2007, Schmid 1998, Matsen 2002, Fredrickson 2002), classical particle physics (Baeurle 2002a, Baeurle 2004), spin glass theory
Spin glass
A spin glass is a magnet with frustrated interactions, augmented by stochastic disorder, where usually ferromagnetic and antiferromagnetic bonds are randomly distributed...
, and electronic structure theory (Rom 1997, Baer 1998).
Calculation of resulting field theories
The resulting field theories are well-suited for the application of effectiveapproximation techniques, like the mean field approximation (Matsen 2002, Fredrickson 2002) or beyond mean field approximation procedures (Baeurle 2007, Baeurle 2006, Baeurle 2007a). A major difficulty arising in the simulation with such field theories is their highly oscillatory nature in case of strong interactions, which leads to the well-known numerical sign problem
Numerical sign problem
The numerical sign problem refers to the difficulty of numerically evaluating the integral of a highly oscillatory function of a large number of variables. Numerical methods fail because of the near-cancellation of the positive and negative contributions to the integral...
(Baeurle 2002, Baeurle 2003). The problem originates from the repulsive part of the interaction potential, which implicates the introduction of the complex factor via the HS transformation. Several analytical and numerical techniques have been developed recently to
alleviate the sign problem in Monte Carlo simulation in an efficient
way (Baeurle 2002, Baeurle 2002a, Baeurle 2003a).