In
lattice field theory
In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a space or spacetime that has been discretised onto a lattice.
Details
Although most lattice field theories are not exactly sol ...
, staggered fermions (also known as Kogut–Susskind fermions) are a
fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks and ...
discretization that reduces the number of
fermion doublers from sixteen to four. They are one of the fastest lattice fermions when it comes to simulations and they also possess some nice features such as a remnant
chiral symmetry
A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, ...
, making them very popular in
lattice QCD calculations. Staggered fermions were first formulated by
John Kogut and
Leonard Susskind
Leonard Susskind (; born June 16, 1940)his 60th birthday was celebrated with a special symposium at Stanford University.in Geoffrey West's introduction, he gives Suskind's current age as 74 and says his birthday was recent. is an American physicis ...
in 1975 and were later found to be equivalent to the discretized version of the
Dirac–Kähler fermion.
Constructing staggered fermions
Single-component basis
The naively discretized
Dirac action
Distributed Research using Advanced Computing (DiRAC) is an integrated supercomputing facility used for research in particle physics, astronomy and cosmology in the United Kingdom. DiRAC makes use of multi-core processors and provides a variety of ...
in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
time with lattice spacing
and
Dirac fields at every lattice point, indexed by
, takes the form
:
Staggered fermions are constructed from this by performing the staggered transformation into a new
basis of fields
defined by
:
Since
Dirac matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\mat ...
square to the identity, this position dependent transformation mixes the fermion
spin components in a way that repeats itself every two lattice spacings. Its effect is to
diagonalize the action in the spinor indices, meaning that the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
ends up splitting into four distinct parts, one for each Dirac spinor component. Denoting one of those components by
, which is
Grassmann variable
In mathematical physics, a Grassmann number, named after Hermann Grassmann (also called an anticommuting number or supernumber), is an element of the exterior algebra over the complex numbers. The special case of a 1-dimensional algebra is known as ...
with no spin structure, the other three components can be dropped, yielding the single-component staggered action
:
where
are unit vectors in the
direction and the staggered sign function is given by
. The staggered transformation is part of a larger class of transformations
satisfying
. Together with a necessary consistency condition on the plaquettes, all these transformations are equivalent to the staggered transformation. Due to fermion doubling, the original naive action described sixteen fermions, but having discarded three of the four copies this new action only describes only four.
Spin-taste basis
To explicitly show that the single-component staggered fermion action describes four Dirac fermions requires blocking the lattice into
hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, p ...
s and reinterpreting the Grassmann fields at the sixteen hypercube sites as the sixteen
degrees of freedom of the four fermions. In analogy to the usage of
flavour in
particle physics
Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, these four fermions are referred to be different tastes of fermions. The blocked lattice sites are indexed by
while for each of these the internal hypercube sites are indexed by
, whose vector components are either zero or one. In this notation the original lattice vector is written as
. The matrices
are used to define the spin-taste basis of staggered fermions
:
The taste index
runs over the four tastes while the spin index
runs over the four spin components. This change of basis turns the one-component action on the lattice with spacing
into the spin-taste action with an effective lattice spacing of
given by
:
Here
and
are shorthand for the symmetrically discretized
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
and
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, respectively. Meanwhile, the
tensor notation separates out the spin and taste matrices as
. Since the
kinetic
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory of gases, Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to i ...
and
mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
terms are diagonal in the taste indices, the action describes four degenerate
Dirac fermions
In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. The vast majority of fermions – perhaps all – fall under this category.
Description
In particle physics, all fermions in the standard model ...
. These interact together in what are known as taste mixing interactions through the second term, which is an
irrelevant dimension five operator that vanishes in the
continuum limit
In mathematical physics and mathematics, the continuum limit or scaling limit of a lattice model (physics), lattice model refers to its behaviour in the limit as the lattice spacing goes to zero. It is often useful to use lattice models to approxi ...
. This action is very similar to the action constructed using four Wilson fermions with the only difference being in the second term tensor structure, which for Wilson fermions is spin and taste diagonal
.
A key property of staggered fermions, not shared by some other lattice fermions such as Wilson fermions, is that they have a remnant chiral
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
in the
massless limit. The remnant symmetry is described in the spin-taste basis by
:
The presence of this remnant symmetry makes staggered fermions especially useful for certain applications since they can describe
spontaneous symmetry breaking
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the ...
and
anomalies. The symmetry also protects massless fermions from gaining a mass upon
renormalization
Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering v ...
.
Staggered fermions are
gauged in the one-component action by inserting link fields into the action to make it
gauge invariant
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie group ...
in the same way that this is done for the naive Dirac lattice action. This approach cannot be implemented in the spin-taste action directly. Instead the interacting single-component action must be used together with a modified spin-taste basis
where
Wilson lines are inserted between the different lattice points within the hypercube to ensure gauge invariance. The resulting action cannot be expressed in a closed form but can be expended out in powers of the lattice spacing, leading to the usual interacting Dirac action for four fermions, together with an infinite series of irrelevant fermion bilinear operators that vanish in the continuum limit.
Momentum-space staggered fermions
Staggered fermions can also be formulated in
momentum space
In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension.
Position space (also real space or coordinate space) is the set of all ''position vectors'' r in space, and h ...
by transform the single-component action into
Fourier space
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
and splitting up the
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice ...
into sixteen blocks. Shifting these to the origin yields sixteen copies of the single-component fermion whose momenta extend over half the Brillouin zone range
. These can be grouped into a
matrix which upon a
unitary transformation
In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
Formal definition
More precisely, ...
and a
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
rescaling, to ensure that the momenta again range over the full