5 QCD on the Lattice
167
For each step in the Markov chain, the conjugate momenta are drawn randomly from
a Gaussian distribution (“momentum refreshment”). The Hamiltonian H [U, ,]
governs the dynamics of the variables U μ (x) and μ (x) with respect to “simulation
time” τ , which parameterizes the evolution of U μ (x) and μ (x) as the simulation
algorithm progresses. The evolution is described by Hamilton’s equations, which
read
d
dτ
U μ (x) = μ (x)U μ (x),
d
dτ
μ (x) = −F G,μ (x) − F F,μ (x),
(5.86)
where
F G,μ (x) =
∂S G [U ]
∂U μ (x)
,
F F,μ (x) =
∂
∂U μ (x)
x∈ E
φ
∗ (x)
(D
†
lat D lat )
−1 φ
(x)
(5.87)
are the forces associated with the gluon and quark fields, respectively. The algorithm
then proceeds by integrating the equations of motion numerically. As in any
numerical integration scheme, the total time interval is divided into a number of
sub-intervals of finite length τ , which is called the step size. Starting from an
initial gauge configuration {U μ (x)} and a set of conjugate momenta { μ (x)}, one
obtains new sets {U
μ (x)}, {
μ (x)} after the integration. In the language of classical
mechanics, the variables U μ (x) and μ (x) evolve along a trajectory in phase
space which connects the initial and final configurations. However, since numerical
integration is not exact, owing to the finite step size, the energy is not conserved.
In the HMC algorithm this is rectified by introducing a global accept/reject step: if
H denotes the energy difference between the initial and final configurations, i.e.
≡ H [U
, ,
] − H [U, ,],
(5.88)
then the new configuration {U
μ (x)} is accepted with probability 7
P {U → U
} = min(1, e
−H ).
(5.89)
In other words, a configuration {U
μ (x)} associated with a large value for the energy
violation H is less likely to be accepted. This final step completes the Monte
Carlo update. The name “Hybrid Monte Carlo” reflects the fact that one combines
a deterministic classical dynamics procedure with a pseudo-random accept/reject
step.
One major problem which has plagued simulations with dynamical quarks over
many years is the fact that the efficiency of the conventional HMC algorithm
7 In practice this is achieved by drawing a random number r, with 0 < r ≤ 1. If r < e −H the new
configuration is rejected.
167
For each step in the Markov chain, the conjugate momenta are drawn randomly from
a Gaussian distribution (“momentum refreshment”). The Hamiltonian H [U, ,]
governs the dynamics of the variables U μ (x) and μ (x) with respect to “simulation
time” τ , which parameterizes the evolution of U μ (x) and μ (x) as the simulation
algorithm progresses. The evolution is described by Hamilton’s equations, which
read
d
dτ
U μ (x) = μ (x)U μ (x),
d
dτ
μ (x) = −F G,μ (x) − F F,μ (x),
(5.86)
where
F G,μ (x) =
∂S G [U ]
∂U μ (x)
,
F F,μ (x) =
∂
∂U μ (x)
x∈ E
φ
∗ (x)
(D
†
lat D lat )
−1 φ
(x)
(5.87)
are the forces associated with the gluon and quark fields, respectively. The algorithm
then proceeds by integrating the equations of motion numerically. As in any
numerical integration scheme, the total time interval is divided into a number of
sub-intervals of finite length τ , which is called the step size. Starting from an
initial gauge configuration {U μ (x)} and a set of conjugate momenta { μ (x)}, one
obtains new sets {U
μ (x)}, {
μ (x)} after the integration. In the language of classical
mechanics, the variables U μ (x) and μ (x) evolve along a trajectory in phase
space which connects the initial and final configurations. However, since numerical
integration is not exact, owing to the finite step size, the energy is not conserved.
In the HMC algorithm this is rectified by introducing a global accept/reject step: if
H denotes the energy difference between the initial and final configurations, i.e.
≡ H [U
, ,
] − H [U, ,],
(5.88)
then the new configuration {U
μ (x)} is accepted with probability 7
P {U → U
} = min(1, e
−H ).
(5.89)
In other words, a configuration {U
μ (x)} associated with a large value for the energy
violation H is less likely to be accepted. This final step completes the Monte
Carlo update. The name “Hybrid Monte Carlo” reflects the fact that one combines
a deterministic classical dynamics procedure with a pseudo-random accept/reject
step.
One major problem which has plagued simulations with dynamical quarks over
many years is the fact that the efficiency of the conventional HMC algorithm
7 In practice this is achieved by drawing a random number r, with 0 < r ≤ 1. If r < e −H the new
configuration is rejected.