Computational Physics

412 Quantum Monte Carlo methods

Figure 12.7. The checkerboard decomposition of the space-time lattice. Two world

lines are shown.

the exponentials, where

∑

Sdenotes a sum over all the spinssiinS:

Z=

∑

Si,S ̄i

〈S 0 |e−

τHodd|S ̄ 0 〉〈S ̄ 0 |e−

τHeven|S 1 〉〈S 1 |e−

τHodd|S ̄ 1 〉

×〈S ̄ 1 |e−

τHeven|S 2 〉···〈SN/ 2 − 1 |e−

τHodd|S ̄N/ 2 − 1 〉

×〈S ̄N/ 2 − 1 |e−

τHeven|S 0 〉. (12.96)

The operators exp(
τHeven)and exp(
τHodd)can be expanded as products of
terms of the form exp(
τHi). Each such term couples the spins around a plaquette of
the space-time lattice and the resulting picture is that of Figure 12.7, which explains
the name ‘checkerboard decomposition’ for this partitioning of the Hamiltonian.
Other decompositions are possible, such as the real-space decomposition [5], but
we shall not go into this here.
The simulation of the system seems straightforward: we have a space-time lattice
with interactions around the shaded plaquettes in Figure 12.7. At each site of the
lattice we have a spinsim, whereidenotes the spatial index andmdenotes the index
along the imaginary-time or inverse-temperature axis. The simulation consists of
attempting spin flips, evaluating the Boltzmann weight before and after the change,
and then deciding to perform the change or not with a probability determined by
the fractions of the Boltzmann weights (before and after). But there is a snake in the
grass. The HamiltoniansHmcommute with the total spin operator,

∑

is

z
i; therefore
the latter is conserved, i.e.

sim+si+1,m=si,m+ 1 +si+1,m+ 1 (12.97)