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The Suzuki-trotter transformation is based on the following result: If is a set of operators that do not necessarily commute with each other, then
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(283) |
this equation is applicable to a quantum system with a partition function given by
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(284) |
qith a Hamiltonian and the inverse temperature. In general can be decomposed into a sum of terms () (a process that is not necessarily uniquely defined [,]), yielding
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(285) |
with
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(286) |
The trace and limit operations have beeen interchanged, and we have defined a step in imaginary time
(in the temperature axis, or Trotter direction). Introducing reslutions of the identity in some appropriate basis, the -th approximant to the partition function can be written as:
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(287) |
where the error is of order
. Next, introducing new resolutions of the identity for each time interval we obtain:
What distinguished one approximation from another is the partition of the Hamiltonian. One select the partition in oder to simplify the evaluation of the matrix elements, avoid sign problems, and effectively implement the conservation rules associated to . In general one uses a partition in real space in terms of local Hamiltonians
that are composed by sums of two-site terms tthat do not overlap. In systems of dimensions ( physical plus the Trotter/time direction) with short range interactions, the Hamiltonian can be written as a sum of interaction terms between nearest neighbors in the lattice:
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(289) |
In (1+1) dimensions, the most natural choice corresponds to the so-called "checkerboard" decomposition, that consiusts of dividing the lattice into two sublattices, one containing the even sites, and the other one, the odd sites. As a consequence, the Hamiltonian can be broken into two pieces and , each of them being the sum of two-site interactions between sites belonging to different sublattices:
We should notice that and are composed by terms that commute with eachother, hence:
As a result, the matrix elements can be obtained by simply solving a two-site problem, which has a very small number of degrees of freedom. In the case of the heisenberg model is is easy to see that:
For each time interval , the operator
is applied first, and then the operator
. That gives rise to a graphic representation as the one shown in Fig., where the horizontal axis represente the spatial direction along the sites of the chain, and the vertical axis represnts the imaginary timer , whcih has been subdivided into segments. The configuration of spins at time correspond to a state in the sum for . the shaded plaquettes correspond to slices of space and time in whcih the spins can interact and we shall call them "interacting plaquettes". The white plaquettes are called "non-interacting" plaquettes. The sum over the intermediate states correspond to summing over all the possible ways to distribute the spins in the spatial direction, for each interval of time. We have connected the -projection of the spin with lines, that are precisely called "world-lines". In the Heisenberg model it is enough with following the trajectory of one kind of spin only, since the others are qutomatically determined. In the case of fermionic systems, onbe assigns a different color to each spin orientation.
For a given interval of time , each matrix element for can be written as:
where the variables are of the ising type and can assume the values . this expression can be rewritten as:
using the equivalent expression for , we finally obtain
In this expression we have replaced the sum over configurations of quantum spins by a sum over states of a -dimensional system of Ising variables. The equation ([]) respresents tha partition function of a two-dimensional Ising system with a 4-spin interaction:
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(295) |
The six possible configurations of 4 spins allowed by the conservation rules are shown in Fig.[]. the sum over states that satisfy the conservation rules is equivalent to summing over all the possible configurations of allowed world lines. Notice that these can be drwan parallel to the lateral sides of a plaquette, or crossing the diagonal of the shaded plaquettes, but never accross the diagonal of a white plaquette.
This graphic representation gives a simple picture about a given configuration and also offers an idea of the ground state of the system. Let us consider the partition function as the trace over eigenstates
,
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(296) |
We can separate into
, and insert a complete set of state in the representation between the two exponentials. In the low temperature limit, and for each time slice we obtain
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(297) |
As a consequence, for each interval of time , the probability of finding a given set of spins
is proportional to the square of the projection of the ground-state on that configuration. This way, the world lines for large offer a picture of the ground state of the system.
Next: Monte Carlo simulation with
Up: World Line Monte Carlo
Previous: World Line Monte Carlo
Adrian E. Feiguin
2009-11-04