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Suzuki-Trotter transformation and the equivalent classical system

The Suzuki-trotter transformation is based on the following result: If ${A_i}$ is a set of operators that do not necessarily commute with each other, then

\begin{displaymath}
e^{A_1+A_2+\cdots +A_p} = \lim_{m\rightarrow \infty} \left(e^{A_1/m}e^{A_2/m}\cdots e^{A_p/m} \right)^m
\end{displaymath} (283)

this equation is applicable to a quantum system with a partition function given by
\begin{displaymath}
Z={\mathrm Tr}(e^{-\beta H})
\end{displaymath} (284)

qith a Hamiltonian $H$ and $\beta=1/T$ the inverse temperature. In general $H$ can be decomposed into a sum of terms $H_i$ ($i=1,2,\cdots$) (a process that is not necessarily uniquely defined [,]), yielding
\begin{displaymath}
Z=\lim_{m\rightarrow \infty} Z^{(m)},
\end{displaymath} (285)

with
\begin{displaymath}
Z^{(m)}={\mathrm Tr}\left(e^{\Delta \tau H_1}e^{\Delta \tau H_2}\cdots e^{\Delta \tau H_p}\right)
\end{displaymath} (286)

The trace and limit operations have beeen interchanged, and we have defined a step in imaginary time $\Delta \tau = \beta / m$ (in the temperature axis, or Trotter direction). Introducing reslutions of the identity in some appropriate basis, the $m$-th approximant to the partition function can be written as:
\begin{displaymath}
Z^{(m)} = \left (\sum_{i_1,i_2,\cdots,i_p} \langle i_1\vert ...
...langle i_p\vert e^{\Delta \tau H_p}\vert i_1\rangle \right)^m,
\end{displaymath} (287)

where the error is of order $(\delta \tau0^2$. Next, introducing new resolutions of the identity for each time interval we obtain:
$\displaystyle Z^{(m)}$ $\textstyle =$ $\displaystyle \sum_{i_1,i_2,\cdots,i_{mp}} \langle i_1\vert e^{\Delta \tau H_1}...
...ert i_3 \rangle \cdots \langle i_p\vert e^{\Delta \tau H_p}\vert i_{p+1}\rangle$  
    $\displaystyle \langle i_{p+1}\vert e^{\Delta \tau H_1}\vert i_{p+2} \rangle \la...
...ngle \cdots \langle i_{2p}\vert e^{\Delta \tau H_p}\vert i_{2p+1}\rangle \cdots$  
    $\displaystyle \langle i_{(m-1)p+1}\vert e^{\Delta \tau H_1}\vert i_{(m-1)p+2} \...
...1)p+3} \rangle \cdots \langle i_{mp}\vert e^{\Delta \tau H_p}\vert i_{1}\rangle$ (288)

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 $H$. In general one uses a partition in real space in terms of local Hamiltonians $h_1,H_2,\cdots H_p$ that are composed by sums of two-site terms tthat do not overlap. In systems of $(d+1)$ dimensions ($d$ 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:

\begin{displaymath}
H=\sum_{i,\delta} H_{i,i+\delta}.
\end{displaymath} (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 $H_1$ and $H_2$, each of them being the sum of two-site interactions between sites belonging to different sublattices:
$\displaystyle H_1$ $\textstyle =$ $\displaystyle \sum_{i\,\, \mathrm{even}} H_{i,i+1}$  
$\displaystyle H_2$ $\textstyle =$ $\displaystyle \sum_{i\,\, \mathrm{odd}} H_{i,i+1}$  

We should notice that $H_1$ and $H_2$ are composed by terms that commute with eachother, hence:
$\displaystyle e^{-\Delta \tau H_1}$ $\textstyle =$ $\displaystyle \prod_{i\,\, \mathrm{even}} e^{-\Delta \tau H_{i,i+1}}$  
$\displaystyle e^{-\Delta \tau H_2}$ $\textstyle =$ $\displaystyle \prod_{i\,\, \mathrm{odd}} e^{-\Delta \tau H_{i,i+1}}$ (290)

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:

$\displaystyle e^{-\Delta \tau H_{i,i+1}} \vert--\rangle$ $\textstyle =$ $\displaystyle e^{-\Delta \tau J/4}\vert--\rangle$  
$\displaystyle e^{-\Delta \tau H_{i,i+1}} \vert++\rangle$ $\textstyle =$ $\displaystyle e^{-\Delta \tau J/4}\vert++\rangle$  
$\displaystyle e^{-\Delta \tau H_{i,i+1}} \vert+-\rangle$ $\textstyle =$ $\displaystyle \left[\cosh(\Delta \tau J/2)\vert+-\rangle + \sinh(\Delta \tau J/2)\vert-+\rangle\right]e^{\Delta \tau J/4}$  
$\displaystyle e^{-\Delta \tau H_{i,i+1}} \vert-+\rangle$ $\textstyle =$ $\displaystyle \left[\sinh(\Delta \tau J/2)\vert+-\rangle + \cosh(\Delta \tau J/2)\vert-+\rangle\right]e^{\Delta \tau J/4}$ (291)

For each time interval $\Delta \tau$, the operator $e^{\Delta \tau H_1}$ is applied first, and then the operator $e^{\Delta \tau H_2}$. 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 $\tau$, whcih has been subdivided into $2m$ segments. The configuration of spins at time $l \Delta \tau$ correspond to a state $\vert i_l\rangle$ in the sum for $Z^{(m)}$. 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 $\uparrow$-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 $l$, each matrix element for $H_1$ can be written as:

$\displaystyle \langle i_l\vert e^{-\Delta \tau H_1}\vert i_{l+1}\rangle$ $\textstyle =$ $\displaystyle \langle i_l\vert e^{-\Delta \tau \prod_{i\,\,{\mathrm even}} H_i{i,i+1}}\vert i_l\rangle$  
  $\textstyle =$ $\displaystyle \prod_{i \,\, {\mathrm even}} \langle i_l\vert e^{-\Delta \tau H_{i,i+1}}\vert i_l\rangle$  
  $\textstyle =$ $\displaystyle \prod_{i \,\, {\mathrm even}} \langle s_{i,l}s_{i+1,l}\vert e^{-\Delta \tau H_{i,i+1}}\vert s_{i,l+1}s_{i+1,l+1}\rangle$ (292)

where the variables $S_{i,l}$ are of the ising type and can assume the values $\pm 1/2$. this expression can be rewritten as:
$\displaystyle \langle i_l\vert e^{-\Delta \tau H_1}\vert i_{l+1}\rangle$ $\textstyle =$ $\displaystyle \exp \left\{\ln \left[\prod_{i \,\, {\mathrm even}} \langle s_{i,...
...ert e^{-\Delta \tau H_{i,i+1}}\vert s_{i,l+1}s_{i+1,l+1}\rangle \right]\right\}$  
  $\textstyle =$ $\displaystyle \exp \left\{ \sum_{i \,\,{\mathrm even}} \ln \langle s_{i,l}s_{i+1,l}\vert e^{-\Delta \tau H_{i,i+1}}\vert s_{i,l+1}s_{i+1,l+1}\rangle \right\}.$ (293)

using the equivalent expression for $H_2$, we finally obtain

$\displaystyle Z^{(m)}$ $\textstyle =$ $\displaystyle \sum_{\left\{i_1,i_2,\cdots,i_{2m}\right\}} \exp \left[ \sum_{{\m...
...i+1,l}\vert e^{-\Delta \tau H_{i,i+1}}\vert s_{i,l+1}s_{i+1,l+1}\rangle \right]$  
  $\textstyle =$ $\displaystyle \sum_{\{states\}} \exp \left[-\beta \sum_{{\mathrm sh.plq}} \left...
...i+1,l}\vert e^{-\Delta \tau H_{i,i+1}}\vert s_{i,l+1}s_{i+1,l+1}\rangle \right]$  
  $\textstyle =$ $\displaystyle \sum_{\{states\}} \exp \left[ -\beta \sum_{{\mathrm sh.plq}} h(i,l)\right].$ (294)

In this expression we have replaced the sum over configurations of quantum spins by a sum over states of a $(1+1)$-dimensional system of Ising variables. The equation ([]) respresents tha partition function of a two-dimensional Ising system with a 4-spin interaction:
\begin{displaymath}
h(i,j) = \left(\frac{1}{\beta}\right) \ln \langle s_{i,l}s_{...
...t e^{-\Delta \tau H_{i,i+1}}\vert s_{i,l+1}s_{i+1,l+1}\rangle.
\end{displaymath} (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 $\vert\alpha\rangle$,

\begin{displaymath}
Z=\sum_\alpha \langle \alpha \vert e^{-\beta H}\vert \alpha \rangle.
\end{displaymath} (296)

We can separate $e^{-\beta H}$ into $e^{-(\beta-\tau)H}e^{-\tau H}$, and insert a complete set of state in the $S^z$ representation between the two exponentials. In the low temperature limit, and for each time slice $\tau$ we obtain
\begin{displaymath}
Z=e^{-\beta E_0}\sum_{\{S_i^z\}} \vert\langle s_1 s_2 \cdots s_N\vert\psi_0\rangle\vert^2.
\end{displaymath} (297)

As a consequence, for each interval of time $\tau$, the probability of finding a given set of spins $\vert s_1s_2\cdots s_N\rangle$ is proportional to the square of the projection of the ground-state on that configuration. This way, the world lines for large $\beta$ offer a picture of the ground state of the system.


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Next: Monte Carlo simulation with Up: World Line Monte Carlo Previous: World Line Monte Carlo
Adrian E. Feiguin 2009-11-04