One-Point Function

The one-point function is the probability of finding an up-spin at a specific position at a specific time. The (brute force) computation for the one-point function requires a large number of summations given the wave function of the system. The number of terms in the summation increases exponentially with the size of the system. Thus, we test our formulas by computing the one-point function. Here is an example for the case of a ring of length 41 with two up-spins that start in the middle of the ring:

The dimension of the system is 820 and, for each time slice, the computation required a summation with roughly \(820^2=672400\) terms. The computation took 5770 seconds using OSU’s cluster

Formulas

We denote the one-point function as follows

\[\rho(x) : \mathbb{Z} \rightarrow [0, 1]\]

which is the quantity described above.

The one-point function is determined by the marginal of the joint probability function for the system. Let \(\Xi (x)\) denote all the configurations that contain the point \(x \in \mathbb{Z}\). Then, the one-point function is given as follow:

\[\begin{align} \rho(x)&= \sum_{\vec{x} \in \Xi} \mathrm{Prob}_{\Psi}(\vec{x})\\ &= \sum_{\vec{x} \in \Xi} \vert \langle \vec{x}\mid \Psi \rangle\vert^2 \end{align}\]