Dynamical spin structure factor

Tags: Dynamical Spin Structure Factor

Intro

We consider the XXZ ring of length \(L\) with \(N\) up-spin particles. Let’s compute the dynamical spin structure factor (DSSF). We follow the formulas and notation from this paper. The DSSF is given by the following formula

\[S^{zz}(q, \omega) = \frac{1}{L} \sum_{j, j' =1}^N e^{- i q (j-j')} \int_{- \infty}^{\infty}dt e^{i \omega t} \langle S_j^z(t) S_{j'}^z(0) \rangle\]

where \(L\) is the length of the ring.

Integrand

First, we need compute the integrand from the previous formula. We have

The z-spin operators \(S_j^{z}\) is a local oprator acting on the \(j^{th}\) spin of the ring. The z-spin operator is given by the following matrix

\[\left(\begin{array}{cc} 1/2 & 0 \\ 0 & -1/2\end{array} \right)\]

on the up-down basis, \(\mid \uparrow \rangle , \mid \downarrow \rangle\). Namely, the z-spin operator returns \(+1/2\) for an up-spin and \(-1/2\) for a down-spin. The, we have

\[S_j^z(t) = e^{i t H /\hbar } S_j^z e^{-i t H /\hbar }\]

where \(H\) is the Hamiltonian of the spin chain.

Warning: This my intepretation of the operator in the integrand, which may be wrong. One should double check that this is what the authors of the paper had in mind.

Action on the wave function

Recall the formual for the wave function

\[\begin{align} \mid \Psi(t) \rangle &= e^{-i t H /\hbar } \mid \Psi(0) \rangle\\ &=\sum_{x \in X} \left(\sum_{[\xi] \in \Xi} \ell(y, \xi) u(\xi, x) e^{-i t E(\xi)} \right) \mid x \rangle \end{align}\]

with \(x = (x_1, x_2, \dots, x_N)\) is the configuration of the up-spins on the ring. We have the following action of the z-spin operator

\[S_j^z : \mid x \rangle \mapsto \begin{cases}\tfrac{1}{2} \mid x\rangle &\quad j \in x \\ - \tfrac{1}{2} \mid x \rangle. &\quad j \notin x \end{cases}.\]

Let us partition the configuration space by the configurations that include an up-spin at position \(j\) and those that don’t. Set

\[X(j) = \{ x \in X \mid j \in x \}.\]

Then,

\[\begin{align} &\left(\langle \Psi(0) \mid e^{i t H/\hbar} S_j^z\right)^{\dagger} = S_j^z e^{-i t H/\hbar} \mid \Psi(0) \rangle\\ &= \frac{1}{2} \left( \sum_{x \in X(j)} \left(\sum_{[\xi] \in \Xi} \ell(y, \xi) u(\xi, x) e^{-i t E(\xi)} \right) \mid x \rangle - \sum_{x \in X(j)^{c}} \left(\sum_{[\xi] \in \Xi} \ell(y, \xi) u(\xi, x) e^{-i t E(\xi)} \right) \mid x \rangle\right)\\ \end{align}\]

Also, we have

\[\begin{align} S_{j'}^z |\Psi(0)\rangle = S_{j'}^z |y\rangle = \begin{cases} \frac{1}{2} \mid y \rangle, &\quad j' \in y \\ -\frac{1}{2} \mid y \rangle, &\quad j' \notin y \end{cases} \end{align}\]

so that

\[\begin{align} & e^{-i t H/\hbar} S_{j'}^z \mid \Psi(0) \rangle\\ &= \frac{(-1)^{j' \notin y}}{2} \sum_{x \in X} \left(\sum_{[\xi] \in \Xi} \ell(y, \xi) u(\xi, x) e^{-i t E(\xi)} \right) \mid x \rangle. \end{align}\]

where \((-1)^{j' \notin y} = -1\) when \(j' \notin y\) and \((-1)^{j' \notin y} = 1\) when \(j' \in y\).

Expected value

We have

\[\begin{align} & \langle S_j^z(t) S_{j'}^z \rangle = \langle \Psi \mid e^{i t H /\hbar } S_j^z e^{-i t H /\hbar } S_{j'}^z \mid \Psi \rangle\\ &= \frac{(-1)^{j' \notin y}}{2} \sum_{x \in X} \frac{(-1)^{j \notin x}}{2} \sum_{[\zeta], [\xi] \in \Xi} \overline{\ell(y, \zeta) u(\zeta, x) } \ell(y, \xi) u(\xi, x) e^{-i t (E(\xi)- E(\zeta) )}. \end{align}\]

Note that the final (summation) expression is real. If you take the complex conjugate of the expersion, the dummy indexes switch roles and the expression stay the same, meaning that the expression is real. This is a good sanity check.

Next Steps

• Compute the \(N=1\) case.