Stage 0: Spin chains 101

Introduction to the Heisenberg model

In this challenge we will focus on simulating QTD of 1-D integrable spin chains with nearest-neighbor interactions. In particular, we will consider the Heisenberg model, which is one of the simplest statistical mechanical models used to study the physics of magnetic systems. We will use the Heisenberg model to study a system’s ferromagnetic properties by computing its time-dependent staggered magnetization.

The Heisenberg model considers a magnetic system comprised by “spins” \(\sigma_j\) (particles with a magnetic dipole moment) placed at every site in a lattice. Here we denote the magnetic north and south pole of each particle by \(\uparrow\) and \(\downarrow\), respectively. We only consider the 1-D case, so our lattice is a “chain.” Thus in the classical picture the state of our system is completely described by a sequence of up and down arrows.

However, as in the related Ising model, we treat the spins quantum mechanically. This means we replace each \(\sigma_j \in \{\uparrow, \downarrow\}\) by a two-level quantum system with state space described by \(\mathbb{C}^2\). We label the computational basis for each spin state-space by

\[\vert {\downarrow} \rangle \leftrightarrow \vert 0 \rangle \quad\text{and}\quad \vert {\uparrow} \rangle \leftrightarrow \vert 1 \rangle.\]

This means the state of an \(N\)-particle system is described by a state-vector \(\vert \psi \rangle \in \left(\mathbb{C}^2\right)^{\otimes N}\).

In the quantum-mechanical picture, the Hamiltonian captures the system dynamics. To define it, we recall the Pauli matrices \(\sigma^x\), \(\sigma^y\), and \(\sigma^z\): they are given by

\[\begin{split}\sigma^x = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad \sigma^y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \quad\text{and}\quad \sigma^z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}.\end{split}\]

In addition, for \(2 \times 2\) complex matrix and any \(1 \leq j \leq N\), we let

\[A_j = I^{\otimes j-1} \otimes A \otimes I^{\otimes N-j}\]

denote the operator that applies \(A\) on the \(j\)th spin and acts trivially on all other sites. With this notation in mind, we are ready to introduce the Hamiltonian for our spin chain.

The Heisenberg model proposes the following Hamiltonian for an \(N\)-particle system:

(1)\[\mathcal{H} = {1 \over 2} \sum_{j=1}^{N - 1} \left( J_x \sigma^x_j \sigma^x_{j+1} + J_y \sigma^y_j \sigma^y_{j+1} + J_z \sigma^z_j \sigma^z_{j+1} \right) + \frac{h}{2} \sum_{j=1}^N \sigma_j^z.\]

The coupling parameters \(J_x, J_y\), and \(J_z\) denote the exchange interaction between adjacent spins along the \(x, y\), and \(z\) directions. In addition, \(h\) denotes the external magnetic field strength. We write \(\mathcal{H} = \mathcal{H}_S + \mathcal{H}_M\) to indicate the total Hamiltonian is the sum of the spin-spin interaction and external magnetic field terms.

We take a moment to comment on nomenclature: it is common to name the model depending on the values of \(J_x, J_y\), and \(J_z\). When all three constants are distinct, the model is known as the Heisenberg XYZ model; in case \(J_x = J_y \neq J_z\), it is the XXZ model; if all three \(J_x = J_y = J_z\), it is known as the XXX model.

Exercise 0.1

Show that the spin interaction terms acting on a common pair of sites along different axes commute. Concretely, let

\[[A, B] = AB - BA\]

denote the commutator of \(A\) and \(B\), let \(S_j^\alpha = \sigma_j^\alpha \sigma_{j+1}^\alpha\) denote the spin-spin interaction term at the \((j, j+1)\)-st sites along the \(\alpha\) direction, and show that

\[[S_j^\alpha, S_j^\beta] = 0.\]

Now show that, conversely, spin interaction terms at adjacent pairs of sites do not commute; that is, prove that

\[[S_j^\alpha, S_{j+1}^\beta] \neq 0.\]

This result will be useful later…