Study note about Heisenberg model

The main references of the study note are Monte Carlo simulation of classical Heisenberg model in three dimensions and Computational Quantum Physics.

Spin-1/2

In quantum mechanics, spin is an intrinsic property of all elementary particles. (Wiki. Spin-1/2) The spin quantum number describing how many symmetrical facets a particle has in one full rotation indicates the intrinsic angular momentum of the particle; a spin of $1/2$ means that the particle must be rotated by two full turns (through $720^{\circ}$) before it has the same configuration as when it started.

The value of spin quantum number $s$ for an elementary particle depends only on the type of particle and cannot be altered in any known way. Spin-statistics theorem indicates that Spin-1/2 objects are all fermions with half-integer spins. All of these fermions with spin quantum number $s = n / 2 \text{ (where } n = 0, 1, 2, \cdots)$ satisfy the Pauli exclusion principle. Pauli exclusion principle states that two or more spin-1/2 particles cannot simultaneously occupy the same quantum state within a system that obeys the laws of quantum mechanics. The spin angular momentum $S$ of any physical system is quantized. The allowed values of $S$ are:

$$S = \hbar\sqrt{s(s+1)} = \frac{h}{2\pi}\sqrt{\frac{n}{2}\frac{n+2}{2}} = \frac{h}{4\pi}\sqrt{n(n+2)}$$

where $h$ is the Planck constant, and $\hbar = h/2\pi$ is the reduced Planck constant.

Pauli Matrices

(Wiki. Pauli Matrices) In mathematical physics and mathematics, the Pauli matrices are a set of three $2 \times 2$ complex matrices that are traceless, Hermitian, involutory and unitary. These matrices with eigenvalues of $\pm 1$ are used to represent the spin operators for spin-1/2 particles. They can be mathematically expressed as:

$$\sigma_{1} = \sigma_{x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \tag{1}$$

$$\sigma_{2} = \sigma_{y} = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \tag{2}$$

$$\sigma_{3} = \sigma_{z} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \tag{3}$$

The determinants and traces of the Pauli matrices are:

$$\text{det } \sigma_{x} = \text{det } \sigma_{y} = \text{det } \sigma_{z} = -1, \quad \text{tr } \sigma_{x} = \text{tr } \sigma_{y} = \text{tr } \sigma_{z} = 0$$

The Pauli matrices satisfy specific commutation relations, which are analogous to the angular momentum commutation relations:

$$[\sigma_{j}, \sigma_{k}] = 2i\epsilon_{jkl}\sigma_{l}$$

where $\epsilon_{jkl}$ is the Levi-Civita symbol, and $j, k, l \in \{x, y, z\}$. $\epsilon_{jkl}$ is 1 if $(j, k, l)$ is an even permutation of $(x, y, z) = (1, 2, 3)$, -1 if it is an odd permutation, and 0 if any index is repeated. As a result, $[\sigma_{j}, \sigma_{k}] \neq [\sigma_{k}, \sigma_{j}]$

Spin Operators

In quantum mechanics, the outcome of a measurement in a quantum system is usually intrusive and not deterministic, and observables are represented by operators. After measuring an observable $A$, the wave function of the system collapses to an eigenvector of operator $\hat{A}$ and the outcome of the measurement is given by the respective eigenvalue. Observable states of the particle are then found by the spin operators along the $x$, $y$, and $z$ axes ($\hat{S}^{x}, \hat{S}^{y}, \hat{S}^{z}$):

$$\hat{S}^{x} = \frac{\hbar}{2} \sigma_{x},\quad \hat{S}^{y} = \frac{\hbar}{2} \sigma_{y},\quad \hat{S}^{z} = \frac{\hbar}{2} \sigma_{z}$$

The spin projection operator $\hat{S}^{z}$ affects a measurement of the spin in the $z$ direction.

As Pauli matrices satisfy specific commutation relations, the spin operators also follow commutation relations:

$$[\hat{S}^{x}, \hat{S}^{y}] = 2i\epsilon_{123} \hat{S}^{z} = i\hbar\sigma_{z}$$

$$[\hat{S}^{y}, \hat{S}^{z}] = 2i\epsilon_{231} \hat{S}^{x} = i\hbar\sigma_{x}$$

$$[\hat{S}^{z}, \hat{S}^{x}] = 2i\epsilon_{312} \hat{S}^{y} = i\hbar\sigma_{y}$$

Since $[\hat{S}^{y}, \hat{S}^{x}] = 2i\epsilon_{213} \hat{S}^{z} = -i\hbar\sigma_{z}$, it is noticeable that $[\hat{S}^{x}, \hat{S}^{y}] \neq [\hat{S}^{y}, \hat{S}^{x}]$. Hence, spin operators do not commute.

The compatibility theorem

Consider two observables, $A \text{ and } B$, represented by the operators $\hat{A}\text{ and }\hat{B}$. (Wiki. Complete set of commuting observables) The compatibility theorem points out “Observables $A \text{ and } B$ are compatible observables” is equivalent to “The operators $\hat{A}\text{ and }\hat{B}$ commute, meaning $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} = 0$”. In addition, if observables $A \text{ and } B$ commute, operators $\hat{A}\text{ and }\hat{B}$ share a common eigenbasis. Hence, if $[\hat{A}, \hat{B}]\neq 0$, it is not possible to measure observables $A, B$ simultaneously without uncertainty.

Simply applies the compatibiity theorem to the spin operators explained above, it is clear that: As spin operators do not commute, it is impossible to measure two orthogonal components (e.g., $\hat{S}_{x}\text{ and }\hat{S}_{y}$) simultaneously with arbitrary precision.

Spin states

A spin-1/2 particle with a spin quantum number $s = 1/2$ has two possible spin projections: $+\frac{1}{2}$ (spin-up) or $-\frac{1}{2}$ (spin-down). The eigenstates of the Pauli matrices represent the spin-up and spin-down states of the particle. For instance, the corresponding eigenstates of $\sigma_{z}$ are:

$$\text{"up": }\vert\uparrow\rangle = |0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \text{"down": }|\downarrow\rangle = |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$

where $|\uparrow\rangle \text{ and } |\downarrow\rangle$ represent the spin-up and spin-down states, respectively. They are also commonly applied as basis states to express the superposition of a spin state. The corresponding mathematical expression is given by:

$$|\psi\rangle = \alpha|\uparrow\rangle + \beta|\downarrow\rangle$$

with $|\alpha|^{2} + |\beta|^{2} = 1$.

Similarly, the corresponding projections of the spin angular momentum along any axis (usually the $z$-axis) can take on only two possible values: $\pm\hbar/2$.

Spin Raising and Lowering Operators

In quantum mechanics, spin raising and lowering operators are essential tools for understanding the behavior of spin systems. For a spin-1/2 particle, the spin raising ($\hat{S}^{+}$) and lowering ($\hat{S}^{-}$) operators are defined as

$$\hat{S}^{+} = \hat{S}^{x} + i\hat{S}^{y}\\ \hat{S}^{-} = \hat{S}^{x} - i\hat{S}^{y}$$

These operators change the state of the spin system by raising or lowering the projections of spin angular momentum along the $z$-axis. They can act on the basis states $|\uparrow\rangle$ and $|\downarrow\rangle$ as follows:

$$\hat{S}^{+}|\downarrow\rangle = \hbar|\uparrow\rangle, \quad \hat{S}^{+}|\uparrow\rangle = 0\\ \hat{S}^{-}|\uparrow\rangle = \hbar|\downarrow\rangle, \quad \hat{S}^{-}|\downarrow\rangle = 0$$

These actions demonstrate that $\hat{S}^{+}$ raises the spin projection from $-\frac{1}{2}$ to $+\frac{1}{2}$, and $\hat{S}^{-}$ lowers the spin projection from $+\frac{1}{2}$ to $-\frac{1}{2}$.

Heisenberg Model

Classical Heisenberg Model

Classical Heisenberg model is a generalization of Ising model where the spin $\vec{s_{i}}$ of site $i$ can point to any direction of euclidean space but not just up or down. Consider a set $\Lambda$ of lattice sites. For each lattice site $k\in\Lambda$, its spin state is represented by $\vec{s_{k}} = (s_{k}^{x}, s_{k}^{y}, s_{k}^{z})$ with $|\vec{s_{k}}| = 1$. The energy of a spin configuration $\vec{s} = \{\vec{s}_{k}\}_{k\in\Lambda}$ is given by the Hamiltonian function $H(\vec{s})$

$$H(\vec{s}) = -\sum_{<ij>}J_{ij}\vec{s_{i}}\vec{s_{j}}$$

where $\vec{s_{i}} = (s_{i}^{x}, s_{i}^{y}, s_{i}^{z}) \text{ and } \vec{s_{j}} = (s_{j}^{x}, s_{j}^{y}, s_{j}^{z})$ are the spin states of site $i$ and $j$ on lattice, respectively. $J_{ij}$ is the interaction between $i$ and $j$, and it satisfies

$$J_{ij} = \begin{cases} 1, & \text{if }i, j \text{ are neighbors}\\ 0, & \text{else.} \end{cases}$$

Quantum Heisenberg Model

In Quantum Heisenberg Model, the spins of the magnetic systems are treated quantum mechanically, by replacing the spin by a quantum operator acting upon the tensor product $({\mathbb{C}^{2}})^{\otimes N}$, of dimension $2^{N}$. Each spin can be in a superposition of quantum states.

For a spin pointing along an arbitrary direction $\vec{e_{k}} =(e^{x}_{k}\quad e^{y}_{k}\quad e^{z}_{k})$ in site $k$, the operator for the spin in this direction is simply:

$$\vec{e_{k}}\cdot\hat{\vec{S}} = \frac{\hbar}{2}(e^{x}_{k}\quad e^{y}_{k}\quad e^{z}_{k})\begin{pmatrix} \sigma_{x} \\ \sigma_{y} \\ \sigma_{z} \end{pmatrix} = \frac{\hbar}{2}(e^{x}_{k}\sigma_{x} + e^{y}_{k}\sigma_{y} + e^{z}_{k}\sigma_{z}) = \frac{\hbar}{2}\begin{pmatrix} e^{z}_{k} & e^{x}_{k} - ie^{y}_{k} \\ e^{x}_{k} + ie^{y}_{k} & -e^{z}_{k} \end{pmatrix}$$

Assume that magnetic interactions only occur between adjacent dipoles and the strength of all these interactions are equal. The corresponding Hamiltonian operator $\hat{H}$ is given by:

$$\hat{H} = -J\sum_{<ij>}\hat{\vec{S}}_{i}\cdot\hat{\vec{S}}_{j} = -J\sum_{<ij>}(\hat{S}^{x}_{i}\hat{S}^{x}_{j} + \hat{S}^{y}_{i}\hat{S}^{y}_{j} + \hat{S}^{z}_{i}\hat{S}^{z}_{j})$$

where $\hat{\vec{S}}_{i}$ and $\hat{\vec{S}}_{j}$ are spin operators for particles at sites $i$ and $j$, respectively. $\hat{S}^{x}_{k}, \hat{S}^{y}_{k}, \hat{S}^{z}_{k}$ are the components of the spin operator for site $k\in\{i, j\}$. $J$ is the coupling constant, and $<ij>$ denotes summation over nearest-neighbor pairs.

Given a choice of real-valued coupling constants $J_{x}, J_{y} \text{ and } J_{z}$, the Hamiltonian is given by

$$\hat{H} = -\sum_{<ij>}(J_{x}\hat{S}^{x}_{i}\hat{S}^{x}_{j} + J_{y}\hat{S}^{y}_{i}\hat{S}^{y}_{j} + J_{z}\hat{S}^{z}_{i}\hat{S}^{z}_{j})$$

If $J_{x} \neq J_{y} \neq J_{z}$, the model is called the Heisenberg XYZ model. In case $J_{x} = J_{y} \neq J_{z}$, the model is called the Heisenberg XXZ model. If $J_{x} = J_{y} = J_{z}$, the model is called the Heisenberg XXX model.