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.

The main references of the study note are Monte Carlo simulation of classical Heisenberg model in three dimensions, Computational Quantum Physics and arXiv:0803.0217.

Ising Model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic ‘spins’ that can be in one of two states: +1 (up) or -1 (down). The spins are arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors.

Mathematical definition of Ising Model

Consider a set $\Lambda$ of lattice sites. For each lattice site $k\in\Lambda$, its spin state is represented by $\sigma_{k} = \{+1, -1\}$. The energy of a spin configuration $\sigma = \{\sigma_{k}\}_{k\in\Lambda}$ is given by the Hamiltonian function $H(\sigma)$:

$$H(\sigma) = -\sum_{<ij>} J_{ij}\sigma_{i}\sigma_{j} - \mu\sum_{j}h_{j}\sigma_{j}\tag{1}$$

where $\sigma_{i}$ and $\sigma_{j}$ are spin states (+1 or -1) of two adjacent sites $i, j \in \Lambda$, respectively. $J_{ij}$ is the interaction between site $i$ and site $j$. $J_{ij}>0$ indicates that the interaction is ferromagnetic, while $J_{ij}<0$ indicates that the interaction is antiferromagnetic. Quantity $\mu$ is the magnetic moment and $h_{j}$ is the external magnetic field interacting with site $j$. $h_{j}>0$ means the spin site $j$ desires to line up in the positive direction, while $h_{j}<0$ means the spin site $j$ desires to line up in the negative direction. The first sum is over pairs of adjacent spins (every pair is counted once). The notation $<ij>$ indicates that sites $i$ and $j$ are nearest neighbors.

The configuration probability is given by the Boltzmann distribution with inverse temperature $\beta = 1/(k_{B}T)\geq 0$:

$$P_{\beta}(\sigma) = \frac{e^{-\beta H(\sigma)}}{Z_{\beta}} = \frac{e^{-H(\sigma)/(k_{B}T)}}{Z_{\beta}}\tag{2}$$

where $k_{B}$ is the Boltzmann constant, $T$ is the temperature of the system, and $Z_{\beta} = \sum_{\sigma}e^{-\beta H(\sigma)}$ is the partition function. The configuration probabilities $P_{\beta }(\sigma)$ represent the probability that (in equilibrium) the system is in a state with configuration $\sigma$. Hence, $P_{\beta}(\sigma)\propto e^{-H(\sigma)/T}$ implies that lower energy states are exponentially more probable than higher energy states, especially at low temperature. As a result, the system tends to evolve over time towards configurations that minimize the total energy.

Simplication

Ignore the external magnetic field $h_{j}$, the corresponding Hamiltionian function of Ising Model becomes:

$$H(\sigma) = -\sum_{<ij>} J_{ij}\sigma_{i}\sigma_{j} \tag{3}$$

(Monte Carlo simulation of classical Heisenberg model in three dimensions) If the system is ferromagnetic, which means $J_{ij}>0$ holds for all adjacent sites in this system; it is obvious that configurations with spins to same direction are less energetic and therefore are more probable in the lower temperatures.

Monte-Carlo simulation of 2D Ising model

Here assumes $L = |\Lambda|$ is the number of sites on the square lattice. Since every spin site has two possible states: +1 (up) or -1 (down), there are $2^{L}$ different states that are possible. Hence, it is difficult to evaluate the Ising Model numerically with a large value of $L$, which motivates the reason for applying Monte Carlo Methods to simulate the Ising Model.

Metropolis-Hastings algorithm

In practice, Metropolis-Hastings algorithm with powerful versatility is the most commonly used Monte Carlo algorithm for the estimation of Ising Model. The main steps of Metropolis algorithm are:

1. Initialize a spin configuration $\sigma$ of $L$ spins.
2. Choose a random site $j$ on lattice, and then flip the spin of $j$ (Here just let $\sigma_{j} = -\sigma_{j}$). Finally record the spin configuration with flipped value of $\sigma_{j}$ as $\sigma^{'}$.
3. Caculate the change in energy $dE$ (Here $dE$ is just equivalent to $H(\sigma^{'}) - H(\sigma)$).
4. If $dE<0$, keep the flipped value. Otherwise, keep the flipped value with probability $e^{-\beta (H(\sigma^{'}) - H(\sigma))}$.
5. Repeat the step 2-4.

Implementation of the Ising Model simulations by Monte Carlo approach can be found in repository “Simulation”.

The main references of this study note are Phys. Rev. Lett. 116, 061102 and Annalen der Physik 529, 1-2. Here I focus on the first Gravitaional-Wave (GW) event GW150914 detected by the LIGO Hanford, WA, and Livingston, LA, observatories on September 14, 2015 at 09:50:45 UTC to illustrate the Binary Black Hole (BBH) mergers.

GW150914 is produced by a Binary Black Hole system which is one kind of typical GW signal sources. As such a system comprises two black holes, it is a real 2-body system. To better describe the BBH system, here I begin with the basic features of its main component - the black holes. In this note, I mainly introduce the properties of the Black holes.

Black Hole

Black Hole is a region of space-time where the gravitational field is so strong that nothing, not even light and other electromagnetic waves, can escape. The boundary of no escape is called the event horizon. No-hair theorem states that a stationary black hole can be completely characterized by only three independent externally observable classical parameters: mass, electric charge, and angular momentum.

One kind of typical black holes is the static black hole described by Schwarzschild solution. Such a static black hole has neither electric charge nor angular momentum. Hence, according to no-hair theorem, mass is the only identity of the static black hole. If there are two Schwarzschild Black Holes with the same mass, but the first black hole was made by collapsing ordinary matter whereas the second was made out of antimatter; the no-hair theorem indicates that they will be completely indistinguishable to an observer outside the event horizon situated at the Schwarzschild radius ($r_{s}$). Schwarzschild radius associated with the mass $m$ is mathematically expressed as:

$$r_{s}(m) = \frac{2Gm}{c^{2}} = 2.95(\frac{m}{M_{\odot}})\text{ km}$$

where $M_{\odot} = 1.99 \times 10^{30}\text{ kg}$ is the solar mass, $G = 6.67 \times 10^{-11} \text{ m}^{3}/\text{kg s}$ is Newton’s gravitational constant, and $c = 2.998 \times 10^{8} \text{ m/s}$ is the speed of light. (Phys. Rev. Lett. 116, 061102) According to the hoop conjecture, if a non-spinning mass is compressed to within its corresponding Schwarzschild radius, then it must form a black hole. Once the black hole is formed, any object that comes within this radius can no longer escape out of it.