The magnetization operator for the ith atom in a lattice containing \(N\) atoms is \(\hat{M}_{i}=\mu \hat{S}_{i, z^{\prime}}\) where \(\mu\) is the magnetic moment and \(\bar{S}_{i, z}\) is the spin of the ith atom. Neglecting interactions between the particles, the Hamiltonian (energy) of the lattice is \(\hat{H}=-\hat{M}_{T} B\), where \(B\) is an applied magnetic field and \(\hat{M}_{T}=\sum_{i=1}^{N} M_{i}\) is the total magnetization of the lattice. Derive an expression for the variance \(\left\langle M_{T}^{2}\right\rangle_{\text {eq in terms of a thermodynamic response function. Which response function is it? }}\)

In physics, a magnetization operator is used to represent the magnetic moment of a particle in a quantum system. For an atom in a lattice, this operator captures how the atom's magnetic moment behaves. More specifically, let's define the magnetization operator for the ith atom in a lattice with N atoms.
The formula given is \[ \hat{M}_{i} = \mu \hat{S}_{i, z^{'} \].
Here, \mu represents the magnetic moment, and \hat{S}_{i, z^{'}} denotes the spin of the ith atom along the z'-axis.
This concept is crucial because it describes how the atom's intrinsic magnetic property (its spin) contributes to the overall magnetization of the system. Each atom in the lattice has its own magnetization operator, and collectively, these operators determine the system's total magnetization.

The Hamiltonian of a system represents its total energy, and it is fundamental in describing the dynamics of quantum systems. For the problem given, the Hamiltonian is expressed as:
\[ \hat{H} = -\hat{M}_T B \].
Here, \hat{M}_T is the total magnetization of the lattice, and B is the external magnetic field applied to the system. To find \hat{M}_T, we sum up the magnetization of all individual atoms in the lattice:
\[ \hat{M}_T = \sum_{i=1}^{N} \hat{M}_{i} \].
This Hamiltonian shows how the system's total energy is influenced by the presence of an external magnetic field. Understanding the Hamiltonian is critical for calculating other properties, such as the response of the system to external changes or the variance in magnetization.

Variance in magnetization quantifies how much the magnetization fluctuates around its average value.
It is given by the formula:
\[\text{Var}(M_{T}) = \left\langle M_{T}^{2} \right\rangle - \left\langle M_{T} \right\rangle^{2} \].
Here, \left\langle M_{T}^{2} \right\rangle denotes the average of the square of the total magnetization, while \left\langle M_{T} \right\rangle represents the average magnetization. These averages are taken over thermal equilibrium, meaning they reflect the system's behavior at a steady state under constant temperature. Understanding the variance helps in predicting the extent of the magnetization fluctuations in response to external influences like temperature or magnetic field.

Magnetic susceptibility \chi is a measure of how much a material becomes magnetized in response to an applied magnetic field.
In the context of thermodynamics, the relationship between susceptibility and magnetization variance is crucial. Under thermal equilibrium, the susceptibility \chi is given by:
\[ \chi = \frac{1}{k_{B} T} \left( \left\langle M_{T}^{2} \right\rangle - \left\langle M_{T} \right\rangle^{2} \right) \].
Here, k_B is the Boltzmann constant, and T is the absolute temperature.
This formula reveals that susceptibility links the system's microscopic properties (variance in magnetization) with macroscopic observables (response to an external field). In simpler terms, it quantifies how sensitive the system's total magnetization is to changes in the external magnetic field.
Combining all the relations, you derive the final expression for the variance of the total magnetization:
\[ \left\langle M_{T}^{2} \right\rangle = k_{B} T \chi + \left \langle M_{T} \right\rangle^{2} \].
Here, the magnetic susceptibility serves as the thermodynamic response function that connects the microscopic spin properties of the atoms to the system's macroscopic behavior.