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I'm no expert but this is what I understand. A disorder is something which generically breaks some invariance of the Hamiltonian or lead to some deviation from the lattice periodicity. When some disorder is present, all points in the lattice are not equivalent. For example, consider the tight-binding model $$H=\sum\limits_{i}\epsilon_0|i\rangle\langle i|+\sum\limits_{i,j}(h_{ij}|i\rangle\langle j|+|j\rangle\langle i|)$$ where $\epsilon_0$ is called the on-site energy and $h_{ij}$ is called the hopping parameter. If $h_{ij}=h\delta_{i+1,j}$, it does not model a disorder. However, if $h_{ij}$ is different for different $i,j$, it models a static disorder$^1$ and breaks the discrete translation symmetry of $H$. Also by making $h$ to be a random function of time $h(t)$ i.e., a stochastic variable one can introduce a dynamic disorder.

Addendum In everyday life, hysteresis in ferromagnets is due to disorder. Disorder in spin systems can lead to frustration or localization of spin waves which might change the response of the magnetic susceptibility.

$^1$ Physically it corresponds to sprinkling some impurities (say, ${\rm Ni}$ atoms) randomly in a crystal of ${\rm Cu}$ atoms. Disorders dramatically affect the properties such as the conductivity of the system due to impurity scattering. In fact, if the degree of randomness is in some sense "large", extended Bloch states can become localized which goes by the name of Anderson localization.

I'm no expert but this is what I understand. A disorder is something which generically breaks some invariance of the Hamiltonian or lead to some deviation from the lattice periodicity. When some disorder is present, all points in the lattice are not equivalent. For example, consider the tight-binding model $$ H=\sum\limits_{i}\epsilon_0|i\rangle\langle i|+\sum\limits_{i,j}h_{ij}(|i\rangle\langle j|+|j\rangle\langle i|)$$ where $\epsilon_0$ is called the on-site energy and $h_{ij}$ is called the hopping parameter. If $h_{ij}=h\delta_{i+1,j}$, it does not model a disorder. However, if $h_{ij}$ is different for different $i,j$, it models a static disorder$^1$ and breaks the discrete translation symmetry of $H$. Also by making $h$ to be a random function of time $h(t)$ i.e., a stochastic variable one can introduce a dynamic disorder.

Addendum In everyday life, hysteresis in ferromagnets is due to disorder. Disorder in spin systems can lead to frustration or localization of spin waves which might change the response of the magnetic susceptibility.

$^1$ Physically it corresponds to sprinkling some impurities (say, ${\rm Ni}$ atoms) randomly in a crystal of ${\rm Cu}$ atoms. Disorders dramatically affect the properties such as the conductivity of the system due to impurity scattering. In fact, if the degree of randomness is in some sense "large", extended Bloch states can become localized which goes by the name of Anderson localization.

I'm no expert but this is what I understand. A disorder is something which generically breaks some invariance of the Hamiltonian or lead to some deviation from the lattice periodicity. When some disorder is present, all points in the lattice are not equivalent. For example, consider the tight-binding model $$H=\sum\limits_{i}\epsilon_0|i\rangle\langle i|+\sum\limits_{i,j}(h_{ij}|i\rangle\langle j|+|j\rangle\langle i|)$$ where $\epsilon_0$ is called the on-site energy and $h_{ij}$ is called the hopping parameter. If $h_{ij}=h\delta_{i+1,j}$, it does not model a disorder. However, if $h_{ij}$ is different for different $i,j$, it models a static disorder$^1$ and breaks the discrete translation symmetry of $H$. Also by making $h$ to be a random function of time $h(t)$ i.e., a stochastic variable one can introduce a dynamic disorder.

$^1$ Physically it corresponds to sprinkling some impurities (say, ${\rm Ni}$ atoms) randomly in a crystal of ${\rm Cu}$ atoms. Disorders dramatically affect the properties such as the conductivity of the system due to impurity scattering. In fact, if the degree of randomness is in some sense "large", extended Bloch states can become localized which goes by the name of Anderson localization.

I'm no expert but this is what I understand. A disorder is something which generically breaks some invariance of the Hamiltonian or lead to some deviation from the lattice periodicity. When some disorder is present, all points in the lattice are not equivalent. For example, consider the tight-binding model $$H=\sum\limits_{i}\epsilon_0|i\rangle\langle i|+\sum\limits_{i,j}(h_{ij}|i\rangle\langle j|+|j\rangle\langle i|)$$ where $\epsilon_0$ is called the on-site energy and $h_{ij}$ is called the hopping parameter. If $h_{ij}=h\delta_{i+1,j}$, it does not model a disorder. However, if $h_{ij}$ is different for different $i,j$, it models a static disorder$^1$ and breaks the discrete translation symmetry of $H$. Also by making $h$ to be a random function of time $h(t)$ i.e., a stochastic variable one can introduce a dynamic disorder.

Addendum In everyday life, hysteresis in ferromagnets is due to disorder. Disorder in spin systems can lead to frustration or localization of spin waves which might change the response of the magnetic susceptibility.

$^1$ Physically it corresponds to sprinkling some impurities (say, ${\rm Ni}$ atoms) randomly in a crystal of ${\rm Cu}$ atoms. Disorders dramatically affect the properties such as the conductivity of the system due to impurity scattering. In fact, if the degree of randomness is in some sense "large", extended Bloch states can become localized which goes by the name of Anderson localization.

I'm no expert but this is what I understand. A disorder is something which generically breaks some invariance of the Hamiltonian or some deviation from the lattice periodicity. When some disorder is present, all points in the lattice are not equivalent. For example, consider the tight-binding model $$H=\sum\limits_{i}\epsilon_0|i\rangle\langle i|+\sum\limits_{i,j}(h_{ij}|i\rangle\langle j|+|j\rangle\langle i|)$$ where $\epsilon_0$ is called the on-site energy and $h_{ij}$ is called the hopping parameter. If $h_{ij}=h\delta_{i+1,j}$, it does not model a disorder. However, if $h_{ij}$ is different for different $i,j$, it models a static disorder$^1$ and breaks the discrete translation symmetry of $H$. Also by making $h$ to be a random function of time $h(t)$ i.e., a stochastic variable one can introduce a dynamic disorder.

$^1$ Physically it corresponds to sprinkling some impurities (say, ${\rm Ni}$ atoms) randomly in a crystal of ${\rm Cu}$ atoms. Disorders dramatically affect the properties such as the conductivity of the system due to impurity scattering. In fact, if the degree of randomness is in some sense "large", extended Bloch states can become localized which goes by the name of Anderson localization.

I'm no expert but this is what I understand. A disorder is something which generically breaks some invariance of the Hamiltonian or lead to some deviation from the lattice periodicity. When some disorder is present, all points in the lattice are not equivalent. For example, consider the tight-binding model $$H=\sum\limits_{i}\epsilon_0|i\rangle\langle i|+\sum\limits_{i,j}(h_{ij}|i\rangle\langle j|+|j\rangle\langle i|)$$ where $\epsilon_0$ is called the on-site energy and $h_{ij}$ is called the hopping parameter. If $h_{ij}=h\delta_{i+1,j}$, it does not model a disorder. However, if $h_{ij}$ is different for different $i,j$, it models a static disorder$^1$ and breaks the discrete translation symmetry of $H$. Also by making $h$ to be a random function of time $h(t)$ i.e., a stochastic variable one can introduce a dynamic disorder.

$^1$ Physically it corresponds to sprinkling some impurities (say, ${\rm Ni}$ atoms) randomly in a crystal of ${\rm Cu}$ atoms. Disorders dramatically affect the properties such as the conductivity of the system due to impurity scattering. In fact, if the degree of randomness is in some sense "large", extended Bloch states can become localized which goes by the name of Anderson localization.