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Condensed matter physicists often use the term disorder. What is a disorder? Is it some point defect or line defect? How are they modelled in a theory?

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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.

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  • $\begingroup$ One should also make the distinction between quenched and unquenched disorder, the former being in ``thermodynamic equilibrium", meaning that macroscopic quantities may be computed by taking an average over the possible disordering. The case of unquenched disorder is of course much harder. $\endgroup$ – Ryan Thorngren Mar 30 '18 at 15:42
  • $\begingroup$ As I said, I'm no expert. Thanks for the information :-) @RyanThorngren $\endgroup$ – SRS Mar 30 '18 at 15:53
  • $\begingroup$ @SRS Nice answer! But could you please expose what you have in mind when you say that hysteresis in ferromagnets is due to disorder? $\endgroup$ – thermomagnetic condensed boson Apr 1 '18 at 20:17
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    $\begingroup$ @no_choice99 Hysteresis requires a sample to have many domains and therefore many domain walls. Spontaneous formation of many domains requires some anisotropy in the material. Simple Ising model or Heisenberg model of ferromagnet do not capture this feature and therefore simulate a single domain. See here en.wikipedia.org/wiki/Magnetic_anisotropy $\endgroup$ – SRS Apr 1 '18 at 20:36

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