Editing Amorphous solid (section)

=== Universal low-temperature properties of amorphous solids ===
At very low temperatures (below 1-10 K), a large family of amorphous solids have various similar low-temperature properties.
Although there are various theoretical models, neither [[glass transition]] nor low-temperature properties of [[glassy solids]] are well understood on the [[fundamental physics]] level.

Amorphous solids is an important area of [[condensed matter physics]] aiming to understand these substances at high temperatures of [[glass transition]] and at low [[temperature]]s towards [[absolute zero]]. From the 1970s, low-temperature properties of amorphous solids were studied experimentally in great detail.<ref name="Stephens2021">{{Cite book |last1=Stephens |first1=Robert B. |first2=Xiao |last2=Liu |title=Low-Energy Excitations in Disordered Solids. A Story of the 'Universal' Phenomena of Structural Tunneling |year=2021 |doi=10.1142/11746|isbn=978-981-12-1724-1 |s2cid=224844997 }}</ref><ref name="Ramos2022">{{Cite book |editor-last1=Ramos |editor-first1=M. |title=Low-Temperature Thermal and Vibrational Properties of Disordered Solids. A Half-Century of Universal "Anomalies" of Glasses |year=2022 |doi=10.1142/q0371|arxiv=2010.02851 |isbn=978-1-80061-257-0 |s2cid=222140882 |last1=Grushin |first1=Adolfo G. }}</ref> For all of these substances, [[specific heat]] has a (nearly) linear dependence as a function of temperature, and [[thermal conductivity]] has nearly quadratic temperature dependence. These properties are conventionally called '''anomalous''' being very different from properties of [[crystalline solid]]s.

On the phenomenological level, many of these properties were described by a collection of tunnelling two-level systems.<ref name="AHV1972">{{Cite journal |last1=Anderson |first1=P.W. |last2=Halperin |first2=B.I. |last3=Varma |first3=C.M |title=Anomalous low-temperature thermal properties of glasses and spin glasses |year=1972 |journal=Philosophical Magazine|volume=25 |issue=1 |pages=1–9|doi=10.1080/14786437208229210|bibcode=1972PMag...25....1A }}</ref><ref name="Phillips1972">{{Cite journal |last1=Phillips |first1=W.A. |title= Tunneling states in amorphous solids |year=1972 |journal=J. Low Temp. Phys., Pp 751 |volume=7 |issue=3–4 |pages=351–360 |doi=10.1007/BF00660072|bibcode=1972JLTP....7..351P |s2cid=119873202 }}</ref> Nevertheless, the microscopic theory of these properties is still missing after more than 50 years of the research.<ref name="Esquinazi1998">{{Cite book |editor-last1=Esquinazi |editor-first1=Pablo |title=Tunneling Systems in Amorphous and Crystalline Solids |year=1998 |doi=10.1007/978-3-662-03695-2|isbn=978-3-642-08371-6 }}</ref>

Remarkably, a '''dimensionless''' quantity of internal friction is nearly universal in these materials.<ref name="Pohl2002">{{Cite journal |last1=Pohl |first1=R.O. |last2=etc |first2=etc |title=Low-temperature thermal conductivity and acoustic attenuation in amorphous solids |year=2002 |journal=Revs. Mod Phys. |volume=74 |issue=1 |pages=991|doi=10.1080/14786437208229210|bibcode=1972PMag...25....1A }}</ref> This quantity is a dimensionless ratio (up to a numerical constant) of the phonon [[wavelength]] to the phonon [[mean free path]]. Since the theory of tunnelling two-level states (TLSs) does not address the origin of the density of TLSs, this theory cannot explain the universality of internal friction, which in turn is proportional to the density of scattering TLSs. The theoretical significance of this important and unsolved problem was highlighted  by [[Anthony Leggett]].<ref name="Leggett1991">{{Cite journal |last1=Leggett |first1=A.J. |title=Amorphous materials at low temperatures: why are they so similar?|year=1991 |journal=Physica B |volume=169 |issue=1–4 |pages=322–327 |doi=10.1016/0921-4526(91)90246-B|bibcode=1991PhyB..169..322L }}</ref>