# Off-diagonal long-range order

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{{Short description|Quantum feature of condensed-matter systems}}

In [condensed matter physics](/source/condensed_matter_physics), an '''off-diagonal long-range order''' ('''ODLRO''') is a feature of [macroscopic quantum phenomena](/source/macroscopic_quantum_phenomena). It refers to off-diagonal elements in the [density matrix](/source/density_matrix) separated in space in a many-body [quantum mechanical](/source/Quantum_mechanics) system. An ODLRO implies correlations between distant particles in the system, indicating quantum behaviour. The concept is analogous to [coherences](/source/Coherence_(physics)) and [higher order coherences](/source/Higher_order_coherence) from [quantum optics](/source/quantum_optics). An ODLRO is an indication of [spontaneous symmetry breaking](/source/spontaneous_symmetry_breaking) in the system.<ref name=":3">{{Cite book |last=Han |first=Rushan |url=https://books.google.com/books?id=lJ15DwAAQBAJ&dq=symmetry+breaking+odlro&pg=PA191 |title=Superconductivity Centennial |date=2018-10-09 |publisher=World Scientific |isbn=978-981-327-315-3 |language=en}}</ref><ref>{{Cite book |last1=Chen |first1=Fong-ching |url=https://books.google.com/books?id=QwGVEAAAQBAJ&dq=symmetry+breaking+odlro&pg=PA315 |title=Festschrift In Honor Of The C N Yang Centenary, A: Scientific Papers |last2=Ge |first2=Mo-lin |last3=Gu |first3=Bin-lin |last4=Phua |first4=Kok Khoo |last5=Young |first5=Kenneth |last6=Zhu |first6=Bang-fen |date=2022-08-03 |publisher=World Scientific |isbn=978-981-12-6416-0 |language=en}}</ref>

ODLRO are different from the usual ('''diagonal''') '''long-range order''' which is the kind of correlations that one finds in [crystals](/source/Crystal) and in many [classical systems](/source/Classical_mechanics).<ref name=":0">{{Cite book |last=Mahan |first=Gerald D. |url=https://books.google.com/books?id=v8du6cp0vUAC&dq=off+diagonal+long+range+order&pg=PA855 |title=Many-Particle Physics |date=1990-03-31 |publisher=Springer Science & Business Media |isbn=978-0-306-43423-5 |language=en}}</ref><ref>{{Cite book |last1=Girvin |first1=Steven M. |url=https://books.google.com/books?id=YYKFDwAAQBAJ&dq=girvin+condensed+matter+odlro&pg=PA535 |title=Modern Condensed Matter Physics |last2=Yang |first2=Kun |date=2019-02-28 |publisher=Cambridge University Press |isbn=978-1-107-13739-4 |language=en}}</ref>

The concept was first introduced by [Oliver Penrose](/source/Oliver_Penrose) in 1951,<ref>{{Cite journal |last=Penrose |first=O. |date=1951-12-01 |title=CXXXVI. On the quantum mechanics of helium II |url=https://doi.org/10.1080/14786445108560954 |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science |volume=42 |issue=335 |pages=1373–1377 |doi=10.1080/14786445108560954 |issn=1941-5982|url-access=subscription }}</ref> and by Penrose and [Lars Onsager](/source/Lars_Onsager) in 1956,<ref>{{Cite journal |last1=Penrose |first1=Oliver |last2=Onsager |first2=Lars |date=1956-11-01 |title=Bose-Einstein Condensation and Liquid Helium |url=https://link.aps.org/doi/10.1103/PhysRev.104.576 |journal=Physical Review |volume=104 |issue=3 |pages=576–584 |doi=10.1103/PhysRev.104.576 |bibcode=1956PhRv..104..576P |url-access=subscription }}</ref> to study [superfluidity](/source/superfluidity) and [Bose–Einstein condensates](/source/Bose%E2%80%93Einstein_condensate).<ref name=":0" /><ref name=":1">{{Cite book |last1=Feng |first1=Duan |url=https://books.google.com/books?id=-iuYN5arHwoC&dq=off+diagonal+long+range+order+penrose+cn+yang&pg=PA483 |title=Introduction to Condensed Matter Physics |last2=Jin |first2=Guojun |date=2005 |publisher=World Scientific |isbn=978-981-238-711-0 |language=en}}</ref> Its mathematical definition in terms of density matrices was done by [C.N. Yang](/source/Yang_Chen-Ning) in 1962,<ref>{{Cite journal |last=Yang |first=C. N. |date=1962-10-01 |title=Concept of Off-Diagonal Long-Range Order and the Quantum Phases of Liquid He and of Superconductors |url=https://link.aps.org/doi/10.1103/RevModPhys.34.694 |journal=Reviews of Modern Physics |volume=34 |issue=4 |pages=694–704 |doi=10.1103/RevModPhys.34.694 |bibcode=1962RvMP...34..694Y |url-access=subscription }}</ref> who coined the term ODLRO and generalized it to other systems like [superconductivity](/source/superconductivity).<ref name=":1" /><ref>{{Cite book |last1=Vollhardt |first1=Dieter |url=https://books.google.com/books?id=jY6yAAAAQBAJ&dq=ferromagnet+diagonal+long+range+order&pg=PA140 |title=The Superfluid Phases of Helium 3 |last2=Wolfle |first2=Peter |date=2013-10-17 |publisher=Courier Corporation |isbn=978-0-486-48631-4 |language=en}}</ref>

== Density matrix and long-range order ==
In terms of [canonical quantization](/source/canonical_quantization), the one-body [density matrix](/source/density_matrix) can be written as<ref name=":1" /><math display="block">\rho (\mathbf r,\mathbf r')=\langle \hat{\psi}(\mathbf r)\hat{\psi}^\dagger(\mathbf r')\rangle</math>where <math display="inline">\langle\cdot\rangle</math> indicates the expectation value of the state of the system, <math display="inline">\hat{\psi}(\mathbf r)</math> is the field operator describing the system at position <math display="inline">\mathbf r</math>. The <math display="inline">\rho(\mathbf r,\mathbf r' )</math> with <math display="inline">\mathbf r\neq\mathbf r'</math> are the off-diagonal elements and  <math display="inline">\rho(\mathbf r,\mathbf r) = n(\mathbf r)</math> is the diagonal element describing the local density. The density matrix is normalized such that integrating over the volume, as<math display="block">\int n(\mathbf r)\mathrm d^3 \mathbf r=N,</math>recovers the number of particles ''N''.

If the <math display="inline">n(\mathbf r)</math> is not constant, then the system has a diagonal long-range order (DLRO). For example, a crystal lattice has a diagonal element <math display="inline"> n(\mathbf r)</math> that oscillates with <math display="inline">\mathbf r</math> (DLRO).<ref name=":1" /> 

== Bosonic systems ==
To understand if a system has a off-diagonal long-range order (ODLRO) one calculates the <math display="inline">\rho(\mathbf r,\mathbf r' )</math> for large separations <math display="inline">|\mathbf r-\mathbf r' |\to\infty</math>.<ref name=":1" /> If the off-diagonal terms <math display="inline">\rho(\mathbf r,\mathbf r' )</math> are not null at long-range, then the systems possesses an off-diagonal long-range order (ODLRO). 

The '''Penrose–Onsager criterion''' stipulates that a if a bosonic system has an ODLRO, the system presents macroscopic quantum behaviour.<ref name=":4">{{Cite book |last1=Deveaud |first1=Benoît |url=https://books.google.com/books?id=aIdePcKrn4oC&dq=odlro+penrose-onsager+criterion&pg=PA457 |title=Quantum Coherence in Solid State Systems |last2=Quattropani |first2=Antonio |last3=Schwendimann |first3=Paolo |date=2009 |publisher=IOS Press |isbn=978-1-60750-039-1 |language=en}}</ref>

For [Bose–Einstein condensates](/source/Bose%E2%80%93Einstein_condensate) it can be shown that below a certain temperature<ref name=":4" /><math display="block">\lim_{|\mathbf r-\mathbf r'|\to\infty}\rho (\mathbf r,\mathbf r')=N/V</math>where ''V'' is the volume. Thus Bose–Einstein condensates possess an ODLRO.<ref>{{Cite book |last1=Proukakis |first1=Nick P. |url=https://books.google.com/books?id=39G6CgAAQBAJ&dq=odlro+penrose-onsager+criterion&pg=PA436 |title=Quantum Gases: Finite Temperature And Non-equilibrium Dynamics |last2=Gardiner |first2=Simon A. |last3=Davis |first3=Matthew |last4=Szymanska |first4=Marzena |date=2013-02-21 |publisher=World Scientific |isbn=978-1-908979-70-4 |language=en}}</ref> The existence of an ODLRO is the consequence of many properties in [superfluidity](/source/superfluidity), like irrotational flow and quantization of vortex.<ref name=":3" /> A system that possesses both DLRO like a crystal and ODLRO is expected to be a [supersolid](/source/supersolid).<ref>{{Cite book |last1=Girvin |first1=Steven M. |url=https://books.google.com/books?id=YYKFDwAAQBAJ&dq=girvin+condensed+matter+odlro&pg=PA535 |title=Modern Condensed Matter Physics |last2=Yang |first2=Kun |date=2019-02-28 |publisher=Cambridge University Press |isbn=978-1-107-13739-4 |language=en}}</ref><ref>{{Cite book |last=Mahan |first=Gerald D. |url=https://books.google.com/books?id=v3FVOH2XK4gC&dq=lasers+odlro&pg=PA240 |title=Condensed Matter in a Nutshell |date=2011 |publisher=Princeton University Press |isbn=978-0-691-14016-2 |language=en}}</ref>

Light can also possess ODLRO, as is the case of coherent sources like [lasers](/source/Laser).<ref>{{Cite journal |last1=Cummings |first1=Frederick W. |last2=Johnston |first2=James R. |date=1966-11-04 |title=Theory of Superfluidity |url=https://link.aps.org/doi/10.1103/PhysRev.151.105 |journal=Physical Review |volume=151 |issue=1 |pages=105–112 |doi=10.1103/PhysRev.151.105 |bibcode=1966PhRv..151..105C |url-access=subscription }}</ref>

== Fermionic systems ==
Systems of [fermion](/source/fermion)s cannot possess a one-body ODLRO.<ref name=":5">{{Cite report |url=https://www.osti.gov/biblio/1343596 |title=The coherent electron. |last1=Peshkin |first1=Murray |last2=Imry |first2=Y. |date=1996-12-31 |publisher=Argonne National Laboratory (ANL) |osti=1343596 |language=English}}</ref> However, when positive interactions are present the formation of [Cooper pairs](/source/Cooper_pair) allows for a [fermionic condensate](/source/fermionic_condensate) with a two-body ODLRO.<ref name=":6">{{Cite book |last=Annett |first=James F. |url=https://books.google.com/books?id=WZcXmBrZIc8C&dq=superconductivity+odlro&pg=PA116 |title=Superconductivity, Superfluids and Condensates |date=2004-03-25 |publisher=OUP Oxford |isbn=978-0-19-850756-7 |language=en}}</ref><ref name=":7">{{Cite book |last1=Chakraborty |first1=Tapash |url=https://books.google.com/books?id=v5nzCAAAQBAJ&dq=odlro+fractional+hall+effect&pg=PA241 |title=The Quantum Hall Effects: Integral and Fractional |last2=Pietiläinen |first2=Pekka |date=2013-03-12 |publisher=Springer Science & Business Media |isbn=978-3-642-79319-6 |language=en}}</ref> In the case of superconductivity, one can define the two-body density matrix as:<ref name=":6" /><math display="block">\rho^{(2)} (\mathbf r_1,\mathbf r_2;\mathbf r_1',\mathbf r_2')=\langle \hat{\psi}_\uparrow(\mathbf r_1)\hat{\psi}_\downarrow(\mathbf r_2)\hat{\psi}^\dagger_\downarrow(\mathbf r_2')\hat{\psi}^\dagger_\uparrow(\mathbf r_1')\rangle</math>where <math display="inline">s=\uparrow,\downarrow</math> indicates the two spin values for a [spin-1/2](/source/spin-1%2F2) particle like the electron. For large range, <math display="inline">|\mathbf r_1-\mathbf r_1' |\to\infty</math> and <math display="inline">|\mathbf r_2-\mathbf r_2' |\to\infty</math>, a finite value indicates an ODLRO.<ref name=":6" /> 

The presence of an  ODLRO indicates macroscopic quantum behaviour, this is the case of superconductivity.<ref name=":5" /> Normal conductors have no ODLRO.<ref name=":5" />

The ODLRO explains flux quantization in superconductors. In a superconducting ring, the [magnetic flux](/source/magnetic_flux) is quantized in units of [magnetic flux quanta](/source/magnetic_flux_quantum) <math>h/2e</math> where <math>h</math> is the [Planck constant](/source/Planck_constant) and <math>e</math> the [elementary charge](/source/elementary_charge), instead of the usual <math>h/e</math> for normal conductors. The necessity of pair ODLRO implies that the basic unit of coherent states in superconductors consists of pair of electrons.<ref name=":5" /> The [Meissner effect](/source/Meissner_effect) is also a consequence of ODLRO.<ref name=":3" />

== Other systems ==
[Anyons](/source/Anyon), particles that are neither bosons or fermions, are expected to be present in the [fractional quantum Hall effect](/source/fractional_quantum_Hall_effect) (FQHE).<ref name=":7" /> The existence of an ODLRO due to anyons has been investigated and calculated under certain conditions to explain the FQHE.<ref name=":7" />

==References==
{{reflist}}

Category:Condensed matter physics
Category:Phases of matter
Category:Bose–Einstein condensates
Category:Superconductivity
Category:Superfluidity

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Adapted from the Wikipedia article [Off-diagonal long-range order](https://en.wikipedia.org/wiki/Off-diagonal_long-range_order) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Off-diagonal_long-range_order?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.