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

In condensed matter physics, an '''off-diagonal long-range order''' ('''ODLRO''') is a feature of macroscopic quantum phenomena. It refers to off-diagonal elements in the density matrix separated in space in a many-body quantum mechanical system. An ODLRO implies correlations between distant particles in the system, indicating quantum behaviour. The concept is analogous to coherences and higher order coherences from quantum optics. An ODLRO is an indication of 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 and in many classical systems.<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 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 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 and Bose–Einstein condensates.<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 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.<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, the one-body 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 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, 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.<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.<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 fermions 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 allows for a 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 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 is quantized in units of magnetic flux quanta <math>h/2e</math> where <math>h</math> is the Planck constant and <math>e</math> the 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 is also a consequence of ODLRO.<ref name=":3" />

== Other systems == Anyons, particles that are neither bosons or fermions, are expected to be present in the 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