词条 | Classical XY model |
释义 |
The classical XY model (sometimes also called classical rotor (rotator) model or O(2) model) is a lattice model of statistical mechanics. It is the special case of the n-vector model for {{math|n {{=}} 2}}. DefinitionGiven a {{mvar|D}}-dimensional lattice {{math|Λ}}, per each lattice site {{math|j ∈ Λ}} there is a two-dimensional, unit-length vector {{math|sj {{=}} (cos θj, sin θj)}} The spin configuration, {{math|s {{=}} (sj)j ∈ Λ}} is an assignment of the angle {{math|−π < θj ≤ π}} for each {{math|j ∈ Λ}}. Given a translation-invariant interaction {{math|Jij {{=}} J(i − j)}} and a point dependent external field , the configuration energy is The case in which {{math|Jij {{=}} 0}} except for {{mvar|ij}} nearest neighbor is called nearest neighbor case. The configuration probability is given by the Boltzmann distribution with inverse temperature {{math|β ≥ 0}}: where {{mvar|Z}} is the normalization, or partition function.[1] The notation indicates the expectation of the random variable {{math|A(s)}} in the infinite volume limit, after periodic boundary conditions have been imposed. General properties
Hence the critical {{mvar|β}} of the XY model cannot be smaller than the double of the critical temperature of the Ising model One dimension
therefore the partition function factorizes under the change of coordinates That gives Finally where is the modified Bessel function of the first kind. The same computation for periodic boundary condition (and still {{math|h {{=}} 0}}) requires the transfer matrix formalism.[4] Two dimensions
Besides, cluster expansion shows that the spin correlations cluster exponentially fast: for instance
but the decay of the correlations is only power law: Fröhlich and Spencer[5] found the lower bound while McBryan and Spencer found the upper bound, for any The fact that at high temperature correlations decay exponentially fast, while at low temperatures decay with power law, even though in both regimes {{math|M(β) {{=}} 0}}, is called Kosterlitz–Thouless transition. The continuous version of the XY model is often used to model systems that possess order parameters with the same kinds of symmetry, e.g. superfluid helium, hexatic liquid crystals. This is what makes them peculiar from other phase transitions which are always accompanied with a symmetry breaking. Topological defects in the XY model lead to a vortex-unbinding transition from the low-temperature phase to the high-temperature disordered phase. Three and higher dimensionsIndependently of the range of the interaction, at low enough temperature the magnetization is positive.
Besides, cluster expansion shows that the spin correlations cluster exponentially fast: for instance
Besides, there exists a 1-parameter family of extremal states, , such that but, conjecturally, in each of these extremal states the truncated correlations decay algebraically. In general, the XY model can be seen as a specialization of Stanley's n-vector model [6] See also
Notes1. ^{{cite book|last1=Chaikin|first1=P.M.|last2=Lubensky|first2=T.C.|title=Principles of Condensed Matter Physics|year=2000|publisher=Cambridge University Press|isbn=978-0521794503|url=https://books.google.com/books?id=P9YjNjzr9OIC&printsec=frontcover&cad=0#v=onepage&q&f=false}} 2. ^{{cite journal|last=Ginibre|first=J.|title=General formulation of Griffiths' inequalities|journal=Comm. Math. Phys.|year=1970|volume=16|issue=4|pages=310–328|doi=10.1007/BF01646537|bibcode = 1970CMaPh..16..310G }} 3. ^{{cite journal| last1=Aizenman| first1=M.|last2=Simon |first2=B.|title=A comparison of plane rotor and Ising models|journal=Phys. Lett. A|year=1980|volume=76|url=http://www.sciencedirect.com/science/article/pii/0375960180904934|doi=10.1016/0375-9601(80)90493-4|bibcode = 1980PhLA...76..281A }} 4. ^{{cite journal|last=Mattis|first=D.C.|title=Transfer matrix in plane-rotator model|journal=Phys. Lett.|year=1984|volume=104 A|url=http://www.sciencedirect.com/science/article/pii/0375960184908168|doi=10.1016/0375-9601(84)90816-8|bibcode = 1984PhLA..104..357M }} 5. ^{{cite journal|last1=Fröhlich|first1=J.|last2=Spencer|first2=T. |title=The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas|journal=Comm. Math. Phys.| year=1981| volume=81|issue=4|pages=527–602|url=http://projecteuclid.org/euclid.cmp/1103920388|doi=10.1007/bf01208273|bibcode = 1981CMaPh..81..527F }} 6. ^{{cite journal|last=Stanley|first=H.E.|title=Dependence of Critical Properties on Dimensionality of Spins|journal=Phys. Rev. Lett.|volume=20|pages=589|doi=10.1103/PhysRevLett.20.589|url=http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.20.589|bibcode = 1968PhRvL..20..589S }} References
Further reading
External links
1 : Lattice models |
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