Berezinskii–Kosterlitz–Thouless transition

The XY model is a two-dimensional vector spin model that possesses U(1) or circular symmetry. This system is not expected to possess a normal second-order phase transition. This is because the expected ordered phase of the system is destroyed by transverse fluctuations, i.e. the Nambu-Goldstone modes associated with this broken continuous symmetry, which logarithmically diverge with system size.

This is a specific case of what is called the Mermin–Wagner theorem in spin systems.

Rigorously the transition is not completely understood, but the existence of two phases was proved by {{harvtxt|McBryan|Spencer|1977}} and {{harvtxt|Fröhlich|Spencer|1981}}.

In the XY model in two dimensions, a second-order phase transition is not seen. However, one finds a low-temperature quasi-ordered phase with a correlation function (see statistical mechanics) that decreases with the distance like a power, which depends on the temperature. The transition from the high-temperature disordered phase with the exponential correlation to this low-temperature quasi-ordered phase is a Kosterlitz–Thouless transition.

It is a phase transition of infinite order.

In the 2-D XY model, vortices are topologically stable configurations. It is found that the high-temperature disordered phase with exponential correlation decay is a result of the formation of vortices. Vortex generation becomes thermodynamically favorable at the critical temperature $T\_\backslash text\{c\}$ of the Kosterlitz–Thouless transition. At temperatures below this, vortex generation has a power law correlation.

Kosterlitz–Thouless transitions is described as a dissociation of bound vortex pairs with opposite circulations, called vortex–antivortex pairs, first described by Vadim Berezinskii. In these systems, thermal generation of vortices produces an even number of vortices of opposite sign. Bound vortex–antivortex pairs have lower energies than free vortices, but have lower entropy as well. In order to minimize free energy, $F=E-TS$, the system undergoes a transition at a critical temperature, $T\_\backslash text\{c\}$. Below $T\_\backslash text\{c\}$, there are only bound vortex–antivortex pairs. Above $T\_\backslash text\{c\}$, there are free vortices.

There is an elegant thermodynamic argument for the Kosterlitz–Thouless transition. The energy of a single vortex is $\backslash kappa\backslash ln(R/a)$, where $\backslash kappa$ is a parameter that depends upon the system in which the vortex is located, $R$ is the system size, and $a$ is the radius of the vortex core. One assumes $R\backslash gg\; a$. In the 2D system, the number of possible positions of a vortex is approximately $(R/a)^2$. From Boltzmann's entropy formula, $S=\; k\_\{\backslash rm\; B\}\; \backslash ln\; W$ (with W is the number of states), the entropy is $S=2k\_\{\backslash rm\; B\}\backslash ln(R/a)$, where $k\_\{\backslash rm\; B\}$ is the Boltzmann constant. Thus, the Helmholtz free energy is

: $F\; =\; E\; -\; TS\; =\; (\backslash kappa\; -\; 2k\_\{\backslash rm\; B\}T)\backslash ln(R/a).$

When $F>0$, the system will not have a vortex. On the other hand, when , entropic considerations favor the formation of a vortex. The critical temperature above which vortices may form can be found by setting $F=0$ and is given by

: $T\_\backslash text\{c\}\; =\; \backslash frac\{\backslash kappa\}\{2k\_\{\backslash rm\; B\}\}.$

The Kosterlitz–Thouless transition can be observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage (I-V) measurements. Above $T\_\backslash text\{c\}$, the relation will be linear $V\; \backslash sim\; I$. Just below $T\_c$, the relation will be $V\; \backslash sim\; I^3$, as the number of free vortices will go as $I^2$. This jump from linear dependence is indicative of a Kosterlitz–Thouless transition and may be used to determine $T\_\backslash text\{c\}$. This approach was used in Resnick et al.{{sfn|Resnick|Garland|Boyd|Shoemaker|1981}} to confirm the Kosterlitz–Thouless transition in proximity-coupled Josephson junction arrays.

The following discussion uses field theoretic methods. Assume a field φ(x) defined in the plane which takes on values in $S^1$, so that $\backslash phi(x)$ is identified with $\backslash phi(x)\; +\; 2\backslash pi$. That is, the circle is realized as $S^1\; =\; \backslash mathbb\{R\}/2\backslash pi\backslash mathbb\{Z\}$.

The energy is given by

: $E\; =\; \backslash int\; \backslash frac\{1\}\{2\}\; \backslash nabla\backslash phi\backslash cdot\backslash nabla\backslash phi\; \backslash ,\; d^2\; x$

and the Boltzmann factor is $\backslash exp\; (-\backslash beta\; E)$.

Taking a contour integral $\backslash textstyle\; \backslash oint\_\backslash gamma\; d\backslash phi\; =\; \backslash oint\_\backslash gamma\; \backslash frac\{d\backslash phi\}\{dx\}dx$ over any contractible closed path $\backslash gamma$, we would expect it to be zero (for example, by the fundamental theorem of calculus. However, this is not the case due to the singular nature of vortices (which give singularities in $\backslash phi$).

To render the theory well-defined, it is only defined up to some energetic cut-off scale $\backslash Lambda$, so that we can puncture the plane at the points where the vortices are located, by removing regions with size of order $1/\backslash Lambda$. If $\backslash gamma$ winds counter-clockwise once around a puncture, the contour integral $\backslash textstyle\; \backslash oint\_\backslash gamma\; d\backslash phi$ is an integer multiple of $2\backslash pi$. The value of this integer is the index of the vector field $\backslash nabla\; \backslash phi$.

Suppose that a given field configuration has $N$ punctures located at $x\_i,\; i=1,\backslash dots,N$ each with index $n\_i=\backslash pm\; 1$. Then, $\backslash phi$ decomposes into the sum of a field configuration with no punctures, $\backslash phi\_0$ and $\backslash textstyle\; \backslash sum\_\{i=1\}^N\; n\_i\backslash arg(z-z\_i)$, where we have switched to the complex plane coordinates for convenience. The complex argument function has a branch cut, but, because $\backslash phi$ is defined modulo $2\backslash pi$, it has no physical consequences.

Now,

:

If $\backslash textstyle\; \backslash sum\_\{i=1\}^N\; n\_i\; \backslash neq\; 0$, the second term is positive and diverges in the limit $\backslash Lambda\; \backslash to\; \backslash infty$: configurations with unbalanced numbers of vortices of each orientation are never energetically favoured.

However, if the neutral condition $\backslash textstyle\; \backslash sum\_\{i=1\}^N\; n\_i=0$ holds, the second term is equal to , which is the total potential energy of a two-dimensional Coulomb gas. The scale L is an arbitrary scale that renders the argument of the logarithm dimensionless.

Assume the case with only vortices of multiplicity $\backslash pm\; 1$. At low temperatures and large $\backslash beta$ the distance between a vortex and antivortex pair tends to be extremely small, essentially of the order $1/\backslash Lambda$. At large temperatures and small $\backslash beta$ this distance increases, and the favoured configuration becomes effectively the one of a gas of free vortices and antivortices. The transition between the two different configurations is the Kosterlitz–Thouless phase transition, and the transition point is associated with an unbinding of vortex-antivortex pairs.

{{div col|colwidth=20em}}

• KTHNY theory
• Goldstone boson
• Composite fermion
• Lambda transition
• Ising model
• Potts model
• Topological defect
• Quantum vortex
• Superfluid film
• Hexatic phase

{{div col end}}

{{Reflist}}

• {{citation|first=В. Л.|last=Березинский|title=Разрушение дальнего порядка в одномерных и двумерных системах с непрерывной группой симметрии I. Классические системы|journal=ЖЭТФ|volume=59|issue=3|year=1970|pages=907–920|language=ru}}. Translation available: {{citation|first=V. L.|last=Berezinskii|title=Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems|journal=Sov. Phys. JETP|volume=32|issue=3|pages=493–500|year=1971|url=http://www.jetp.ras.ru/cgi-bin/dn/e_032_03_0493.pdf|bibcode = 1971JETP...32..493B }}
• {{citation|first=В. Л.|last=Березинский|title=Разрушение дальнего порядка в одномерных и двумерных системах с непрерывной группой симметрии II. Квантовые системы|journal=ЖЭТФ|volume=61|issue=3|year=1971|pages=1144–1156|language=ru}}. Translation available: {{citation|first=V. L.|last=Berezinskii|title=Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group II. Quantum systems|journal=Sov. Phys. JETP|volume=34|issue=3|pages=610–616|year=1972|url=http://www.jetp.ras.ru/cgi-bin/dn/e_034_03_0610.pdf|bibcode = 1972JETP...34..610B }}
• {{citation|first1=J. M.|last1=Kosterlitz|first2=D. J.|last2=Thouless|title=Ordering, metastability and phase transitions in two-dimensional systems|journal=Journal of Physics C: Solid State Physics|volume=6|issue=7|pages=1181–1203|year=1973|doi=10.1088/0022-3719/6/7/010|bibcode = 1973JPhC....6.1181K }}
• {{citation|first1=O.|title=On the decay of correlations in SO(n)-symmetric ferromagnets|last1=McBryan|first2=T.|last2=Spencer|journal=Commun. Math. Phys.|volume=53|issue=3|page=299|year=1977|bibcode = 1977CMaPh..53..299M |doi = 10.1007/BF01609854 |s2cid=119587247}}
• B. I. Halperin, D. R. Nelson, Phys. Rev. Lett. 41, 121 (1978)
• A. P. Young, Phys. Rev. B 19, 1855 (1979)
• {{citation | first1=D.J.|last1=Resnick|first2=J.C.| last2=Garland |first3=J.T.| last3=Boyd|first4=S.| last4=Shoemaker|first5=R.S.|last5=Newrock|title=Kosterlitz Thouless Transition in Proximity Coupled Superconducting Arrays|journal=Phys. Rev. Lett.|volume=47|issue=21|page=1542|year=1981|bibcode = 1981PhRvL..47.1542R |doi = 10.1103/PhysRevLett.47.1542 }}
• {{citation|first1=Jürg|last1=Fröhlich|first2=Thomas|last2=Spencer|title=The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas|journal=Comm. Math. Phys.|volume=81|year=1981|issue=4|pages=527–602|doi=10.1007/bf01208273|bibcode = 1981CMaPh..81..527F |s2cid=73555642|url=http://projecteuclid.org/euclid.cmp/1103920388 }}
• {{citation|author= Z. Hadzibabic|title=Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas|journal=Nature|volume=41|issue=7097|year=2006|pages=1118–21|doi=10.1038/nature04851|pmid=16810249|arxiv = cond-mat/0605291 |bibcode = 2006Natur.441.1118H |s2cid=4314014|display-authors=etal}}
• {{citation|author= M. Mondal|title=Role of the vortex-core energy on the Beresinkii-Kosterlitz-Thouless transition in thin films of NbN|journal=Phys. Rev. Lett.|volume=107|issue=21|page=217003|year=2011|doi=10.1103/PhysRevLett.107.217003|pmid=22181915|arxiv = 1108.0912 |display-authors=etal|bibcode=2011PhRvL.107u7003M|s2cid=34729666}}

• J.V. Jose, 40 Years of Berezinskii–Kosterlitz–Thouless Theory, World Scientific, 2013, {{ISBN|978-981-4417-65-5}}
• H. Kleinert, Gauge Fields in Condensed Matter, Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1–742, [https://archive.today/20060514143926/http://www.worldscibooks.com/physics/0356.htm World Scientific (Singapore, 1989)]; Paperback {{ISBN|9971-5-0210-0}} '' (also available online: [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html Vol. I]. Read pp. 618–688);
• H. Kleinert, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, [https://web.archive.org/web/20080315225354/http://www.worldscibooks.com/physics/6742.html World Scientific (Singapore, 2008)] (also available online: [http://www.physik.fu-berlin.de/~kleinert/re.html#B9 here])

{{Authority control}}

{{DEFAULTSORT:Kosterlitz-Thouless Transition}}

Category:Statistical mechanics

Category:Lattice models

Category:Phase transitions