Kosterlitz-Thouless transition
Encyclopedia
In statistical mechanics
, a part of mathematical physics
, the Kosterlitz–Thouless transition, or Berezinsky–Kosterlitz–Thouless transition, is a kind of phase transition
that appears in the XY model
in 2 spatial dimensions. The XY model
is a 2-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 Goldstone modes (see Goldstone boson
) 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.
The transition is named for John M. Kosterlitz
, David J. Thouless, and Vadim L'vovich Berezinskiĭ (Вади́м Льво́вич Берези́нский).
Rigorously the transition is not completely understood, but the existence of two phases was proved by and .
(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.
are topologically stable configurations. It is found that the high-temperature disordered phase with exponential correlation is a result of the formation of vortices. Vortex generation becomes thermodynamically favorable at the critical temperature
of the KT transition. At temperatures below this, Vortex generation has a power law correlation.
Many systems with KT transitions involve the dissociation of bound anti-parallel vortex pairs, called vortex–antivortex pairs, into unbound vortices rather than vortex generation. In these systems, thermal generation of vortices produces an even number 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,
,the system undergoes a transition at a critical temperature, 
. Below 
, there are only bound vortex–antivortex pairs. Above Tc, there are free vortices.
, where 
is a parameter depending upon the system the vortex is in, 
is the system size, and 
is the radius of the vortex core. We assume
. The number of possible positions of any vortex in the system is approximately 
. From Boltzmann's law, the entropy
is
, where 
is Boltzmann's constant. Thus, the Helmholtz free energy
is
When
, the system will not have a vortex. However when 
, the conditions are sufficient for a vortex to be in the system. We define the transition temperature for 
. Thus, the critical temperature 
is
Vortices are able to form above this critical temperature, but not below. The KT transition can be observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage (I-V) measurements. Above
, the relation will be linear 
. Just below 
, the relation will be 
, as the number of free vortices will go as 
. This jump from linear dependence is indicative of a KT transition and may be used to determine 
. This approach was used in Resnic et al. to confirm the KT transition in proximity-coupled Josephson junction arrays.
The energy is given by
and the Boltzmann factor
is exp(−βE).
If we take the contour integral
over any closed path γ, we would expect it to be zero if γ is contractible, which is what we would expect for a planar curve. But here is the catch. Assume the XY theory has a UV cutoff which requires some UV completion. Then, we can have punctures in the plane, holes so to speak so that if γ is a closed path which winds once around the puncture, 
is only an integer multiple of 2π. These punctures are called vortices and if γ is a closed path which only winds once counterclockwise around the puncture and its winding number about any other puncture is zero, then the integer multiplicity can be attached to the vortex itself. Let's say a field configuration has n punctures at xi, i = 1, ..., n with multiplicities ni. Then, φ decomposes into the sum of a field configuration with no punctures, φ0 and 
where we have switched to the complex plane coordinates for convenience. The latter term has branch cuts, of course, but since φ is only defined modulo 2π they are unphysical.
Now,
It's easy to see that unless
, the second term is positive infinite, making the Boltzmann factor zero which means that we can forget all about it.
When
, the second term is equal to 
.
This is nothing other than a Coulomb gas. The scale L contributes nothing but a constant.
Let's look at the case with only one vortex of multiplicity one and one vortex of multiplicity -1. At low temperatures, i.e. large β, because of the Boltzmann factor, the vortex–antivortex pair tends to be extremely close to one another. In fact, their separation would be around the cutoff scale. With more vortex–antivortex pairs, we have a collection of vortex-antivortex dipoles. At large temperatures, i.e. small β, the probability distribution swings the other way around and we have a plasma of vortices and antivortices. The phase transition between the two is the Kosterlitz–Thouless phase transition.
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, a part of mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, the Kosterlitz–Thouless transition, or Berezinsky–Kosterlitz–Thouless transition, is a kind of phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
that appears in the XY model
XY model
The classical XY model is a model of statistical mechanics. It is the special case of the n-vector model for n=2.-Definition:...
in 2 spatial dimensions. The XY model
XY model
The classical XY model is a model of statistical mechanics. It is the special case of the n-vector model for n=2.-Definition:...
is a 2-dimensional vector spin model that possesses U(1) or circular symmetry. This system is not expected to possess a normal second-order phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
. This is because the expected ordered phase of the system is destroyed by transverse fluctuations, i.e. the Goldstone modes (see Goldstone boson
Goldstone boson
In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...
) associated with this broken continuous symmetry
Continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the...
, which logarithmically diverge with system size.
This is a specific case of what is called the Mermin–Wagner theorem in spin systems.
The transition is named for John M. Kosterlitz
John M. Kosterlitz
John Michael Kosterlitz is a professor of physics at Brown University. He received the Lars Onsager Prize from the American Physical Society in 2000, and the Maxwell Medal and Prize from the British Institute of Physics in 1981, for his work on the Kosterlitz-Thouless transition. He is a Fellow of...
, David J. Thouless, and Vadim L'vovich Berezinskiĭ (Вади́м Льво́вич Берези́нский).
Rigorously the transition is not completely understood, but the existence of two phases was proved by and .
KT Transition: disordered phases with different correlations
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 functionCorrelation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
(see statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
) 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
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
of infinite order.
Role of vortices
In the 2D XY model, vorticesQuantum vortex
In physics, a quantum vortex is a topological defect exhibited in superfluids and superconductors. Superfluids and superconductors are states of matter without friction. They exist only at very low temperatures. The existence of these quantum vortices was independently predicted by Richard Feynman...
are topologically stable configurations. It is found that the high-temperature disordered phase with exponential correlation is a result of the formation of vortices. Vortex generation becomes thermodynamically favorable at the critical temperature
of the KT transition. At temperatures below this, Vortex generation has a power law correlation.Many systems with KT transitions involve the dissociation of bound anti-parallel vortex pairs, called vortex–antivortex pairs, into unbound vortices rather than vortex generation. In these systems, thermal generation of vortices produces an even number 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,
,the system undergoes a transition at a critical temperature, . Below , there are only bound vortex–antivortex pairs. Above Tc, there are free vortices.Informal description
There is a very elegant thermodynamic argument for the KT transition. The energy of a single vortex is of the form, where is a parameter depending upon the system the vortex is in, is the system size, and is the radius of the vortex core. We assume
. The number of possible positions of any vortex in the system is approximately . From Boltzmann's law, the entropyEntropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
is
, where is Boltzmann's constant. Thus, the Helmholtz free energyHelmholtz free energy
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...
is
When
, the system will not have a vortex. However when , the conditions are sufficient for a vortex to be in the system. We define the transition temperature for . Thus, the critical temperature isVortices are able to form above this critical temperature, but not below. The KT transition can be observed experimentally in systems like 2D Josephson junction arrays by taking current and voltage (I-V) measurements. Above
, the relation will be linear . Just below , the relation will be , as the number of free vortices will go as . This jump from linear dependence is indicative of a KT transition and may be used to determine . This approach was used in Resnic et al. to confirm the KT transition in proximity-coupled Josephson junction arrays.Rigorous analysis
We have a field φ over the plane which takes on values in S1. For convenience, we work with its universal cover R instead but identify any two values of φ(x) which differs by an integer multiple of 2π.The energy is given by
and the Boltzmann factor
Boltzmann factor
In physics, the Boltzmann factor is a weighting factor that determines the relative probability of a particle to be in a state i in a multi-state system in thermodynamic equilibrium at temperature T...
is exp(−βE).
If we take the contour integral
over any closed path γ, we would expect it to be zero if γ is contractible, which is what we would expect for a planar curve. But here is the catch. Assume the XY theory has a UV cutoff which requires some UV completion. Then, we can have punctures in the plane, holes so to speak so that if γ is a closed path which winds once around the puncture, is only an integer multiple of 2π. These punctures are called vortices and if γ is a closed path which only winds once counterclockwise around the puncture and its winding number about any other puncture is zero, then the integer multiplicity can be attached to the vortex itself. Let's say a field configuration has n punctures at xi, i = 1, ..., n with multiplicities ni. Then, φ decomposes into the sum of a field configuration with no punctures, φ0 and where we have switched to the complex plane coordinates for convenience. The latter term has branch cuts, of course, but since φ is only defined modulo 2π they are unphysical.Now,
It's easy to see that unless
, the second term is positive infinite, making the Boltzmann factor zero which means that we can forget all about it.When
, the second term is equal to .This is nothing other than a Coulomb gas. The scale L contributes nothing but a constant.
Let's look at the case with only one vortex of multiplicity one and one vortex of multiplicity -1. At low temperatures, i.e. large β, because of the Boltzmann factor, the vortex–antivortex pair tends to be extremely close to one another. In fact, their separation would be around the cutoff scale. With more vortex–antivortex pairs, we have a collection of vortex-antivortex dipoles. At large temperatures, i.e. small β, the probability distribution swings the other way around and we have a plasma of vortices and antivortices. The phase transition between the two is the Kosterlitz–Thouless phase transition.
See also
- Goldstone bosonGoldstone bosonIn particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...
- Ising modelIsing modelThe Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
- Lambda transitionLambda transitionThe λ universality class is probably the most important group in condensed matter physics. It regroups several systems possessing strong analogies, namely, superfluids, superconductors and smectics...
- Potts modelPotts modelIn statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
- Quantum vortexQuantum vortexIn physics, a quantum vortex is a topological defect exhibited in superfluids and superconductors. Superfluids and superconductors are states of matter without friction. They exist only at very low temperatures. The existence of these quantum vortices was independently predicted by Richard Feynman...
- Superfluid filmSuperfluid filmSuperfluidity and superconductivity are macroscopic manifestations of quantum mechanics. There is considerable interest, both theoretical and practical, in these quantum phase transitions. There has been a tremendous amount of work done in the field of phase transitions and critical phenomenon in...
- Topological defectTopological defectIn mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of...
Books
- H. KleinertHagen KleinertHagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...
, Gauge Fields in Condensed Matter, Vol. I, " SUPERFLOW AND VORTEX LINES", pp. 1–742, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online: Vol. I. Read pp. 618–688); - H. KleinertHagen KleinertHagen Kleinert is Professor of Theoretical Physics at the Free University of Berlin, Germany , at theWest University of Timişoara, at thein Bishkek. He is also of the...
, Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation, World Scientific (Singapore, 2008) (also available online: here)