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Kosterlitz-Thouless (KT) Phase Transition An Introduction Johar M. Ashfaque Conformal field theories arise near a phase transition when there are no scale involved. One usually talks about thermal phase transitions but a lot of the recent interest in condensed matter physics has been around quantum phase transitions which are phase transitions that happen as the coupling parameter is varied. In this case it turns out that the phase transition as well as the new phase arising from it, the quantum critical phase can be described in terms of the modifications of a conformal field theory. I.
THERMAL PHASE TRANSITIONS AND CONFORMAL FIELD THEORY
Thermal phase transitions occur at a critical temperature TC . Near the critical point, for T → TC we have the critical behaviour for the thermodynamic quantities that characterize the material. For a second order phase transition, the second derivatives of the thermodynamic potential C, χm , κe for example blow C up in a specific way, namely as power laws in T −T : TC −α T − TC C∝ , TC −γm T − TC χm ∝ , TC −γe T − TC κe ∝ . TC This behaviour is due to the fact that the correlation length that is the distance over which correlation functions of physical quantities is non-trivial, blows up in a similar way −ν T − TC . ξ∝ TC The 2-point correlation functions at small distances go like 1 e−r/ξ rd−2+η but ξ would eventually become much larger than the lattice spacing a in which case we can adopt an effective continuum field theory description. G(r) ∝
A.
Enter Conformal Field Theory
Conformal field theories can be defined as the Euclidean theories (on the spatial dimensions) near a phase transition in condensed matter systems but their applications extend beyond that for example string theory which is defined as a conformal field theory on the Minkowskian 2-dimensional world-sheet of the string. A conformal field theory is a theory in flat space invariant under a conformal transformation which is a generalization of the scale transformation. A scale transformation multiplies the metric with a constant ds2 −→ ds02 = λ2 ds2 whereas a conformal transformation is a transformation xµ → x0µ still in flat space that multiplies the metric with a conformal factor ds2 −→ ds02 = Ω2 (x)ds2 . Note that conformal transformations are transformations on flat space.
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CONFORMAL FIELD THEORY IN TWO DIMENSIONS
In d > 2 the solution to the condition ds02 = â„Ś2 (x)ds2 that defines conformal transformations gives the symmetry group SO(d+1, 1) for d Euclidean dimensions and SO(d, 2) for (d − 1, 1) Minkowski dimensions. A.
d=2
In general, we would define general relativity tensors with covariant indices by their transformation law. But in two dimensions we can form complex coordinates z = z1 + iz2 , z = z1 − iz2 and express everything in terms of them. Then it is easy to see that a conformal transformation would just be a holomorphic transformation z 0 = f (z),
z 0 = f (z)
since then ds2 = dz 0 dz 0 = |f 0 (z)|2 dzdz ≥ â„Ś2 (z, z)dzdz. In two dimensions expressed in complex coordinates, a general relativity tensor would be an object with an integer number of z and z indices. But since conformal transformations can be embedded in general coordinate transformations we should take advantage of the definition of general relativity tensors and define a notion of tensor for conformal transformations only. Such a tensor is called a primary field of dimensions (h, h0 ) and it transforms as a general relativity tensor with h number of z indices and h0 indices of z indices 0 h 0 h0 dz dz 0 . Tz...zz...z = Tz...zz...z dz dz But unlike the case of the general relativity tensors, we don’t need to have integer values for h and h0 0 since they are not associated with a physical number of indices. The primary fields are denoted by φ(h,h ) (z, z). The infinitesimal transformation of a primary field under the transformation z 0 = z + (z) is δ φ(z) = (z)âˆ‚Ď†(z) + h∂ (z)φ(z) + h.c.. III.
QUANTUM PHASE TRANSITIONS
In quantum phase transitions, generically the phases are different topologically. Again, like in the case of the thermal phase transitions, the system in the vicinity of the quantum phase transition is described by a conformal field theory since there is no dimensional scale available. A classic example of such a transition is the insulator-type II superconductor phase transition that occurs as we change the chemical doping g at T = 0. As we move to non-zero temperature, the phase transition point g = gC opens up into an entire phase bounded to the left (small g) by the KT phase transition and to the right by a crossover. A crossover is a smooth change in the thermodynamical potential, unlike a phase transition which means a discontinuity in an n-th derivative of the thermodynamic potential. This phase is known as the quantum critical phase and thus corresponds to putting the conformal field theory. This phase is conformal, strongly coupled since the coupling is close to gC which is large and therefore is poorly understood using conventional condensed theory methods.
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A.
KT Phase Transition
The KT transition in 2 + 1 dimensions is a transition between a state with bound vortex-antivortex pairs (when single vortices disappear) at low temperature and a state with unpaired single vortices at high temperature. The transition can be realized physically in systems modeled by the rotor model known also as the O(2) model or the XY model. For example, Josephson junction arrays, disordered superconductors or granular flows are examples of such systems. The low temperature or low coupling phase is an ordered phase described by power law correlations and at high temperature or large couplings we can have exponential correlations with correlation length ξ∝√
1 . T − TC
However, the order here is known as topological order defined by the presence or absence of vortices. There is a simple semi-quantitative argument for the existence of a phase transition. For a 2 + 1-dimensional system of size R with vortices of vortex core size a, the energy of a vortex is found to be E ∼ κ log
R a
for R a where κ is a constant that depends on the system. On the other hand, the number of possible positions for the vortex are N∼
R2 a2
which leads to the Boltzmann entropy S = kB log N = 2kB log
R . a
Then the free energy of the system is F = U − T S = (κ − 2kB T ) ln
R . a
We see that for low temperature we have F > 0 which means that the single vortices are not preferred so indeed the low temperature phase has only bound vortex pairs. Physically, we understand that the energy of a single vortex diverges as log R a but bound pairs have finite energy as the vortex and antivortex attract each other as if they are particles of opposite charges with a 1r force (log potential) but the total
4 pair is neutral. On the other hand, for high temperatures we have F < 0 which means that the single vortices are stable and we have a gas of unbound vortices. The phase transition temperature or KT temperature TKT between the two phases is TC = TKT =
B.
κ . 2kB
An Example
As an example of the KT transition, consider the Ising model in a transverse magnetic field that is the H field along the x direction perpendicular to z. Specifically X z H = −J (gσix + σjz σj+1 ). i
Then we see that at g = 0 we have the usual ferromagnetic theory whereas at g = ∞ we have all spins oriented along the x direction. That means that somewhere in between there must be a phase transition. In fact, the phase transition occurs at g = 1. IV.
IMPORTANT CONCEPTS
• Near usual, thermal, phase transitions thermodynamic quantities characterizing the material diverge as power laws giving critical behaviour. It is due to the divergence of the correlation length ξ. • At the phase transition, we have an effective continuum field theory with conformal invariance or conformal field theory. • The conformal field theory is determined by the operator product expansions (OPEs). • A CFT primary field is the analog of general relativity tensors with a continuous number indices. • In d > 2, the conformal group is SO(d + 1, 1) in the Euclidean case and SO(d, 2) in the Minkowski case. In two dimensions the conformal group is infinite and conformal transformations are holomorphic ones. • The energy-momentum tensor in two dimensions is characterized by a holomorphic T (z) whose moments satisfy the Virasoro algebra, an infinite dimensional algebra with a central charge. • Representations of the Virasoro algebra are found from the highest weight states, corresponding to primary operators acted upon by L−n giving descendent fields. • The minimal unitary series is labeled by an m with c = c(m) and the conformal dimensions hp,q (m) and starts with the Ising model. • Quantum phase transitions are phase transitions occurring in a coupling g as opposed to temperature thus being governed by quantum fluctuations instead of thermal fluctuations. A conformal field theory describes the transition point. • At non-zero T the quantum phase transition point opens up into a quantum critical phase, strongly coupled and poorly understood that is a T 6= 0 version of the conformal field theory. • The KT phase transition corresponds to a transition in T between bound vortex-antivortex pairs at low T and single vortices for high T .