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Journal of Physics C Solid State Physics

Crossover in the two-dimensional Coulomb gas

A. P. Young, Tomas Bohr

Published Jul 20, 1981• 29 citations
Publisher

Abstract

The screening length xi and the singular part of the free energy of the two-dimensional Coulomb gas are calculated for small fugacity, y, in the vicinity of the transition temperature, T= 1 / 4 , from Kosterlitz's renormalisation-group equations. The results are expressed in terms of a single 'scaling function'. As special cases the authors recover known behaviour in (i) the critical region and (ii) the low-density, high-temperature limit where xi approximately A(T)y - nu with nu =2T/(4T-1). They also obtain the new result that, as T to 1 / 4 , A(T) has the singular form A(T) approximately (T- 1 / 4 ) v .

DOI
10.1088/0022-3719/14/20/010

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The screening length xi and the singular part of the free energy of the two-dimensional Coulomb gas are calculated for small fugacity, y, in the vicinity of the transition temperature, T= 1 / 4 , from Kosterlitz's renormalisation-group equations. The results are expressed in terms of a single 'scaling function'. As special cases the authors recover known behaviour in (i) the critical region and (ii) the low-density, high-temperature limit where xi approximately A(T)y - nu with nu =2T/(4T-1). They also obtain the new result that, as T to 1 / 4 , A(T) has the singular form A(T) approximately (T- 1 / 4 ) v .
A systematic renormalisation group technique for studying the 2D sine-Gordon theory (Coulomb gas, XY model) near its phase transition is presented. The new results are (a) higher order terms in the flow equations beyond those of Kosterlitz (1974) give rise to a new universal quantity; (b) this in turn gives the universal form as well as the relative coefficient of the next-to-leading term in the correlation function of the XY model; (c) the free energy (1PI vacuum sum) is calculated after the singularity at beta 2 =4 pi is treated; (d) vortices with multiple charges are shown to be irrelevant; (e) symmetry breaking fields are analysed systematically. The main ideas that the sine-Gordon theory can be defined as a double expansion in alpha (fugacity) and delta = beta 2 /8 pi -1 (distance from the critical temperature at alpha =0). Wave-function and coupling constant ( alpha ) renormalisations are necessary and sufficient, around beta 2 =8 pi where cos phi acquires dimension 2, for functions with elementary SG fields. This gives rise to renormalisation of beta . The renormalisability is proved to the order calculated in the context of the SG theory, and in general, by using the equivalence to the Thirring-Schwinger model. The renormalised beta 2 plays a role analogous to the dimension in a phi 4 theory-8 pi being the critical dimension. beta 2 >8 pi gives an infrared asymptotically free theory which leads to the well-known fixed line. The infrared properties are understood by analogy with the non-linear sigma model.
A new set of renormalization equations for the two-dimensional Coulomb gas is derived and solved numerically. These equations give a new picture of the Kosterlitz-Thouless transition. A new temperature T∗{T}^{\mathrm{*}}T∗ is found. Above T∗{T}^{\mathrm{*}}T∗ the value of the dielectric constant at the transition is ${\ensuremath{\epsilon}}_{c}$=1/(4Tc{T}_{c}Tc​) as before whereas below T∗{T}^{\mathrm{*}}T∗ the new result ${\ensuremath{\epsilon}}_{c}1/(41/(41/(4{T}_{c}$) is obtained. The singular behavior at T∗{T}^{\mathrm{*}}T∗ is explored. The results are translated into a renormalization-group flow diagram. An interesting consequence is that the universal jump prediction for the Kosterlitz-Thouless transition turns into a nonuniversal jump prediction for critical temperatures below T∗{T}^{\mathrm{*}}T∗.
Renormalisation group recursion relations and matching techniques are used to evaluate to O( epsilon ) the crossover scaling function describing the competition between isotropic short-range and dipolar critical behaviours in the two-point correlation function of a magnetic system above T c in zero field. The scaling function, obtained in closed form, interpolates between the small-q (q xi <<1) and large-q (q xi >>1) limits where asymptotic expressions are written. The results for the effective critical exponents are compared with neutron scattering measurements.
The quantum critical behavior of the (2 + 1)-dimensional Gross-Neveu model in the vicinity of its zero-temperature critical point is considered.The model is known to be renormalisable in the large-N limit, which offers the possibility to obtain expressions for various thermodynamic functions in closed form.We have used the concept of finite-size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling (g)-temperature (T ) plane.These are given by T ∼ |g -gc|, where gc denotes the critical coupling at zero temperature.According to our analysis no spontaneous symmetry breaking survives at finite temperature.We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitude √ 5 3 π.The scaling function of the singular part of the free energy is found to exhibit a maximum at ln 2 2π corresponding to one of the crossover lines.The critical amplitude of the singular part of the free energy is given by the universal number 1 3 1 2π ζ(3) -Cl2 π 3 = -0.274543 . .., where ζ(z) and Cl2(z) are the Riemann zeta and Clausen's functions, respectively.Interpreted in terms of the thermodynamic Casimir effect, this result implies an attractive Casimir "force".This study is expected to be useful in shedding light on a broader class of four fermionic models.
A new and more direct derivation is given of an extended set of renormalization equations for the two-dimensional Coulomb gas. Our equations are equivalent to those proposed by Minnhagen, but are cast in the form of ordinary renormalization equations. It turns out that the difference from the usual renormalization equations is due to the fact that the boundary conditions to be used are of a mixed type: two of them appear as initial conditions; the remaining ones as conditions at the end of a trajectory. The striking new features found by Minnhagen, such as the nonuniversal jump in the inverse dielectric constant and the presence of a first-order transition at higher densities, are a direct result of the possibility (due to the mixed boundary conditions) of having crossing trajectories. These crossing trajectories were also found by Minnhagen et al., but only in the high-temperature region as multiple solutions of their integral equations. Finally, we show that the rather ad hoc criterion adopted by Minnhagen and co-workers for selecting the equilibrium solution indeed corresponds to the one with the smaller free energy, at least in the cases considered by these authors.
The renormalization group equations for the sine-Gordon system with a short wave-length cutoff in two dimensions are derived by means of a momentum-shell recursion method. In a special limit they agree with the Kosterlitz renormalization group equations which describe the critical properties of the phase transitions of the two-dimensional (2-D) neutral Coulomb gas with single charges ±1 and of the 2-D isotropic X-Y model in the absence of an external field. The results, therefore, give rise to a possibility that certain classes of phase transitions in low dimensions can be described in a unified way with the sine-Gordon Hamiltonian.