
Teitel Group Theoretical Statistical and Condensed Matter Physics
Superconductor ModelsUniformly Frustrated XY ModelThe uniformly frustrated XY model is defined by the Hamiltonian
A_{ij} are fixed and do not fluctuate. The sum of the A_{ij} around any closed path of the network is 2π times the total number of flux quanta of magnetic field penetrating the path. For a regular periodic network and uniform magnetic field, the sum of the A_{ij} around any unit cell transverse to the field is the constant 2πf, where the uniform frustration f is the number of flux quanta per unit cell. The presence of a nonzero f, induces vortices in the phases θ_{i}, with a density equal to f vortices per unit cell. A vortex of integer strength n is said to penetrate a given cell if the sum of the phase differences θ_{i}  θ_{j}] (restricting this difference to the interval (π, π]) going around that cell is 2πn. For the case of a three dimesional network, the vortices will take the form of continuous lines that thread the system parallel to the applied magnetic field. The above Hamiltonian directly describes an array of Josephson junctions, where each term in the sum is the energy of the Josephson junction on bond <ij> of the network. Alternatively, it can be taken as the Hamiltonian of a lattice model of a superconductor, obtained from the usual LandauGinzburg free energy functional with the following apporximations: (i) the amplitude of the wavefunction is constant; this is the London apporximation; (ii) the continuum is discretized to a discrete periodic grid of sites; the grid spacing is associated with the finite core radius of a vortex; (iii) fluctuations in the internal magnetic field are ignored; this is good when the magnetic penetration length is much greater than the intervortex spacing, and so can be taken as infinite. The resulting Hamiltonian, given above, represents the kinetic energy of the flowing supercurrents, suitably symmetrized to maintain the invarience with respect to 2π rotations of the phases θ_{i}. Randomness can be added to the model in two natural ways. One may make random variations in the couplings J_{ij}. Changing a particular J_{ij} primarily effects the core energy for a vortex to sit in a cell bordered by the bond <ij>. One may therefore think of this as a random pinning model. Alternatively, one may make random variations in the vector potential terms A_{ij}. In the context of the Josephson junction array, such randomness can arise from geometrical distortions of the bonds of the network about perfect periodic positions. The extreme case of A_{ij} uniformly random in the interval (π,π] is known as the "gauge glass" model. In a three dimensional system, one can include the effect of magnetic field fluctuations and a finite magnetic penetration length λ by adding to the Hamiltonian the term,
where the sum is over all plaquettes α of the lattice, Δ× is the lattice curl operator, A_{ij} gives the internal magnetic field which is now allowed to fluctuate, and A_{ij}^{ext} gives the externally applied magnetic field which is fixed.
For analytic convenience, in particular for making duality transformations
to the 2D Coulomb gas model and the
3D interacting loop model, one
often replaces the cosine function of the XY Hamiltonian with the Villain,
or periodic Gaussian, function. The Villain function V(θ) has the same
symmetries as the cosine, and is defined as below:
The phase angle θ_{i} can be thought of as specifying the direction of a fixed length spin s_{i} lying in a plane, for example the xy plane. This is the origin of the name "XY" model. When the magnetic field is zero, with A_{ij} = 0, the ground state will have all spins aligned, with each bond in a state of minimum energy. When the field is finite, with A_{ij} ≠ 0, the ground state will have spins twisting up to create vortices; it is no longer possible for each bond to be in a state of minimum energy, hence the name "frustrated". When the magnetic field is uniform, the model is "uniformly frustrated". For further details concerning the identification of the uniformly frustrated XY model with fluctuating superconductors, see Phys. Rev. B 47, 359 (1993) and Phys. Rev. B 55, 15197 (1997).
2D Coulomb Gas Model
Starting with the two dimensional uniformly frustrated XY model with the Villain interaction and constant couplings J_{ij} = J, one can make an exact duality transformation (see José et al, Phys. Rev. B 16, 1217 (1977)) to an equivalent lattice Coulomb gas of interacting charges, given by the Hamiltonian:
Here i are the discrete sites of a grid dual to that of the original XY model, i.e. they sit at the centers of the cells of the XY model network. n_{i} is the integer vorticity on site i, and is the thermally fluctuating degree of freedom. 2πf_{i} is the circulation of the XY model's A_{ij} around cell i, and is fixed. For the uniformly frustrated XY model, the f_{i} = f are all equal. G(r) is the lattice 2D Coulomb interaction for the grid of sites i, with G(r) ~ lnr for large r. The sum is over all pairs of grid sites i and j. The above Hamiltonian is thus one of interacting unit charges n_{i} on a fixed background charge distribution f_{i}. Keeping the energy finite requires overal charge neutrality, ∑_{i}(n_{i}f_{i}) = 0. For further details, see Phys. Rev. B 51, 6551 (1995).
3D Interacting Loop Model
Starting with the three dimensional uniformly frustrated XY model with the Villain interaction and constant couplings J_{ij} = J, one can make an exact duality transformation to an equivalent lattice model of interacting vortex loops, given by the Hamiltonian:
Here i are the discrete sites of a grid dual to that of the original XY model, i.e. they sit at the centers of the cells of the XY model network, and µ label the bond directions of this dual grid. n_{iµ} is the integer vorticity on bond µ emanating from site i, and is the thermally fluctuating degree of freedom. The variables n_{iµ} are divergenceless, and hence they form continuous vortex lines that ultimately close back upon themselves ("loops"). 2πf_{iµ} is the circulation of the XY model's A_{ij} around plaquette µ at dual site i, and is fixed. For the uniformly frustrated XY model, the f_{iµ} = fδ_{µ,z}. G_{µ}(r) is the lattice 3D Coulomb interaction for the grid of sites i, with G_{µ}(r) ~ 1/r for large r. The sum is over all pairs of grid sites i and j, and all bond directions µ. Adding magnetic field fluctuations to the model, the Hamiltonian above gets changed in the following ways. The f_{iµ} now give the flux of the applied magnetic field h, and the interaction G_{µ}(r) is now a screened Coulomb interaction, with screening length set by the finite magnetic penetration length λ. For further details, see Phys. Rev. B 55, 15197 (1997). 
University of Rochester Department of Physics and Astronomy P.O. Box 270171 • 500 Wilson Boulevard • Rochester, NY 146270171 phone: (585) 2754351 • fax: (585) 2733237 

This page was last updated: Fri, Sep 30, 2016; 2:34:04 PM 
This page is located at: http://www.pas.rochester.edu/~stte/teitel/resPages/models.html 