Models

Teitel Group
Theoretical Statistical and Condensed Matter Physics

Superconductor Models

Uniformly Frustrated XY Model

The uniformly frustrated XY model is defined by the Hamiltonian

H = −

∑
<ij>

J_ijcos(θ_i-θ_j -A_ij)

Here θ_i is the thermally fluctuating phase of the superconducting wavefunction on node i of a network of sites, and the sum is over all nearest neighbor bonds <ij> of the network. J_ij is the coupling of bond <ij> and A_ij = (2π/Φ₀) ∫_i^jA·dr is the integral of the magnetic vector potential A across bond <ij>, where Φ₀ = hc/2e is the flux quantum. The

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 non-zero 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 Landau-Ginzburg 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 inter-vortex 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,

Jλ²

∑
α

|Δ×(A_ij-A_ij^ext)|²,

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:

e^-V(θ)/T =

∞
∑
n = -∞

e^{-J(θ - 2πn)²/2T}

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:

H =

πJ

∑
i,j

(n_i-f_i)G(r_i-r_j)(n_j-f_j)

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) ~ ln|r| 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:

H =

πJ

∑
i,j,µ

(n_iµ-f_iµ)G_µ(r_i-r_j)(n_jµ-f_jµ)

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).

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