Helicity Modulus Research Articles

Minimal renormalization without ϵ-expansion: Amplitude functions for O( n) symmetric systems in three dimensions below Tc

Massive field theory at fixed dimension d < 4 is combined with the minimal subtraction scheme to calculate the amplitude functions of thermodynamic quantities for the O( n) symmetric Φ 4 model below T c in two-loop order. Goldstone singularities arising at an intermediate stage in the calculation of O( n) symmetric quantities are shown to cancel among themselves leaving a finite result in the limit of zero external field. From the free energy we calculate the amplitude functions in zero field for the order parameter, specific heat and helicity modulus (superfluid density) in three dimensions. We also calculate the q 2 part of the inverse of the wavenumber-dependent transverse susceptibility χT( q) which provides an independent check of our result for the helicity modulus. The two-loop contributions to the superfluid density and specific heat below T c turn out to be comparable in magnitude to the one-loop contributions, indicating the necessity of higher-order calculations and Padé-Borel type resummations.

Monte Carlo calculation of longitudinal and transverse resistivities in a model type-II superconductor

We study the effect of a transport current on the vortex-line lattice in isotropic type-II superconductors in the presence of strong thermal fluctuations by means of 'driven-diffusion' Monte Carlo simulations of a discretized London theory with finite magnetic penetration depth. We calculate the current-voltage (I-V) characteristics for various temperatures, for transverse as well as longitudinal currents I. From these characteristics, we estimate the linear resistivities R_xx=R_yy and R_zz and compare these with equilibrium results for the vortex-lattice structure factor and the helicity moduli. From this comparison a consistent picture arises, in which the melting of the flux-line lattice occurs in two stages for the system size considered. In the first stage of the melting, at a temperature T_m, the structure factor drops to zero and R_xx becomes finite. For a higher temperature T_z, the second stage takes place, in which the longitudinal superconducting coherence is lost, and R_zz becomes finite as well. We compare our results with related recent numerical work and experiments on cuprate superconductors.

Quantum phase fluctuations in an array of mesoscopic Josephson junctions

The effects of macroscopic ordering in a system of mesoscopic Josephson junctions are investigated by the quantum Monte Carlo simulation technique (using path integrals). The phase diagram of the system in the T-q plane (q is the dimensionless quantum parameter \(q = 2{e \mathord{\left/ {\vphantom {e {\sqrt {JC_0 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {JC_0 } }}\), where J is the Josephson coupling constant and C0 is the self-capacitance of the granules) is investigated in detail. An analysis of the behavior of the relative root-mean-square phase shifts, as well as the helicity and vorticity moduli, demonstrates the need to employ these two quantities as the parameters which most completely reflect the character of the topological phase transition in the quantum system under consideration. Two methods are proposed for calculating the vorticity modulus: 1) a modification of the Gibbs-Bogolyubov variational principle for calculating the free energy change in response to alteration of the type of boundary conditions; 2) calculation of the response to the introduction of an infinitesimal magnetic flux at some point in the system. The calculations confirm the absence of reentrant melting and phase transitions of a non-Kosterlitz-Thouless type in the region of strong quantum phase fluctuations q>1.

Helicity modulus and fluctuating type-II superconductors: Elastic approximation and numerical simulations

We develop the helicity modulus as a criterion for superconducting order in the mixed phase of a fluctuating type II superconductor. We show that there is a duality relation between this helicity modulus and the superfluid density of a system of analog 2D bosons. We show that the vortex line lattice exhibits a perfect Meissner effect with respect to a shearing perturbation of the applied magnetic field, and this becomes our creterion for "longitudinal superconductivity" parallel to the applied field. We present arguments based on the 2D boson analogy, as well as the results of numerical simulations, that suggest that longitudinal superconductivity can persist into the vortex line liquid state for systems of finite thickness, comparable to those commonly found in experiments.

Single-Electron Box and the Helicity Modulus of an Inverse SquareXYModel

We calculate the average number of electrons on a metallic single-electron-box as a function of the gate voltage for arbitrary values of the tunneling conductance. In the vicinity of the plateaus the problem is equivalent to calculating the helicity modulus of a classical inverse square XY-model in one dimension. By a combination of perturbation theory, a two-loop renormalization group calculation and a Monte-Carlo simulation in the intermediate regime we provide a complete description of the smearing of the Coulomb staircase at zero temperature with increasing conductance.

Helicity modulus, quantum fluctuations, and superconducting transition in a Josephson junction, array

The effect of quantum fluctuations on a phase transition in a two-dimensional Josephson junction array is studied in terms of the two-dimensional XY model. A self-consistent harmonic approximation is used to calculate the linear response of the system to a perturbation by a uniform phase gradient (helicity modulus γ) as a function of dimensionless temperature Θ and of a quantum parameter q appearing due to the finite capacitance of each junction. Calculation of this quantity has permitted us to find the dependence of the Kosterlitz-Thouless transition temperature on q, which within a broad range of q variation is in agreement with the results of a quantum Monte Carlo simulation.

Universal jump in the helicity modulus of the two-dimensional quantum XY model

The helicity modulus of the S=1/2 XY model is precisely estimated through a world line quantum Monte Carlo method enhanced by a cluster update algorithm. The obtained estimates for various system sizes and temperatures are well fitted by a scaling form with L replaced by \log(L/L_0), which is inferred from the solution of the Kosterlitz renormalization group equation. The validity of the Kosterlitz-Thouless theory for this model is confirmed.

Monte Carlo study of the Villain version of the fully frustrated XY model

The fully frustrated XY model with Villain interaction on a square lattice is studied by means of Monte Carlo simulations. On the basis of the universal jump condition it is argued that there are two distinct transitions in the model, corresponding to the loss of XY order and ${\mathrm{Z}}_{2}$ order, respectively. The Kosterlitz-Thouless (KT) transition is analyzed by finite-size scaling of the helicity modulus at lattices of size L=32 through 128, giving ${\mathrm{T}}_{\mathrm{KT}}$\ensuremath{\approx}0.8108(1). The vorticity-vorticity correlation function is used to determine two different characteristic lengths, the ${\mathrm{Z}}_{2}$ correlation length \ensuremath{\xi}, and the screening length \ensuremath{\lambda}, associated with the KT transition and free vortices. The temperature dependence of \ensuremath{\xi} is examined in order to determine ${\mathrm{T}}_{\mathrm{c}}$ and the correlation length exponent, \ensuremath{\nu}. The exponent is found to be consistent with the two-dimensional (2D) Ising value, \ensuremath{\nu}=1, and the obtained critical temperature is ${\mathrm{T}}_{\mathrm{c}}$=0.8225(5). The determinations of both \ensuremath{\xi} and \ensuremath{\nu} are done carefully, first applying the techniques to the 2D Ising model, which serves as a convenient testing ground.

Finite-size scaling of the helicity modulus of the two-dimensional O(3) model

Using Monte Carlo methods, we compute the finite-size scaling function of the helicity modulus $\Upsilon$ of the two-dimensional O(3) model and compare it to the low temperature expansion prediction. From this, we estimate the range of validity for the leading terms of the low temperature expansion of the finite-size scaling function and for the low temperature expansion of the correlation length. Our results strongly suggest that a Kosterlitz-Thouless transition at a temperature $T > 0$ is extremely unlikely in this model.

The critical behavior of the 3D X- Y model and its relation with fractal properties of the vortex excitations

The relation between the fractal dimensionality of the vortex excitations and the critical exponents of the vortex phase transition in the 3D X- Y model is studied. The helicity modulus of the model is expressed via the average “vortex moment” of the system. The average “vortex moment” is described in terms of the polymer theory. Using this approach we derive RG equations of the 3D X- Y model in the low fugacity limit, considering the vortex loop as a polymer chain with critical exponent νv p. The critical exponents νv of the correlation length coincide with νv p.

Topological phase transition in 2D quantum Josephson array

Path Integral Quantum Monte Carlo simulation is used to study thermodynamic properties and a phase diagram of 2D quantum Josephson array, described by 2 + 1 XY model. The helicity and vorticity moduli, correlation function of phases and other characteristics of the system as functions of quantum parameter q and temperature T are studied ( q= h √JC ), J is the Josephson coupling constant, C is the intragrain capacitance). Quantum fluctuation induced superconductor-normal phase transition is studied in detail through the use of behavior of above-mentioned quantities. No discontinuous or reentrant phase transition in q- T plane is found. Analysis of the vorticity and the renormalized coupling constant leads to the conclusion that the whole line of phase transition is of Kosterlitz-Thouless type.

Simulations of current driven three-dimensional Josephson junction arrays

We present Langevin simulations of anisotropic three-dimensional Josephson arrays driven by an external current. We study the superconducting phase transition at zero magnetic field, by analyzing both the helicity modulus and the voltage. We find that there is a well defined phase transition at any finite current smaller than the Josephson critical current. The critical temperature T c( I) obtained decreases linearly with current I. We also find that the onset of voltage occurs at temperatures smaller than T c( I). We discuss the consequences of this result for the interpretation of recent experiments in the flux transformer configuration.

Criticality of the fully frustrated XY model: A study using Monte Carlo hard-spin mean-field theory (abstract)

The number and types of phase transitions occurring in the two-dimensional fully frustrated XY model have remained controversial in spite of over a decade of attention. In this study, we have developed a Monte Carlo hard-spin mean-field approach for models with a continuous spin variable and applied it to this system. From our calculations of the chirality, helicity modulus, specific heat, and sublattice magnetizations, we find that reflection and rotational symmetries are broken at the same temperature. The chirality, order parameter for the reflection symmetry breaking, exhibits power-law dependence on T−Tc with a critical exponent of 0.246±0.006. This result is decisively different from the mean-field value of 0.500, which would be obtained for a transition in the Ising universality class using this method. This finding enables us to rule out the possibility of two separate transitions, too closely spaced in temperature to be resolved by our calculation. The magnitude of the magnetization on each sublattice also shows power-law behavior as a function of temperature in the critical region, with a critical exponent of 0.126±0.002.

Monte Carlo studies of helicity modulus and heat capacity of a coupled XY model in two dimensions.

Monte Carlo studies of helicity modulus and heat capacity of a coupled XY model in two dimensions.

Phase dynamics in percolating Josephson junction arrays

The temperature-dependent helicity modulus Γ(T) of site-diluted triangular arrays of proximity-effect coupled Pb/Cu/Pb Josephson junctions has been extracted from measurements of the complex ac sheet conductance performed in the frequency range 0.1 Hz to 20 kHz in zero magnetic field. At low frequencies Γ(T) exhibits the characteristic superfluid drop predicted by the Kosterlitz-Thouless theory, demonstrating that a disordered array behaves like a homogeneous two-dimensional system at length scales-larger than the percolation correlation length.

Helicity modulus near a three-state Potts-like transition

By employing Monte Carlo simulations, both the helicity modulus and the heat capacity of a coupled classical XY model in two dimensions have been calculated. The model is based on the Hamiltonian proposed by Bruinsma and Aeppli [Phys. Rev. Lett. 48, 1625 (1982)]. For the first time, the model enables us to study the temperature dependence of the helicity modulus and obtain the associated critical exponent near a three-state Potts transition.

Phase transitions in the uniformly frustrated XY model with frustration parameter f=1/3 studied with use of the microcanonical Monte Carlo technique.

We study the phase transition of the frustrated XY model in the square lattice with frustration f=1/3. The system has a discrete ${\mathit{Z}}_{3}$\ifmmode\times\else\texttimes\fi{}${\mathit{Z}}_{2}$ symmetry and a continuous U(1) symmetry. We study the system via the microcanonical Monte Carlo technique. We have found that the system has three kinds of phase transitions. At the temperature 0.208J/${\mathit{k}}_{\mathit{B}}$, the system has a Kosterlitz-Thouless transition with a larger-than-universal jump in the helicity modulus. At the temperature 0.215J/${\mathit{k}}_{\mathit{B}}$, the system has a vortex-lattice melting transition related to discrete ${\mathit{Z}}_{2}$ symmetry. At some higher temperature 0.219J/${\mathit{k}}_{\mathit{B}}$, the system has a second-order vortex-lattice melting transition related to ${\mathit{Z}}_{3}$ symmetry. It is also found that the second transition gives a weak anomaly of the specific heat while the third transition gives rise to the divergence in the specific heat.

Three-dimensional to two-dimensional crossover in layered high-Tc superconductors.

We calculate vortex lattice properties in a highly anisotropic layered high-${\mathit{T}}_{\mathit{c}}$ superconductor using Monte Carlo simulations. The superconducting order parameter is expanded in products of lowest Landau level states in the ab plane and tight-binding Bloch states in the c direction. The phase diagram is then a universal function of a dimensionless effective temperature scrT and effective interlayer coupling \ensuremath{\eta}, both of which depend on temperature T and magnetic field B. We characterize the vortex lattice by the helicity modulus (superfluid density) \ensuremath{\Upsilon} in the c direction, and shear modulus \ensuremath{\mu} in the ab plane. There appears to be no phase transition separating two-dimensional (2D) and 3D solids. Instead, the 3D to 2D transition is manifested by a smooth crossover of \ensuremath{\Upsilon} from nearly mean-field 3D behavior \ensuremath{\Upsilon}\ensuremath{\sim}\ensuremath{\eta} at large \ensuremath{\eta}, to a 2D, fluctuation-dominated regime \ensuremath{\Upsilon}\ensuremath{\sim}${\mathrm{\ensuremath{\eta}}}^{2}$ at small \ensuremath{\eta}. In the limit \ensuremath{\eta}\ensuremath{\rightarrow}0 the shear modulus smoothly approaches a finite value characteristic of a purely 2D vortex lattice. We discuss the possibility that this dimensional crossover may account for the disappearance of the neutron scattering peaks in ${\mathrm{BiSr}}_{2}$${\mathrm{Ca}}_{2}$${\mathrm{CuO}}_{8+\mathrm{\ensuremath{\delta}}}$ at high fields.

Monte Carlo analysis of the two-dimensional XY model. II. Comparison with the Kosterlitz renormalization-group equations.

The temperature scale of relevance for the vortex excitations in the two-dimensional XY model is identified. The size dependence of the helicity modulus at constant vortex temperature is then obtained by Monte Carlo simulations. The Monte Carlo data are then compared with the results for the length-dependent dielectric function from the Kosterlitz renormalization-group equations. The agreement is excellent for temperatures up to and slightly above ${\mathit{T}}_{\mathit{c}}$. The critical temperature is determined to be ${\mathit{T}}_{\mathit{c}}$=0.892 13(10). The temperature dependence of the characteristic length right above ${\mathit{T}}_{\mathit{c}}$ is also obtained and found to be in agreement with recent results from high-temperature series expansions.

Flux-lattice melting and depinning in the weakly frustrated two-dimensional XY model.

Monte Carlo simulations of the frustrated two-dimensional (2D) XY model were carried out at small commensurate values of the frustration f. For f=1/30 a single transition was observed at which phase coherence (finite helicity modulus) and vortex lattice orientational order vanish together. For f=1/56 a new phase in which phase coherence is absent but orientational order persists was observed. Where compariosn is possible, the results are in detailed agreement with the behavior of the lattice Coulomb gas model of vortices. It is argued that the helicity modulus of the frustrated 2D XY model vanishes for any finite temperature in the limit of weak frustration f.