Vortex Free Energy Research Articles

Formula omitted] lattice gauge theory in [formula omitted] dimensions: Critical couplings from twisted boundary conditions and universality

We present a precision determination of the critical coupling βc for the deconfinement transition in pure SU(2) gauge theory in 2+1 dimensions. This is possible from universality, by intersecting the center vortex free energy as a function of the lattice coupling β with the exactly known value of the interface free energy in the 2D Ising model at criticality. The method allows us to find critical couplings with unprecedented numerical accuracy. Results for lattices with different numbers of sites Nt along the Euclidean time direction are used to determine how β varies with temperature for a given Nt around the deconfinement transition.

A theorem concerning twisted and untwisted partition functions in [formula omitted] and [formula omitted] lattice gauge theories

In order to get a clue to understanding the volume-dependence of vortex free energy (which is defined as the ratio of the twisted against the untwisted partition function), we investigate the relation between vortex free energies defined on lattices of different sizes. An equality is derived through a simple calculation which equates a general linear combination of vortex free energies defined on a lattice to that on a smaller lattice. The couplings in the denominator and in the numerator however shows a discrepancy, and we argue that it vanishes in the thermodynamic limit. Comparison between our result and the work of Tomboulis is also presented. In the appendix we carefully examine the proof of quark confinement by Tomboulis and summarize its loopholes.

Vortex free energy and deconfinement in center-blind discretizations of Yang-Mills theories

Maximal 't Hooft loops are studied in SO(3) lattice gauge theory at finite temperature T. Tunneling barriers among twist sectors causing loss of ergodicity for local update algorithms are overcome through parallel tempering, enabling us to measure the vortex free energy F and to identify a deconfinement transition at some $\beta_A^{crit}$. The behavior of F below $\beta_A^{crit}$ shows however striking differences with what is expected from discretizations in the fundamental representation.

Center vortex model for the infrared sector ofSU(3)Yang-Mills theory: Vortex free energy

The vortex free energy is studied in the random vortex world-surface model of the infrared sector of $SU(3)$ Yang-Mills theory. The free energy of a center vortex extending into two spatial directions, which is introduced into Yang-Mills configurations when acting with the 't Hooft loop operator, is verified to furnish an order parameter for the deconfinement phase transition. It is shown to exhibit a weak discontinuity at the critical temperature, corresponding to the weak first-order character of the transition.

Vortex structure of the vacuum and confinement

We discuss the recently published numerical computations of the vortex free energy. They dramatically demonstrate that the excitation probability for a sufficiently thick vortex approaches unity at large beta, i.e. that thick vortices ‘condense’ in the vacuum. This is known to imply confinement. An analytical approach for exploring this phenomenon is also outlined.

Vortices and confinement in hot and cold D = 2+1 gauge theories

We calculate the variation with temperature of the vortex free energy in D=2+1 SU(2) lattice gauge theories. We do so both above and below the deconfining transition at T=Tc. We find that this quantity is zero at all T for large enough volumes. For T<Tc this observation is consistent with the fact that the phase is linearly confining; while for T>Tc it is consistent with the conventional expectation of `spatial' linear confinement. In small spatial volumes this quantity is shown to be non-zero. The way it decreases to zero with increasing volume is shown to be controlled by the (spatial) string tension and it has the functional form one would expect if the vortices being studied were responsible for the confinement at low T, and for the `spatial' confinement at large T. We also discuss in detail some of the direct numerical evidence for a non-zero spatial string tension at high T, and we show that the observed linearity of the (spatial) potential extends over distances that are large compared to typical high-T length scales.

Vortex condensation and mass gap generation in two-dimensional principal chiral models

We consider the two-dimensional $\mathrm{SU}(N)$ principal chiral model and discuss a vortex condensation mechanism which could explain the existence of a non-zero mass gap at arbitrarily small values of the coupling constant. The mechanism is an analogue of the vortex condensation mechanism of confinement in 4D non-Abelian gauge theories. We formulate a sufficient condition for the mass gap to be non-vanishing in terms of the behavior of the vortex free energy. The $\mathrm{SU}(2)$ model is studied in detail. In one dimension we calculate the vortex free energy exactly. An effective model for the center variables of the spin configurations of the 2D $\mathrm{SU}(2)$ model is proposed and the $Z(2)$ correlation function is derived in this model. We define a $Z(2)$ mass gap in both the full and effective model and argue that they should coincide whenever the genuine mass gap is non-zero. We show via Monte Carlo simulations of the $\mathrm{SU}(2)$ model that the $Z(2)$ mass gap reproduces the full mass gap with perfect accuracy. We also test this mechanism in the positive link model which is an analogue of the positive plaquette model in gauge theories and find excellent agreement between the full and the $Z(2)$ mass gap.

Thermally induced magnetic vortex nucleation near the surface of type II superconductor

The free energy of a magnetic vortex with an arbitrary core shape near the surface of bulk superconductor is calculated. The contribution of the configurational entropy of the vortex core to the vortex free energy is estimated, as well as to the saddle-point free energy barrier which determines the probability of transformation of small virtual vortex nucleus into large vaible vortex under the large thermally induced fluctuations of the core shape.

Vortex fluctuations and interlayer coupling in cuprates.

It was observed 1 that all magnetization M(T) curves in cuprates at different H@Hcr cross at a temperature T* and M5M*, where M and Hcr are the reversible magnetization with H parallel to the c axis and a crossover field, respectively. Immediately after its discovery, this field-independent magnetization was taken as evidence for two-dimensional ~2D! vortex fluctuations. Experimental data further demonstrated that M* and T* serve as fundamental scales for M and T in the fluctuation regime. The data in a Bi2:2:1:2 sample 2 show that an additional term of 2TM * /T * in the expression ]M/] lnH5f 0/[32p 2 l ab (T)# of the London model is needed to account for the vortex-fluctuation effects, where f0 and lab are the flux quantum and the penetration depth in the a,b plane, respectively. The scaling fit of M in Bi2:2:2:3 ~Ref. 3! demonstrates that the ratio M/ M* is the proper dimensionless variable in the proposed scaling expression. It was also observed that the specific heat of Bi2:2:1:2 at various fields peaks at T*. 4 These observations led to much theoretical activity. The effects of vortex-configuration entropy were considered by Bulaevskii et al. ~the BLK model!. 5 There, a vortex line is treated as a stack of 2D vortex pancakes with an average separation of s, and is modeled as 2D classical particles of size apj ab , held through Josephson coupling, where jab and a are the coherence length in the a,b plane and a numerical parameter ;1, respectively. Decoupling of the pancakes occurs above a crossover field Hcr , and leads to an additional configuration-entropy term in the vortex free energy F, which is proportional to the density 1/s of the 2D pancake sheets. As a result, the derivative

Vortex fluctuations in layered superconductors and thin films.

We determine the free energy of a vortex line within the Ginzburg-Landau formalism. Besides the usual translational modes we also take account of self-fluctuations of the order parameter in the calculation of the vortex free energy. The latter are due to the internal structure of a vortex line and lead to an observable downward renormalization of the lower critical-field line [ital H][sub [ital c]1]([ital T]) in strongly layered superconductors. In thin films the self-fluctuations of the vortex lead to a reduction in the power-law exponent of the current-voltage characteristic. We argue that in a three-dimensional system the spontaneous creation of vortex lines cannot occur.

A NOTE ON INTERFACES WITH PERIODIC BOUNDARIES

The partition funtion of a fluctuating interface is argued to approach a universal form at large area, including the numerical coefficient, provided that the interface has periodic boundary conditions und continuum behavior. This is demonstrated by explicit calculations, using effective actions with different regularisations. The result is applied to an analysis of the vortex free energy in SU(2) lattice gauge theory.

The Order Parameter Length Scale of a Vortex Line in a Type II Superconductor

In this paper, we show that the scale of spatial variation of the order parameter ~ in extreme type II superconductors has a temperature dependence other than that of the temperature dependent Ginzburg Landau coherence length ~o( T). Furthermore for temperatures in the vicinity of the critical temperature Tc our results indicate that ~/~o( T) decreases with decreasing temperature. When an external magnetic field is applied to a type II superconductor, vortex lines form provided that the magnetic field is below the upper critical field H e2• 1 ) For temperatures just below the superconducting transition temperature T e, or for a magnetic field just below H e2, such vortices can be analysed within the Ginzburg Landau (GL) framework. 2 ) For temperatures further below Te the Eilenberger and Bogoliubov equations have been investigated, through interative or variational methods. 3 )-5) Kramer et a1. 3 ) and Gygi et a1. 4 ) find that the scale of spatial variation of the order parameter divided by the GL temperature dependent coherence length decreases with decreasing temperature. On the other hand Dorsey et a1. 5 ) claim that there is no appreciable temperature dependence in this quantity. Instead of formulating the problem in terms of Green's functions and the order parameter we adopt a path integral approach. Here the fermion fields associated with the electrons are integrated out, resulting in a formal expression for the free energy in terms of a functional determinant. The order parameter configuration is then given by minimisation of the free energy. In this paper we will be concerned with the quantity ?.=~/J[~o(T), where ~ is the scale of spatial variation of the order parameter and ~o( T) is the GL temperature dependent coherence length. Any temperature dependence of?. indicates that ~ has a temperature dependence that is different from the GL variation. This work aims to determine whether? has any temperature dependence. The vortex free energy, relative to that of the uniform configuration, at finite temperature, T=l//3, can be written as a ratio of two path integrals over Grassmann fields 6

The vortex free energy in the screening phase of the Z(2) Higgs model

The vortex free energy was proposed to distinguish between the confinement and the Higgs phase (in the sense of 't Hooft) in lattice gauge theory, when matter fields are present that transform according to an arbitrary representation of the gauge group. In this paper I consider the Z(2) Higgs model and calculate the vortex free energy in the screening part of the confining/screening phase of Fradkin and Shenker. The result does not agree with the expected behavior that corresponds to the structure of the phase diagram. Therefore the vortex free energy is no longer a good indicator for confinement when matter fields transform non-trivially under the center of the gauge group (such as Z(2) Higgs scalars).

Three-dimensional Z( N) lattice gauge models at large N

An upper bound is derived for the Z(2) vortex free energy in three-dimensional Z( N) lattice gauge models ( N even). This is proved that the inverse temperature β c( N) is unbounded as N goes to infinity, and to argue that it goes at least as √ N is it is proved that the three-dimensional SU( N) model confines static quarks as N → ∞ with β SU( N) fixed.

Monopoles, vortices and confinement

An exact relation is established between an SO(3) lattice gauge theory model without monopoles, and a corresponding SU(2) model. Elimination of the monopoles (and their strings) leads to a substantial lowering of the entropy of thin vortices and a corresponding decrease of the string tension for low β. This is revealed by approximate calculations of the vortex free energy and is confirmed by Monte Carlo data. The value of the physical transition temperature to “hot gluon soup” is also lowered considerably.

A disorder parameter that test for confinement in gauge theories with quark fields

We propose to use a suitably defined vortex free energy as a disorder parameter in gauge field theories with matter fields. It is supposed to distinguish between the confinement phase, massless phase(s) and Higgs phase where they exist. The matter fields may transform according to an arbitrary representation of the gauge group. We compute the vortex free energy by series expansion for a Z 2 Higgs model and for SU(2) lattice models with quark or Higgs fields in the fundamental representation at strong coupling (confinement phase), and for the Z 2 Higgs model in the range of validity of low-temperature expansions (Higgs phase). The results are in agreement with the expected behavior.

High-temperature expansions for the free energy of vortices and the string tension in lattice gauge theories

We derive high-temperature cluster expansions for the free energy of vortices in SU(2) and Z 2 lattice gauge theories in 3 and 4 dimensions. The expected behaviour of the vortex free energy is verified. It obeys an area law behaviour. The coefficient of the area is shown to be equal to the string tension between static quarks. We calculate its expansion up to 12th order. For SU(2) in 4 dimensions the result is compared with Monte Carlo calculations of Creutz and is in good agreement at strong and intermediate coupling.

Vortex free energy and string tension at strong and intermediate coupling

We present results of high temperature expansions up to order g −24 for the vortex free energy respectively string tension in pure lattice gauge theories with gauge groups SU(2) and Z 2 in 3 and 4 dimensions. For SU(2) in 4 dimensions the result is compared with Monte Carlo calculations of Creutz and is in good agreement. An intermediate coupling region is seen, where the string tension smoothly interpolates between strong coupling and weak coupling behaviour.

Vortex patterns and energies in a rotating superfluid

The two-dimensional vortex patterns that occur in a rotating cylinder of superfluid $^{4}\mathrm{He}$ are systematically ordered for numbers of vortices $N=1,2,\dots{},30,37,50$ using a prescription for their free energy that is independent of angular velocity and is based upon the justified omission of images. Barrier energies between patterns of the same $N$ and of neighboring $N$ are discussed. A new derivation of the vortex free energy for perfect square and triangular lattices gives the result in terms of $\ensuremath{\Gamma}(x)$. Patterns that are expected to display high triangular symmetry are studied up to $N=217$, but circular distortion strongly reduces the region of triangular symmetry even in an unbounded fluid, as shown by the scattering structure factor. According to calculations on arrays containing over one million vortices, the destabilizing velocity at the vortex position $R$ in a finite circular region of a perfect triangular lattice is proportional to ${(\frac{R}{{R}_{0}})}^{5}$ where ${R}_{0}$ is the radius of the circular region.