Confined subdiffusion in three dimensions

Three-dimensional （3D） Fick＇s diffusion equation and fractional diffusion equation are solved for different reflecting boundaries. We use the continuous time random walk model （CTRW） to investigate the time-averaged mean square dis- placement （MSD） of a 3D single particle trajectory. Theoretical results show that the ensemble average of the time-averaged MSD can be expressed analytically by a Mittag-Leffler function. Our new expression is in agreement with previous formu- las in two limiting cases： （^-δ2） ~ △1 in short lag time and （^-δ2} ~ △1 -α in long lag time. We also simulate the experimental data of mRNA diffusion in living E. coli using a 3D CTRW model under confined and crowded conditions. The simulation results are well consistent with experimental results. The calculations of power spectral density （PSD） further indicate the subdiffsive behavior of an individual trajectory.     

Superinsulator–superconductor duality in two dimensions

For nearly a half century the dominant orthodoxy has been that the only effect of the Cooper pairing is the state with zero resistivity at finite temperatures, superconductivity. In this work we demonstrate that by the symmetry of the Heisenberg uncertainty principle relating the amplitude and phase of the superconducting order parameter, Cooper pairing can generate the dual state with zero conductivity in the finite temperature range, superinsulation. We show that this duality realizes in the planar Josephson junction arrays (JJA) via the duality between the Berezinskii–Kosterlitz–Thouless (BKT) transition in the vortex–antivortex plasma, resulting in phase-coherent superconductivity below the transition temperature, and the charge-BKT transition occurring in the insulating state of JJA and marking formation of the low-temperature charge-BKT state, superinsulation. We find that in disordered superconducting films that are on the brink of superconductor–insulator transition the Coulomb forces between the charges acquire two-dimensional character, i.e. the corresponding interaction energy depends logarithmically upon charge separation, bringing the same vortex-charge-BKT transition duality, and realization of superinsulation in disordered films as the low-temperature charge-BKT state. Finally, we discuss possible applications and utilizations of superconductivity–superinsulation duality.     

Shape dynamics in 2 + 1 dimensions

Shape Dynamics is a formulation of General Relativity where refoliation invariance is traded for local spatial conformal invariance. In this paper we explicitly construct Shape Dynamics for a torus universe in 2 + 1 dimensions through a linking gauge theory that ensures dynamical equivalence with General Relativity. The Hamiltonian we obtain is formally a reduced phase space Hamiltonian. The construction of the Shape Dynamics Hamiltonian on higher genus surfaces is not explicitly possible, but we give an explicit expansion of the Shape Dynamics Hamiltonian for large CMC volume. The fact that all local constraints are linear in momenta allows us to quantize these explicitly under a certain assumption on the kinematic Hilbert space, and the quantization problem for Shape Dynamics turns out to be equivalent to reduced phase space quantization. We consider the large CMC-volume asymptotics of conformal transformations of the wave function. We then discuss the similarity of Shape Dynamics on the 2-torus with the explicitly constructible strong gravity Shape Dynamics Hamiltonian in higher dimensions.     

Ising gauge theory in 3.999… dimensions

We numerically study Ising gauge theories in non-integer dimensions below four dimensions using fractals. We find indications that the first-order transition of the d = 4 theory becomes second order for d = 4 − ϵ for arbitrarily small non-zero ϵ. This suggests that the upper critical dimension of abelian gauge theories is four.     

Inverse participation ratio in 2+ε dimensions

The averaged moments of the eigenfunctions (including the inverse participation ratio) of a particle in a random potential are considered near the mobility edge. The exponents of the power laws are given in an-expansion in one-loop order for ad=2+ dimensional system. The calculation is based on a recent formulation of the mobility edge problem which maps it onto a model of interacting matrices.Work supported in part by the Sonderforschungsbereich 123 (Stochastic Mathematic Models) of the Deutsche Forschungsgemeinschaft     

Gödel type metrics in three dimensions

We show that the Gödel type metrics in three dimensions with arbitrary two dimensional background space satisfy the Einstein-perfect fluid field equations. We also show that there exists only one first order partial differential equation satisfied by the components of fluid’s velocity vector field. We then show that the same metrics solve the field equations of the topologically massive gravity where the two dimensional background geometry is a space of constant negative Gaussian curvature. We discuss the possibility that the Gödel type metrics to solve the Ricci and Cotton flow equations. When the vector field u ^μ is a Killing vector field, we came to the conclusion that the stationary Gödel type metrics solve the field equations of the most possible gravitational field equations where the interaction lagrangian is an arbitrary function of the electromagnetic field and the curvature tensors.     

Is there a c-theorem in four dimensions?

The difficulties of extending Zamolodchikov's c-theorem to dimensions d ≠ 2 are discussed. It is shown that, for d even, the one-point function of the trace of the stress tensor on the sphere, S^d, when suitably regularized, defines a c-function, which, at least to one loop order, is decreasing along RG trajectories and is stationary at RG fixed points, where it is proportional to the usual conformal anomaly. It is shown that the existence of such a c-function, if it satisfies these properties to all orders, is consistent with the expected behavior of QCD in four dimensions.     

Conformally invariant σ models in 2n dimensions

The conformally invariant σ-model of Felsager and Leinaas in four dimensions (n=2) is generalised to 2n dimensions, and the unit-charge instantons are presented. This hierarchy of σ-models starts with the O(3) σ-model (n=1). The relationship of these models with the generalised YM systems for even n=2p is pointed out.     

Locally supersymmetric σ-model with Wess-Zumino term in two dimensions and critical dimensions for strings

We construct an N = 1 locally supersymmetric σ-model with a Wess-Zumino term coupled to supergravity in two dimensions. If one takes the σ-model manifold to be the product of d-dimensional Minkowski space M^d and a group manifold G, and if the radius of G is quantized in appropriate units of the string tension, then the model describes a Neveu-Schwarz-Ramond (NSR)-type string moving on M_d × G. (Our model generalizes earlier work of refs. [1,2] which do not contain a Wess-Zumino term and that of refs. [5,6] which is not locally supersymmetric.) The zweibein and the gravitino field equations yield constraints which generalize those of the NSR model to the case of a non-abelian group manifold. In particular, the fermionic constraint contains a new term trilinear in the fermionic fields. We quantize the theory in the light-cone gauge and derive the critical dimensions. We compute the mass spectrum of a closed string moving on M_d × G and show that massless fermions do not arise for non-abelian G for the spinning string, in agreement with the result of Friedan and Shenker [22].     

Heterotic σ-models and conformal supergravity in two dimensions

The (1, 0) and (2, 0) type heterotic σ-models with Wess-Zumino term are coupled to conformal supergravity in two dimensions. There are no new restrictions on the σ-model manifolds in addition to those which arise in the globally supersymmetric cases. In the (1, 0) case possible isometries of the scalar manifold are gauged. A derivation of d = 2 conformal supergravity based on the super Lie algebra OSp(2, N)⊕OSp(2, N) (N = 1, 2) is given.     

Optimal 1 → M phase-covariant cloning in three dimensions

In this paper, we derive the explicit transformations of the optimal 1 → 3, 4, 5 phase-covariant cloning in three dimensions, and then generalize them to the cases of 1 → M = 3n, 3n + 1, 3n + 2(n ≥ 1 integer) cloning. The clone fidelities are coincident with the theoretical bounds found.     

Phase transitions in the generalized spin-one-half Falicov–Kimball model in two dimensions

The extrapolation of small-cluster exact-diagonalisation calculations and the Monte-Carlo method is used to study the spin-one-half Falicov–Kimball model extended by the spin-dependent Coulomb interaction (J) between the localized f and itinerant d electrons as well as the on-site Coulomb interaction (U _ff ) between the localized f electrons. It is shown that in the symmetric case the ground-state phase diagram of the model has an extremely simple structure that consists of only two phases, and namely, the charge-density-wave (CDW) phase and the spin-density-wave (SDW) phase. The nonzero temperature studies showed that these phases persist also at finite temperatures. The critical temperature T _c for a transition from the low-temperature ordered phases to the high-temperature disordered phase is calculated numerically for various values of J and U _ff .     

A dual non-Abelian Yang–Mills amplitude in four dimensions

We derive an explicit formula for the vertex amplitude of dual SU(2)$\mathit{SU}(2)$ Yang–Mills theory in four dimensions on the lattice, and provide an efficient algorithm (of order j⁴$j^4$) for its computation. This opens the way for both numerical and analytical development of dual methods, previously limited to the case of three dimensions.     

N=2 superstrings in more than two dimensions?

For a string propagating in a Parisi-Sourlas superspace the critical dimension equals the difference in the number of positive-and negative-dimensional coordinates. In this way the dimension of the Minkowski subspace can be increased. Here we apply this to the N=2 superstring, with D_c=2 and find anomaly-free N=2 superstrings in all positive even dimensions. Nontrivial theories can be constructed from these N=2 theories by truncation: In a Parisi-Sourlas superspace with a ten-dimensional Minkowski subspace we find the N=1 NSR superstring, and with a four-dimensional Minkowski subspace we find an N=1 superstring, classically related to the D=10 NSR superstring by a canonical transformation.