On certain finiteness questions in the arithmetic of modular forms

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.Comment: 25 pages; v2: one of the conjectures from v1 now proved; v3: restructered parts of the article; v4: minor corrections and change

Size distributions of shocks and static avalanches from the Functional Renormalization Group

Interfaces pinned by quenched disorder are often used to model jerky self-organized critical motion. We study static avalanches, or shocks, defined here as jumps between distinct global minima upon changing an external field. We show how the full statistics of these jumps is encoded in the functional-renormalization-group fixed-point functions. This allows us to obtain the size distribution P(S) of static avalanches in an expansion in the internal dimension d of the interface. Near and above d=4 this yields the mean-field distribution P(S) ~ S^(-3/2) exp(-S/[4 S_m]) where S_m is a large-scale cutoff, in some cases calculable. Resumming all 1-loop contributions, we find P(S) ~ S^(-tau) exp(C (S/S_m)^(1/2) -B/4 (S/S_m)^delta) where B, C, delta, tau are obtained to first order in epsilon=4-d. Our result is consistent to O(epsilon) with the relation tau = 2-2/(d+zeta), where zeta is the static roughness exponent, often conjectured to hold at depinning. Our calculation applies to all static universality classes, including random-bond, random-field and random-periodic disorder. Extended to long-range elastic systems, it yields a different size distribution for the case of contact-line elasticity, with an exponent compatible with tau=2-1/(d+zeta) to O(epsilon=2-d). We discuss consequences for avalanches at depinning and for sandpile models, relations to Burgers turbulence and the possibility that the above relations for tau be violated to higher loop order. Finally, we show that the avalanche-size distribution on a hyper-plane of co-dimension one is in mean-field (valid close to and above d=4) given by P(S) ~ K_{1/3}(S)/S, where K is the Bessel-K function, thus tau=4/3 for the hyper plane.Comment: 34 pages, 30 figure

From Doubled Chern-Simons-Maxwell Lattice Gauge Theory to Extensions of the Toric Code

We regularize compact and non-compact Abelian Chern-Simons-Maxwell theories on a spatial lattice using the Hamiltonian formulation. We consider a doubled theory with gauge fields living on a lattice and its dual lattice. The Hilbert space of the theory is a product of local Hilbert spaces, each associated with a link and the corresponding dual link. The two electric field operators associated with the link-pair do not commute. In the non-compact case with gauge group $\mathbb{R}$, each local Hilbert space is analogous to the one of a charged "particle" moving in the link-pair group space $\mathbb{R}^2$ in a constant "magnetic" background field. In the compact case, the link-pair group space is a torus $U(1)^2$ threaded by $k$ units of quantized "magnetic" flux, with $k$ being the level of the Chern-Simons theory. The holonomies of the torus $U(1)^2$ give rise to two self-adjoint extension parameters, which form two non-dynamical background lattice gauge fields that explicitly break the manifest gauge symmetry from $U(1)$ to $\mathbb{Z}(k)$. The local Hilbert space of a link-pair then decomposes into representations of a magnetic translation group. In the pure Chern-Simons limit of a large "photon" mass, this results in a $\mathbb{Z}(k)$-symmetric variant of Kitaev's toric code, self-adjointly extended by the two non-dynamical background lattice gauge fields. Electric charges on the original lattice and on the dual lattice obey mutually anyonic statistics with the statistics angle $\frac{2 \pi}{k}$. Non-Abelian $U(k)$ Berry gauge fields that arise from the self-adjoint extension parameters may be interesting in the context of quantum information processing.Comment: 38 pages, 4 figure

Graphical Tensor Product Reduction Scheme for the Lie Algebras so(5) = sp(2), su(3), and g(2)

We develop in detail a graphical tensor product reduction scheme, first described by Antoine and Speiser, for the simple rank 2 Lie algebras so(5) = sp(2), su(3), and g(2). This leads to an efficient practical method to reduce tensor products of irreducible representations into sums of such representations. For this purpose, the 2-dimensional weight diagram of a given representation is placed in a "landscape" of irreducible representations. We provide both the landscapes and the weight diagrams for a large number of representations for the three simple rank 2 Lie algebras. We also apply the algebraic "girdle" method, which is much less efficient for calculations by hand for moderately large representations. Computer code for reducing tensor products, based on the graphical method, has been developed as well and is available from the authors upon request.Comment: 43 pages, 18 figure

Magnetic properties of antiferromagnetically coupled CoFeB/Ru/CoFeB

This work reports on the thermal stability of two amorphous CoFeB layers coupled antiferromagnetically via a thin Ru interlayer. The saturation field of the artificial ferrimagnet which is determined by the coupling, J, is almost independent on the annealing temperature up to more than 300 degree C. An annealing at more than 325 degree C significantly increases the coercivity, Hc, indicating the onset of crystallization.Comment: 4 pages, 3 figure

On the scaling behaviour of cross-tie domain wall structures in patterned NiFe elements

The cross-tie domain wall structure in micrometre and sub-micrometre wide patterned elements of NiFe, and a thickness range of 30 to 70nm, has been studied by Lorentz microscopy. Whilst the basic geometry of the cross-tie repeat units remains unchanged, their density increases when the cross-tie length is constrained to be smaller than the value associated with a continuous film. This occurs when element widths are sufficiently narrow or when the wall is forced to move close to an edge under the action of an applied field. To a very good approximation the cross-tie density scales with the inverse of the distance between the main wall and the element edge. The experiments show that in confined structures, the wall constantly modifies its form and that the need to generate, and subsequently annihilate, extra vortex/anti-vortex pairs constitutes an additional source of hysteresis.Comment: 4 pages, 5 figures, accepted for publication in Europhysics Letters (EPL

Roughness and critical force for depinning at 3-loop order

A $d$-dimensional elastic manifold at depinning is described by a renormalized field theory, based on the Functional Renormalization Group (FRG). Here we analyze this theory to 3-loop order, equivalent to third order in $\epsilon=4-d$, where $d$ is the internal dimension. The critical exponent reads $\zeta = \frac \epsilon3 + 0.04777 \epsilon^2 -0.068354 \epsilon^3 + {\cal O}(\epsilon^4)$. Using that $\zeta(d=0)=2^-$, we estimate $\zeta(d=1)=1.266(20)$, $\zeta(d=2)=0.752(1)$ and $\zeta(d=3)=0.357(1)$. For Gaussian disorder, the pinning force per site is estimated as $f_{\rm c}= {\cal B} m^{2}\rho_m + f_{\rm c}^0$, where $m^2$ is the strength of the confining potential, $\cal B$ a universal amplitude, $\rho_m$ the correlation length of the disorder, and $f_{\rm c}^0$ a non-universal lattice dependent term. For charge-density waves, we find a mapping to the standard $\phi^4$-theory with $O(n)$ symmetry in the limit of $n\to -2$. This gives $f_{\rm c} = \tilde {\cal A}(d) m^2 \ln (m) + f_{\rm c}^0$, with $\tilde {\cal A}(d) = -\partial_n \big[\nu(d,n)^{-1}+\eta(d,n)\big]_{n=-2}$, reminiscent of log-CFTs.Comment: 24 pages, 17 figure

Antiferromagnetically coupled CoFeB/Ru/CoFeB trilayers

This work reports on the magnetic interlayer coupling between two amorphous CoFeB layers, separated by a thin Ru spacer. We observe an antiferromagnetic coupling which oscillates as a function of the Ru thickness x, with the second antiferromagnetic maximum found for x=1.0 to 1.1 nm. We have studied the switching of a CoFeB/Ru/CoFeB trilayer for a Ru thickness of 1.1 nm and found that the coercivity depends on the net magnetic moment, i.e. the thickness difference of the two CoFeB layers. The antiferromagnetic coupling is almost independent on the annealing temperatures up to 300 degree C while an annealing at 350 degree C reduces the coupling and increases the coercivity, indicating the onset of crystallization. Used as a soft electrode in a magnetic tunnel junction, a high tunneling magnetoresistance of about 50%, a well defined plateau and a rectangular switching behavior is achieved.Comment: 3 pages, 3 figure

Statics and dynamics of elastic manifolds in media with long-range correlated disorder

We study the statics and dynamics of an elastic manifold in a disordered medium with quenched defects correlated as r^{-a} for large separation r. We derive the functional renormalization-group equations to one-loop order, which allow us to describe the universal properties of the system in equilibrium and at the depinning transition. Using a double epsilon=4-d and delta=4-a expansion, we compute the fixed points characterizing different universality classes and analyze their regions of stability. The long-range disorder-correlator remains analytic but generates short-range disorder whose correlator exhibits the usual cusp. The critical exponents and universal amplitudes are computed to first order in epsilon and delta at the fixed points. At depinning, a velocity-versus-force exponent beta larger than unity can occur. We discuss possible realizations using extended defects.Comment: 16 pages, 11 figures, revtex

Perturbation Theory for Fractional Brownian Motion in Presence of Absorbing Boundaries

Fractional Brownian motion is a Gaussian process x(t) with zero mean and two-time correlations ~ t^{2H} + s^{2H} - |t-s|^{2H}, where H, with 0<H<1 is called the Hurst exponent. For H = 1/2, x(t) is a Brownian motion, while for H unequal 1/2, x(t) is a non-Markovian process. Here we study x(t) in presence of an absorbing boundary at the origin and focus on the probability density P(x,t) for the process to arrive at x at time t, starting near the origin at time 0, given that it has never crossed the origin. It has a scaling form P(x,t) ~ R(x/t^H)/t^H. Our objective is to compute the scaling function R(y), which up to now was only known for the Markov case H=1/2. We develop a systematic perturbation theory around this limit, setting H = 1/2 + epsilon, to calculate the scaling function R(y) to first order in epsilon. We find that R(y) behaves as R(y) ~ y^phi as y -> 0 (near the absorbing boundary), while R(y) ~ y^gamma exp(-y^2/2) as y -> oo, with phi = 1 - 4 epsilon + O(epsilon^2) and gamma = 1 - 2 epsilon + O(epsilon^2). Our epsilon-expansion result confirms the scaling relation phi = (1-H)/H proposed in Ref. [28]. We verify our findings via numerical simulations for H = 2/3. The tools developed here are versatile, powerful, and adaptable to different situations.Comment: 16 pages, 8 figures; revised version 2 adds discussion on spatial small-distance cutof