Gr\"obner bases and syzygies on bimodules

We propose a new more efficient method for the computation of two-sided Gr\"obner bases of ideals and bimodules shifting the problem to the enveloping algebra. Arising from the ideas this method involves, we introduce the notion of two-sided syzygy, which reveals to be useful in the computation of the intersection of bimodules. Further applications are left for a sequel.Comment: 14 pages, 4 algorithms, submitted to J. Symb. Compu

Understanding and enhancing superconductivity in FeSe/STO by quantum size effects

Superconductivity in one-atom-layer iron selenide (FeSe) on a strontium titanate (STO) substrate is enhanced by almost an order of magnitude with respect to bulk FeSe. There is recent experimental evidence suggesting that this enhancement persists in FeSe/STO nano-islands. More specifically, for sizes $L \sim 10$ nm, the superconducting gap is a highly non-monotonic function of $L$ with peaks well above the bulk gap value. This is the expected behavior only for weakly-coupled metallic superconductors such as Al or Sn. Here we develop a theoretical formalism to describe these experiments based on three ingredients: Eliashberg theory of superconductivity in the weak coupling limit, pairing dominated by forward scattering and periodic orbit theory to model spectral fluctuations. We obtain an explicit analytical expression for the size dependence of the gap that describes quantitatively the experimental results with no free parameters. This is a strong suggestion that superconductivity in FeSe/STO is mediated by STO phonons. We propose that, since FeSe/STO is still a weakly coupled superconductor, quantum size effects can be used to further enhance the bulk critical temperature in this interface.Comment: 20 pages, 2 figures, added references and corrected typo

Interplay of classical and "quantum" capacitance in a one dimensional array of Josephson junctions

Even in the absence of Coulomb interactions phase fluctuations induced by quantum size effects become increasingly important in superconducting nano-structures as the mean level spacing becomes comparable with the bulk superconducting gap. Here we study the role of these fluctuations, termed "quantum capacitance", in the phase diagram of a one-dimensional (1D) ring of ultrasmall Josephson junctions (JJ) at zero temperature by using path integral techniques. Our analysis also includes dissipation due to quasiparticle tunneling and Coulomb interactions through a finite mutual and self capacitance. The resulting phase diagram has several interesting features: A finite quantum capacitance can stabilize superconductivity even in the limit of only a finite mutual-capacitance energy which classically leads to breaking of phase coherence. In the case of vanishing charging effects, relevant in cold atom settings where Coulomb interactions are absent, we show analytically that superfluidity is robust to small quantum finite-size fluctuations and identify the minimum grain size for phase coherence to exist in the array. We have also found that the renormalization group results are in some cases very sensitive to relatively small changes of the instanton fugacity. For instance, a certain combination of capacitances could lead to a non-monotonic dependence of the superconductor-insulator transition on the Josephson coupling.Comment: 11 pages, 4 figure

Number theory, periodic orbits and superconductivity in nano-cubes

We study superconductivity in isolated superconducting nano-cubes and nano-squares of size $L$ in the limit of negligible disorder, $\delta/\Delta_0 \ll 1$ and $k_F L \gg 1$ for which mean-field theory and semiclassical techniques are applicable, with $k_F$ the Fermi wave vector, $\delta$ the mean level spacing and $\Delta_0$ the bulk gap. By using periodic orbit theory and number theory we find explicit analytical expressions for the size dependence of the superconducting order parameter. Our formalism takes into account contributions from both the spectral density and the interaction matrix elements in a basis of one-body eigenstates. The leading size dependence of the energy gap in three dimensions seems to be universal as it agrees with the result for chaotic grains. In the region of parameters corresponding to conventional metallic superconductors, and for sizes $L \gtrsim 10$nm, the contribution to the superconducting gap from the matrix elements is substantial ($\sim 20\%$). Deviations from the bulk limit are still clearly observed even for comparatively large grains $L \sim 50$nm. These analytical results are in excellent agreement with the numerical solution of the mean-field gap equation.Comment: 10 pages, 3 figures, 2 appendice

Strong enhancement of bulk superconductivity by engineered nanogranularity

It is now well established, both theoretically and experimentally, that very small changes in the size of isolated nanograins lead to substantial nonmonotonic variations, and sometimes enhancement, of the mean-field spectroscopic gap of conventional superconductors. A natural question to ask, of broad relevance for the theory and applications of superconductivity, is whether these size effects can also enhance the critical temperature of a bulk granular material composed of such nanograins. Here we answer this question affirmatively. We combine mean-field, semiclassical, and percolation techniques to show that engineered nanoscale granularity in conventional superconductors can enhance the critical temperature by up to a few times compared to the nongranular bulk limit. This prediction is valid for three-dimensional and also quasi-two-dimensional samples, provided the thickness is much larger than the grain size. Our model takes into account an experimentally realistic distribution of grain sizes in the array, charging effects, tunnelling by quasiparticles, and limitations related to the proliferation of thermal fluctuations for sufficiently small grains.Comment: 11 pages, 5 figure

Phase coherence in one-dimensional superconductivity by power-law hopping

In a one-dimensional (1D) superconductor, zero temperature quantum fluctuations destroy phase coherence. Here we put forward a mechanism which can restore phase coherence: power-law hopping. We study a 1D attractive-U Hubbard model with power-law hopping by Abelian bosonization and density-matrix renormalization group (DMRG) techniques. The parameter that controls the hopping decay acts as the effective, non-integer spatial dimensionality $d_{eff}$. For real-valued hopping amplitudes we identify analytically a range of parameters for which power-law hopping suppress fluctuations and restore superconducting long-range order for any $d_{eff} > 1$. A detailed DMRG analysis fully supports these findings. These results are also of direct relevance to quantum magnetism as our model can be mapped onto a S=1/2 XXZ spin-chain with power-law decaying couplings, which can be studied experimentally by cold ion-trap techniques.Comment: 8 pages, 2 figures. New version with new figures, new references, clarified discussion on the variational method and an Appendix for detail

Universal quantum constraints on the butterfly effect

Lyapunov exponents, a purely classical quantity, play an important role in the evolution of quantum chaotic systems in the semiclassical limit. We conjecture the existence of an upper bound on the Lyapunov exponents that contribute to the quantum motion, namely, even in the semiclassical limit only a limited range of Lyapunov exponents, bounded from above, are important for the quantum evolution. This is a universal feature in any quantum system or quantum field theory, including those with a gravity dual. It has its origin in the finite size of the Hilbert space that is available to an initial quasi-classical configuration. An upper bound also exists in the limit of an infinite Hilbert space provided that the system is in contact with an environment, for instance a thermal bath. An important consequence of this result is a universal quantum bound on the maximum growth rate of the entanglement entropy at zero and finite temperature.Comment: 5 page

Inhomogenous pairing and enhancement of superconductivity in large Sn nanograins

A substantial enhancement of the superconducting gap was recently reported in clean, large ~30nm, and close to hemispherical Sn grains. A satisfactory explanation of this behaviour is still missing as shell effects caused by fluctuations of the spectral density or surface phonons are negligible in this region. Here we show that this enhancement is caused by spatial inhomogeneities of the Cooper's pairs density of probability. In the mean field approach that we employ these inhomogeneities are closely related to the eigenstates of the one-body problem, namely, a particle in a hemispherical shaped potential. The parameter free theoretical prediction agrees well with the experimental results. A similar enhancement is predicted for other weakly coupled superconductors.Comment: 5 pages, 1 figure, proceedings conference "Quantum in Complex Matter:Superconductivity, Magnetism and Ferroelectricity" Ischia, May 27th - June 1st 201

Universality and Thouless energy in the supersymmetric Sachdev-Ye-Kitaev Model

We investigate the supersymmetric Sachdev-Ye-Kitaev (SYK) model, $N$ Majorana fermions with infinite range interactions in $0+1$ dimensions. We have found that, close to the ground state $E \approx 0$, discrete symmetries alter qualitatively the spectral properties with respect to the non-supersymmetric SYK model. The average spectral density at finite $N$, which we compute analytically and numerically, grows exponentially with $N$ for $E \approx 0$. However the chiral condensate, which is normalized with respect the total number of eigenvalues, vanishes in the thermodynamic limit. Slightly above $E \approx 0$, the spectral density grows exponential with the energy. Deep in the quantum regime, corresponding to the first $O(N)$ eigenvalues, the average spectral density is universal and well described by random matrix ensembles with chiral and superconducting discrete symmetries. The dynamics for $E \approx 0$ is investigated by level fluctuations. Also in this case we find excellent agreement with the prediction of chiral and superconducting random matrix ensembles for eigenvalues separations smaller than the Thouless energy, which seems to scale linearly with $N$. Deviations beyond the Thouless energy, which describes how ergodicity is approached, are universality characterized by a quadratic growth of the number variance. In the time domain, we have found analytically that the spectral form factor $g(t)$, obtained from the connected two-level correlation function of the unfolded spectrum, decays as $1/t^2$ for times shorter but comparable to the Thouless time with $g(0)$ related to the coefficient of the quadratic growth of the number variance. Our results provide further support that quantum black holes are ergodic and therefore can be classified by random matrix theory.Comment: 24 pages, 6 figures, added reference

Shape resonances and shell effects in thin-film multiband superconductors

We study analytically the evolution of superconductivity in clean quasi-two-dimensional multiband supercon- ductors as the film thickness enters the nanoscale region by mean-field and semiclassical techniques. Tunneling into the substrate and finite lateral size effects, which are important in experiments, are also considered in our model. As a result, it is possible to investigate the interplay between quantum coherence effects, such as shape resonances and shell effects, with the potential to enhance superconductivity, and the multiband structure and the coupling to the substrate that tend to suppress it. The case of magnesium diboride, which is the conventional superconductor with the highest critical temperature, is discussed in detail. Once the effect of the substrate is considered, we still observe quantum size effects such as the oscillation of the critical temperature with the thickness but without a significant enhancement of superconductivity. In thin films with a sufficiently longer superconducting coherence length, it is, however, possible to increase the critical temperature above the bulk limit by tuning the film thickness or lateral size.Comment: 11 pages, 8 figure