Electrically contacted molecules are considered to be the ultimate current switches. Their typical dimensions of about 1 nm would allow to design the smallest possible transistor units. However, contrary to solid state circuitry their conductance properties can be decisively influenced by the harmonic degrees of freedom (local phonons). Surprisingly, there are parameter regimes, in which the presence of such seemingly disturbing effects might in fact become an advantage making such a device intrinsically bistable. We discuss these possibilities using analytical as well as numerical techniques and make predictions for the transport properties of contacted molecules in the relevant regimes and make contact to such fundamental multi-particle phenomena as Kondo effect.

References:

J. Klatt, L. Mühlbacher, and A. Komnik, submitted to PRB (2015)
K.F. Albrecht, H. Wang, L. Mühlbacher, M. Thoss, and A. Komnik Phys. Rev. B 86, 081412(R) (2012)
S. Maier, T. L. Schmidt, and A. Komnik, Phys. Rev. B 83, 085401 (2011)

One of the ways to manipulate artificially created quantum systems is to reverse their dynamics. Our ability to do this is limited by the phenomenon of chaos. In classical systems, chaos implies exponential sensitivity to small perturbations. It is to be shown in this presentation that nonintegrable lattices of spins 1/2, which are often considered to be chaotic, are not exponentially sensitive to small perturbations [1]. This result is obtained by comparing the responses of chaotic lattices of classical spins and nonintegrable lattices of spins 1/2 to imperfect reversal of spin dynamics known as Loschmidt echo. In the classical case, Loschmidt echoes exhibit exponential sensitivity to small perturbations characterized by twice the value of the largest Lyapunov exponent of the system. In the case of spins 1/2, Loschmidt echoes are only power-law sensitive to small perturbations. Our findings imply that it is impossible to define Lyapunov exponents for lattices of spins 1/2 even in the macroscopic limit. At the same time, the above absence of exponential sensitivity to small perturbations is an encouraging news for the efforts to create quantum simulators. The power-law sensitivity of spin 1/2 lattices to small perturbations is predicted to measurable in nuclear magnetic resonance experiments.

[1] B. V. Fine, T. A. Elsayed, C. M. Kropf and A. S. de Wijn, Phys. Rev. E 89, 012923 (2014).

Recent progress in the design and fabrication of artificial two-dimensional (2D) materials paves the way for the experimental realization of electron systems moving on plane fractals. In this work, we present the results of computer simulations for the conductance and optical absorption spectrum of a 2D electron gas roaming on a Sierpinski carpet, i.e. a plane fractal with Hausdorff dimension intermediate between one and two. We find that the conductance is sensitive to the spatial location of the leads and that it displays fractal fluctuations whose dimension is compatible with the Hausdorff dimension of the sample. Very interestingly, electrons in this fractal display a broadband optical absorption spectrum, which possesses sharp "molecular" peaks at low photon energies.

We investigate the zero-temperature properties of a diluted homogeneous Bose gas made of \(N\) particles interacting via a two-body square-well potential by performing Monte Carlo simulations. We tune the interaction strength to achieve arbitrary positive values of the scattering length and compute by Monte Carlo quadrature the energy per particle $E/N$ and the condensate fraction \(N_0/N\) of this system by using a Jastrow ansatz for the many-body wave function which avoids the formation of the self-bound ground-state and describes instead a (metastable) gaseous state with uniform density.
In the unitarity limit, where the scattering length diverges while the range of the inter-atomic potential is much smaller than the average distance between atoms, we find a finite energy per particle (\(E/N=0.70\ \hbar^2(6\pi^2n)^{2/3}/2m\), with \(n\) the number density) and a quite large condensate fraction (\(N_0/N=0.83\)) [1].
Starting from the obtained equation of state we study also the frequencies of the monopole and quadrupole oscillations of the gas trapped in a isotropic harmonic potential within Density Functional Theory in the Local Density approximation. We include also the damping effect of three-body losses on such modes [2].
Prompted by the very recent experimental data of \(^{85}\)Rb atoms at unitarity [3] we focus on the momentum distribution as a function of time. Our results suggest that at unitarity, a quasi-stationary momentum distribution is reached at low momenta after a long transient, contrary to what found experimentally for large momenta which equilibrate on a time scale shorter than the one for three body losses [4].

References
1. M. Rossi, L. Salasnich, F. Ancilotto and F. Toigo, Phys. Rev. A 89, 041602(R) (2014)
2. M. Rossi, F. Ancilotto, L. Salasnich and F. Toigo, arXiv:1408.3945 (accepted in EPJ)
3. P. Makotyn, C.E. Klauss, D.L. Goldberger, E.A. Cornell and D.S. Jin, Nature Phys. 10, 116 (2014)
4. F. Ancilotto, M. Rossi, L. Salasnich and F. Toigo, arXiv:1501.05491 (accepted in FBSY)

We explore [1] the conditions on a pair interaction for the validity of the Vlasov equation to describe the dynamics of an interacting \(N\) particle system in the large \(N\) limit. Using a coarse-graining in phase space of the exact Klimontovich equation for such a system, we evaluate the scalings with \(N\) of the terms describing the corrections to the Vlasov equation for the coarse-grained one particle phase space density. Considering an interaction with radial pair force \(F(r)\sim1/ra\), regulated to a bounded behavior below a "softening" scale \(l\), we find that there is an essential qualitative difference between the cases \( a < d \) (i.e. the spatial dimension) and \(a > d\) , i.e., depending on the the integrability at large distances of \(F(r)\). For \( a < d \) the corrections to the Vlasov dynamics for a given coarse-grained scale are essentially insensitive to the softening parameter \(l\), while for \(a>d\) the corrections are directly regulated by \(l\), i.e. by the small scale properties of the interaction, in agreement with the Chandrasekhar approach [2]. This gives a simple physical criterion for a basic distinction between long-range (\( a < d \)) and short range (\(a>d\)) interactions, different from the thermodynamic one (\( a < d-1 \) or \( a > d-1 \)). This alternative classification, based purely on dynamical arguments, is relevant notably to understanding the conditions for the existence of so-called quasi-stationary states in long-range interacting systems.

[1] A. Gabrielli et al., Phys. Rev. E, 90, 062910 (2014).
[2] A. Gabrielli et al., Phys. Rev. Lett., 115, 210602 (2010).

The dynamics of a one-dimensional membrane confined between two walls is derived from an hydrodynamic model. The main ingredients of the evolution equation are the adhesion potential with the confining walls and the bending rigidity of the membrane. The resulting dynamics is frozen, with the membrane decomposing in adhesion patches on the two walls. If the system is strongly perturbed, a transition from frozen dynamics to coarsening may be observed.

In collaboration with Thomas Le Goff and Olivier Pierre-Louis (CNRS-Lyon).

I study matrix product states (MPS) and matrix product operators (MPO) from the point of view of computational complexity. MPS are pure states whose coefficients in an orthonormal tensor basis are transition amplitudes in an auxiliary, virtual Hilbert space, while MPO are operators with coefficients in an orthonormal operator tensor basis being represented as trantion amplitudes in an auxiliary Hilbert space. In particular, I show that measuring transition amplitudes of MPS or expectations of local operators of density matrices described by a MPO allows to encode, in the auxiliary space, quantum circuits with the additional power of general linear operators, and thus solve very hard computational problems. I will exemplify the above result with cluster state (MPS) and steady states of boundary dissipated quantum spin chains (MPO). This result deepens the knowledge of hardly implementable quantum problems. As a byproduct of the above construction and after a very short introduction to complexity classes, I prove an important result of computational complexity. The latter states that quantum Turing machine with postselection and bounded error probability with subroutines, that exactly solve problems probabilistically solvable by the same type of machine, can be simulated without the exact subroutines. This result, together with the equivalence between quantum Turing machine with postselection and bounded error probability and probabilistic Turing machines with unbounded error probability, implies the collapse of a hierarchy of classical computational classes that was not yet proven.

The 1D Kitaev chain is a prototypical example of a system which shows topological order, signalled by the presence of an odd number of spatially separated Majorana zero modes. Depending on the Hamiltonian parameters, such a system undergoes a transition from a topologically trivial to a non-trivial phase. We show that a suitably engineered decoherence may indeed enact a similar transition to a topologically non-trivial phase, starting from an otherwise trivial one. Such a phenomenon is the result of the interplay between Hamiltonian and dissipative interactions. These findings are relevant to applications in solid state physics as well as in the context of cold-atoms.

In collaboration with Fabio Anzà, Davide Valenti, Bernardo Spagnolo

We present a model of percolation mimicking the self-healing dynamics of a smart grid. While in the case of random graphs it is possible to work out an analytic solution, in the case of two dimensional networks we must resort to numerical simulations. Our findings hint that for planar lattices duality plays a key role yet to be understood. Finally, by to tackling the problem of being connected to a source node, we find the existence of two separate solutions that we study by applying cavity methods and recursive equations.

We develop a rigorous large deviation theory for random vectors within an exactly solvable lattice gas model. The rate function is completely characterized. The occurrence of discontinuous phase transitions leads to a rate function which is affine on part of its domain.

We construct an approximation to the free energy of the Ising model on fractal lattices of dimension smaller than two, in the case of zero external magnetic field. The result is obtained as the limit of the exact free energies of the Ising model on periodic approximations. The free energies are computed using a generalization of the combinatorial method of Feynman and Vodvickenko.
As a first application, we compute estimates to the critical temperature for many different Sierpinski carpets and we compare the results with known Monte Carlo estimates. The results show that our method is capable of determining the critical temperature with, possibly, arbitrary accuracy and paves the way to determine \(T_c\) for any fractal of dimension below two.
The singularity of the free energy is logarithmic at the critical point, thus \(\alpha = 0\), for any periodic approximation. We also compute the correlation length as a function of the temperature and extract the relative critical exponent, we find \(\nu=1\) for all periodic approximation, as expected from Universality.

We study \(O(N)\) models with power-law interactions by using functional renormalization group methods: we show that both in Local Potential Approximation (LPA) and in LPA' their critical exponents can be computed from the ones of the corresponding short-range \(O(N)\) models at an effective fractional dimension. In LPA such effective dimension is given by \(D_{\rm eff}=2d/\sigma\), where \(d\) is the spatial dimension and \(d+\sigma\) is the exponent of the power-law decay of the interactions. In LPA' the prediction by Sak [Phys. Rev. B 8, 1 (1973)] for the critical exponent \(\eta\) is retrieved and an effective fractional dimension \(D'_{\rm eff}\) is obtained. Using these results we determine the existence of multicritical universality classes of long-range \(O(N)\) models and we present analytical predictions for the critical exponent \(\nu\) as a function of \(\sigma\) and \(N\): explicit results in 2 and 3 dimensions are given. Finally, we propose an improved LPA" approximation to describe the full theory space of the models where both short-range and long-range interactions are present and competing: a long-range fixed point is found to branch from the short-range fixed point at the critical value \(\sigma_∗=2−\eta_{\rm SR} \) (where \(\eta_{\rm SR}\) is the anomalous dimension of the short-range model), and to subsequently control the critical behavior of the system for \(\sigma <\sigma_∗\).

In many social and information systems, the network of the interactions is generated by the agents activity. The temporal evolution of the connectivity pattern and the dynamics taking place on the network might be strictly coupled, as they evolve on the same time scales (information diffusion, sexual transmitted diseases etc.). Besides, most of these systems manifest memory effect on the agent dynamics. Given these difficulties, the description of the processes behind the networks evolution is a challenging task that we can now tackle using large, high definition datasets.
We present a generalized version of the model found in Karsai et al. [Sci. Rep. 4 2014]: here, the temporal evolution of the single agent’s egocentric network is encoded in a reinforcement process where the creation of new edges by an active agent is discouraged. We propose an analytical approach to the problem and provide an asymptotic solution for the evolving degree distribution and for the average degree of the network as a function of the memory parameters. The introduced model is tested against numerical simulations and several time-resolved datasets: a striking agreement between the model predictions and both the numerical and real data is found.

A problem typically encountered when studying complex systems is the limitedness of the information available on their topology, which hinders our understanding of their structure and of the dynamical processes taking place on them. A paramount example is provided by financial networks, whose data are privacy-protected: banks publicly disclose only their aggregate exposure towards other banks, keeping individual exposures towards each single bank secret. Yet, the estimation of systemic risk strongly depends on the detailed structure of the interbank network. The resulting challenge is that of using aggregate information to statistically reconstruct a network and correctly predict its higher-order properties. Standard approaches either generate unrealistically dense networks, or fail to reproduce the observed topology by assigning homogeneous link weights. Here we develop an improved reconstruction method based on statistical mechanics concepts, that makes use of the empirical link density in a highly nontrivial way. Technically, the novelty of our approach lies in the preliminary estimation of node degrees from empirical node strengths and link density, followed by a maximum-entropy inference based on the combination of empirical strengths and estimated degrees. Our method is successfully tested on the international trade network and the interbank money market, and represents a valuable tool for gaining insights on privacy-protected or partially-accessible systems.