Clustering with shallow trees

We propose a new method for hierarchical clustering based on the optimisation of a cost function over trees of limited depth, and we derive a message--passing method that allows to solve it efficiently. The method and algorithm can be interpreted as a natural interpolation between two well-known approaches, namely single linkage and the recently presented Affinity Propagation. We analyze with this general scheme three biological/medical structured datasets (human population based on genetic information, proteins based on sequences and verbal autopsies) and show that the interpolation technique provides new insight.Comment: 11 pages, 7 figure

Quantum fluctuations can promote or inhibit glass formation

The very nature of glass is somewhat mysterious: while relaxation times in glasses are of sufficient magnitude that large-scale motion on the atomic level is essentially as slow as it is in the crystalline state, the structure of glass appears barely different than that of the liquid that produced it. Quantum mechanical systems ranging from electron liquids to superfluid helium appear to form glasses, but as yet no unifying framework exists connecting classical and quantum regimes of vitrification. Here we develop new insights from theory and simulation into the quantum glass transition that surprisingly reveal distinct regions where quantum fluctuations can either promote or inhibit glass formation.Comment: Accepted for publication in Nature Physics. 22 pages, 3 figures, 1 Tabl

Localization and Glassy Dynamics Of Many-Body Quantum Systems

When classical systems fail to explore their entire configurational space, intriguing macroscopic phenomena like aging and glass formation may emerge. Also closed quanto-mechanical systems may stop wandering freely around the whole Hilbert space, even if they are initially prepared into a macroscopically large combination of eigenstates. Here, we report numerical evidences that the dynamics of strongly interacting lattice bosons driven sufficiently far from equilibrium can be trapped into extremely long-lived inhomogeneous metastable states. The slowing down of incoherent density excitations above a threshold energy, much reminiscent of a dynamical arrest on the verge of a glass transition, is identified as the key feature of this phenomenon. We argue that the resulting long-lived inhomogeneities are responsible for the lack of thermalization observed in large systems. Such a rich phenomenology could be experimentally uncovered upon probing the out-of-equilibrium dynamics of conveniently prepared quantum states of trapped cold atoms which we hereby suggest

Defining the effective temperature of a quantum driven system from current-current correlation functions

We calculate current-current correlation functions and find an expression for the zero-frequency noise of multiterminal systems driven by harmonically time-dependent voltages within the Keldysh non-equilibrium Green's functions formalism. We also propose a fluctuation-dissipation relation for current-current correlation functions to define an effective temperature. We discuss the behavior of this temperature and compare it with the local temperature determined by a thermometer and with the effective temperature defined from a single-particle fluctuation-dissipation relation. We show that for low frequencies all the definitions of the temperature coincide.Comment: 11 pages, 5 figure

On the relation between kinetically constrained models of glass dynamics and the random first-order transition theory

In this paper we revisit and extend the mapping between two apparently different classes of models. The first class contains the prototypical models described-at the mean-field level-by the random first-order transition (RFOT) theory of the glass transition, called either the 'random XORSAT problem' (in the information theory community) or the 'diluted p-spin model' (in the spin glass community), undergoing a single spin-flip Glauber dynamics. The models in the second class are kinetically constrained models (KCM): their Hamiltonian is that of independent spins in a constant magnetic field, hence their thermodynamics is completely trivial, but the dynamics is such that only groups of spins can flip together, thus implementing a kinetic constraint that induces a non-trivial dynamical behavior. A mapping between some representatives of these two classes has been known for a long time. Here we formally prove this mapping at the level of the master equation, and we apply it to the particular case of Bethe lattice models. This allows us to show that an RFOT model can be mapped exactly into a KCM. However, the natural order parameter for the RFOT model, namely the spin overlap, is mapped into a very complicated non-local function in the KCM. Therefore, if one were to study the KCM without knowing the mapping onto the RFOT model, one would guess that its physics is quite different from the RFOT one. Our results instead suggest that these two apparently different descriptions of the glass transition are, at least in some cases, closely related

Electrostatic solution of massless quenches in Luttinger liquids

The study of non-equilibrium dynamics of many-body systems after a quantum quench received a considerable boost and a deep theoretical understanding from the path integral formulation in imaginary time. However, the celebrated problem of a quench in the Luttinger parameter of a one dimensional quantum critical system (massless quench) has so far only been solved in the real-time Heisenberg picture. In order to bridge this theoretical gap and to understand on the same ground massive and massless quenches, we study the problem of a gaussian field characterized by a coupling parameter K within a strip and a different one K-0 in the remaining two semi-infinite planes. We give a fully analytical solution using the electrostatic analogy with the problem of a dielectric material within a strip surrounded by an infinite medium of different dielectric constant, and exploiting the method of charge images. After analytic continuation, this solution allows us to obtain all the correlation functions after the quench within a path integral approach in imaginary time, thus recovering and generalizing the results in real time. Furthermore, this imaginary-time approach establishes a remarkable connection between the quench and the famous problem of the conductivity of a Tomonaga-Luttinger liquid coupled to two semi-infinite leads: the two are in fact related by a rotation of the spacetime coordinates

Leggett's bound for amorphous solids

We investigate the constraints on the superfluid fraction of an amorphous solid following from an upper bound derived by Leggett. To accomplish this, we use as input density profiles generated for amorphous solids in a variety of different manners including by investigating Gaussian fluctuations around classical results. These rough estimates suggest that, at least at the level of the upper bound, there is not much difference in terms of superfluidity between a glass and a crystal characterized by the same Lindemann ratio. Moreover, we perform path integral Monte Carlo simulations of distinguishable helium-4 rapidly quenched from the liquid phase to very low temperature, at the density of the freezing transition. We find that the system crystallizes very quickly, without any sign of intermediate glassiness. Overall our results suggest that the experimental observations of large superfluid fractions in helium-4 particles after a rapid quench correspond to samples evolving far from equilibrium, instead of being in a stable glass phase. Other scenarios and comparisons to other results on the super-glass phase are also discussed

The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective

Among various algorithms designed to exploit the specific properties of quantum computers with respect to classical ones, the quantum adiabatic algorithm is a versatile proposition to find the minimal value of an arbitrary cost function (ground state energy). Random optimization problems provide a natural testbed to compare its efficiency with that of classical algorithms. These problems correspond to mean field spin glasses that have been extensively studied in the classical case. This paper reviews recent analytical works that extended these studies to incorporate the effect of quantum fluctuations, and presents also some original results in this direction. (c) 2012 Elsevier B.V. All rights reserved