A brief description of my research -- side projects
The paper numbers below refer to my publication list
Quantum disordered systems, quantum glasses, quantum optimization problems
Divulgative paper: 48
Review paper: 63
This research project is focused on quantum disordered and frustrated system. The idea was to apply the methods that have been developed in the context of classical glasses to quantum systems. The main application are: (1) disordered bosons (e.g. disordered Helium 4), in order to describe possible superglass phases; (2) quantum spin glasses, mainly in connection to quantum optimization problems.
A detailed description of my work in this field can be found here.
A very complete description can be found in the (rejected) ERC Starting Grant project AQUAMAN.
We formulated a first attempt to describe quantum glasses in paper [32], which was selected as an "Editor Suggestion" in Phys.Rev.B and has been reviewed by Z.Nussinov in Physics 1, 40 (2008). Quoting from Nussinov's review:
"Glasses are liquids that have ceased to flow on experimentally measurable time scales. By constrast, superfluids flow without any resistance. The existence of a phase characterized by simultaneous glassiness and superfluidity may seem like a clear contradiction. However, in this paper, Biroli, Chamon, and Zamponi prove that this is not so and illustrate theoretically the possibility of a superglass phase. This phase forms an intriguing amorphous counterpart to the supersolid phase that has seen a surge of interest in recent years. Within a supersolid phase, superfluidity can occur without disrupting crystalline order. Interacting quantum particles can indeed form such a superglass phase at very low temperatures and high density, and the work of Biroli et al. confirms the earlier suggestion by Boninsegni, Prokofev, and Svistunov and an investigation by Wu and Philips of such a phase. The superglass phase is characterized by an amorphous density profile, yet at the same time a finite fraction of the particles flow without any resistance as if they were superfluid. Thus the superglass constitutes a glassy counterpart to the supersolid phase. The approach invoked by Biroli et al. to prove the existence of a superglass is particularly elegant. It relies on mapping viscous classical systems, whose properties are well known, to new many-body quantum systems. In realizing the link between classical and quantum systems to gain insight into the quantum many-body phases, Biroli et al. nicely add an important new result to earlier investigations that built on such similar insights elsewhere."
See also my talk "A new glassy phase: the superglass".
The main series of papers concern the quantum cavity method. This method (in the version originally proposed by Laumann, Scardicchio and Sondhi) has been developped by us in papers [33, 38] for spin models and bosonic models respectively. This method is designed to treat quantum glassy phases of models defined on Bethe lattices, that is, lattices that are locally tree-like. We applied it to Bosonic models that display a quantum glass phase in [39], and to several quantum optimization problems in [41,45].
See also my talks "A quantum cavity method", "Bose-Einstein condensation in quantum glasses", "A solvable model of quantum random optimization problems".
Statistical mechanics of nonequilibrium stationary states
Review paper: 25
A detailed description of my work in this field can be found here.The fluctuation relation connects positive and negative fluctuations of entropy production in non-equilibrium dissipative systems. It can be considered as a generalization of the second law of thermodynamics to mesoscopic length and time scales. Also, it is a generalization of the Green-Kubo relation to arbitrarily large driving forces. For these reasons, it attracted a lot of interest in the last decade, since it was first proposed by Evans, Morriss, and Cohen and proven by Gallavotti and Cohen for a class of smooth dynamical systems. Unfortunately, its numerical verification in realistic systems is far from being easy, since one has to measure large deviations of the entropy production from its average value over mesoscopic length and time scales, and such events are very rare. For this reason, in most cases, this relation has been tested only for small deviations (or in other words, close to equilibrium) where, however, it reduces to the Green-Kubo relation, or linear response theory. In paper [11] we were able to test this relation, in a realistic model of a driven fluid, at a much higher level of precision. This was made possible by combining a careful choice of the parameters, in order to maximize nonlinear effecs, and a detailed analysis of finite time corrections to the fluctuation relation. This result is important since it has been the first (and for the moment, unique) case in which the fluctuation relation has been tested with high accuracy beyond linear response theory in a realistic model.
Physics of biomolecules
In paper [35] we formulated a dynamical model of DNA mechanical unzipping under the action of a force. The model includes the motion of the fork in the sequence-dependent landscape, the trap acting on the bead, and the polymeric components of the molecular construction: unzipped single strands of DNA, and linkers.
Paper [44] results from a collaboration with the group of Ralf Jockers at institut Cochin. They characterized the molecular complex of the melatonin MT1 receptor, which directly couples to Gi proteins and the regulator of G-protein signaling (RGS) 20. The molecular organization of the ternary MT1/Gi/RGS20 complex was monitored in its basal and activated state by bioluminescence resonance energy transfer between probes inserted at multiple sites of the complex. I collaborated in the elaboration of a geometrical model of the spatial arrangement of the different proteins, compatible with measured distances between the probes.
Optimization problems
Book chapter: 30
This research subject is based on the successful application of statistical mechanics that have been achieved starting from the 1990s. A partial review of these results is paper [30].
In paper [23], we studied the K-SAT problem, which is the archetype of combinatorial optimization problems, in the limit where the ratio of number of clauses and number of variables is large. We showed that in this limit, if the instances are drawn uniformly at random (random K-SAT), the problem can be solved in polynomial time (O(log(N)), where N is the number of variables) by a simple message passing algorithm called Warning Propagation.
In paper [28] we studied the performances of a class of stochastic heuristic search algorithms on some random constraint satisfaction problems. We showed that, for a large class of heuristics, the (heuristic-dependent) largest ratio of constraints per variables for which a search algorithm is likely to find solutions is smaller than the critical ratio above which solutions are clustered and highly correlated.
Random surfaces and quantum gravity
Lorentzian Dynamical Triangulations are a particular class of random discretized surfaces that may be relevant for quantum gravity. In paper [27] we studied a model of LDT in 2+1 dimensions. We impose an additional notion of order on the 2-dimensional "spatial" slices (corresponding to a 1+1+1 structure) that simplifies the combinatorial problem of counting geometries just enough to enable us to calculate the transfer matrix between boundary states labelled by the area of the spatial slices. This is done by rewriting the partition function of the model as the partition function of a product of random matrices. We then solve the random matrix problem using standard techniques such as the replica method. In this way we can identify a critical point and investigate the continuum limit around this point, in particular calculating the quantum Hamiltonian of the continuum theory. This is the first time in dimension larger than two that a Hamiltonian has been derived from such a model by mainly analytical means, and might open the way for a better understanding of scaling and renormalization issues in these models.
Propagation of light in disordered media
The aim of this study was to analytically investigate the dynamics of the excited modes in multi-mode laser cavities, in the presence of a disordered amplifying medium. We developed a statistical approach to mode-locking transitions of multi-mode laser cavities. We showed that, in presence of disorder, the equations for the interacting modes can be mapped onto a statistical model exhibiting a spin glass transition, with the average mode-energy playing the role of inverse temperature. The transition is the disordered analog of phase-locking of modes in standard multi-mode lasers. However, due to disorder, the locking happens in random values of the phases ("random mode locking"), leading to complicated interference patterns in the emitted light. We also show that the mode locking is accompanied by slow dynamics, aging, and history dependent responses, as usual in spin glasses. The method is quite general and can be applied to other nonlinear equations, for instance to the Gross-Pitaevskii equation describing the dynamics of Bose-Einstein condensates. These predictions can therefore be tested in experiments on random lasers and disordered cold atomic gases.
Topology of potential energy surfaces
This series of papers was an attempt to use the notion of "inherent structures", inherited from the physics of glasses, to describe standard phase transitions of first or second order. In particular, we tried to relate singularities in the topology of the potential energy surface, manifested by the properties of its minima and saddle points, to the presence of a phase transition. Despite some interesting results on specific models, we failed to obtain a general relation between topology and thermodynamics.