Criticality in statistical physics naturally emerges at isolated points in the phase diagram. Jamming of spheres is not an exception: varying density, it is the critical point that separates the unjammed phase where spheres do not overlap and the jammed phase where they cannot be arranged without overlaps. The same remains true in more general constraint satisfaction problems with continuous variables (CCSP) where jamming coincides with the (protocol dependent) satisfiability transition point. In this work we show that by carefully choosing the cost function to be minimized, the region of criticality extends to occupy a whole region of the jammed phase. As a working example, we consider the spherical perceptron with a linear cost function in the unsatisfiable (UNSAT) jammed phase and we perform numerical simulations which show critical power laws emerging in the configurations obtained minimizing the linear cost function. We develop a scaling theory to compute the emerging critical exponents.

We analyze the topological deformations of the ground state manifold of a quantum spin-1/2 in a magnetic field H = H(sin theta cos phi, sin theta sin phi cos theta) induced by a coupling to an ohmic quantum dissipative environment at zero temperature. From Bethe ansatz results and a variational approach, we confirm that the Chern number associated with the geometry of the reduced spin ground state manifold is preserved in the delocalized phase for alpha < 1. We report a divergence of the Berry curvature at alpha(c) = 1 for magnetic fields aligned along the equator theta = pi/2. This divergence is caused by the complete quenching of the transverse magnetic field by the bath associated with a gap closing that occurs at the localization Kosterlitz-Thouless quantum phase transition in this model. Recent experiments in quantum circuits have engineered nonequilibrium protocols to access topological properties from a measurement of a dynamical Chern number defined via the out-of-equilibrium spin expectation values. Applying a numerically exact stochastic Schrodinger approach we find that, for a fixed field sweep velocity theta(t) = vt, the bath induces a crossover from ( quasi) adiabatic to nonadiabatic dynamical behavior when the spin bath coupling a increases. We also investigate the particular regime H/omega(c) << v/H << 1 with large bath cutoff frequency.c, where the dynamical Chern number vanishes already at alpha = 1/2. In this regime, the mapping to an interacting resonance level model enables us to analytically describe the behavior of the dynamical Chern number in the vicinity of alpha = 1/2. We further provide an intuitive physical explanation of the bath-induced breakdown of adiabaticity in analogy to the Faraday effect in electromagnetism. We demonstrate that the driving of the spin leads to the production of a large number of bosonic excitations in the bath, which strongly affect the spin dynamics. Finally, we quantify the spin-bath entanglement and formulate an analogy with an effective model at thermal equilibrium. Disciplines Condensed Matter Physics | Physics CommentsThis article is published as Henriet, Loïc, Antonio Sclocchi, Peter P. Orth, and Karyn Le Hur. "Topology of a dissipative spin: Dynamical Chern number, bath-induced nonadiabaticity, and a quantum dynamo effect." Physical Review B 95, no. 5 (2017) We analyze the topological deformations of the ground state manifold of a quantum spin-1/2 in a magnetic field H = H (sin θ cos φ, sin θ sin φ, cos θ ) induced by a coupling to an ohmic quantum dissipative environment at zero temperature. From Bethe ansatz results and a variational approach, we confirm that the Chern number associated with the geometry of the reduced spin ground state manifold is preserved in the delocalized phase for α < 1. We report a divergence of the Berry curvature at α c = 1 for magnetic fields aligned along the equator θ = π/2. This divergence is caused by the complete quenching of the transverse magnetic field by the bath associated with a gap closing that occurs a...

We show that soft spheres interacting with a linear ramp potential when overcompressed beyond the jamming point fall in an amorphous solid phase which is critical, mechanically marginally stable and share many features with the jamming point itself. In the whole phase, the relevant local minima of the potential energy landscape display an isostatic contact network of perfectly touching spheres whose statistics is controlled by an infinite lengthscale. Excitations around such energy minima are non-linear, system spanning, and characterized by a set of non-trivial critical exponents. We perform numerical simulations to measure their values and show that, while they coincide, within numerical precision, with the critical exponents appearing at jamming, the nature of the corresponding excitations is richer. Therefore, linear soft spheres appear as a novel class of finite dimensional systems that self-organize into new, critical, marginally stable, states.

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We investigate the properties of local minima of the energy landscape of a continuous non-convex optimization problem, the spherical perceptron with piecewise linear cost function and show that they are critical, marginally stable and displaying a set of pseudogaps, singularities and non-linear excitations whose properties appear to be in the same universality class of jammed packings of hard spheres. The piecewise linear perceptron problem appears as an evolution of the purely linear perceptron optimization problem that has been recently investigated in [1]. Its cost function contains two non-analytic points where the derivative has a jump. Correspondingly, in the non-convex/glassy phase, these two points give rise to four pseudogaps in the force distribution and this induces four power laws in the gap distribution as well. In addition one can define an extended notion of isostaticity and show that local minima appear again to be isostatic in this phase. We believe that our results generalize naturally to more complex cases with a proliferation of non-linear excitations as the number of non-analytic points in the cost function is increased.

Recently, several theories including the replica method made predictions for the generalization error of Kernel Ridge Regression. In some regimes, they predict that the method has a 'spectral bias': decomposing the true function f * on the eigenbasis of the kernel, it fits well the coefficients associated with the O(P) largest eigenvalues, where P is the size of the training set. This prediction works very well on benchmark data sets such as images, yet the assumptions these approaches make on the data are never satisfied in practice. To clarify when the spectral bias prediction holds, we first focus on a one-dimensional model where rigorous results are obtained and then use scaling arguments to generalize and test our findings in higher dimensions. Our predictions include the classification case f (x) =sign(x 1 ) with a data distribution that vanishes at the decision boundary p(x) ∼ x χ 1 . For χ > 0 and a Laplace kernel, we find that (i) there exists a cross-over ridge λ * d,χ (P) ∼ P − 1 d+χ such that for λ λ * d,χ (P), the replica method applies, but not for λ λ * d,χ (P), (ii) in the ridge-less case, spectral bias predicts the correct training curve exponent only in the limit d → ∞.

Recently, we showed that optimization problems, both in infinite as well as in finite dimensions, for continuous variables and soft excluded volume constraints, can display entire isostatic phases where local minima of the cost function are marginally stable configurations endowed with non-linear excitations [, ]. In this work we describe an athermal adiabatic algorithm to explore with continuity the corresponding rough high-dimensional landscape. We concentrate on a prototype problem of this kind, the spherical perceptron optimization problem with linear cost function (hinge loss). This algorithm allows to ‘surf’ between isostatic marginally stable configurations and to investigate some properties of such landscape. In particular we focus on the statistics of avalanches occurring when local minima are destabilized. We show that when perturbing such minima, the system undergoes plastic rearrangements whose size is power law distributed and we characterize the corresponding critical exponent. Finally we investigate the critical properties of the unjamming transition, showing that the linear interaction potential gives rise to logarithmic behavior in the scaling of energy and pressure as a function of the distance from the unjamming point. For some quantities, the logarithmic corrections can be gauged out. This is the case of the number of soft constraints that are violated as a function of the distance from jamming which follows a non-trivial power law behavior.

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