Non-equilibrium fixed points of coupled Ising models

Driven-dissipative systems are expected to give rise to non-equilibrium phenomena that are absent in their equilibrium counterparts. However, phase transitions in these systems generically exhibit an effectively classical equilibrium behavior in spite of their non-equilibrium origin. In this paper, we show that multicritical points in such systems lead to a rich and genuinely non-equilibrium behavior. Specifically, we investigate a driven-dissipative model of interacting bosons that possesses two distinct phase transitions: one from a high- to a low-density phase---reminiscent of a liquid-gas transition---and another to an antiferromagnetic phase. Each phase transition is described by the Ising universality class characterized by an (emergent or microscopic) $\mathbb{Z}_2$ symmetry. They, however, coalesce at a multicritical point, giving rise to a non-equilibrium model of coupled Ising-like order parameters described by a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry. Using a dynamical renormalization-group approach, we show that a pair of non-equilibrium fixed points (NEFPs) emerge that govern the long-distance critical behavior of the system. We elucidate various exotic features of these NEFPs. In particular, we show that a generic continuous scale invariance at criticality is reduced to a discrete scale invariance. This further results in complex-valued critical exponents and spiraling phase boundaries, and it is also accompanied by a complex Liouvillian gap even close to the phase transition. As direct evidence of the non-equilibrium nature of the NEFPs, we show that the fluctuation-dissipation relation is violated at all scales, leading to an effective temperature that becomes "hotter" and "hotter" at longer and longer wavelengths. Finally, we argue that this non-equilibrium behavior can be observed in cavity arrays with cross-Kerr nonlinearities.Comment: 19+11 pages, 7+9 figure

Probing analytical and numerical integrability: The curious case of (AdS5×S5)η(AdS_5\times S^5)_{\eta}

Motivated by recent studies related to integrability of string motion in various backgrounds via analytical and numerical procedures, we discuss these procedures for a well known integrable string background $(AdS_5\times S^5)_{\eta}$. We start by revisiting conclusions from earlier studies on string motion in $(\mathbb{R}\times S^3)_{\eta}$ and $(AdS_3)_{\eta}$ and then move on to more complex problems of $(\mathbb{R}\times S^5)_{\eta}$ and $(AdS_5)_{\eta}$. Discussing both analytically and numerically, we deduce that while $(AdS_5)_{\eta}$ strings do not encounter any irregular trajectories, string motion in the deformed five-sphere can indeed, quite surprisingly, run into chaotic trajectories. We discuss the implications of these results both on the procedures used and the background itself.Comment: 31 pages, 3 figures, references updated, analysis for Spiky strings in section (4.1) have been revised, version to appear in JHE

Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields

In this paper we analyse the integrability of a dynamical system describing the rotational motion of a rigid satellite under the influence of gravitational and magnetic fields. In our investigations we apply an extension of the Ziglin theory developed by Morales-Ruiz and Ramis. We prove that for a symmetric satellite the system does not admit an additional real meromorphic first integral except for one case when the value of the induced magnetic moment along the symmetry axis is related to the principal moments of inertia in a special way.Comment: 39 pages, 4 figures, missing bibliography was adde

Dissipative quantum mechanics beyond Bloch-Redfield: A consistent weak-coupling expansion of the ohmic spin boson model at arbitrary bias

We study the time dynamics of the ohmic spin boson model at arbitrary bias $\epsilon$ and small coupling $\alpha$ to the bosonic bath. Using perturbation theory and the real-time renormalization group (RG) method we present a consistent zero-temperature weak-coupling expansion for the time evolution of the reduced density matrix one order beyond the Bloch-Redfield solution. We develop a renormalized perturbation theory and present an analytical solution covering the whole range from small to large times, including further results for exponentially small or large times. Resumming all secular terms in all orders of perturbation theory we find exponential decay for all terms of the time evolution. We determine the preexponential functions and find slowly varying logarithmic terms with the renormalized Rabi frequency $\Omega$ as energy scale together with strongly varying parts falling off asymptocially as $1/t$ in leading order, in contrast to the unbiased case. Resumming all logarithmic terms in all orders of perturbation theory via real-time RG we find the correct renormalized tunneling and a power-law behaviour for the oscillating modes with exponent crossing over from $2\alpha$ for exponentially small times to a bias-dependent value $2\alpha \epsilon^2/\Omega^2$ for exponentially large times. Furthermore, we present a degenerate perturbation theory to calculate consistently the purely decaying mode one order beyond Bloch-Redfield.Comment: 27 pages, 2 figure

Finding nonlocal Lie symmetries algorithmically

Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and non-local symmetries) algorithmically. The procedure is based on an idea arising from the formal equivalence between the total derivative operator and the vector field associated with the 2ODE over its solutions (Cartan vector field). Basically, from the formal representation of a Lie symmetry it is possible to extract information that allows to use this symmetry practically (in the 2ODE integration process) even in cases where the formal operation cannot be performed, i.e., in cases where the symmetry is nonlocal. Furthermore, when the 2ODE in question depends on parameters, the procedure allows an analysis that determines the regions of the parameter space in which the integrable cases are located