Failure of Universality in Noncompact Lattice Field Theories

The nonuniversal behavior of two noncompact nonlinear sigma models is described. When these theories are defined on a lattice, the behavior of the order parameter (magnetization) near the critical point is sensitive to the details of the lattice definition. This is counter to experience and to expectations based on the ideas of universality.Comment: 24 pages, REVTeX version 3.0 with 4 embedded figures, provided separately in compressed-uuencoded postscript packed in a self-extracting csh script produced with uufiles. To appear in J. Math. Phys

Non-monotonicity in the quantum-classical transition: Chaos induced by quantum effects

The transition from classical to quantum behavior for chaotic systems is understood to be accompanied by the suppression of chaotic effects as the relative size of $\hbar$ is increased. We show evidence to the contrary in the behavior of the quantum trajectory dynamics of a dissipative quantum chaotic system, the double-well Duffing oscillator. The classical limit in the case considered has regular behavior, but as the effective $\hbar$ is increased we see chaotic behavior. This chaos then disappears deeper into the quantum regime, which means that the quantum-classical transition in this case is non-monotonic in $\hbar$.Comment: 4 pages; presentation modified significantly to demonstrate that quantum effects are indeed responsible for the `anomalous' chaos. 2 figures adde

Permutation entropy of indexed ensembles: Quantifying thermalization dynamics

We introduce `PI-Entropy' $\Pi(\tilde{\rho})$ (the Permutation entropy of an Indexed ensemble) to quantify mixing due to complex dynamics for an ensemble $\rho$ of different initial states evolving under identical dynamics. We find that $\Pi(\tilde{\rho})$ acts as an excellent proxy for the thermodynamic entropy $S(\rho)$ but is much more computationally efficient. We study 1-D and 2-D iterative maps and find that $\Pi(\tilde{\rho})$ dynamics distinguish a variety of system time scales and track global loss of information as the ensemble relaxes to equilibrium. There is a universal S-shaped relaxation to equilibrium for generally chaotic systems, and this relaxation is characterized by a \emph{shuffling} timescale that correlates with the system's Lyapunov exponent. For the Chirikov Standard Map, a system with a mixed phase space where the chaos grows with nonlinear kick strength $K$, we find that for high $K$, $\Pi(\tilde{\rho})$ behaves like the uniformly hyperbolic 2-D Cat Map. For low $K$ we see periodic behavior with a relaxation envelope resembling those of the chaotic regime, but with frequencies that depend on the size and location of the initial ensemble in the mixed phase space as well as $K$. We discuss how $\Pi(\tilde{\rho})$ adapts to experimental work and its general utility in quantifying how complex systems change from a low entropy to a high entropy state.Comment: 7 pages, 6 figure

Parametric hypersensitivity in many-body bath-mediated transport: The quantum Rabi model

We demonstrate that non-equilibrium steady states of the dissipative Rabi model show dramatic spikes in transport rates over narrow parameter ranges. Similar results are found for the Holstein and Dicke models. This is found to be due to avoided energy level crossings in the corresponding closed systems, and correlates with spikes in the entanglement entropy of key eigenstates, a signature of strong mixing and resonance among system degrees of freedom. Further, contrasting the Rabi model with the Jaynes-Cummings model reveals this behavior as being related to quantum integrability.Comment: 5 pages, 4 figure

The Quantized O(1,2)/O(2)×Z2O(1,2)/O(2)\times Z_2 Sigma Model Has No Continuum Limit in Four Dimensions. I. Theoretical Framework

The nonlinear sigma model for which the field takes its values in the coset space $O(1,2)/O(2)\times Z_2$ is similar to quantum gravity in being perturbatively nonrenormalizable and having a noncompact curved configuration space. It is therefore a good model for testing nonperturbative methods that may be useful in quantum gravity, especially methods based on lattice field theory. In this paper we develop the theoretical framework necessary for recognizing and studying a consistent nonperturbative quantum field theory of the $O(1,2)/O(2)\times Z_2$ model. We describe the action, the geometry of the configuration space, the conserved Noether currents, and the current algebra, and we construct a version of the Ward-Slavnov identity that makes it easy to switch from a given field to a nonlinearly related one. Renormalization of the model is defined via the effective action and via current algebra. The two definitions are shown to be equivalent. In a companion paper we develop a lattice formulation of the theory that is particularly well suited to the sigma model, and we report the results of Monte Carlo simulations of this lattice model. These simulations indicate that as the lattice cutoff is removed the theory becomes that of a pair of massless free fields. Because the geometry and symmetries of these fields differ from those of the original model we conclude that a continuum limit of the $O(1,2)/O(2)\times Z_2$ model which preserves these properties does not exist.Comment: 25 pages, no figure

Experimental signatures of the quantum-classical transition in a nanomechanical oscillator modeled as a damped driven double-well problem

We demonstrate robust and reliable signatures for the transition from quantum to classical behavior in the position probability distribution of a damped double-well system using the Qunatum State Diffusion approach to open quantum systems. We argue that these signatures are within experimental reach, for example in a doubly-clamped nanomechanical beam.Comment: Proceedings of the conference FMQT 1

The Quantized O(1,2)/O(2)×Z2O(1,2)/O(2)\times Z_2 Sigma Model Has No Continuum Limit in Four Dimensions. II. Lattice Simulation

A lattice formulation of the $O(1,2)/O(2)\times Z_2$ sigma model is developed, based on the continuum theory presented in the preceding paper. Special attention is given to choosing a lattice action (the ``geodesic'' action) that is appropriate for fields having noncompact curved configuration spaces. A consistent continuum limit of the model exists only if the renormalized scale constant $\beta_R$ vanishes for some value of the bare scale constant~$\beta$. The geodesic action has a special form that allows direct access to the small-$\beta$ limit. In this limit half of the degrees of freedom can be integrated out exactly. The remaining degrees of freedom are those of a compact model having a $\beta$-independent action which is noteworthy in being unbounded from below yet yielding integrable averages. Both the exact action and the $\beta$-independent action are used to obtain $\beta_R$ from Monte Carlo computations of field-field averages (2-point functions) and current-current averages. Many consistency cross-checks are performed. It is found that there is no value of $\beta$ for which $\beta_R$ vanishes. This means that as the lattice cutoff is removed the theory becomes that of a pair of massless free fields. Because these fields have neither the geometry nor the symmetries of the original model we conclude that the $O(1,2)/O(2)\times Z_2$ model has no continuum limit.Comment: 32 pages, 7 postscript figures, UTREL 92-0