The Information Geometry of the Ising Model on Planar Random Graphs

It has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterisation of the phase structure, particularly in the case where there are two such parameters -- such as the Ising model with inverse temperature $\beta$ and external field $h$. In various two parameter calculable models the scalar curvature ${\cal R}$ of the information metric has been found to diverge at the phase transition point $\beta_c$ and a plausible scaling relation postulated: ${\cal R} \sim |\beta- \beta_c|^{\alpha - 2}$. For spin models the necessity of calculating in non-zero field has limited analytic consideration to 1D, mean-field and Bethe lattice Ising models. In this letter we use the solution in field of the Ising model on an ensemble of planar random graphs (where $\alpha=-1, \beta=1/2, \gamma=2$) to evaluate the scaling behaviour of the scalar curvature, and find ${\cal R} \sim | \beta- \beta_c |^{-2}$. The apparent discrepancy is traced back to the effect of a negative $\alpha$.Comment: Version accepted for publication in PRE, revtex

The quantum brachistochrone problem for non-Hermitian Hamiltonians

Recently Bender, Brody, Jones and Meister found that in the quantum brachistochrone problem the passage time needed for the evolution of certain initial states into specified final states can be made arbitrarily small, when the time-evolution operator is taken to be non-Hermitian but PT-symmetric. Here we demonstrate that such phenomena can also be obtained for non-Hermitian Hamiltonians for which PT-symmetry is completely broken, i.e. dissipative systems. We observe that the effect of a tunable passage time can be achieved by projecting between orthogonal eigenstates by means of a time-evolution operator associated with a non-Hermitian Hamiltonian. It is not essential that this Hamiltonian is PT-symmetric

Martingale Models for Quantum State Reduction

Stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory. Here we examine the correspondence of the two theories in detail, making a systematic use of the methods of martingale theory. An analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred. We consider the general projection postulate of L\"uders applicable in the case of a possibly degenerate eigenvalue spectrum, and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state. We also analyse the associated Lindblad equation for the evolution of the density matrix, and obtain an exact time-dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the L\"uders density matrix. Finally, we apply Girsanov's theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric Brownian motions, thereby constructing an explicit unravelling of the Lindblad equation.Comment: 30 pages LaTeX. Submitted to Journal of Physics

Thermodynamic curvature and black holes

I give a relatively broad survey of thermodynamic curvature $R$, one spanning results in fluids and solids, spin systems, and black hole thermodynamics. $R$ results from the thermodynamic information metric giving thermodynamic fluctuations. $R$ has a unique status in thermodynamics as being a geometric invariant, the same for any given thermodynamic state. In fluid and solid systems, the sign of $R$ indicates the character of microscopic interactions, repulsive or attractive. $|R|$ gives the average size of organized mesoscopic fluctuating structures. The broad generality of thermodynamic principles might lead one to believe the same for black hole thermodynamics. This paper explores this issue with a systematic tabulation of results in a number of cases.Comment: 27 pages, 10 figures, 7 tables, 78 references. Talk presented at the conference Breaking of Supersymmetry and Ultraviolet Divergences in extended Supergravity, in Frascati, Italy, March 27, 2013. v2 corrects some small problem

The Information Geometry of the Spherical Model

Motivated by previous observations that geometrizing statistical mechanics offers an interesting alternative to more standard approaches,we have recently calculated the curvature (the fundamental object in this approach) of the information geometry metric for the Ising model on an ensemble of planar random graphs. The standard critical exponents for this model are alpha=-1, beta=1/2, gamma=2 and we found that the scalar curvature, R, behaves as epsilon^(-2),where epsilon = beta_c - beta is the distance from criticality. This contrasts with the naively expected R ~ epsilon^(-3) and the apparent discrepancy was traced back to the effect of a negative alpha on the scaling of R. Oddly,the set of standard critical exponents is shared with the 3D spherical model. In this paper we calculate the scaling behaviour of R for the 3D spherical model, again finding that R ~ epsilon^(-2), coinciding with the scaling behaviour of the Ising model on planar random graphs. We also discuss briefly the scaling of R in higher dimensions, where mean-field behaviour sets in.Comment: 7 pages, no figure

Quantum Chaos Versus Classical Chaos: Why is Quantum Chaos Weaker?

We discuss the questions: How to compare quantitatively classical chaos with quantum chaos? Which one is stronger? What are the underlying physical reasons

Misleading signatures of quantum chaos

The main signature of chaos in a quantum system is provided by spectral statistical analysis of the nearest neighbor spacing distribution and the spectral rigidity given by $\Delta_3(L)$. It is shown that some standard unfolding procedures, like local unfolding and Gaussian broadening, lead to a spurious increase of the spectral rigidity that spoils the $\Delta_3(L)$ relationship with the regular or chaotic motion of the system. This effect can also be misinterpreted as Berry's saturation.Comment: 4 pages, 5 figures, submitted to Physical Review

The Impact of Isospin Breaking on the Distribution of Transition Probabilities

In the present paper we investigate the effect of symmetry breaking in the statistical distributions of reduced transition amplitudes and reduced transition probabilities. These quantities are easier to access experimentally than the components of the eigenvectors and were measured by Adams et al. for the electromagnetic transitions in ^{26}Al. We focus on isospin symmetry breaking described by a matrix model where both, the Hamiltonian and the electromagnetic operator, break the symmetry. The results show that for partial isospin conservation, the statistical distribution of the reduced transition probability can considerably deviate from the Porter-Thomas distribution.Comment: 16 pages, 8 figures, submitted to PR

Geometrothermodynamics of black holes

The thermodynamics of black holes is reformulated within the context of the recently developed formalism of geometrothermodynamics. This reformulation is shown to be invariant with respect to Legendre transformations, and to allow several equivalent representations. Legendre invariance allows us to explain a series of contradictory results known in the literature from the use of Weinhold's and Ruppeiner's thermodynamic metrics for black holes. For the Reissner-Nordstr\"om black hole the geometry of the space of equilibrium states is curved, showing a non trivial thermodynamic interaction, and the curvature contains information about critical points and phase transitions. On the contrary, for the Kerr black hole the geometry is flat and does not explain its phase transition structure.Comment: Revised version, to be published in Gen.Rel.Grav.(Mashhoon's Festschrift

Correlation functions of scattering matrix elements in microwave cavities with strong absorption

The scattering matrix was measured for microwave cavities with two antennas. It was analyzed in the regime of overlapping resonances. The theoretical description in terms of a statistical scattering matrix and the rescaled Breit-Wigner approximation has been applied to this regime. The experimental results for the auto-correlation function show that the absorption in the cavity walls yields an exponential decay. This behavior can only be modeled using a large number of weakly coupled channels. In comparison to the auto-correlation functions, the cross-correlation functions of the diagonal S-matrix elements display a more pronounced difference between regular and chaotic systems