Energy Landscape Statistics of the Random Orthogonal Model

The Random Orthogonal Model (ROM) of Marinari-Parisi-Ritort [MPR1,MPR2] is a model of statistical mechanics where the couplings among the spins are defined by a matrix chosen randomly within the orthogonal ensemble. It reproduces the most relevant properties of the Parisi solution of the Sherrington-Kirckpatrick model. Here we compute the energy distribution, and work out an extimate for the two-point correlation function. Moreover, we show exponential increase of the number of metastable states also for non zero magnetic field.Comment: 23 pages, 6 figures, submitted to J. Phys.

Ground states for a class of deterministic spin models with glassy behaviour

We consider the deterministic model with glassy behaviour, recently introduced by Marinari, Parisi and Ritort, with \ha\ $H=\sum_{i,j=1}^N J_{i,j}\sigma_i\sigma_j$, where $J$ is the discrete sine Fourier transform. The ground state found by these authors for $N$ odd and $2N+1$ prime is shown to become asymptotically dege\-ne\-ra\-te when $2N+1$ is a product of odd primes, and to disappear for $N$ even. This last result is based on the explicit construction of a set of eigenvectors for $J$, obtained through its formal identity with the imaginary part of the propagator of the quantized unit symplectic matrix over the $2$-torus.Comment: 15 pages, plain LaTe

Thermodynamical Limit for Correlated Gaussian Random Energy Models

Let \{E_{\s}(N)\}_{\s\in\Sigma_N} be a family of $|\Sigma_N|=2^N$ centered unit Gaussian random variables defined by the covariance matrix $C_N$ of elements \displaystyle c_N(\s,\tau):=\av{E_{\s}(N)E_{\tau}(N)}, and H_N(\s) = - \sqrt{N} E_{\s}(N) the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition $N=N_1+N_2$, and all pairs (\s,\t)\in \Sigma_N\times \Sigma_N: c_N(\s,\tau)\leq \frac{N_1}{N} c_{N_1}(\pi_1(\s),\pi_1(\tau))+ \frac{N_2}{N} c_{N_2}(\pi_2(\s),\pi_2(\tau)) where \pi_k(\s), k=1,2 are the projections of \s\in\Sigma_N into $\Sigma_{N_k}$. The condition is explicitly verified for the Sherrington-Kirckpatrick, the even $p$-spin, the Derrida REM and the Derrida-Gardner GREM models.Comment: 15 pages, few remarks and two references added. To appear in Commun. Math. Phy

Statistics of energy levels and zero temperature dynamics for deterministic spin models with glassy behaviour

We consider the zero-temperature dynamics for the infinite-range, non translation invariant one-dimensional spin model introduced by Marinari, Parisi and Ritort to generate glassy behaviour out of a deterministic interaction. It is shown that there can be a large number of metatastable (i.e., one-flip stable) states with very small overlap with the ground state but very close in energy to it, and that their total number increases exponentially with the size of the system.Comment: 25 pages, 8 figure

Effective mapping of spin-1 chains onto integrable fermionic models. A study of string and Neel correlation functions

We derive the dominant contribution to the large-distance decay of correlation functions for a spin chain model that exhibits both Haldane and Neel phases in its ground state phase diagram. The analytic results are obtained by means of an approximate mapping between a spin-1 anisotropic Hamiltonian onto a fermionic model of noninteracting Bogolioubov quasiparticles related in turn to the XY spin-1/2 chain in a transverse field. This approach allows us to express the spin-1 string operators in terms of fermionic operators so that the dominant contribution to the string correlators at large distances can be computed using the technique of Toeplitz determinants. As expected, we find long-range string order both in the longitudinal and in the transverse channel in the Haldane phase, while in the Neel phase only the longitudinal order survives. In this way, the long-range string order can be explicitly related to the components of the magnetization of the XY model. Moreover, apart from the critical line, where the decay is algebraic, we find that in the gapped phases the decay is governed by an exponential tail multiplied by algebraic factors. As regards the usual two points correlation functions, we show that the longitudinal one behaves in a 'dual' fashion with respect to the transverse string correlator, namely both the asymptotic values and the decay laws exchange when the transition line is crossed. For the transverse spin-spin correlator, we find a finite characteristic length which is an unexpected feature at the critical point. We also comment briefly the entanglement features of the original system versus those of the effective model. The goodness of the approximation and the analytical predictions are checked versus density-matrix renormalization group calculations.Comment: 28 pages, plain LaTeX, 2 EPS figure

Deterministic spin models with a glassy phase transition

We consider the infinite-range deterministic spin models with Hamiltonian $H=\sum_{i,j=1}^N J_{i,j}\sigma_i\sigma_j$, where $J$ is the quantization of a chaotic map of the torus. The mean field (TAP) equations are derived by summing the high temperature expansion. They predict a glassy phase transition at the critical temperature $T\sim 0.8$.Comment: 8 pages, no figures, RevTex forma

On c=1c=1 critical phases in anisotropic spin-1 chains

Quantum spin-1 chains may develop massless phases in presence of Ising-like and single-ion anisotropies. We have studied c=1 critical phases by means of both analytical techniques, including a mapping of the lattice Hamiltonian onto an O(2) nonlinear sigma model, and a multi-target DMRG algorithm which allows for accurate calculation of excited states. We find excellent quantitative agreement with the theoretical predictions and conclude that a pure Gaussian model, without any orbifold construction, describes correctly the low-energy physics of these critical phases. This combined analysis indicates that the multicritical point at large single-ion anisotropy does not belong to the same universality class as the Takhtajan-Babujian Hamiltonian as claimed in the past. A link between string-order correlation functions and twisting vertex operators, along the c=1 line that ends at this point, is also suggested.Comment: 9 pages, 3 figures, svjour format, submitted to Eur. Phys. J.

Egorov property in perturbed cat map

We study the time evolution of the quantum-classical correspondence (QCC) for the well known model of quantised perturbed cat maps on the torus in the very specific regime of semi-classically small perturbations. The quality of the QCC is measured by the overlap of classical phase-space density and corresponding Wigner function of the quantum system called quantum-classical fidelity (QCF). In the analysed regime the QCF strongly deviates from the known general behaviour in particular it decays faster then exponential. Here we study and explain the observed behavior of the QCF and the apparent violation of the QCC principle.Comment: 12 pages, 7 figure

An infinite step billiard

A class of non-compact billiards is introduced, namely the infinite step billiards, i.e. systems of a point particle moving freely in the domain Ω = ∪n∈ℕ[n,n + 1] × [0, p_n], with elastic reflections on the boundary; here p_0 = 1, p_n > 0 and pn ↘ 0. After describing some generic ergodic features of these dynamical systems, we turn to a more detailed study of the example p_n = 2^{-n}. Playing an important role in this case are the so-called escape orbits, that is, orbits going to +∞ monotonically in the X-velocity. A fairly complete description of them is given. This enables us to prove some results concerning the topology of the dynamics on the billiard