Session U22 - Statistical Physics, Critical Behavior and Phase Transitions.
ORAL session, Thursday morning, March 25
513B, Palais des Congres

[U22.001] Existence of solutions to the Bethe-Ansatz equations for the one dimensional Hubbard model on a finite lattice.

In this work, a proof of the existence of solutions to the Bethe-Ansatz equations for the one-dimensional Hubbard model on a finite lattice is presented. The well known solution of the model in the thermodynamic limit, by Lieb and Wu, requires the existence of this solution for finite systems. Continuity of the energy with respect to the interaction strength and other properties of the solution are also discussed.

[U22.002] Beyond mean-field universality in the random solid state

At the mean-field level, the random solid state exhibits certain strikingly universal features, most notably the (scaled) distribution of localization lengths, near the solidification critical point [1]. By identifying the appropriate Goldstone and non-Goldstone fluctuations [2], we construct a field-theoretic description of the random solid state. We use a renormalization-group approach and expansion around six dimensions to investigate the robustness of the mean-field universality of the distribution of localization lengths, once the effects of fluctuations are incorporated, again near the critical point. We also use this approach to address aspects of elasticity beyond the Gaussian approximation, as well as order-parameter correlations in the random solid state.

[1] W. Peng, H. E. Castillo, P. M. Goldbart, A. Zippelius, Phys. Rev. B 57, 893 (1998). [2] S. Mukhopadhyay, P. M. Goldbart, A. Zippelius, cond-mat/0310664.

[U22.003] Fluctuating random solids in low dimensions

The amorphous solid state associated with the random localization of particles is examined with regard to the effects of Goldstone fluctuations [1]. These long-wavelength, low-energy fluctuations dominate physical properties deep within the solid state, particularly in low dimensions. As such fluctuations are identified with shear deformations, they furnish an expression for the shear modulus in terms of the order parameter. In addition, Goldstone fluctuations produce a constant shift in the distribution of localization lengths, relative to the mean-field distribution. This shift diverges in two dimensions: thus fluctuations restore the broken translational symmetry and particles are no longer truly localized. Nevertheless, order-parameter correlations decay algebraically and the shear modulus remains finite, so that---as with crystalline solids---two-dimensional random solids exhibit quasi-long range, albeit random, order.

[1] S. Mukhopadhyay, P. M. Goldbart, A. Zippelius, cond-mat/0310664.

[U22.004] Numerical simulations of the propagation of a wave through a random non-linear medium

We studied the propagation of a signal in a non-linear two-dimensional random medium, through real space numerical simulations. The non-linearity of the medium was included by considering a dependence on the effectiveness of the individual randomly placed scatterers with the local intensity of the radiation in the medium. The numerical simulations, as opposed to transmission measurements in the laboratory, allow to study the evolution of the system as well as determining the distribution of scattering intensities in the stationary state. We have found an interesting, quite general, segregating behavior in the system between a region where the propagation is diffusive and a region where it is quasi-ballistic, with a very sharp boundary between those regions. We have studied the dependence of the size of these regions with the geometry of the system and the intensity of the incident wave.

[U22.005] Effects of aperiodic sequences on first-order phase transitions

We study the effects of aperiodic sequences on first-order phase transitions. The model we treat is the Blume-Capel one, with Hamiltonian: -\beta H = \sum_ J_i,j S_i S_j- \Delta \sum_i S_i^2, where S_i=\pm 1,0, the first sum is over first-neighbor sites on a lattice and the second one is over all sites. At low temperatures (high J), this model presents a symmetry-breaking first-order phase transition. This model is studied on a Bethe lattice, where an exact solution is possible. We use two-letter substitutional sequences (one with bounded fluctuatuions and the other one with unbounded fluctuations) and the value for J_i,j on a given generation of the Bethe lattice depends on the letter on the corresponding aperiodic sequence. Bethe lattices with coordination number z=3 and \infty are studied, representing low-dimensional systems and the mean field model, respectively. The limit of stability of the paramagnetic and ferromagnetic phases were determined, with the location of the tricritical point. The introduction of the aperiodic sequence is relevant for the unbounded sequence, with a stronger effect on the Bethe lattice with z=3. This sequence diminishes the region where the transition is of first-order but, contrarily to random disorder, does not replace the entire first-order line by a continuous one.

[U22.006] Exact ground-state energies of one-dimensional random-field Ising models

We study the one-dimensional random field Ising model where the random field is chosen to be +h with probability p and -h with probability 1-p. Using the zero-temperature transfer matrix method proposed by Kadowaki et al, we calculate exactly the ground-state energy and show that it is piecewise linear as a function of random-field strength. We also consider two kinds of extension, the one-dimensional model with three possible values of field strength \h,0,-h\ with probabilities p, q and 1-p-q respectively and 2-leg Ising spin ladder model in bimodal random field. For small field-strength region, we find that these two extensions are equivalent. This means that two spins on the same rung point to the same direction when the random-field strength is small.

[U22.007] Universal mean moment rate profiles of earthquake ruptures: Theoretical predictions and analysis of observed data

Earthquake statistics exhibit a number of power law distributions including the Gutenberg-Richter frequency-size statistics and the Omori law for aftershock decay rates. In search for a model with minimal ingredients to render correct predictions on long spatio-temporal scales we discuss a simple model that is able to approximately capture the fault-dependent observed scaling behaviour of frequency size distributions for main shocks in narrow, heterogeneous fault regions, as well as the statistics of after shocks observed mostly in nearby off-fault regions. Since measured scaling exponents typically have too large statistical error bars to rule out certain models, we also compute universal moment rate shape functions as a stronger test of the theory against observations. Near a critical point, the universal theoretical results have exact corresponding counterparts in models of magnetization avalanches in magnetic systems driven by a slowly increasing external magnetic field. Just like in these magnetic systems, we find that our analysis for earthquakes shows overall agreement between theory and observations, however with a discrepancy in one universal scaling function for moment-rates, which is symmetric under time reversal in the model but asymmetric in the observations. The results point to the existence of deep connections between the physics earthquakes and of avalanches in different types of systems and provide new target signals for earthquake studies.

[U22.008] Hysteresis in a one-dimensional random field Ising model with second-nearest-neighbor interactions

The zero-temperature Ising model with random local fields has been used to describe hysteresis. This model is clearly an oversimplification of real systems, but is does incorporate some of the physics associated with hysteresis. These nclude randomness, non-equilibrium and many-body interactions. Despite its simplifications, this model is quite difficult to solve. Exact results have been obtained for nearest-neighbor interactions on one-dimensional and Bethe lattices. Mean field and renormalization group results have also been obtained.

The new result presented here is an exact calculation for a one-dimensional lattice with second-nearest-neighbor interactions. The complexity of the solution suggest that exact results for more realistic cases could be very difficult to obtain.

[U22.009] Scaling theory of random field Ising model driven out of equilibrium

The random field Ising model (RFIM) has proven fertile ground for investigations of dirty systems that exhibit second order phase transitions. There are two categorically distinct regimes where critical behavior is observed in the RFIM: In equilibrium and far from equilibrium. We will present a scaling theory that describes hysteretic properties of driven RFI systems that start in the equilibrium state but fall out of equilibrium due to diverging relaxation times. Where hysteresis and scaling exists we argue that the system can be described by universal exponents associated with the zero temperature non-equilibrium RFIM.

[U22.010] The generalized Wilsonian renormalization group applied to a \phi^4-theory

In an earlier work we used control and interpolation theory methods to rigorously establish how to coarse grain systems based on their linear response. Much of the motivation for the real space renormalization group (RG) and Wilsonian RG is based on the observation that many physical systems are approximately isotropic and demonstrate separation of time and spatial scales. For such systems, local coarse graining the system variables is a natural and intuitive choice of RG transformation. We apply our new coarse graining methods to provide a straightforward way to justify this practice and shed light on how to appropriately construct RG transformations for more general systems. In particular, we renormalize a \phi^4 field theory that is amenable to our methods but not the standard Wilsonian RG.

[U22.011] Inversion of Specific Heat to Obtain Phonon Spectra from Real Data

Efficient and practical inversion of real specific heat data to obtain the corresponding phonon spectrum has been an elusive goal of condensed matter physics at least since Montroll's work on this problem in 1942. We report success in this quest using Dai's exact formula to regularize the inversion, which is then carried out by means of an odd Hermite expansion of the phonon spectrum, coupled with a least squares fit to the specific heat data using singular value decomposition. We exhibit successful numerical inversions which constitute stringent tests of our methods: the Debye specific heat, with the challenging problem posed by the phonon spectrum discontinuity at the cutoff frequency, as well as other simulated specific heats, and real data for YBa_2Cu_3O_6.92. This method satisfies the inversion uniqueness and existence theorem term by term and is found to be sufficiently powerful and efficient to be generally useful for the analysis of experimental data of adequate accuracy and range.

[U22.012] Statistical theory of high-gain free-electron laser saturation

We propose a new approach, based on statistical mechanics, to predict the saturated state of a single-pass, high-gain free-electron laser (FEL). In analogy with the violent relaxation process in self-gravitating systems and in the Euler equation of 2D turbulence, the initial relaxation of the laser can be described by the statistical mechanics of an associated Vlasov equation. Laser field intensity and electron bunching parameter reach a quasi-stationary value that are well described by a Vlasov stationary state if the number of electrons N is sufficiently large. Finite N effects (granularity) finally drive the system to Boltzmann-Gibbs statistical equilibrium, but this occurs on times that are unphysical (i.e. excessively long undulators). All theoretical predictions are successfully tested in finite N numerical experiments.

[U22.013] Regular and Anomalous Diffusion in Classical Models of Small Polaron Motion in Organic Solids

Although many quantum mechanical calculations of small polaron transport have been performed, they often start from simplified models whose global properties are not well understood. For example, the Holstein Molecular Model often used as a basis for understanding charge transport in organic solids, e.g., describes a particle in a tight-binding band coupled to local optical phonons. For this model it is not really known if the mean-square displacement of an initially localized polaron is ballistic, diffusive, subdiffusive, or self-trapped. Since the combined quantum state space for any such model contains many vibrational degrees of freedom, exact numerical calculations exploring this issue are impossible. In this talk we introduce a simple classical analog of the Holstein polaron model, whose exact dynamical behavior we explore using numerically exact solutions to the classical equations of motion. The one-dimensional version of this model appears diffusive, but is actually marginally subdiffusive, with a diffusion coefficient that decreases logarithmically with time. This is a peculiar feature of the 1D density of states, which gives rise to a non-negligible distribution of particles with velocity near zero. Motion in higher dimensions is diffusive. We describe these results along with calculations that explore the dissipative behavior that occurs in the presence of a uniform electric field.

[U22.014] Quantum dynamics of the formation of a polaron quasiparticle

Time resolved optical pump, terahertz probe experiments on CMR manganites suggest the optical liberation of free electrons and subsequent retrapping as polarons. We study this process theoretically. Starting with a bare electron, the time evolution of the many-body wave function is calculated by integrating the time-dependent Schrödinger equation. The quantum dynamical nature of the phonons is preserved and the full many-body wavefunction is obtained essentially without approximation. We compute the time evolution of the electron and phonon densities and the electron-phonon correlation functions for the Holstein model. Physical consequences for polaron formation dynamics are discussed.

This work was supported by the US DOE.

Part U of program listing