Spherical Spin Models Research Articles

Quenched dynamics of classical isolated systems: the spherical spin model with two-body random interactions or the Neumann integrable model

We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body random interactions. In the statistical physics framework, the potential energy is of the so-called p = 2 kind, closely linked to the scalar field theory. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable model. We take initial conditions from the Boltzmann equilibrium measure at a temperature that can be above or below the static phase transition, typical of a disordered (paramagnetic) or of an ordered (disguised ferromagnetic) equilibrium phase. We subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian, obtained from an instantaneous global rescaling of the elements in the interaction random matrix. In the limit of infinitely many degrees of freedom, , we identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We next set the analysis of the system with finite number of degrees of freedom in terms of N non-linearly coupled modes. We argue that in the limit the modes decouple at long times. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with the fluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the N − 1 integrals of motion, notably, their scaling with N, and we use them to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs–Boltzmann behaviour of the global observables. We elaborate on the role played by these constants of motion after the quench and we briefly discuss the possible description of the asymptotic dynamics in terms of a generalised Gibbs ensemble.

Novel Self-Assembled Morphologies from Isotropic Interactions

We present results from particle simulations with isotropic medium range interactions in two dimensions. At low temperature novel types of aggregated structures appear. We show that these structures can be explained by spontaneous symmetry breaking in analytic solutions to an adaptation of the spherical spin model. We predict the critical particle number where the symmetry breaking occurs and show that the resulting phase diagram agrees well with results from particle simulations.

MAP estimation algorithm for phase response curves based on analysis of the observation process

Many research groups have sought to measure phase response curves (PRCs) from real neurons. However, methods of estimating PRCs from noisy spike-response data have yet to be established. In this paper, we propose a Bayesian approach for estimating PRCs. First, we analytically obtain a likelihood function of the PRC from a detailed model of the observation process formulated as Langevin equations. Then we construct a maximum a posteriori (MAP) estimation algorithm based on the analytically obtained likelihood function. The MAP estimation algorithm derived here is equivalent to the spherical spin model. Moreover, we analytically calculate a marginal likelihood corresponding to the free energy of the spherical spin model, which enables us to estimate the hyper-parameters, i.e., the intensity of the Langevin force and the smoothness of the prior.

Quantum spherical spin model on hypercubic lattices

We present an alternative treatment to the quantum spherical spin model on $(d\ensuremath{\ge}2)$-dimensional hypercubic lattices, focusing on the effects of quantum $(g)$ and thermal $(T)$ fluctuations, under a uniform magnetic field $h$, on the correlation function, correlation length, entropy, specific heat, and energy gap in the excitation spectrum. Explicit expressions for such quantities are provided close to the $d\ensuremath{\ge}2$ quantum ($g={g}_{c}$, $T=0$) and $d\ensuremath{\ge}3$ thermal $[T={T}_{c}(g)]$ phase transitions in $h=0$, including the low-$T$ quantum regimes near the quantum critical point. In particular, the calculation of the correlation function and correlation length generalizes the results on the $g=0$ classical spherical model. At $T=0$, the zero-field system is gapless at and below ${g}_{c}$; however, a gap opens in the quantum-disordered ground state, $g>{g}_{c}$. Conversely, the null gap for $T\ensuremath{\le}{T}_{c}(g\ensuremath{\ne}0)$ becomes finite as $T\ensuremath{\rightarrow}{T}_{c}(g\ensuremath{\ne}0{)}^{+}$; thus, quantum fluctuations suppress the critical prefactors of observables near ${T}_{c}(g\ensuremath{\ne}0)$, though they are irrelevant to the universality class shared with the gapless classical spherical model. The results on the entropy and specific heat in $g\ensuremath{\ne}0$ circumvent the drawback in classical spherical models concerning the third law of thermodynamics, as $T\ensuremath{\rightarrow}0$.

Quantum spherical spin model on theAB2chain

A quantum spherical spin model with an antiferromagnetic coupling $J$ on the $A{B}_{2}$ chain is studied, whose topology is of interest in the context of ferrimagnetic polymers and oxocuprates. A ferrimagnetic long-range order is found at the only critical point $g=T=h=0$, where $T$ denotes the temperature, $h$ the magnetic field, and $g$ the quantum coupling constant in energy units. The approach to the critical point, with diverging correlation length $\ensuremath{\xi}$ and cell susceptibility ${\ensuremath{\chi}}_{\text{cell}}$, is characterized through several paths in the ${g,T,h}$ parameter space: for $(T∕J)\ensuremath{\rightarrow}0$ and $g=h=0$, ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕{T}^{2}$, as also found in several classical and quantum spherical and Heisenberg models; for $(h∕J)\ensuremath{\rightarrow}0$ and $g=T=0$, ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕h$; and for $(g∕J)\ensuremath{\rightarrow}0$ and $T=h=0$, $\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}1∕\sqrt{g}$, thus evidencing an essential singularity due to quantum fluctuations. In any path chosen the relation ${\ensuremath{\chi}}_{\text{cell}}\ensuremath{\sim}{\ensuremath{\xi}}^{2}$ is satisfied. For finite $g$ and $T$ a field-induced short-range ferrimagnetism occurs to some extent in the ${g,T,h}$ space, as confirmed by the analysis of the local spin averages, cell magnetization with a rapid increase for very low fields, and spin-spin correlation functions. The asymptotic limits of the correlation functions are also provided with respect to $g$, $T$, $h$, and spin distance. The analysis of the entropy and specific heat reveals that the quantum fluctuations fix the well-known drawback of classical spherical models concerning the third law of thermodynamics.

Quantum spherical spin models.

A recently introduced class of quantum spherical spin models is considered in detail. Since the spherical constraint already contains a kinetic part, the Hamiltonian need not have kinetic term. As a consequence, situations with or without momenta in the Hamiltonian can be described, which may lead to different symmetry classes. Two models that show this difference are analyzed. Both models are exactly solvable and their phase diagram is analyzed. A transversal external field leads to a phase transition line that ends in a quantum critical point. The two considered symmetries of the Hamiltonian considered give different critical phenomena in the quantum critical region. The model with momenta is argued to be analog to the large-N limit of an SU(N) Heisenberg ferromagnet, and the model without momenta shares the critical phenomena of an SU(N) Heisenberg antiferromagnet.

Phase transitions in nonextensive spin systems.

The spherical spin model with infinite-range ferromagnetic interactions is investigated analytically in the framework of nonextensive thermostatics generalizing the Boltzmann-Gibbs statistical mechanics. We show that for repulsive correlations, a weak-ferromagnetic phase develops. There is a tricritical point separating para-, weak-ferro, and ferro regimes. The transition from paramagnetic to weak-ferromagnetic phase is an unusual first-order phase transition in which a discontinuity of the averaged order parameter appears, even for finite number of spins. This result puts in a different way the question of the stability of critical phenomena with respect to the long-ranged correlations.

Dynamics of ferromagnetic spherical spin models with power law interactions: exact solution

We solve the Langevin dynamics of d-dimensional ferromagnetic spherical models with interactions that decay with distance as r −( d+ σ) . The long time dynamics of correlations and responses are studied in detail in the different dynamical regimes and the validity of fluctuation–dissipation relations (or its violation) are shown. In particular, we show that the fluctuation–dissipation ratio X( t+ t w , t w ) is asymptotically a function only of the waiting time t w in the aging regime and that X→0 as t w →∞. The results are valid in any finite dimension d and for 0< σ<2 where short-range behavior is recovered. We also solve the T=0 Cahn–Hilliard dynamics of this model (conserved order parameter). An analysis of the multiscaling behavior of the autocorrelation function is presented.

Resonant quantum coherence of magnetization at excited states in nanospin systems with different crystal symmetries

We investigate the non-equilibrium dynamics of spherical spin models with two-spin interactions. For the exactly solvable models of the d-dimensional spherical ferromagnet and the spherical Sherrington-Kirkpatrick (SK) model the asymptotic dynamics has for large times and large waiting times the same formal structure. In the limit of large waiting times we find in both models an intermediate time scale, scaling as a power of the waiting time with an exponent smaller than one, and thus separating the time-translation-invariant short-time dynamics from the aging regime. It is this time scale on which the fluctuation-dissipation theorem is violated. Aging in these models is similar to that observed in spin glasses at the level of correlation functions, but different at the level of response functions, and thus different at the level of experimentally accessible quantities like thermoremanent magnetization.

Frustrated spherical spin model and walk statistics on Bethe lattices

It is known that the free energy of the spherical model of Berlin and Kac [Phys. Rev. 86, 821 (1952)], with nearest-neighbor couplings on an Euclidean lattice, can be expressed in terms of generating functions of walks on the same lattice. We show that this relationship is valid on any discrete geometrical structure (graph) and can be exploited to solve the model on non-Euclidean lattices, such as a variety of infinite tree graph. We show next that this technique of solution can be employed even when interactions are extended beyond nearest neighbors and frustrated, provided one defines distance in terms of shortest path on the graph between two sites. Implications of this definition for Euclidean lattices is discussed. We then compute the phase diagram of the spherical model with next-nearest-neighbor competing interactions on the usual Bethe lattice, and the thermodynamical behavior of ferromagnet and antiferromagnet on Bethe cactuses, that is, trees whose branches are symplexes.