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Cluster mean-field dynamics of the long-range interacting Ising chain

G. Piccitto

Published Sep 29, 2020• 0 citations

Abstract

In this thesis we study the dynamics of the long-range interacting Ising chain using a cluster mean-field approach, a mean-field theory that accounts for short range correlations. The principal result is that, if the system is isolated from an external environment, the model exhibits two different behaviors depending on the range of the interactions. Whenever the system is truly long-range, it exhibits a sharp mean-field dynamical phase transition from a dynamical ferromagnetic to a dynamical paramagnetic phase. Reducing the range of the interactions a critical region, showing hypersensitivity to initial conditions, appears. Interestingly, this chaotic region shares the same physics of a classical tossed coin that is allowed to bounce on the floor. In a second part of the work we derive the cluster mean-field equations of motion describing the dynamics of a fully connected Ising chain connected to an external bath. In particular we focused on the dynamics in presence of a dissipation generated by string of Glauber operators acting on one or more sites of the chain. This model, in presence of global dissipative processes, exhibits persistent oscillations in time revealing the existence of a boundary time-crystal and we studied the stability of the boundary time-crystal showing that global dissipative processes are a key ingredient for their existence.

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In this thesis we study the dynamics of the long-range interacting Ising chain using a cluster mean-field approach, a mean-field theory that accounts for short range correlations. The principal result is that, if the system is isolated from an external environment, the model exhibits two different behaviors depending on the range of the interactions. Whenever the system is truly long-range, it exhibits a sharp mean-field dynamical phase transition from a dynamical ferromagnetic to a dynamical paramagnetic phase. Reducing the range of the interactions a critical region, showing hypersensitivity to initial conditions, appears. Interestingly, this chaotic region shares the same physics of a classical tossed coin that is allowed to bounce on the floor. In a second part of the work we derive the cluster mean-field equations of motion describing the dynamics of a fully connected Ising chain connected to an external bath. In particular we focused on the dynamics in presence of a dissipation generated by string of Glauber operators acting on one or more sites of the chain. This model, in presence of global dissipative processes, exhibits persistent oscillations in time revealing the existence of a boundary time-crystal and we studied the stability of the boundary time-crystal showing that global dissipative processes are a key ingredient for their existence.

We investigate the effect of short-range correlations on the dynamical phase diagram of quantum many-body systems with long-range interactions. Focusing on Ising spin chains with power-law decaying interactions and accounting for short-range correlations by a cluster mean field theory we show that short-range correlations are responsible for the emergence of a chaotic dynamical region. Analyzing the fine details of the phase diagram, we show that the resulting chaotic dynamics bears close analogies with that of a tossed coin.

Abstract Quantum many body systems with long range interactions are known to display many fascinating phenomena experimentally observable in trapped ions, Rydberg atoms and polar molecules. Among these are dynamical phase transitions which occur after an abrupt quench in spin chains with interactions decaying as and whose critical dynamics depend crucially on the power : for systems with the transition is sharp while for it fans out in a chaotic crossover region. In this paper we explore the fate of critical dynamics in Ising chains with long-range interactions when the transverse field is ramped up with a finite speed. While for abrupt quenches we observe a chaotic region that widens as is increased, the width of the crossover region diminishes as the time of the ramp increases, suggesting that chaos will disappear altogether and be replaced by a sharp transition in the adiabatic limit.

We study the influence of Ohmic dissipation on the random transverse-field Ising chain by means of large-scale Monte-Carlo simulations. To this end, we first map the Hamiltonian onto a classical Ising model with long-range

1/τ^2

interaction in the time-like direction. We then apply the highly efficient cluster algorithm proposed by Luijten and Blöte for system with long-range interactions. Our simulations show that Ohmic dissipation destroys the infinite-randomness quantum critical point of the dissipationless system. Instead, the quantum phase transition between the paramagnetic and ferromagnetic phases is smeared. We compare our results to recent predictions of a strong-disorder renormalization group approach, and we discuss generalizations to higher dimensions as well as experiments.

The phase diagram of a two-sited magnetic Ising chain, with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}},$ is studied. In this investigation, the finite-range scaling technique is employed and a proper transfer matrix is developed. The critical temperature of the chain, as a function of each type of interaction, i.e., the interaction between the similar sites and different sites, in both the classical and nonclassical regions is calculated. The results indicate that the critical temperature is strongly affected by the interaction range parameter \ensuremath{\sigma} for both types of interaction, but its behavior is quite different for each of the interactions. The behavior of the system is explained by calculating the strength of each interaction using the mean-field approximation. The study of the critical exponent of the correlation length indicates that the exponent is, within a good approximation, independent of the interaction constants.

I consider the dynamics of a soft-spin Ising magnet, subject to the time-varying external magnetic field h(t)=h cos\ensuremath{\Omega}t. The system is modeled with a time-dependent Ginzburg-Landau equation and is studied at a mean-field level. The time-averaged magnetization (scrM) acts as the order parameter and divides the temperature-h plane into two phases. In contrast to a previous result that predicted a dynamical tricritical point separating a line of continuous and discontinuous transitions in scrM, I find that the transition is always continuous. The previous work utilized Glauber dynamics which, it is here argued, becomes inapplicable near the phase boundary line due to a critical slowing down.

The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a “ferrimagnetic” phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points. 1