arXiv:0710.5644 · Physical Review B

Thermodynamics from a scaling Hamiltonian

Luis Alberto del Pino, P. Troncoso, Sergio Curilef

Published Nov 8, 2007• 10 citations

Abstract

There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance $r$ as $1∕{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. Here, we review old nonextensive scaling. In addition, we propose a Hamiltonian scaled by $2\frac{{(N∕2)}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1}{1\ensuremath{-}\ensuremath{\alpha}}$ that explicitly includes symmetry of the lattice and dependence on the size $N$ of the system. The approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.

DOI: 10.1103/physrevb.76.172402

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arXiv:0710.5644 · Physical Review B

Thermodynamics from a scaling Hamiltonian

Luis Alberto del Pino, P. Troncoso, Sergio Curilef

Published Nov 8, 2007• 10 citations

arXiv HTML PDF

Abstract

r

1∕{r}^{\ensuremath{\alpha}}

for

\ensuremath{\alpha}\ensuremath{\geqslant}0

. Here, we review old nonextensive scaling. In addition, we propose a Hamiltonian scaled by

2\frac{{(N∕2)}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1}{1\ensuremath{-}\ensuremath{\alpha}}

that explicitly includes symmetry of the lattice and dependence on the size

N

of the system. The approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.

DOI: 10.1103/physrevb.76.172402

Related papers

Critical temperature of a chain of long range interacting ferromagnets

Luis Alberto del Pino, P. Troncoso, Sergio Curilef

The thermodynamic behavior of systems with long range interactions is anomalous, because there are some problems about defining the thermodynamic limit. A way to solve the problem is to use scaled thermodynamic quantities. In this work, we use a nonextensive scaling into Hamiltonian and characterize transitions between two different magnetic ordering phases. The critical temperature is estimated by Binder method. Ferromagnetic long range interactions are included in a special Hamiltonian through a power law that decays at large interparticle distance r as r -α for α ≥ 0. In addition, we improve the known nonextensive scaling and obtain the critical temperature for several values of α.The thermodynamic behavior of systems with long range interactions is anomalous, because there are some problems about defining the thermodynamic limit. A way to solve the problem is to use scaled thermodynamic quantities. In this work, we use a nonextensive scaling into Hamiltonian and characterize transitions between two different magnetic ordering phases. The critical temperature is estimated by Binder method. Ferromagnetic long range interactions are included in a special Hamiltonian through a power law that decays at large interparticle distance r as r -α for α ≥ 0. In addition, we improve the known nonextensive scaling and obtain the critical temperature for several values of α.

Journal of Physics Conference SeriesNov 1, 20082 citations

Long-range interactions and nonextensivity in ferromagnetic spin models

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A numerically efficient transfer matrix (TM) approach is introduced to investigate the long-range Ising spin chain. Results obtained within this procedure are primarily used to verify the Tsallis scaling hypothesis for long-range systems with an alpha power-law decay of the interaction constants, both in the extensive (alpha>1) and nonextensive (alpha<1) regimes. Results for finite-size systems, taking into account all interactions between spins up to 24 sites apart, show that the conjecture is satisfied with a very good precision (less than 0.004%) for all temperature intervals. This TM procedure is further used to investigate several other thermodynamic and critical properties of this system, and it may also be extended to similar one-dimensional long-range systems.A numerically efficient transfer matrix (TM) approach is introduced to investigate the long-range Ising spin chain. Results obtained within this procedure are primarily used to verify the Tsallis scaling hypothesis for long-range systems with an alpha power-law decay of the interaction constants, both in the extensive (alpha>1) and nonextensive (alpha<1) regimes. Results for finite-size systems, taking into account all interactions between spins up to 24 sites apart, show that the conjecture is satisfied with a very good precision (less than 0.004%) for all temperature intervals. This TM procedure is further used to investigate several other thermodynamic and critical properties of this system, and it may also be extended to similar one-dimensional long-range systems.

Physical Review EFeb 25, 200525 citations

Long-range interactions and periodic boundary conditions in ferromagnetic spin models

Sergio Curilef

We present a possible way to study long-range interacting particles in finite-infinite systems with periodic boundary conditions. A symmetrical lattice and its contributions overall space are being used. In this context, we study the Ising model with ferromagnetic interaction that decays as a 1/r a law. We verify by Monte Carlo heat-bath simulations in the D (dimension) = 1 case that the thermodynamics quantities scale in a way proposed by Tsallis, and we confirm that mean field theory is exact in the last model for all 0 < a < D suggested by Cannas and Tamarit.We present a possible way to study long-range interacting particles in finite-infinite systems with periodic boundary conditions. A symmetrical lattice and its contributions overall space are being used. In this context, we study the Ising model with ferromagnetic interaction that decays as a 1/r a law. We verify by Monte Carlo heat-bath simulations in the D (dimension) = 1 case that the thermodynamics quantities scale in a way proposed by Tsallis, and we confirm that mean field theory is exact in the last model for all 0 < a < D suggested by Cannas and Tamarit.

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Overcoming System-Size Limitations in Spin Glasses

Helmut G. Katzgraber, Mathias Körner, Frauke Liers +1 more

In order to overcome the limitations of small system sizes in spin-glass simulations, we investigate the one-dimensional Ising spin chain with power-law interactions.The model has the advantage over traditional higher-dimensional Hamiltonians in that a large range of system sizes can be studied.In addition, the universality class of the model can be changed by tuning the power law exponent, thus allowing us to scan from the mean-field to long-range and short-range universality classes.We illustrate the advantages of this model by studying the nature of the spin glass state where our results hint towards a replica symmetry breaking scenario.We also compute ground-state energy distributions and show that mean-field and non-mean-field models are intrinsically different. * )Recently Krzakala and Martin, as well as Palassini and Young, suggest an intermediate picture in which droplets have a fractal surface, and their energy is finite in the thermodynamic limit.See Ref. 5) and references therein for details.* * ) Note that we explicitly distinguish between θ , the stiffness exponent for droplet excitations, and θ, the exponent for domain walls, because DP predicts θ = θ , which we want to test here.In order to overcome the limitations of small system sizes in spin-glass simulations, we investigate the one-dimensional Ising spin chain with power-law interactions.The model has the advantage over traditional higher-dimensional Hamiltonians in that a large range of system sizes can be studied.In addition, the universality class of the model can be changed by tuning the power law exponent, thus allowing us to scan from the mean-field to long-range and short-range universality classes.We illustrate the advantages of this model by studying the nature of the spin glass state where our results hint towards a replica symmetry breaking scenario.We also compute ground-state energy distributions and show that mean-field and non-mean-field models are intrinsically different. * )Recently Krzakala and Martin, as well as Palassini and Young, suggest an intermediate picture in which droplets have a fractal surface, and their energy is finite in the thermodynamic limit.See Ref. 5) and references therein for details.* * ) Note that we explicitly distinguish between θ , the stiffness exponent for droplet excitations, and θ, the exponent for domain walls, because DP predicts θ = θ , which we want to test here.

arXiv:cond-mat/0509469Progress of Theoretical Physics SupplementJan 1, 200511 citations

r

1∕{r}^{\ensuremath{\alpha}}

for

\ensuremath{\alpha}\ensuremath{\geqslant}0

. Here, we review old nonextensive scaling. In addition, we propose a Hamiltonian scaled by

2\frac{{(N∕2)}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1}{1\ensuremath{-}\ensuremath{\alpha}}

that explicitly includes symmetry of the lattice and dependence on the size

N

N

being the number of spins) we propose a generalization of the Curie-Weiss model, for which the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit is well defined for all $\ensuremath{\alpha}>~0$. We conjecture that mean-field theory is exact in the last model for all $0<~\ensuremath{\alpha}<~d$. This conjecture is supported by Monte Carlo heat-bath simulations in the

d=1

case. Moreover, we confirm a recently conjectured scaling by Tsallis that allows for a unification of extensive ($\ensuremath{\alpha}>d$) and nonextensive ($0<~\ensuremath{\alpha}<~d$) regimes.

A detailed investigation of the scaling properties of the fully finite

\mathcal{O}(n)

systems, under periodic boundary conditions, with long-range interaction, decaying algebraically with the interparticle distance r like ${r}^{\ensuremath{-}d\ensuremath{-}\ensuremath{\sigma}},$ below their upper critical dimension, is presented. The computation of the scaling functions is done to one loop order in the nonzero modes. The results are obtained in an expansion of powers of $\sqrt{\ensuremath{\varepsilon}},$ where $\ensuremath{\varepsilon}=2\ensuremath{\sigma}\ensuremath{-}d$ up to $\mathcal{O}({\ensuremath{\varepsilon}}^{3/2}).$ The thermodynamic functions are found to depend upon the scaling variable ${z=RU}^{\ensuremath{-}1/2}{L}^{2\ensuremath{-}\ensuremath{\eta}\ensuremath{-}\ensuremath{\varepsilon}/2},$ where R and U are the coupling constants of the constructed effective theory, and L is the linear size of the system. Some simple universal results are obtained.