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Critical behavior of the Ising model with long range interactions

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 Added by Marco Picco
 Publication date 2012
  fields Physics
and research's language is English
 Authors Marco Picco




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We present results of a Monte Carlo study for the ferromagnetic Ising model with long range interactions in two dimensions. This model has been simulated for a large range of interaction parameter $sigma$ and for large sizes. We observe that the results close to the change of regime from intermediate to short range do not agree with the renormalization group predictions.



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139 - R.T. Scalettar 2004
Statistical mechanical models with local interactions in $d>1$ dimension can be regarded as $d=1$ dimensional models with regular long range interactions. In this paper we study the critical properties of Ising models having $V$ sites, each having $z$ randomly chosen neighbors. For $z=2$ the model reduces to the $d=1$ Ising model. For $z= infty$ we get a mean field model. We find that for finite $z > 2$ the system has a second order phase transition characterized by a length scale $L={rm ln}V$ and mean field critical exponents that are independent of $z$.
We study the phase diagram and critical properties of quantum Ising chains with long-range ferromagnetic interactions decaying in a power-law fashion with exponent $alpha$, in regimes of direct interest for current trapped ion experiments. Using large-scale path integral Monte Carlo simulations, we investigate both the ground-state and the nonzero-temperature regimes. We identify the phase boundary of the ferromagnetic phase and obtain accurate estimates for the ferromagnetic-paramagnetic transition temperatures. We further determine the critical exponents of the respective transitions. Our results are in agreement with existing predictions for interaction exponents $alpha > 1$ up to small deviations in some critical exponents. We also address the elusive regime $alpha < 1$, where we find that the universality class of both the ground-state and nonzero-temperature transition is consistent with the mean-field limit at $alpha = 0$. Our work not only contributes to the understanding of the equilibrium properties of long-range interacting quantum Ising models, but can also be important for addressing fundamental dynamical aspects, such as issues concerning the open question of thermalization in such models.
We study critical behavior of the diluted 2D Ising model in the presence of disorder correlations which decay algebraically with distance as $sim r^{-a}$. Mapping the problem onto 2D Dirac fermions with correlated disorder we calculate the critical properties using renormalization group up to two-loop order. We show that beside the Gaussian fixed point the flow equations have a non trivial fixed point which is stable for $0.995<a<2$ and is characterized by the correlation length exponent $ u= 2/a + O((2-a)^3)$. Using bosonization, we also calculate the averaged square of the spin-spin correlation function and find the corresponding critical exponent $eta_2=1/2-(2-a)/4+O((2-a)^2)$.
93 - Jamir Marino 2021
We show that spatial resolved dissipation can act on Ising lattices molding the universality class of their critical points. We consider non-local spin losses with a Liouvillian gap closing at small momenta as $propto q^alpha$, with $alpha$ a positive tunable exponent, directly related to the power-law decay of the spatial profile of losses at long distances. The associated quantum noise spectrum is gapless in the infrared and it yields a class of soft modes asymptotically decoupled from dissipation at small momenta. These modes are responsible for the emergence of a critical scaling regime which can be regarded as the non-unitary analogue of the universality class of long-range interacting Ising models. In particular, for $0<alpha<1$ we find a non-equilibrium critical point ruled by a dynamical field theory ascribable to a Langevin model with coexisting inertial ($proptoomega^2$) and frictional ($proptoomega$) kinetic coefficients, and driven by a gapless Markovian noise with variance $propto q^alpha$ at small momenta. This effective field theory is beyond the Halperin-Hohenberg description of dynamical criticality, and its critical exponents differ from their unitary long-range counterparts. Furthermore, by employing a one-loop improved RG calculation, we estimate the conditions for observability of this scaling regime before incoherent local emission intrudes in the spin sample, dragging the system into a thermal fixed point. We also explore other instances of criticality which emerge for $alpha>1$ or adding long-range spin interactions. Our work lays out perspectives for a revision of universality in driven-open systems by employing dark states supported by non-local dissipation.
159 - R. M. Liu 2016
In this work, we investigate the phase transitions and critical behaviors of the frustrated J1-J2-J3 Ising model on the square lattice using Monte Carlo simulations, and particular attention goes to the effect of the second next nearest neighbor interaction J3 on the phase transition from a disordered state to the single stripe antiferromagnetic state. A continuous Ashkin-Teller-like transition behavior in a certain range of J3 is identified, while the 4-state Potts-critical end point [J3/J1]C is estimated based on the analytic method reported in earlier work [Jin et al., Phys. Rev. Lett. 108, 045702 (2012)]. It is suggested that the interaction J3 can tune the transition temperature and in turn modulate the critical behaviors of the frustrated model. Furthermore, it is revealed that an antiferromagnetic J3 can stabilize the staggered dimer state via a phase transition of strong first-order character.
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