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Register a new userCritical Properties of an Ising Model with Dilute Long-range Interactions
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Added by
Richard T. Scalettar
Publication date
2004
fields
Physics
and research's language is
English
Authors
R.T. Scalettar
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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 $z$.
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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.
We study the statistical properties of Ising spin chains with finite (although arbitrary large) range of interaction between the elements. We examine mesoscopic subsystems (fragments of an Ising chain) with the lengths comparable with the interaction range. The equivalence of the Ising chains and the multi-step Markov sequences is used for calculating different non-additive statistical quantities of a chain and its fragments. In particular, we study the variance of fluctuating magnetization of fragments, magnetization of the chain in the external magnetic field, etc. Asymptotical expressions for the non-additive energy and entropy of the mesoscopic fragments are derived in the limiting cases of weak and strong interactions.
Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature $T_c$ for a one-dimensional Ising spin glass model with diluted long-range interactions. In this model, the probability $P_{ij}$ that an edge ${i,j}$ has nonvanishing interaction falls as a power-law with chord distance, $P_{ij}propto1/R_{ij}^{2sigma}$, and we study a range of values of $sigma$ with $1/2<sigma<1$. We consider a correlation function $C_{4}(r,t)$. A dynamic correlation length that shows power-law growth with time $xi(t)propto t^{1/z}$ can be identified in the data and, for large time $t$, $C_{4}(r,t)$ decays as a power law $r^{-alpha_d}$ with distance $r$ when $rll xi(t)$. The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent $alpha_d$ differentiates between a trivial MMAS ($alpha_d=0$), as expected in the droplet picture of spin glasses, and a nontrivial MMAS ($alpha_d e 0$), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero $alpha_d$ even in the regime $sigma >2/3$ which corresponds to short-range systems below six dimensions. For $sigma < 2/3$, the decay exponent $alpha_d$ follows the RSB prediction for the decay exponent $alpha_s = 3 - 4 sigma$ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches $alpha_d=1-sigma$ in the regime $2/3<sigma<1$; however, it deviates from both lines in the vicinity of $sigma=2/3$.
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.
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