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arXiv:0808.3860 · Journal of Physics A Mathematical and Theoretical

Non-additive properties of finite 1D Ising chains with long-range interactions

S. S. Apostolov, Z. А. Maizelis, OLEG V. USATENKO, V. A. Yampol’skiı̆

Published Feb 4, 2009• 4 citations
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Abstract

We study the statistical properties of Ising spin chains with a finite (although arbitrarily large) range of interaction among the elements. We examine mesoscopic subsystems (fragments of an Ising chain) with 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 the chain and its fragments. In particular, we study the variance of the fluctuating magnetization of fragments, magnetization of the chain in an external magnetic field, etc. Asymptotical expressions for the non-additive energy and entropy of mesoscopic fragments are derived in the limiting cases of weak and strong interactions.

DOI
10.1088/1751-8113/42/9/095004

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We study the statistical properties of Ising spin chains with a finite (although arbitrarily large) range of interaction among the elements. We examine mesoscopic subsystems (fragments of an Ising chain) with 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 the chain and its fragments. In particular, we study the variance of the fluctuating magnetization of fragments, magnetization of the chain in an external magnetic field, etc. Asymptotical expressions for the non-additive energy and entropy of mesoscopic fragments are derived in the limiting cases of weak and strong interactions.
We analyze frustrated states of the one-dimensional dilute Ising chain with charged interacting impurities of two types with mapping of the system to some Markov chain. We perform classification and reveal two types of Markov chains: periodic with period 2 and aperiodic. Frustrated phases with various types of chains have different properties. In phases with periodic Markov chains, long-range order is observed in the sublattice while another sublattice remains disordered. This results in a conjunction of the non-zero residual entropy and the infinite correlation length. In frustrated phases with aperiodic chains, there is no long-range order, and the correlation length remains finite. It is shown that under the magnetic field the most significant change in the spin chain structure corresponds to the change of the Markov chain type.
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.
We investigate entanglement properties of N initially disentangled spins, embedded in a ring or d-dimensional lattice. The spins interact via some long-range Ising-type interaction. A description in terms of generalized valence bond solids allows us, for arbitrarily large N, to efficiently calculate reduced density operators of up to ten particles. From these density operators we can compute many quantities, for example the bipartite entanglement S_L between a block of L neighboring spins and the remaining system, lower and upper bounds on the localizable entanglement, and higher order correlation functions. We vary the distance dependence of the interaction and investigate how this variation affects the scaling of the entropy S_L as L grows. We provide a sufficient condition when S_L saturates. In other cases, we find that the entanglement length diverges. For special configurations, we determine S_L analytically for all L in the limit N to infinity.
The phase diagram and critical exponent of the correlation length for an Ising chain with a long-range interaction in the form of ${1/r}^{1+\ensuremath{\sigma}}$ for S=1/2,1,3/2,2S=1/2,1,3/2,2S=1/2,1,3/2,2 is calculated by the finite-range scaling technique. The spin dependence of the critical temperature and critical exponent $\ensuremath{\nu}$ in the classical and nonclassical regions is also studied. It is observed that, with a good approximation, the critical exponent of the correlation length is independent of the magnitude of spin in both the classical and nonclassical regions.
We consider N initially disentangled spins, embedded in a ring or d-dimensional lattice of arbitrary geometry, which interact via some long-range Ising-type interaction. We investigate relations between entanglement properties of the resulting states and the distance dependence of the interaction in the limit N-->infinity. We provide a sufficient condition when bipartite entanglement between blocks of L neighboring spins and the remaining system saturates and determine S(L) analytically for special configurations. We find an unbounded increase of S(L) as well as diverging correlation and entanglement length under certain circumstances. For arbitrarily large N, we can efficiently calculate all quantities associated with reduced density operators of up to ten particles.
A long-range Ising chain, which has ferromagnetic 1/r 3 type interactions, has been studied by exact numerical calculation of thermodynamic properties of the sequence of systems of N spins (N<or=30). By expressing the sequence in terms of a series and using extrapolation methods such as the Pade approximant, the authors have found a consistent way to obtain critical temperatures which is accurate to 0.4% for the very long-range equivalent neighbour model. The authors have estimated the high-temperature susceptibility and the low- temperature long-range correlations of the infinite chain and thence have made estimates for the critical exponents gamma and beta . Although the nature of the phase transition appears to be classical for 1<s<or approximately=1.3, it is certainly not classical for 1.6<s<2; in particular, as s approaches 2 from below, gamma increases regularly to about 2 or greater and beta decreases regularly to near 0.