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

Abstract

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.

Thermodynamics from first principles: correlations and nonextensivity

“…See [53] for details. Aside from such trivial maximally-correlated cases and the long-range Ising models [23,40,[44][45][46]56] (corresponding to the third row of Table I as long as subsystems are macroscopic), we are not aware of any previous thermostatistic formalism that can describe nontrivial long-range-interacting systems, as in the last row, by finding the equilibrium distribution.…”

“…It would be interesting to examine the thermodynamics of low-dimensional long-range Ising-type models [23,40,[44][45][46][56][57][58][59], which exhibit phase transitions under certain conditions [57][58][59]. In the context of our formalism, such phase transitions are driven by the set of control parameters given above as {α X } (and which include the temperature through a global function [55]).…”

Finding the ground states of symmetric infinite-dimensional Hamiltonians: explicit constrained optimizations of tensor networks

“…Let us rst consider long-range Hamiltonians of the general form i> j,a=x,y,z J a r α i j Ŝa i Ŝa j , where r ij denotes the distance between spins (or some other form of subsystems) i and j, and α identi es the range of interactions. In the last four decades, such models have been consistently in the center of the attention due to exhibiting rich phase diagrams [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], relevance to experimental cavity-mediated Bose-Einstein condensate [7,18,25,26] or trapped ions [27,28] quantum simulators, and the emergence of nonextensive thermostatistics [3,4,14,[29][30][31][32][33][34][35][36]-see [35] for an extended review. The extreme case of in nite (global or all-to-all) range interactions corresponds to α = 0 and, also, receives a great amount of attention as evident from references [1][2][3][4][5][6][7][8][9][10].…”

Thermodynamics from first principles: Correlations and nonextensivity

“…See [56] for details. Aside from such trivial maximally-correlated cases and the long-range Ising models [23,40,[44][45][46]58] (corresponding to the third row of Table I as long as subsystems are macroscopic), we are not aware of any previous thermostatistic formalism that can describe nontrivial long-range-interacting systems, as in the last row, by finding the equilibrium distribution.…”

“…It would be interesting to examine the thermodynamics of low-dimensional long-range Ising-type models [23,40,[44][45][46][58][59][60][61], which exhibit phase transitions under certain conditions [59][60][61]. In the context of our formalism, such phase transitions are driven by the set of control parameters given above as {a X , b X } (and which include the temperature through a global function [2]).…”