For more than two decades, exact and explicit solutions for the initial isothermal transverse susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) of the quantum-mechanical Ising model ferromagnet have been known on the honeycomb, square, and triangular lattices but not as yet on the kagom\'e lattice which is the only other regular lattice in two dimensions. Recently, however, exact solutions have been found for localized even-number correlations of the ferromagnetic kagom\'e Ising model, rendering its ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) exactly calculable. Similar to the other three regular lattices, the resulting continuous curve for ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T)/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0) shows both a weak (energy-type) singularity at the critical temperature ${T}_{c}$ where ${\ensuremath{\chi}}_{\ensuremath{\perp}}$( ${T}_{c}$ )/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1426. . . and a rounded maximum at ${T}_{\mathrm{max}>{T}_{c}}$ with the parameters of this maximum given by ${\ensuremath{\chi}}_{\ensuremath{\perp}}^{\mathrm{max}}$/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1945. . . at ${T}_{\mathrm{max}}$ / ${T}_{c}$ =1.0994. . . . The exact solution for ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) of the decorated-kagom\'e Ising-model ferromagnet is also obtained having modified features illustrating the effects of the decoration spins (singly decorated bonds) upon ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T).For more than two decades, exact and explicit solutions for the initial isothermal transverse susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) of the quantum-mechanical Ising model ferromagnet have been known on the honeycomb, square, and triangular lattices but not as yet on the kagom\'e lattice which is the only other regular lattice in two dimensions. Recently, however, exact solutions have been found for localized even-number correlations of the ferromagnetic kagom\'e Ising model, rendering its ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) exactly calculable. Similar to the other three regular lattices, the resulting continuous curve for ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T)/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0) shows both a weak (energy-type) singularity at the critical temperature ${T}_{c}$ where ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(${T}_{c}$)/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1426. . . and a rounded maximum at ${T}_{\mathrm{max}>{T}_{c}}$ with the parameters of this maximum given by ${\ensuremath{\chi}}_{\ensuremath{\perp}}^{\mathrm{max}}$/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1945. . . at ${T}_{\mathrm{max}}$/${T}_{c}$=1.0994. . . . The exact solution for ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) of the decorated-kagom\'e Ising-model ferromagnet is also obtained having modified features illustrating the effects of the decoration spins (singly decorated bonds) upon ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T).

Physical review. B, Condensed matterJun 1, 198716 citations

The Statistics of the Ising Ferromagnet.

Alfred Levitas

Jan 1, 19570 citations

Exact ground-state properties of disordered Ising systems

Joachim Esser, U. Nowak, Klaus D. Usadel

Exact ground states are calculated with an integer optimization algorithm for two- and three-dimensional site-diluted Ising antiferromagnets in a field (DAFF) and random field Ising ferromagnets (RFIM), the latter with Gaussian- and bimodal-distributed random fields. We investigate the structure and the size distribution of the domains of the ground state and compare it to earlier results from Monte Carlo (MC) simulations for finite temperature. Although DAFF and RFIM are thought to be in the same universality class we found differences between these systems as far as the distribution of domain sizes is concerned. In the limit of strong disorder for the DAFF in two and three dimensions the ground states consist of domains with a broad size distribution that can be described by a power law with exponential cutoff. For the RFIM this is only true in two dimensions while in three dimensions above the critical field where long-range order breaks down the system consists of two infinite interpenetrating domains of up and down spins---the system is in a two-domain state. For DAFF and RFIM the structure of the domains of finite size is fractal and the fractal dimensions for the DAFF and the RFIM agree within our numerical accuracy supporting that DAFF and RFIM are in the same universality class. Also, the DAFF ground-state properties agree with earlier results from MC simulations in the whole whereas there are essential differences between our exact ground-state calculations and earlier MC simulations for the RFIM which suggested that there are differences between the fractality of domains in RFIM and DAFF. Additionally, we show that for the case of higher disorder there are strong deviations from Imry-Ma-type arguments for RFIM and DAFF in two and three dimensions.Exact ground states are calculated with an integer optimization algorithm for two- and three-dimensional site-diluted Ising antiferromagnets in a field (DAFF) and random field Ising ferromagnets (RFIM), the latter with Gaussian- and bimodal-distributed random fields. We investigate the structure and the size distribution of the domains of the ground state and compare it to earlier results from Monte Carlo (MC) simulations for finite temperature. Although DAFF and RFIM are thought to be in the same universality class we found differences between these systems as far as the distribution of domain sizes is concerned. In the limit of strong disorder for the DAFF in two and three dimensions the ground states consist of domains with a broad size distribution that can be described by a power law with exponential cutoff. For the RFIM this is only true in two dimensions while in three dimensions above the critical field where long-range order breaks down the system consists of two infinite interpenetrating domains of up and down spins---the system is in a two-domain state. For DAFF and RFIM the structure of the domains of finite size is fractal and the fractal dimensions for the DAFF and the RFIM agree within our numerical accuracy supporting that DAFF and RFIM are in the same universality class. Also, the DAFF ground-state properties agree with earlier results from MC simulations in the whole whereas there are essential differences between our exact ground-state calculations and earlier MC simulations for the RFIM which suggested that there are differences between the fractality of domains in RFIM and DAFF. Additionally, we show that for the case of higher disorder there are strong deviations from Imry-Ma-type arguments for RFIM and DAFF in two and three dimensions.

arXiv:cond-mat/9612022Physical review. B, Condensed matterMar 1, 199770 citations

Note on the Critical Behavior of Ising Ferromagnets

Ryuzo Abe

Progress of Theoretical PhysicsJul 1, 196776 citations

{T}_{c}

where ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(

{T}_{c}

)/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1426. . . and a rounded maximum at

{T}_{\mathrm{max}>{T}_{c}}

with the parameters of this maximum given by ${\ensuremath{\chi}}_{\ensuremath{\perp}}^{\mathrm{max}}$/${\ensuremath{\chi}}_{\ensuremath{\perp}}$(0)=1.1945. . . at

{T}_{\mathrm{max}}

{T}_{c}

=1.0994. . . . The exact solution for ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T) of the decorated-kagom\'e Ising-model ferromagnet is also obtained having modified features illustrating the effects of the decoration spins (singly decorated bonds) upon ${\ensuremath{\chi}}_{\ensuremath{\perp}}$(T).