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Solid State Communications

Spin and energy coupling in the Ising model with a transverse field: One dimension, T=∞

David L. Huber, T. N. Tommet

Published Dec 1, 1973• 6 citations
Publisher
DOI
10.1016/0038-1098(73)90012-4

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Dynamical correlation functions for the transverse (along the direction of the field) spin and energy densities are calculated for the one-dimensional spin-\textonehalf{} Ising model with a transverse field $H=\ensuremath{-}\ensuremath{\Gamma}{\ensuremath{\Sigma}}_{i}{S}_{i}^{x}\ensuremath{-}J{\ensuremath{\Sigma}}_{i}{S}_{i}^{z}{S}_{i+1}^{z}$. Explicit results are obtained for $〈{{S}^{x}(q, \ensuremath{\omega}){S}^{x}(\ensuremath{-}q)}〉$, $〈{E(q, \ensuremath{\omega}), E(\ensuremath{-}q)}〉$, and their Fourier transforms in the limits $T=\ensuremath{\infty}$ and $T=0$ and for the autocorrelation functions $〈{{S}_{i}^{x}, {S}_{i}^{x}(t)}〉$ and $〈{{E}_{i}, {E}_{i}(t)}〉$ in the same limits. The results are discussed with special attention given to the coupling of the transverse spin component and energy densities.

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Dynamics and Statistical Mechanics of the Classical Ising Model in a Transverse Field

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Dynamics and the approximate thermodynamical properties of the classical Ising model in a transverse field (CIMTF) are studied. This is done by elucidating a close correspondence of the CIMTF to a nonlinear lattice field in which each atom, while coupled harmonically with its neighbours, sits on on-site anharmonic potential. It is shown that when the ratio of the Ising interaction to a transverse field-strength exceeds a certain critical value, the on-site potential is of double-well type with saturable nonlinearity. Domain wall modes and magnon modes in continuous one-dimensional CIMTF are studied for both systems to substantiate the correspondence. By exploiting this, a brief study is made on the approximate properties of the free energy and of the dynamical structure factor S ( k ,ω) of longitudinal spin fluctuations of the CIMTF. These quantities are shown to be composed of two contributions, one from domain-wall modes and the other from magnon modes, which manifest themselves as a central peak and side peaks in S ( k ,ω), respectively.

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Dynamics of the Ising Model in a Transverse Field: Spin 1/2.

T. N. Tommet

•Jan 1, 1974•0 citations
Dynamical correlation functions for the transverse (along the direction of the field) spin and energy densities are calculated for the one-dimensional spin-\textonehalf{} Ising model with a transverse field $H=\ensuremath{-}\ensuremath{\Gamma}{\ensuremath{\Sigma}}_{i}{S}_{i}^{x}\ensuremath{-}J{\ensuremath{\Sigma}}_{i}{S}_{i}^{z}{S}_{i+1}^{z}$. Explicit results are obtained for $〈{{S}^{x}(q, \ensuremath{\omega}){S}^{x}(\ensuremath{-}q)}〉$, $〈{E(q, \ensuremath{\omega}), E(\ensuremath{-}q)}〉$, and their Fourier transforms in the limits $T=\ensuremath{\infty}$ and T=0T=0T=0 and for the autocorrelation functions 〈Six,Six(t)〉〈{{S}_{i}^{x}, {S}_{i}^{x}(t)}〉〈Six​,Six​(t)〉 and 〈Ei,Ei(t)〉〈{{E}_{i}, {E}_{i}(t)}〉〈Ei​,Ei​(t)〉 in the same limits. The results are discussed with special attention given to the coupling of the transverse spin component and energy densities.
We show in a simple exactly solvable toy model that a properly designed impulse perturbation can transiently cool down low-energy degrees of freedom at the expense of high-energy ones that heat up. The model consists of two infinite-range quantum Ising models: one, the high-energy sector, with a transverse field much bigger than the other, the low-energy sector. The finite-duration perturbation is a spin exchange that couples the two Ising models with an oscillating coupling strength. We find a cooling of the low-energy sector that is optimized by the oscillation frequency in resonance with the spin exchange excitation. After the perturbation is turned off, the Ising model with a low transverse field can even develop a spontaneous symmetry breaking despite being initially above the critical temperature.
Dynamics and the approximate thermodynamical properties of the classical Ising model in a transverse field (CIMTF) are studied. This is done by elucidating a close correspondence of the CIMTF to a nonlinear lattice field in which each atom, while coupled harmonically with its neighbours, sits on on-site anharmonic potential. It is shown that when the ratio of the Ising interaction to a transverse field-strength exceeds a certain critical value, the on-site potential is of double-well type with saturable nonlinearity. Domain wall modes and magnon modes in continuous one-dimensional CIMTF are studied for both systems to substantiate the correspondence. By exploiting this, a brief study is made on the approximate properties of the free energy and of the dynamical structure factor S ( k ,ω) of longitudinal spin fluctuations of the CIMTF. These quantities are shown to be composed of two contributions, one from domain-wall modes and the other from magnon modes, which manifest themselves as a central peak and side peaks in S ( k ,ω), respectively.