Frustration Properties of the 1D Ising Model

Abstract

We analyze frustration properties of the Ising model for a 1D monatomic equidistant lattice in an external uniform magnetic field considering exchange interactions of atomic spins at the sites of the first (nearest) and second neighbors. Exact analytic expressions for thermodynamic and magnetic characteristics of the system, as well as for the Fourier transform of the pair spin–spin correlation function and magnetic diffused scattering wavevectors, were obtained by the method of the Kramers–Wannier transfer matrix considering the exchange interaction between spins at the nearest-neighbor sites in an external magnetic field, as well as the exchange interaction between spins at the second-neighbor sites in zero magnetic field. The criteria for the emergence of magnetic frustration in the presence of competition not only between exchange-interaction energies, but also between these energies and the external magnetic field energy are formulated. The frustration points and the values of frustration magnetic fields depending on the magnitudes and signs of the exchange interaction are determined. The distinguishing features of this model in the frustration regime and its vicinity are analyzed. The set of characteristic features inherent in observables for systems with magnetic frustrations is considered and analyzed. It is shown that the inclusion of competing exchange interactions at the first- and second-neighbor sites makes it possible to describe the behavior of the magnetic diffused scattering wavevector for commensurate, incommensurate, and lock-in structures.

APPENDIX

Here, we write exact analytic expressions for the entropy and magnetization at zero temperature in the case of competing exchange interactions between spins at the sites of the first and second neighbors (88) for the Ising spin chain in an external magnetic field.

Thus, in the case of the antiferro–antiferromagnetic variant of parameters of the exchange interactions of spins at the sites of the first and second neighbors (J₁ < 0, J₂ < 0) in a frustrating magnetic field H_AA0 (112), the zero-temperature entropy and magnetization are given by

$$\begin{gathered} \mathop {\lim }\limits_{T \to 0} S({{H}_{{AA0}}}) = \ln \left[ {\frac{1}{3}\left( {1 + {{\vartheta }_{0}} + \frac{1}{{{{\vartheta }_{0}}}}} \right)} \right] \approx 0.382, \\ {{\vartheta }_{0}} = \sqrt[3]{{\frac{{\sqrt {({{3}^{3}} + {{2}^{2}}){{3}^{3}}} + {{3}^{3}} + 2}}{2}}}, \\ \end{gathered} $$

((118))

$$\mathop {\lim }\limits_{T \to 0} M({{H}_{{AA0}}}) = \frac{1}{3}\left( {1 + {{\nu }_{0}} + \frac{{{{2}^{2}}}}{{{{3}^{3}} + {{2}^{2}}}}\frac{1}{{{{\nu }_{0}}}}} \right) \approx 0.611,$$

((119))

$${{\nu }_{0}} = {{\left( {\frac{2}{{{{3}^{3}} + {{2}^{2}}}}} \right)}^{{2/3}}}\sqrt[3]{{\sqrt {({{3}^{3}} + {{2}^{2}}){{3}^{3}}} + ({{3}^{3}} + {{2}^{2}})}},$$

respectively. In frustrating magnetic field H_AA1 (113), we have

$$\begin{gathered} \mathop {\lim }\limits_{T \to 0} S({{H}_{{AA1}}}) = \ln \left[ {\frac{1}{3}\left( {{{\vartheta }_{1}} + \frac{3}{{{{\vartheta }_{1}}}}} \right)} \right] \approx 0.281, \\ {{\vartheta }_{1}} = \sqrt[3]{{\frac{{\sqrt {({{3}^{3}} - {{2}^{2}}){{3}^{3}}} + {{3}^{3}}}}{2}}}, \\ \end{gathered} $$

((120))

$$\mathop {\lim }\limits_{T \to 0} M({{H}_{{AA1}}}) = \frac{1}{3}\left( {1 + {{\nu }_{1}} - \frac{{{{2}^{2}}}}{{{{3}^{3}} - {{2}^{2}}}}\frac{1}{{{{\nu }_{1}}}}} \right) \approx 0.177,$$

((121))

$${{\nu }_{1}} = {{\left( {\frac{2}{{{{3}^{3}} - {{2}^{2}}}}} \right)}^{{2/3}}}\sqrt[3]{{\sqrt {({{3}^{3}} - {{2}^{2}}){{3}^{3}}} - ({{3}^{3}} - {{2}^{2}})}},$$

while in frustrating magnetic field H_AA2 (115), we get

$$\begin{gathered} \mathop {\lim }\limits_{T \to 0} S({{H}_{{AA2}}}) = \ln \left[ {{{\zeta }_{2}} + \frac{1}{2}\sqrt {\frac{1}{{{{\zeta }_{2}}}} - {{{(2{{\zeta }_{2}})}}^{2}}} } \right] \approx 0.199, \\ {{\zeta }_{2}} = \frac{1}{2}\sqrt {\frac{{{{\vartheta }_{2}}}}{3} - \frac{4}{{{{\vartheta }_{2}}}}} , \\ \end{gathered} $$

((122))

$${{\vartheta }_{2}} = \sqrt[3]{{\frac{{\sqrt {({{3}^{3}} + {{2}^{8}}){{3}^{3}}} + {{3}^{3}}}}{2}}},$$

$$\begin{gathered} \mathop {\lim }\limits_{T \to 0} M({{H}_{{AA2}}}) \\ = - {{\eta }_{2}} + \sqrt {\frac{{{{3}^{2}}}}{{{{3}^{3}} + {{2}^{8}}}} - \eta _{2}^{2} + \frac{2}{{{{3}^{3}} + {{2}^{8}}}}\frac{1}{{{{\eta }_{2}}}}} \approx 0.159, \\ \end{gathered} $$

((123))

$${{\eta }_{2}} = \sqrt {\frac{3}{{{{3}^{3}} + {{2}^{8}}}} + \frac{1}{{{{2}^{2}}}}\frac{{{{\nu }_{2}}}}{3} - \frac{{{{2}^{8}}}}{{{{{({{3}^{3}} + {{2}^{8}})}}^{2}}}}\frac{1}{{{{\nu }_{2}}}}} ,$$

$${{\nu }_{2}} = {{2}^{{2/3}}}\frac{{{{2}^{3}}}}{{{{3}^{3}} + {{2}^{8}}}}\sqrt[3]{{\sqrt {({{3}^{3}} + {{2}^{8}}){{3}^{3}}} + {{3}^{3}}}}.$$

In the ferro–antiferromagnetic variant of the exchange interaction parameters for spins at the site of the first and second neighbors (J₁ > 0, J₂ < 0) in frustrating magnetic field H_FA (117), the zero-temperature entropy and magnetization of the system are given by

$$\begin{gathered} \mathop {\lim }\limits_{T \to 0} S({{H}_{{FA}}}) \\ = \ln \left[ {{{\zeta }_{3}} + \frac{1}{4} + \frac{1}{2}\sqrt {\frac{1}{{{{2}^{3}}}}\frac{1}{{{{\zeta }_{3}}}} - {{{(2{{\zeta }_{3}})}}^{2}} + \frac{3}{4}} } \right] \approx 0.322, \\ \end{gathered} $$

((124))

$${{\zeta }_{3}} = \frac{1}{2}\sqrt {\frac{{{{\vartheta }_{3}}}}{3} - \frac{4}{{{{\vartheta }_{3}}}} + \frac{1}{4}} ,$$

$${{\vartheta }_{3}} = \sqrt[3]{{\frac{{\sqrt {({{3}^{3}} + {{2}^{8}}){{3}^{3}}} - {{3}^{3}}}}{2}}},$$

$$\begin{gathered} \mathop {\lim }\limits_{T \to 0} M({{H}_{{FA}}}) \\ = {{\eta }_{3}} + \sqrt {\frac{{{{3}^{2}}}}{{{{3}^{3}} + {{2}^{8}}}} - \eta _{3}^{2} + \frac{2}{{{{3}^{3}} + {{2}^{8}}}}\frac{1}{{{{\eta }_{3}}}}} \approx 0.397, \\ \end{gathered} $$

((125))

$${{\eta }_{3}} = \sqrt {\frac{3}{{{{3}^{3}} + {{2}^{8}}}} + \frac{1}{{{{2}^{2}}}}\frac{{{{\nu }_{3}}}}{3} - \frac{{{{2}^{8}}}}{{{{{({{3}^{3}} + {{2}^{8}})}}^{2}}}}\frac{1}{{{{\nu }_{3}}}}} ,$$

$${{\nu }_{3}} = {{2}^{{2/3}}}\frac{{{{2}^{3}}}}{{{{3}^{3}} + {{2}^{8}}}}\sqrt[3]{{\sqrt {({{3}^{3}} + {{2}^{8}}){{3}^{3}}} + {{3}^{3}}}}.$$