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Physical review. B, Condensed matter

Exact solutions for perpendicular susceptibilities of kagomé and decorated-kagomé Ising models

Jeremiah H. Barry, Mahfuza Khatun

Published Jun 1, 1987• 16 citations
Publisher

Abstract

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).

DOI
10.1103/physrevb.35.8601

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For more than two decades, exact and explicit solutions for the initial isothermal transverse susceptibility \ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​(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 \ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​(T) exactly calculable. Similar to the other three regular lattices, the resulting continuous curve for \ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​(T)/\ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​(0) shows both a weak (energy-type) singularity at the critical temperature Tc{T}_{c}Tc​ where \ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​(Tc{T}_{c}Tc​)/\ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​(0)=1.1426. . . and a rounded maximum at Tmax>Tc{T}_{\mathrm{max}>{T}_{c}}Tmax>Tc​​ with the parameters of this maximum given by \ensuremathχ\ensuremath⊥max{\ensuremath{\chi}}_{\ensuremath{\perp}}^{\mathrm{max}}\ensuremathχ\ensuremath⊥max​/\ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​(0)=1.1945. . . at Tmax{T}_{\mathrm{max}}Tmax​/Tc{T}_{c}Tc​=1.0994. . . . The exact solution for \ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​(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 \ensuremathχ\ensuremath⊥{\ensuremath{\chi}}_{\ensuremath{\perp}}\ensuremathχ\ensuremath⊥​(T).
A two-dimensional Ising model on a particular topological structure, a ``triangles-in-triangles'' Kagom\'e structure, which is called a triangular-Kagom\'e lattice, is investigated. In this model, we consider two interaction parameters Jaa{J}_{aa}Jaa​ and Jab{J}_{ab}Jab​, corresponding to the interactions among spins on the inner sites and between spins on the inner and vertex sites of the triangular in the Kagom\'e lattice, respectively. By summing over all spins at inner sites in the partition function, we arrive at the partition function of the Kagom\'e lattice. The effective interaction of the corresponding Kagom\'e lattice is always ferromagnetic, even for antiferromagnetic Jaa{J}_{aa}Jaa​. The critical properties of the system depend only on ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$. When ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}>\ensuremath{-}1$, the system has long range order at low temperature. However, when ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$, the partition function of the triangular-Kagom\'e lattice can be related to that of the Kagom\'e lattice with effective interaction at a temperature higher than its critical temperature. Therefore, the system will be in paramagnetic phase at all temperatures for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$. The phase diagram for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$ is given exactly.
A transverse magnetic field $\mathrm{\ensuremath{\Gamma}}$ is known to induce antiferromagnetic three-sublattice order of the Ising spins ${\ensuremath{\sigma}}^{z}$ in the triangular lattice Ising antiferromagnet at low enough temperature. This low-temperature order is known to melt on heating in a two-step manner, with a power-law ordered intermediate temperature phase characterized by power-law correlations at the three-sublattice wave vector Q\mathbf{Q}Q: $\ensuremath{\langle}{\ensuremath{\sigma}}^{z}(\stackrel{P\vec}{R}){\ensuremath{\sigma}}^{z}(0)\ensuremath{\rangle}\ensuremath{\sim}cos(\mathbf{Q}\ifmmode\cdot\else\textperiodcentered\fi{}\stackrel{P\vec}{R})/{|\stackrel{P\vec}{R}|}^{\ensuremath{\eta}(T)}$ with the temperature-dependent power-law exponent $\ensuremath{\eta}(T)\ensuremath{\in}(1/9,1/4)$. Here, we use a quantum cluster algorithm to study the ferromagnetic easy-axis susceptibility ${\ensuremath{\chi}}_{u}(L)$ of an $L\ifmmode\times\else\texttimes\fi{}L$ sample in this power-law ordered phase. Our numerical results are consistent with a recent prediction of a singular LLL dependence ${\ensuremath{\chi}}_{u}(L)\ensuremath{\sim}{L}^{2\ensuremath{-}9\ensuremath{\eta}}$ when $\ensuremath{\eta}(T)$ is in the range (1/9,2/9)(1/9,2/9)(1/9,2/9). This finite-size result implies, via standard scaling arguments, that the ferromagnetic susceptibility ${\ensuremath{\chi}}_{u}(B)$ to a uniform field BBB along the easy axis is singular at intermediate temperatures in the small BBB limit, ${\ensuremath{\chi}}_{u}(B)\ensuremath{\sim}{|B|}^{\ensuremath{-}\frac{4\ensuremath{-}18\ensuremath{\eta}}{4\ensuremath{-}9\ensuremath{\eta}}}$ for $\ensuremath{\eta}(T)\ensuremath{\in}(1/9,2/9)$, although there is no ferromagnetic long-range order in the low temperature state. Additionally we establish similar two-step melting behavior (via a study of the order parameter susceptibility ${\ensuremath{\chi}}_{\mathbf{Q}}$) in the case of the ferrimagnetic three-sublattice ordered phase which is stabilized by ferromagnetic next-neighbor couplings (J2{J}_{2}J2​) and confirm that the ferromagnetic susceptibility obeys the predicted singular form in the associated power-law ordered phase.
The Kubo linear response theory is applied to the quantum mechanical Ising model to obtain a general expression which is valid for all regular lattices for the perpendicular response function in terms of spin correlations. For the honeycomb and the square lattices, exact expressions are found for the frequency-dependent initial perpendicular susceptibility, [Formula: see text] valid for all temperatures. For the honeycomb lattice, the present result for zero frequency agrees with Fisher's static calculation, whereas there is disagreement in the case of the square lattice. The discrepancy is resolved by using an extended calculation which explicitly illustrates the difference between isothermal and adiabatic susceptibilities.
The two-dimensional Ising model on a distorted kagome lattice is studied by means of exact solutions and the tensor renormalization-group (TRG) method. The zero-field phase diagrams are obtained, where three phases such as ferromagnetic, ferrimagnetic, and paramagnetic phases, along with the second-order phase transitions, have been identified. The TRG results are quite accurate and reliable in comparison to the exact solutions. In a magnetic field, the magnetization (m)(m)(m), susceptibility, and specific heat are studied by the TRG algorithm, where the m=1/3m=1/3m=1/3 plateaux are observed in the magnetization curves for some couplings. The experimental data of susceptibility for the complex $\text{Co}{({\text{N}}_{3})}_{2}(\text{bpg})\ensuremath{\cdot}{\text{DMF}}_{4/3}$ are fitted with the TRG results, giving the couplings of the complex J=22 KJ=22\text{ }\text{K}J=22 K and ${J}^{\ensuremath{'}}=33\text{ }\text{K}$.
By combining all available information it is possible to extend the numerical knowledge of the high- and low-temperature zero-field isothermal susceptibility of the Ising model. Estimates of the third most singular term of the ferromagnetic singularity are obtained, using both direct estimates and an extension of the Generalized Law of Corresponding States. A recent conjecture of Barouch, McCoy and Wu (1973) concerning the equality of the constant term in the asymptotic expansion of the high- and low-temperature ferromagnetic susceptibility is confirmed numerically. Finally, a non-physical critical exponent of the low-temperature susceptibility of the triangular lattice is identified.