Frustrated Ising Model on a Diamond Hierarchical Lattice

Abstract

A frustrated Ising model on a diamond hierarchical lattice is studied. We obtain the exact partition function of this model and calculate the transition temperature, specific heat, entropy, magnetization, and ferromagnetic correlation function. Depending on the magnitude of a parameter giving the frustration, there exist three types of ground states: ferromagnetic, classical spin-liquid with highly developed short-range order, and paramagnetic. The dependence of the zero-temperature entropy on the frustration parameter has an infinite number of steps. The temperature dependence of the specific heat exhibits many peaks with decreasing temperature and entropy loss. The dominant spin configurations at low temperatures are also specified.KEYWORDS: classical spin-liquid ground state, frustration, diamond hierarchical lattice, Ising model, phase transition IntroductionMany magnetic systems with competing interactions exhibit frustration that leads to multiple ground states called spin glass or spin liquid even at the classical level. These interesting properties are attributable to the delicate balance of frustrated spin-spin interactions. If we can obtain the exact partition function of a model with frustrated magnetic phenomena, we can obtain a better understanding of complicated frustrated phenomena.In statistical physics, few exactly soluble models with phase transitions are known, such as two-dimensional Ising model, 1 eight-vertex model, 2 one-dimensional van der Waals gas model, 3 and a hierarchical model. 4 Since Berker and Ostlund proposed a hierarchical model related to the renormalization group method, 4, 5 many different models on hierarchical lattices have been proposed and developed.5-8 The magnetic and thermodynamic behaviors have been studied, 9-11 and the distribution of the zeros of the partition function and the critical exponents have also been obtained.10, 12 Using a hierarchical model with competing ferro-and antiferromagnetic interactions, McKay et al. studied the spin-glass behavior 13 and Nogueria et al. investigated the local magnetization.14 However, to the best of our knowledge, no hierarchical model describes the spin-liquid ground state. In this study, we consider a hierarchical Ising model with frustrated interactions that lead to the classical spinliquid ground state. This paper is organized as follows. We introduce our diamond hierarchical lattice and frustrated Ising model in §2 and describe our recursion relations in §3. Thermodynamic and ground state properties are described in §4 and §5, respectively. In §6, the temperature evolution of dominant spin configurations at low temperatures is discussed. In §7, we summarize the results obtained in this study.

Exact Solutions on the Ground States of Ising Models in Magnetic Fields with Frustration on a Diamond Hierarchical Lattice

“…Next, we discuss the typical features of the resultant magnetization and entropy curve for the case of a weak AF diagonal coupling, 3 2 ≤ α < 2. The zero-field ground state for 3 2 ≤ α < 2 is the spin liquid state [3]. We show a calculated result in Fig.…”

“…Recently, we have derived exact solution of the system with multistep magnetization [2]. We consider a frustrated Ising model on diamond hierarchical lattices [3]. The diamond hierarchical lattices are constructed via infinite iteration procedures and have vertices with coordination numbers that increase with increased stage as shown in Fig.1.…”

Infinitely Multiple Steps in Magnetization of Ferro- and Antiferromagnetic Ising Models with Frustration on a Diamond Hierarchical Lattice

“…19 An additional condition 3K 2 ≤ B appears in the course of calculations. In the case of B < 2K, we also get the highest term in a 2 , b 2 , and c 2 .…”

“…19 As the initial values for the recursion formulas, we have Table XI. Typical features of the resultant magnetization curves are as follows.…”

Haldane Phases and Ferrimagnetic Phases with Spontaneous Translational Symmetry Breakdown in Distorted Mixed Diamond Chains with Spins 1 and 1/2

“…14, the MDC is related to other important models of frustrated magnetism such as the dimer-plaquette model, [23][24][25][26][27][28] frustrated Heisenberg ladders, 29) hybrid diamond chains consisting of Heisenberg bonds and Ising bonds, 30,31) and an Ising model on a hierarchical diamond lattice. 32) Among them, the dimerplaquette chain with ferromagnetic interplaquette interaction reduces to the MDC in the limit of strong interplaquette interaction. 28) Thus far, in spite of the theoretical relevance of the MDC, no materials described by the MDC have been found.…”