Topological Disorder in Spin Models on Hierarchical Lattices

The magnetic properties of the mixed spin-$\frac{1}{2}$ and spin-$\frac{7}{2}$ Ising model with a crystal-field in a longitudinal magnetic field are investigated on the Bethe lattice using exact recursion relations. The ground-state phase diagram is constructed. The temperature-dependent one is displayed in the case of uniform crystal-field on the $(k_{\text{B}}T/|J|, D/|J|)$ plane in the absence of the external constraint for lattice coordination numbers $z = 3, 4, 6$. The o...

Recently, it has been shown that, when the dimension of a graph turns out to be infinite dimensional in a broad sense, the upper critical surface and the corresponding critical behavior of an arbitrary Ising spin glass model defined over such a graph, can be exactly mapped on the critical surface and behavior of a non random Ising model. A graph can be infinite dimensional in a strict sense, like the fully connected graph, or in a broad sense, as happens on a Bethe lattice an...

Collective organization in matter plays a significant role in its expressed physical properties. Typically, it is detected via an order parameter, appropriately defined for each given system's observed emergent patterns. Recent developments in information theory, however, suggest quantifying collective organization in a system- and phenomenon-agnostic way: decompose the system's thermodynamic entropy density into a localized entropy, that solely contained in the dynamics at a...

The finite lattice method of series expansion is generalised to the $q$-state Potts model on the simple cubic lattice. It is found that the computational effort grows exponentially with the square of the number of series terms obtained, unlike two-dimensional lattices where the computational requirements grow exponentially with the number of terms. For the Ising ($q=2$) case we have extended low-temperature series for the partition functions, magnetisation and zero-field su...

We develop a transfer matrix method to compute exactly the spin-spin correlation functions of Bethe lattice spin models in the external magnetic field h and for any temperature T. We first compute the correlation function for the most general spin - S Ising model, which contains all possible single-ion and nearest-neighbor pair interactions. This general spin - S Ising model includes the spin-1/2 simple Ising model and the Blume-Emery-Griffiths (BEG) model as special cases. F...

We present a large deviations theory of the spin-spin correlation functions in the Random Field Ising Model on the Bethe lattice, both at finite and zero temperature. Rare events of atypically correlated variables are particularly important at the critical point: the phase transition is driven by few pairs of strongly correlated spins, while the majority remains basically uncorrelated. At the zero temperature critical point the number of spin pairs correlated over a distance ...

We show that the simple update approach proposed by Jiang et. al. [H.C. Jiang, Z.Y. Weng, and T. Xiang, Phys. Rev. Lett. 101, 090603 (2008)] is an efficient and accurate method for determining the infinite tree tensor network states on the Bethe lattice. Ground state properties of the quantum transverse Ising model and the Heisenberg XXZ model on the Bethe lattice are studied. The transverse Ising model is found to undergo a second-order quantum phase transition with a diverg...

In this thesis, we consider some spin effects in QCD and recurrence lattices with multi-site exchanges. Main topic of our manuscript are critical phenomena in spin systems defined on the recurrence lattices. Main tool of our approach is the method of recursive (hierarchical) lattices. We apply the method of dynamical mapping (or recursive lattices) for investigation of magnetic properties of the fluid and solid $^3$He, phase transitions in crystals and macromolecules. First, ...

We study the dynamical low temperature behaviour of the Ising spin glass on the Bethe lattice. Starting from Glauber dynamics we propose a cavity like Ansatz that allows for the treatment of the slow (low temperature) part of dynamics. Assuming a continuous phase transitions and ultrametricity with respect to long time scales we approach the problem perturbatively near the critical temperature. The theory is formulated in terms of correlation-response-functions of arbitrary o...

We exactly solve the ferromagnetic spin-1/2 Ising model on the Bethe lattice in the presence of an external magnetic field by means of the equations of motion method within the Green's function formalism. In particular, such an approach is applied to an isomorphic model of localized Fermi particles interacting via an intersite Coulomb interaction. A complete set of eigenoperators is found together with the corresponding eigenvalues. The Green's functions and the correlation f...