The phase diagram and the order parameters of the exactly solvable quantum 1D model are analysed. The model in its spin representation is the dimerized XY spin chain in the presence of uniform and staggered transverse fields. In the fermionic representation this model is the dimerized noninteracting Kitaev chain with a modulated chemical potential. The model has a rich phase diagram which contains phases with local and nonlocal (string) orders. We have calculated within the same systematic framework the local order parameters (spontaneous magnetization) and the nonlocal string order parameters, along with the topological winding numbers for all domains of the phase diagram. The topologically nontrivial phase is shown to have a peculiar oscillating string order with the wavenumber q = π/2, awaiting for its experimental confirmation.

Thermodynamics of the short-range model of spin ice magnets in a field is considered in the BethePeierls approximation. The results obtained for [111], [100] and [011] The discovery of spin ice compounds [1,2] has opened a wide perspective in the studies of real geometrically frustrated magnets with their reach physics stemming from the macroscopically degenerate ground states. The more so as they can be described by the relatively simple Ising model with the nearest-neighbour exchange on the pyrochlore lattice. This is due to the lucky chance that strong dipole interactions in these compounds have a negligible effect on the low-energy excitations of the Ising moments directed along the lines connecting the centres of corner-sharing tetrahedra [3]. So low-temperature physics of spin-ices can be adequately captured by the short-range Ising model except for the ultra low temperatures where the equilibrium properties may be unobservable [4].Such model predicts the absence of phase transitions in zero field in accordance with experiments in the (established) acknowledged spin ice compounds [1,2]. Meanwhile a wealth of more or less sharp anomalies in the applied magnetic fields H of different directions is observed in their thermodynamic parameters [5][6][7][8][9][10][11][12][13][14]. Some of these anomalies are interpreted as the field-induced transitions while others are thought to indicate the crossover between the regions with different types of collective spin fluctuations. The notion of such regions originates from the Villain's idea of lowtemperature "spin-liquid" state in frustrated magnets [15] where spin fluctuations are strongly correlated being confined mostly to the ground states' subspace. In contrast the high temperature region features uncorrelated spin fluctuations thus representing a genuine paramagnet. While there is no true phase transition between paramagnet and spin-liquid state still the temperature dividing these "quasi-phases" can be pointed out -it is the temperature T m of specific heat maximum in its temperature dependence C(T) [4]. Indeed, this maximum indicates the more or less sharp drop of entropy due to the confinement of spin fluctuations at low T. One may hope that such definition of T m can justify the notion of "pseudotransition" between the "quasi-phases" with different types of spin fluctuations and may help to quantify in the framework of rigorous theory the regions where various spin-liquid states exist.Implicitly the notions of "quasi-phases" and "pseudo-transition" are widely used to interpret heuristically the observed field-induced anomalies of C(T) in spin ices and to identify the regions belonging to the different spin-liquid states on the H-T planes [9][10][11][12][13]. Yet it is important to discriminate the "pseudo-transitions" and the ordinary ones as the microscopic models describing first of them would not have any singular point but only the crossover regions. The more so as these crossovers can grow more and more sharp at low T and in the vicinity of critical ...

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Phenomenological thermodynamic theory describing the properties of metastable states in disordered ferromagnets and ferroelectrics with frustrative random interactions is developed and its ability to describe various nonergodic phenomena in real crystals is demonstrated.