**Theme: **Theoretical study of frustrated spin model on a bipartite lattice

**Supervisor: **Prof. RNDr. Andrej Bobák, DrSc.

**Annotation:** Generally, the frustration is generated by the competition of different kinds of interaction and/or by the lattice geometry. As a result, in the ground state all bonds are not fully satisfied. Thus in the frustrated Ising spin systems, a number of spins behave as free spins. The ground-state of frustrated spin systems is therefore highly degenerate and new induced symmetries give rise to spectacular and often unexpected behaviors at finite temperatures. Well-known example is the Ising model on the antiferromagnetic triangular lattice. On the other hand, the honeycomb lattice is a derivative of the triangular lattice and is obtained by periodically removing 1/3 of the spins from a triangular lattice. Since a honeycomb lattice antiferromagnet with only nearest-neighbor exchange interactions is considered as a bipartite lattice, the ground state exhibits long-range ordering. The system becomes frustrated like the square lattice, if the next-nearest-neighbor exchange interactions are considered. However, the spin fluctuations are expected to be larger for the honeycomb lattice than the square lattice because coordination number z = 3 in the honeycomb lattice is smaller than that of z = 4 in the square lattice. Hence, it is interesting to study the magnetic properties on the honeycomb lattice under frustrating interactions, which are still not well understood at present.

**Literature:**

1. H.T. Diep (Ed.), Frustrated Spin Systems, World Scientific, London, 2004.

2. C. Lacroix, P. Mendels, F. Mila (Ed.), Introduction to Frustrated Magnetism, Springer, Berlin-Heidelberg, 2011.

**Theme: **Theoretical investigation of localized spin models with many-body interactions

**Supervisor: **Doc. RNDr. Michal Jaščur, CSc.

**Annotation:** Theoretical investigation of many-body interactions represents a very hard mathematical problem even in the case of moderate number of interacting bodies. We are therefore frequently forced to limit ourselves to the case of pair (two-body) interactions only. The localized spin models, however, represent outstanding physical systems that enable to incorporate more complex interaction terms into Hamiltonian. The investigation of multi-spin interactions in such models is very important for understanding magnetic properties of various systems. Moreover, the detailed study of multi-spin interactions in these systems enables to illuminate some fundamental properties of many-body interactions in Nature. The models with multi-spin interactions exhibit very interesting magnetic behavior which is related to competing effects between exchange interactions and crystal field. The subject of the proposed thesis will be focused on the exact study of all relevant physical properties of localized spin models on decorated one and two dimensional lattices.

**Literature:**

1. M.E. Fisher, Phys. Rev. 113 (1959) 969.

2. L. Onsager, Phys . Rev. 65 (1944) 117.

3. T. Iwashita, N. Uryu, J. Phys. C. 17 (1984) 855.

4. C. Lacroix, P. Mendels, F. Mila (Ed.), Introduction to Frustrated Magnetism, Springer,

Berlin-Heidelberg, 2011.

**Theme: **Quantum entanglement in unconventional states of exactly solved spin models

**Supervisor: **Doc. RNDr. Jozef Strečka, PhD.

**Annotation: **Quantum entanglement belongs to the most intriguing consequences of the quantum mechanics, which is highly contra-intuitive and remains at a forefront of the current research interest. Among the most challenging issues related to this subject matter is to investigate the thermal entanglement through different measures (e.g. concurrence, fidelity, etc.), which quantify how the quantum entanglement evolves with rising temperature. The main focus of this dissertation thesis is to examine different measures of the thermal entanglement in the exactly solved lattice-statistical spin models. In particular, the considerable attention will be devoted to an open questions whether or not the thermal entanglement may coexist with a spontaneous long-range order and how it changes across quantum critical points.

**Literature:**

1. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.

2. L. Amico, R. Fazio, A. Osterloh, V. Vedral, Entanglement in many-body systems, Reviews of Modern Physics 80 (2008) 517-575.

Theme: Theoretical investigation of correlated spin and electron systems

**Supervisor: **Doc. RNDr. Jozef Strečka, PhD.

**Annotation: **The dissertation thesis is focused on theoretical investigation of various collective phenomena in the exactly solved correlated spin and electron systems. In particular, the main attention will be aimed at the exactly solvable spin-electron models, in which finite cluster of electrons are merely coupled to few localized Ising spins placed at lattice sites of one- or two-dimensional regular lattices. The results obtained by this study will be used for an explanation of the quantum origin of striking fractional magnetization plateaus and the enhanced magnetocaloric effect occurring during the adiabatic demagnetization.

**Literature:**

1. C. Lacroix, P. Mendels, F. Mila, Introduction to Frustrated Magnetism: Materials, Experiments, Theory, Springer, Berlin, 2011.

2. M.S.S. Pereira, F.A.B.F. de Moura, M.L. Lyra, Physical Review B 77 (2008) 024402.

**Theme: **Theoretical investigation of phase separation in binary and ternary liquid mixtures using exactly solvable lattice-statistical models

**Supervisor: **Doc. RNDr. Jozef Strečka, PhD.

**Annotation: **The phase separation of binary and ternary liquid mixtures will be investigated within the framework of the lattice-statistical models such as the generalized Frenkel-Louis and Lin-Taylor models. In particular, the main emphasis will be laid by accounting for the multiparticle (e.g. three-body) interactions, which may play a crucial role in determining possible non-universal critical behavior. Besides, the dissertation thesis will focus on a problem of reentrant miscibility of some binary and ternary liquid mixtures with highly orientation-dependent forces.

**Literature:**

1. T. Narayanan and A. Kumar, Reentrant phase transitions in multicomponent liquid mixtures, Physics Reports 249 (1994) 135-218.

2. J. Strečka, L. Čanová, M. Jaščur, Investigation of phase separation within the generalized Lin–Taylor model for a binary liquid mixture of large hexagonal and small triangular particles, Molecular Physics 104 (2006) 3831-3839.

**Theme: **Fluctuation theorems in non-markovian stochastic systems

**Supervisor: **Prof. RNDr. Vladimír Lisý, DrSc.

**Annotation: **When the time and space scales decrease, violations of the Second Law of Thermodynamics are observed in stochastic systems. An extension of this law is given by the Evans and Searles fluctuation theorem, which also provides its first proof. The theorem is formulated for the probabilities of a generalized entropy production. The Jarzynski equality and Crooks theorem relate the probabilities for the work at the transition of a system from its initial to a final state. These theorems belong to the greatest discoveries in statistical physics and attract thus an extraordinary attention of both theorists and experimentalists. The interpretations of the experiments carried out so far however do not take into account that the systems can possess various kinds of memory. The dissertation will be oriented on the description of non-markovian stochastic systems, presumably within the generalized Langevin theory. The theory will be applied to concrete experiments realized on micro- and nanosystems, with the aim to verify the fluctuation theorems.

**Literature:**

1. D.J. Evans, D.J. Searles, The Fluctuation Theorem. Advances in Physics 51 (2002) 1529-1585.

2. K. Sekimoto, Stochastic Energetics. Lect. Notes in Physics 799, Springer, Berlin- Heidelberg, 2010.