Oleg Derzhko

Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine
We start from recalling generally known topics of the phase transition theory: phase transitions of the first and the second order in classical systems at nonzero temperature, the Onsager solution of the square-lattice Ising model, critical behaviour of the physical quantities, universality, scaling, renormalization group. Then we turn to the basic concepts of quantum phase transition theory discussing the experiment of Bitko, Rosenbaum and Aeppli (1996) on Ising system (LiHoF4) placed hi a magnetic field transverse to the magnetic axis and the phase diagram of the Ising spin model in the plane temperature - transverse field. Onedimensional spin- i Ising model hi a transverse field is a simplest model exhibiting the second-order quantum phase transition. We discuss a relation of that model to the square-lattice Ising model and present the ‘old’ results of rigorous calculation derived by Pfeuty (1970). The essential tool hi this solution is the Jordan-Wigner fermionization. We briefly explain how the fermionic description can be introduced and thus how the results of Pfeuty (and some other results) were derived. We contrast quantum and classical transverse Ising chains emphasizing that the zero-temperature continuous phase transition driven by the transverse field occius in the quantum chain only.
The second part of the lecture deals with the effects of regular alternation of the Hamiltonian parameters (i.e., the intersite exchange interaction and on-site field) on the quantum phase transition. We elaborate an approach based on continued fractions for rigorous calculation of the thermodynamic quantities. The spin correlation functions for regularly alternating transverse Ising chains can be obtained numerically. We discuss hi detail the case of a chain of period 2 comparing exact analytical and exact numerical results for the ground state properties. Moreover, we demonstrate how the ground state (and therefore all spin correlation functions) can derived for special parameter values. We complete the analysis of the effects of regular alternation examining the low-temperature behaviour of the specific heat. We sketch the phase diagram for a chain of period 3. We end up with conclusions emphasizing the effects of regular alternation on the second-order quantum phase transition hi the transverse Ising chain.
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