Quantum Phase Transitions in Alternating Transverse Ising Chains

Abstract

This chapter is devoted to a discussion of quantum phase transitions in regularly alternating spin-1 2 Ising chain in a transverse field. After recalling some generally-known topics of the classical (temperature-driven) phase transition theory and some basic concepts of the quantum phase transition theory I pass to the statistical mechanics calculations for a one-dimensional spin-1 2 Ising model in a transverse field, which is the simplest possible system exhibiting the continuous quantum phase transition. The essential tool for these calculations is the Jordan-Wigner fermionization. The latter technique being completed by the continued fraction approach permits to obtain analytically the thermodynamic quantities for a "slightly complicated" model in which the intersite exchange interactions and on-site fields vary regularly along a chain. Rigorous analytical results for the ground-state and thermodynamic quantities, as well as exact numerical data for the spin correlations computed for long chains (up to a few thousand sites) demonstrate how the regularly alternating bonds/fields effect the quantum phase transition. I discuss in detail the case of period 2, swiftly sketch the case of period 3 and finally summarize emphasizing the effects of periodically modulated Hamiltonian parameters on quantum phase transitions in the transverse Ising chain and in some related models.

The Robustness and Vulnerability of a Complex Adaptive System With Co-Evolving Agent Behavior and Local Structure

“…al., 2000). Similar conclusions have been obtained by Derzhko, (2004); Berche et al, (2004); Nogueira et al, (2008); and Barenblatt et al (1996). Their results show that inhomogeneous random complex networks are robust if attacked randomly; however, there exists a critical probability c when attacked intentionally.…”

“…Common conclusion shows that there exists certain probabilities p and c corresponding to random attack and intentional attack respectively, such that the CAS would be destroyed sharply if another ¢ p is less than p or another ¢ c is larger than c ; p and c are all called corresponding critical probability (Afra, 2004;Brauer, et al, 2010;Baxter, et al, 2011;2009;Fumiya & Kousuke, 2011;Callway, et al, 2000;Albert, et al, 2000). According to studies of Derzhko, (2004); Berche, et al, (2004) ;Nogueira, et al, (2008), andBarenblatt, et al, (1996), the inhomogeneous random complex network is robust if it is attacked randomly. However, there exists a critical probability c if it is attacked intentionally.…”

“…In this paper, we focused on how the critical percolation probability p 0 and c 0 are affected by the game radius r and agents' memory of past games. If l = 1 , r L = = ∞ 1, and the Boolean game plays a dominant role, this system degenerates into a scale-free random complex network, which is also robust and vulnerable when the system is attacked randomly and intentionally, respectively (see Albert et al, 2000;Misra et al, 2010;Wang & Rong, 2009;Derzhko, 2004;Bollobàs et al, 2008;Bollobas & Riordan, 2007;Bowles, Baxter et al, 2011;Xiao Zang, et. al., 2020;Viktor Novičenko & Irmantas Ratas, 2018).…”

Contributions to theory of quantum spin models: 60th anniversary of Oleg Derzhko

“…All of us read with great pleasure new research papers by Oleg, some of which have already become cornerstones of the modern quantum theory of magnetism. The pedagogical mastery of Oleg can be corroborated by a few excellent review articles [28,38] and book chapters [39][40][41][42] available due to their unique style of writing even to nonexperts, as well as the pedagogical article about quantum phase transitions from a variational mean-field perspective made in collaboration with Professor Johannes Richter [43] already included in the Master and PhD study programmes at several universities from Ukraine and abroad.…”

Quantum phase transitions: a variational mean-field perspective

“…Rather, external pressure, magnetic field or competing terms in the Hamiltonian etc may be appropriate to tune the strength of quantum fluctuations, i.e., these parameters can drive the transition at zero temperature. The simplest example showing a quantum phase transition is the spin-1/2 Ising ferromagnet in a transverse magnetic field [5][6][7] å å…”