Quantum phase transitions in alternating transverse Ising chains: Analytical and numerical results

Abstract

We examine the ground state properties of the s = 1 2 transverse Ising chain with regularly alternating bonds and fields using exact analytical results and exact numerical data for long (up to N = 900) and short (N = 20) chains. For a given period of alternation the system may exhibit a series of quantum phase transitions, where the number of transitions depends on the concrete set of the parameters of the Hamiltonian. The critical behaviour for the nonuniform and the uniform chain is the same. (1) (s α n = 1 2 σ α n , σ α n are the Pauli matrices attached to the site n, Ω is the on-site transverse field, 2I is the exchange interaction between neighbouring spins) is known to be the simplest system exhibiting a quantum phase transition.

Quantum Phase Transitions in Alternating Transverse Ising Chains

“…It is interesting to note that short chains (N = 20) are already sufficient to reproduce a correct orderparameter behavior in the quantum Ising phase away from the quantum critical point, but not in the quantum paramagnetic phases. 26 The chain length of up to N = 900 sites is clearly sufficient to see a sharp transition in the order parameter (Figs. 4b, 4e) and even to extract reliable results for ξ x from the long-distance behavior of s Finally, the low-temperature dependence of the specific heat c at different Ω (≥ 0) obtained from the exact analytical expression (85) confirms the existence of either two phase transitions (Fig.…”

“…24,25 Moreover, for certain sets of the Hamiltonian parameters the exact results may be obtained for the spin correlation functions. 26 In all other cases, spin correlations may be explored numerically at a high precision level considering sufficiently long chains up to a few thousand sites. 26,27 It is appropriate to mention here the old studies 28,29 referring to the chains of period 2 which were extended further to the one-dimensional anisotropic XY model on superlattices.…”

Thermodynamic properties of the mixed spin-1/2 and spin-1 Ising chain with both longitudinal and transverse single-ion anisotropies

“…is the Hamiltonian of the reduced subsystem, which is equivalent to an S ¼1/2 Ising chain of length l n ¼x n À x n À 1 À 3 with alternating transverse fields [21,29,30]. For convenience in Eq.…”

“…Varieties of models describing such quantum mixed spin chains have been investigated [1][2][3][4][5][6], among which much attention has been directed to the two-sublattice mixed spin-S A and spin-S B (S A aS B ) Ising systems, particularly the mixed-spin, spin-1/2 and spin-1 Ising system. This system has been extensively studied by both theoretical and numerical techniques, such as the renormalization group technique [7,8], the high-temperature series expansions [9], the effective-field theory [10][11][12][13][14][15][16][17][18][19], the mean-field theory [20][21][22], the spin-wave theory [3,23], Green's function approach [25], the DMRG methods [3,23], and the Monte Carlo simulations [24]. Although delightful results were acquired from the above investigations, exact analytical solution for this model will continue to be of great value for us to understand the essential physics of this quantum mixed spin chain model.…”

Exact results of a dimerized S=1 Ising chain with single-ion anisotropy

“…So it is important to investigate the quantum phase transitions due to the interplay of dimerization and singleion anisotropy in S4 1 2 Ising lattice models. Recently, Derzhko et al [22][23][24][25][26][27] discussed the quantum phase transitions in regularly alternating spin- 1 2 Ising chain in a transverse field. For a given period of alternation the system exhibits a series of quantum phase transitions, where the number of transitions depends on the concrete set of the parameters of the Hamiltonian.…”