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Quantum Phase Transition and Entanglement of Bi-Spin System

Tosihide H. YOSIDA, Ji-Xuan Hou

Published May 1, 2007• 0 citations
OpenAlex

Abstract

In this paper we study the quantum phase transition and entanglement in s1=1/2 and s2=1 spin pair system by the exact diagonalization method. We show that, for this exactly solvable quantum bi-spin system, entanglement appears before quantum phase transition and disappears after it. Moreover, we show that the von Neumann entropy, as a measure of entanglement, can reveal quantum phase transition in this system.

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In this paper we study the quantum phase transition and entanglement in s1=1/2 and s2=1 spin pair system by the exact diagonalization method. We show that, for this exactly solvable quantum bi-spin system, entanglement appears before quantum phase transition and disappears after it. Moreover, we show that the von Neumann entropy, as a measure of entanglement, can reveal quantum phase transition in this system.
We show that, for an exactly solvable quantum spin model, a discontinuity in the first derivative of the ground-state concurrence appears in the absence of a quantum phase transition. It is opposed to the popular belief that the nonanalyticity property of ground-state concurrence can be used to determine quantum phase transitions. We further point out that the analyticity property of the ground-state concurrence in general can be more intricate than that of the ground-state energy. Thus there is no one-to-one correspondence between quantum phase transitions and the nonanalyticity property of the concurrence. Moreover, we show that the von Neumann entropy, as another measure of entanglement, cannot reveal quantum phase transitions in the present model. Therefore, in order to link with quantum phase transitions, some other measures of entanglement are needed.
Classical phase transitions have a quantum counterpart, where the mechanism which controls the phase changing is not the temperature anymore, but quantum fluctuations, as a consequence of Heisenberg uncertainty principle. In this monograph, we approach this subject by studying entanglement, a kind of correlation exclusive to quantum systems. By using the Ising model under a transverse magnetic field, we will show how entanglement can be used to detect quantum phase transitions in interacting systems. We will focus our studies in two main entanglement measures: negativity, which measures correlations between pair of spins inside the system, and the von Neumann entropy, to measure correlations between blocks of spins. We will discuss the main aspects of each type of these measures, and show their frailties when aplied to studies on more complex phase transition models
Spin-orbital entanglement in quantum spin-orbital systems is quantified by a specifically reduced von Neumann entropy and is calculated for the ground state of a coupled spin-orbital chain with $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$ symmetry. By analyzing the discontinuity and local extreme of the reduced entropy, we deduce a rich phase diagram describing quantum phase transitions in this system with complex correlations between multiple degrees of freedom.
The characterization of an infinite-order quantum phase transition (QPT) by entanglement measures is analyzed. To this aim, we consider two closely related solvable spin-1/2 chains, namely, the Ashkin-Teller and the staggered XXZ\mathit{XXZ}XXZ models. These systems display a distinct pattern of eigenstates but exhibit the same thermodynamics, that is, the same energy spectrum. By performing exact diagonalization, we investigate the behavior of pairwise and block entanglement in the ground state of both models. In contrast with the XXZ\mathit{XXZ}XXZ chain, we show that pairwise entanglement fails in the characterization of the infinite-order QPT in the Ashkin-Teller model, although it can be achieved by analyzing the distance of the pair state from the separability boundary. Concerning block entanglement, we show that both XXZ\mathit{XXZ}XXZ and Ashkin-Teller models exhibit identical von Neumann entropies as long as a suitable choice of blocks is performed. Entanglement entropy is then shown to be able to identify the quantum phase diagram, even though its local extremes (either maximum or minimum) may also appear in the absence of any infinite-order QPT.