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Physical Review E

Identifying a correlated spin fluctuation in an entangled spin chain subject to a quantum phase transition

Kaoru Shimizu, 康弘 都倉

Published Dec 28, 2015• 0 citations
Publisher

Abstract

This paper presents a theoretical framework for analyzing the quantum fluctuation properties of a quantum spin chain subject to a quantum phase transition. We can quantify the fluctuation properties by examining the correlation between the fluctuations of two neighboring spins subject to the quantum uncertainty. To do this, we first compute the reduced density matrix \ensuremath{\rho} of the spin pair from the ground state |\ensuremath{\Psi}⟩ of a spin chain, and then identify the quantum correlation part ${\ensuremath{\rho}}_{q}$ embedded in \ensuremath{\rho}. If the spin chain is translationally symmetric and characterized by a nearest-neighbor two-body spin interaction, we can determine uniquely the form of ${\ensuremath{\rho}}_{q}$ as $W|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}\ensuremath{\langle}\mathrm{\ensuremath{\Phi}}|$ with the weight $W$ $\ensuremath{\le}1$, and quantify the fluctuation properties using the two-spin entangled state $|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}$. We demonstrate the framework for a transverse-field quantum Ising spin chain and indicate its validity for more general spin chain models.

DOI
10.1103/physreve.92.062143

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This paper presents a theoretical framework for analyzing the quantum fluctuation properties of a quantum spin chain subject to a quantum phase transition. We can quantify the fluctuation properties by examining the correlation between the fluctuations of two neighboring spins subject to the quantum uncertainty. To do this, we first compute the reduced density matrix \ensuremath{\rho} of the spin pair from the ground state |\ensuremath{\Psi}⟩ of a spin chain, and then identify the quantum correlation part \ensuremathρq{\ensuremath{\rho}}_{q}\ensuremathρq​ embedded in \ensuremath{\rho}. If the spin chain is translationally symmetric and characterized by a nearest-neighbor two-body spin interaction, we can determine uniquely the form of \ensuremathρq{\ensuremath{\rho}}_{q}\ensuremathρq​ as W∣\ensuremathΦ\ensuremath⟩\ensuremath⟨\ensuremathΦ∣W|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}\ensuremath{\langle}\mathrm{\ensuremath{\Phi}}|W∣\ensuremathΦ\ensuremath⟩\ensuremath⟨\ensuremathΦ∣ with the weight WWW \ensuremath≤1\ensuremath{\le}1\ensuremath≤1, and quantify the fluctuation properties using the two-spin entangled state ∣\ensuremathΦ\ensuremath⟩|\mathrm{\ensuremath{\Phi}}\ensuremath{\rangle}∣\ensuremathΦ\ensuremath⟩. We demonstrate the framework for a transverse-field quantum Ising spin chain and indicate its validity for more general spin chain models.
We studied the entangled state for a one-dimensional S=1/2S=1/2S=1/2 antiferromagnetic quantum spin chain in a transverse field. We calculate the ground state using the density matrix renormalization group and discuss how the entangled state changes around a quantum phase transition (QPT) point. By analyzing concurrence C(ρ)C(ρ)C(ρ) for two-qubit density matrix ρρρ after the Lewenstein-Sanpera decomposition, $ρ=Λρ_s + (1-Λ) ρ_e $, where ρsρ_sρs​ is a separable density matrix and ρeρ_eρe​ is a pure entangled state obtained by a linear combination of Bell states, we find singular behaviors both in C(ρe)C(ρ_e)C(ρe​) and 1−Λ1-Λ1−Λ at the QPT point. C(ρe)C(ρ_e)C(ρe​) includes the effects of quantum fluctuations, which manifest the competition between the antiferromagnetic spin fluctuation and the effect of transverse field in the transverse Ising model. The quantum fluctuation shows the singular maximum at the QPT point as expected from the general picture of phase transition. In contrast, 1−Λ1-Λ1−Λ reveals the singular minimum at QPT point.
The behavior of Ising chains with the spin-spin interaction value λ in a transverse magnetic field of constant intensity ( h = 1) is considered. For a chain of infinite length, exact analytical formulas are obtained for the second central moment (dispersion) of the entropy operator Ŝ = -lnρ with reduced density matrix ρ, which corresponds to a semi-infinite part of the model chain occurring in the ground state. In the vicinity of a critical point λ c = 1, the entanglement entropy fluctuation Δ S (defined as the square root of dispersion) diverges as Δ S ∼ [ln(1/|1 − λ|)] 1/2 . For the known behavior of the entanglement entropy S , this divergence results in that the relative fluctuation δ S = Δ S / S vanishes at the critical point, that is, a state with almost nonfluctuating entanglement is attained.
We investigate superpositions of quantum Ising ground states generated through a quantum interaction, in which the nearest-neighbor coupling strength is promoted to an operator acting on a central auxiliary spin. We prepare the central spin in a superposition, which allows the chain to evolve coherently under different effective interaction strengths. The states are prepared by adiabatic evolution of the joint system comprising the central spin and the many-body spin chain, followed by a measurement on the central spin and post-selection of a specific outcome. It is possible to prepare superpositions of transverse-field quantum Ising ground states belonging to the three magnetic phases, namely, the paramagnetic, ferromagnetic, and anti-ferromagnetic phases. For a spin-1/2 central system, it is possible to prepare superpositions of ground states belonging to any two phases, while for the spin-1 case, the superpositions may contain three ground states, each belonging to one of the three phases. We analytically derive expressions for the magnetization, spin correlations, and entanglement of such states. The magnetization and spin correlations show distinct features in different superposition sectors. The nearest-neighbor entanglement is largest where the transverse field and Ising interaction compete, and its behavior differs from that of the magnetic observables. We also derive the quantum Fisher information relevant to estimating the Ising interaction strength. And it exhibits multiple peaks near the critical points of the constituent ground states, with the maximum value showing Heisenberg scaling, for the parameters and system sizes examined.
We consider a one-dimensional spin chain for which the ground state is the cluster state, capable of functioning as a quantum computational wire when subjected to local adaptive measurements of individual qubits, and investigate the robustness of this property to local and coupled (Ising-type) perturbations. We investigate the ground state both by identifying suitable correlation functions as order parameters, as well as numerically using a variational method based on matrix product states. We find that the model retains an infinite localizable entanglement length for Ising and local fields up to a quantum phase transition, but that the resulting entangled state is not simply characterized by a Pauli correction based on the measurement results.
Using a renormalization group approach, we solve the time evolution of random Ising spin chains with generic interactions starting from initial states of arbitrary energy. As a function of the Hamiltonian parameters, the system is tuned through a dynamical transition, similar to the ground-state critical point, at which the local spin correlations establish true long-range temporal order. In the state with a dominant transverse field, a spin that starts in an up state loses its orientation with time, while in the ``ordered'' state it never does. As in ground-state quantum phase transitions, the dynamical transition has unique signatures in the entanglement properties of the system. When the system is initialized in a product state, the entanglement entropy grows as log(t)\mathrm{log}(t)log(t) in the two ``phases,'' while at the critical point it grows as ${\mathrm{log}}^{\ensuremath{\alpha}}(t)$, with $\ensuremath{\alpha}$ a universal number. This universal entanglement growth requires generic (``integrability breaking'') interactions to be added to the pure transverse field Ising model.
The entanglement and quantum correlation measures have been investigated for the ground state of a spin chain with a Kitaev-type exchange interactions on alternating bonds, along with a transverse magnetic field. There is a macroscopic degeneracy in the ground state for zero magnetic field, implying a quantum critical point. In this model, many entanglement and correlation measures, except the macroscopic multispecies entanglement measure, do not show true singular behavior in the vicinity of the critical point, as seen in the transverse-Ising model ground state and related models. We compute the pair concurrence measure of entanglement, the pair quantum discord to track the quantum correlations, and a global entanglement measure and a multispecies entanglement measure to investigate multiparty entanglement, both analytically and numerically. The nearest-neighbor concurrence shows a peak structure as a function of magnetic field, near the critical point, for various values of the ratio of interaction strengths, but its derivative does not show a singular behavior close to the critical point. A similar behavior is shown by the quantum discord and the global entanglement. The multispecies entanglement shows a clear signature of the critical point behavior, and the first-order derivative shows a divergence near zero magnetic field, for equal interaction strengths.