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arXiv:1310.3715 · Physical Review Letters

Proposal for Verification of the Haldane Phase Using Trapped Ions

I. Cohen, Alex Retzker

Published Jan 31, 2014• 46 citations
arXivHTMLPDF

Abstract

A proposal to use trapped ions to implement spin-one XXZ antiferromagnetic chains as an experimental tool to explore the Haldane phase is presented. We explain how to reach the Haldane phase adiabatically, demonstrate the robustness of the ground states to noise in the magnetic field and Rabi frequencies, and propose a way to detect them using their characteristics: an excitation gap and exponentially decaying correlations, a nonvanishing nonlocal string order, and a double degenerate entanglement spectrum. Scaling up to higher dimensions and more frustrated lattices, we obtain richer phase diagrams, and we can reach spin liquid phase, which can be detected by its entanglement entropy which obeys the boundary law.

DOI
10.1103/physrevlett.112.040503

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A proposal to use trapped ions to implement spin-one XXZ antiferromagnetic chains as an experimental tool to explore the Haldane phase is presented. We explain how to reach the Haldane phase adiabatically, demonstrate the robustness of the ground states to noise in the magnetic field and Rabi frequencies, and propose a way to detect them using their characteristics: an excitation gap and exponentially decaying correlations, a nonvanishing nonlocal string order, and a double degenerate entanglement spectrum. Scaling up to higher dimensions and more frustrated lattices, we obtain richer phase diagrams, and we can reach spin liquid phase, which can be detected by its entanglement entropy which obeys the boundary law.
Defects in frustrated antiferromagnetic spin chains are universally present in geometrically frustrated systems. We consider the defects of the one-dimensional, spin-sss XXZ chain with single-ion anisotropy on a periodic chain with NNN sites that was famously studied by Haldane. For NNN odd, the antiferromagnetic model is frustrated, and the ground state must include a soliton defect. We consider the Heisenberg interaction perturbatively and determine the corresponding perturbative solitonic ground state. Then we compute the entanglement spectrum, entanglement entropy (EE), capacity of entanglement (CE), and spin correlations in the solitonic ground state. For weak frustration, we find an algebraic violation of the area law for the EE consistent with recent results on weakly frustrated chains. Our analysis then moves beyond the weak frustration regime, and we obtain a novel extensive scaling law for the EE when strong frustration prevails, signaling large entanglement, and failure of the quasiparticle interpretation in this regime. Enhanced frustration results in less total correlations but relatively more nonlocal correlations.
String correlations are investigated in an infinite-size XXZ spin-1 chain. By using the infinite matrix product state representation, we calculate a long-range string order. In the XY phase, the string correlations decay within a relatively very large lattice distance, which makes a finite-size study difficult to verify the non-existence of the string order. Thus, in the Haldane phase, the non-vanishing string correlations in the limit of a very large distance allow to characterize the phase boundaries to the XY phase and the Neel phase, which implies that the transverse long-range string order is the order parameter for the Haldane phase. In addition, the singular behaviors of the von Neumann entropy and the fidelity per lattice site are shown to capture clearly the phase transition points that are consistent with the results from the string order. The estimated critical points including a BKT transition from the XY phase to the Haldane phase agree well with the previous results: Δc2=0Δ_{c2} = 0Δc2​=0 for the XY-Haldane phase transition and Δc3=1.185Δ_{c3} = 1.185Δc3​=1.185 for the Haldane-N'eel phase transition from the density renormalization group. From a finite-entanglement scaling of the von Neumann entropy with respect to the truncation dimension, the central charges are found to be c≃1.0c \simeq 1.0c≃1.0 at Δc2=0Δ_{c2} = 0Δc2​=0 and c≃0.5c \simeq 0.5c≃0.5 at Δc3=1.185Δ_{c3} = 1.185Δc3​=1.185, respectively, which shows that the XY-Haldane phase transition at Δc2=0Δ_{c2} = 0Δc2​=0 belongs to the Heisenberg universality class, while the Haldane-Neel phase transition at Δc2=1.185Δ_{c2} = 1.185Δc2​=1.185 belongs to the two-dimensional classical Ising universality class. It is also shown that, the long-range order parameters and the von Neumann entropy, as well as the fidelity per site approach, can be applied to characterize quantum phase transitions as a universal phase transition indicator for one-dimensional lattice many-body systems.
We propose an optical lattice setup to investigate spin chains and ladders. Electric and magnetic fields allow us to vary at will the coupling constants, producing a variety of quantum phases including the Haldane phase, critical phases, quantum dimers, etc. Numerical simulations are presented showing how ground states can be prepared adiabatically. We also propose ways to measure a number of observables, like energy gap, staggered magnetization, end-chain spins effects, spin correlations, and the string-order parameter.
The quantum simulation of complex many body systems holds promise for understanding the origin of emergent properties of strongly correlated systems, such as high-Tc superconductors and spin liquids. Cold atomic systems provide an almost ideal platform for quantum simulation due to their excellent quantum coherence, initialization and readout properties, and their ability to support several forms of interactions. In this thesis, I present experiments on the quantum simulation of long range Ising models in the presence of transverse magnetic fields with a chain of up to sixteen ultracold 171-Yb+ ions trapped in a linear radio frequency Paul trap. Two hyperfine levels in each of the 171-Yb+ ions serve as the spin-1/2 systems. We detect the spin states of the individual ions by observing state-dependent fluorescence with single site resolution, and can directly measure any possible spin correlation function. The spin-spin interactions are engineered by applying dipole forces from precisely tuned lasers whose beatnotes induce stimulated Raman transitions that couple virtually to collective phonon modes of the ion motion. The Ising couplings are controlled, both in sign and strength with respect to the effective transverse field, and adiabatically manipulated to study various aspects of this spin model, such as the emergence of a quantum phase transition in the ground state and spin frustration due to competing antiferromagnetic interactions. Spin frustration often gives rise to a massive degeneracy in the ground state, which can lead to entanglement in the spin system. We detect and characterize this frustration induced entanglement in a system of three spins, demonstrating the first direct experimental connection between frustration and entanglement. With larger numbers of spins we also vary the range of the antiferromagnetic couplings through appropriate laser tunings and observe that longer range interactions reduce the excitation energy and thereby frustrate the ground state order. This system can potentially be scaled up to study a wide range of fully connected spin networks with a few dozens of spins, where the underlying theory becomes intractable on a classical computer.
Entanglement of the ground states in the S = 1/2 XXZ chain, dimerized Heisenberg spin chain, two-leg spin ladders as well as S = 1 anisotropic Haldane chain is analysed using the entanglement entropy between a selected sublattice of spins and the rest of the system. In particular, we reveal that quantum phase transition points/boundaries may be identified based on the analysis on the local extreme of this sublattice entanglement entropy, which is illustrated to be superior over the concurrence scenario and may enable us to explore quantum phase transitions in many other systems including higher dimensional ones.