Quantum Fisher Information in semiclassical magnon systems

The gist Quantum Fisher Information (QFI) serves as a momentum-resolved probe for quantum entanglement in condensed matter systems, revealing how entanglement depth and magnon squeezing relate to emergent quantum phases.

Interpretation of QFI in Spin Systems

1. QFI quantifies the momentum-dependent degree of quantum entanglement within a wave function when examining finite sized spin systems In this study we elaborate on the meaning and interpretation of QFI in condensed matter by examining simple theoretical spin systems It is shown that QFI quantifies the momentum-dependent degree of magnon squeezing in the ground state within semiclassical frustrated spin models.

It is important to note that QFI provides a lower bound for entanglement depth, and when nQFI > m, the spin system is at least m + 1 partite entangled Furthermore, in the context of linear spin wave theory (LSWT), QFI quantifies the momentum-dependent degree of magnon squeezing in the ground state.

Momentum Dependence and Entanglement Depth

1. The normalized QFI (nQFI) depends on the chosen momentum transfer Q, which is given by nQFI[Q] = fQ[Q]/4S squared This dependence indicates whether the wave function involves a distribution of different basis states encoding different periodicities, and thus an extended quantum correlation length.

The maximal nQFI as a bound of entanglement depth is obtained when one chooses the momentum transfer Q where nQFI is maximal, which is illustrated in Fig. 1(a). As N grows, the QFI peak around Q = π gets sharper so that as N → ∞ the peak becomes a delta function, reflecting that for an antiferromagnetic (AFM) S = 1/2 Ising chain in the thermodynamic limit, S(Q, ω) ∼ δ(ω) · δ(Q − π).

Semiclassical Magnon Systems and Criticality

1. In gapless antiferromagnets, the spectral weight exhibits a divergence around Q = (1/2, 1/2) for dominant nearest-neighbor exchange (δ = 0). This S(Q, ω) ∼ 1/ω intensity divergence corresponds to an instability where the Bogoliubov transform also diverges.

This QFI divergence means that the square lattice antiferromagnet is actually highly entangled already in its magnetically ordered regime (at zero temperature) that is well described in the semiclassical approximation. The hyperbolic rotation angle of the Bogoliubov transform, which encodes the matrix rotation from the Néel ordered basis, scales with the magnitude of nQFI at the same wave-vector.

Emergent Quantum Phases and QFI Divergence

1. When emergent quantum phases appear, QFI diverges occur over multiple wavevectors in momentum space. This behavior of diverging entanglement depth near quantum critical points is not unique to the square lattice, but occurs also in several frustrated LSWT models.

In all examples where a putative emergent quantum phase appears, QFI diverges at the momentum of a gapless Goldstone mode within the ordered, antiferromagnetic phase. This suggests that QFI is able to distinguish quantum phase transitions with emergent quantum phases from quantum phase transitions without emergent quantum phases.

Limitations and Nuances

1. A diverging QFI in the semiclassical models does not necessarily imply diverging entanglement depth in the quantum limit since the exotic quantum phases may have finite QFI with respect to the same observable. The exotic phases may still be highly entangled in a different basis or measure.

However, QFI is more than just a bound on entanglement depth; its Q-dependence reveals the degree of collective quantum superposition at various wavevectors. The study suggests that momentum-resolved QFI accessible with neutron spectroscopy may be a unique probe for exotic phenomena.

Conclusion

In summary, enhanced nQFI at a wave-vector translates directly to stronger quasiparticle squeezing and therefore enhanced entanglement for quasiparticles of the same wave-vector. The study highlights that QFI probes entangled quantum superposition at different wavevectors. Its enhancement over many momenta indicates proximity to emergent quantum phases of matter where ordinary magnon physics breaks down. This work suggests momentum-resolved QFI may serve as a witness of proximity to emergent quantum phases and breakdown of static magnetic order in materials such as strangemetals, cuprates, and heavy-fermions.

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Wolfgang Simeth, Pontus Laurell, Allen Scheie

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Quantum Fisher Information (QFI) is a powerful spectroscopic tool to witness many-body quantum entanglement in solid state materials, but it is not always obvious how to relate it to other characteristics of condensed matter systems In this study we elaborate on the meaning and interpretation of QFI in condensed matter by examining simple theoretical spin systems. We use finite sized spin systems to illustrate that QFI quantifies the momentum- dependent degree of quantum entanglement (that is the entanglement depth at a given wave vector) within a wave function. We subsequently use semiclassical frustrated spin models to show that, in the context of linear spin wave theory (LSWT), QFI quantifies the momentum-dependent degree of magnon squeezing in the ground state. In antiferromagnets with a zero-energy Goldstone mode, LSWT breaks down and QFI (and entanglement depth) diverges at the magnetic ordering wave vector. We also show examples of emergent quantum phases in frustrated spin systems that do not appear in classical phase diagram. When these emergent phases are approached, QFI diverges across multiple wave vectors in momentum space. Taken together, QFI is not only helpful as a lower bound for entanglement depth, but serves as a momentum-resolved probe of entanglement that offers a novel perspective on quantum critical phenomena.

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To begin, let us briefly review the definitions of QFI. For a pure state Ψ⟩, the QFI density with respect to an observable O is given by fQ:= 4/N · (⟨Ψ O†O Ψ⟩ − ⟨Ψ O Ψ⟩ 2), (1) where N denotes the number of spins in the system. That is, fQ is an operator-dependent quantity. In the context of spin entanglement, we are interested in observables OQ = Sα(Q) = P j S α j exp(iQ · rj). At T = 0 this can be written as the energy-integrated inelastic magnetic scattering Sαα(Q, E), via the relation fQ[Q] = 4 Z ∞ 0+ dE Sαα(Q, E) (2). The normalized QFI (nQFI) is given by nQFI[Q] = fQ/4S squared (3) and provides–for each wave vector Q–a lower bound on the entanglement-depth in the spin system. If nQFI > m, where m is a non-negative integer, the spin system is at least m + 1 partite entangled. As we are considering finite-sized spin system, m must be a divisor of N for this bound to hold.

1. The improved AI system can perform momentum-resolved entanglement diagnostics in bulk materials by utilizing neutron spectroscopy to measure QFI as a novel perspective on quantum critical phenomena. This allows for the distinction between entangled antiferromagnets from emergent quantum phases by analyzing whether QFI diverges across multiple wavevectors in momentum space.

2. The system can diagnose the nature of ground states in frustrated spin systems by calculating and interpreting the momentum-dependent QFI, specifically identifying when it serves as a harbinger of emergent quantum phases. This capability is crucial for understanding phases that do not appear in classical phase diagram[s].

3. The AI can assess the degree of collective quantum superposition within a wave function by analyzing the amplitude of nQFI, which witnesses the full entanglement depth nQFI = N for states like GHZ and cat states. This allows for quantifying how correlations contribute to the variance of O, providing a measure beyond just counting basis states.

4. The system can predict whether a zero-temperature phase transition will exhibit a line-diverging nQFI, as it can distinguish between systems where QFI diverges across multiple wavevectors and those where it does not, such as the field-driven quantum dimer ladder or cubic pyrochlore ferromagnet.

5. The AI can provide a rigorous theoretical framework for interpreting LSWT results by applying the Bogoliubov transformation, recognizing that in gapless antiferromagnets, QFI and therefore entanglement depth diverges to infinity as well. This allows for the interpretation of semiclassical models as strongly squeezed ground states even when they appear unentangled classically.