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arXiv:1206.1062 · Physical Review B

von Neumann entropy spectra and entangled excitations in spin-orbital models

Wen‐Long You, Andrzej M. Oleś, Peter Horsch

Published Sep 10, 2012• 38 citations
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Abstract

We consider the low-energy excitations of one-dimensional spin-orbital models which consist of spin waves, orbital waves, and joint spin-orbital excitations. Among the latter we identify strongly entangled spin-orbital bound states and spin-orbital quasiparticle states which appear as peaks in the von Neumann entropy spectral function introduced in this work. We present the scaling of the von Neumann entropy with system size and find a qualitatively different behavior for the bound state and the quasiparticle---the strong entanglement of these states is manifested by a universal logarithmic scaling of the von Neumann entropy with system size, while the entropy saturates for other spin-orbital excitations. We suggest that spin-orbital entanglement can be experimentally explored by the measurement of the dynamical spin-orbital correlations using resonant inelastic x-ray scattering, where strong spin-orbit coupling associated with the core hole plays a role.

DOI
10.1103/physrevb.86.094412

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We consider the low-energy excitations of one-dimensional spin-orbital models which consist of spin waves, orbital waves, and joint spin-orbital excitations. Among the latter we identify strongly entangled spin-orbital bound states and spin-orbital quasiparticle states which appear as peaks in the von Neumann entropy spectral function introduced in this work. We present the scaling of the von Neumann entropy with system size and find a qualitatively different behavior for the bound state and the quasiparticle---the strong entanglement of these states is manifested by a universal logarithmic scaling of the von Neumann entropy with system size, while the entropy saturates for other spin-orbital excitations. We suggest that spin-orbital entanglement can be experimentally explored by the measurement of the dynamical spin-orbital correlations using resonant inelastic x-ray scattering, where strong spin-orbit coupling associated with the core hole plays a role.
We investigate the phase diagram and the spin-orbital entanglement of a one-dimensional $\mathrm{SU}(2)\phantom{\rule{0.16em}{0ex}}\ensuremath{\bigotimes}\phantom{\rule{0.16em}{0ex}}\text{XXZ}$ model with SU(2) spin exchange and anisotropic XXZ\text{XXZ}XXZ orbital exchange interactions and negative exchange coupling constant. As a unique feature, the spin-orbital entanglement entropy in the entangled ground states increases here linearly with system size. In the case of Ising orbital interactions, we identify an emergent phase with long-range spin-singlet dimer correlations triggered by a quadrupling of correlations in the orbital sector. The peculiar translational-invariant spin-singlet dimer phase has finite von Neumann entanglement entropy and survives when orbital quantum fluctuations are included. It even persists in the isotropic $\mathrm{SU}(2)\phantom{\rule{0.16em}{0ex}}\ensuremath{\bigotimes}\phantom{\rule{0.16em}{0ex}}\text{SU}$(2) limit. Surprisingly, for finite transverse orbital coupling, the long-range spin-singlet correlations also coexist in the antiferromagnetic spin and alternating orbital phase making this phase also unconventional. Moreover, we also find a complementary orbital singlet phase that exists in the isotropic case but does not extend to the Ising limit. The nature of entanglement appears essentially different from that found in the frequently discussed model with positive coupling. Furthermore, we investigate the collective spin and orbital wave excitations of the disentangled ferromagnetic-spin/ferro-orbital ground state and explore the continuum of spin-orbital excitations. Interestingly, one finds among the latter excitations two modes of exciton bound states. Their spin-orbital correlations differ from the remaining continuum states and exhibit logarithmic scaling of the von Neumann entropy with increasing system size. We demonstrate that spin-orbital excitons can be experimentally explored using resonant inelastic x-ray scattering, where the strongly entangled exciton states can be easily distinguished from the spin-orbital continuum.
Entanglement plays a central role in quantum technologies, yet its characterization and control in materials remain challenging. Recent developments in spectrum-based entanglement witnesses have enabled new strategies for quantifying many-body entanglement in macroscopic materials. Here, we develop a protocol for detecting spin-orbital entanglement using experiment-accessible resonant inelastic x-ray scattering (RIXS). Central to our approach is the construction of a Hermitian generator from experimentally measurable spectra, which allows us to compute the quantum Fisher information (QFI) available in spin--orbital systems. The resulting QFI provides upper bounds for kkk-producible states and thus serves as a robust witness of spin-orbital entanglement. To account for realistic experimental limitations, we further extend our framework to include relaxed QFI bounds applicable to measurements lacking full polarization resolution.
To demonstrate the role played by the von Neumann entropy (vNE) spectra in quantum phase transitions we investigate the one-dimensional anisotropic SU(2) spin–orbital model with negative exchange parameter. In the case of classical Ising orbital interactions we discover an unexpected novel phase with Majumdar–Ghosh-like spin–singlet dimer correlations triggered by spin–orbital entanglement (SOE) and having orbital correlations, while all the other phases are disentangled. For anisotropic XXZ orbital interactions both SOE and spin–dimer correlations extend to the antiferro-spin/alternating-orbital phase. This quantum phase provides a unique example of two coupled order parameters which change the character of the phase transition from first-order to continuous. Hereby we have established the vNE spectral function as a valuable tool to identify the change of ground state degeneracies and of the SOE of elementary excitations in quantum phase transitions.
We investigate the distribution of the entanglement of the one-dimensional single-hole Hubbard model (HM) and study the relationship between the entanglement and quantum phase transition in the model. The von Neumann entropy of a block with neighbouring spins L for a single-hole HM is calculated using the density-matrix renormalization group. The distributions of the entanglement entropy in the ground state, as a function of block length, show a dramatic effect, i.e. effectively decoupling with the centres, no matter how the Coulomb interaction u >0 or u <0. Contrarily, for the Coulomb interaction u = 0 or close to zero, the entanglement entropy in the single-hole model reaches a saturation value for a certain block size. For a fixed size L = 40, the ground state entanglement entropy measure, as a function of u , shows a peak corresponding to the critical quantum phase transition.
The von Neumann entanglement entropy of exact valence-bond ground states is studied in two frustrated one-dimensional spin chains: the spin-1/2 Majumdar-Ghosh (MG) model and the spin-3/2 J1-J2-J3 chain in its fully dimerized (FD) and partially dimerized (PD) phases. Using matrix-product-state representations, the entropy is computed as a function of system size for three complementary bipartitions - half-chain, single-site, and pairwise - under both open and periodic boundary conditions. In all cases, the entropy saturates to a finite constant in the thermodynamic limit, confirming area-law behavior. The saturation values, extracted via finite-size scaling, are directly related to the underlying virtual-spin bond structure. The MG model and FD phase exhibit similar entanglement behavior, differing primarily in saturation magnitude determined by spin value and bond multiplicity, and both display even-odd oscillations and exponential convergence with system size. In contrast, the PD phase shows qualitatively distinct signatures, including multiple half-chain saturation values depending on the bond type at the cut, asymmetric edge contributions in the single-site entropy, and a multi-band structure in the pairwise entropy reflecting the coexistence of single- and double-singlet bonds. These results establish entanglement entropy as a robust signature of frustrated bond architecture, enabling clear distinction among dimerized phases with different spin magnitude, bond multiplicity, and dimerization patterns.
Interactions between elementary excitations in quasi-one-dimensional antiferromagnets are of experimental relevance and their quantitative theoretical treatment has been a theoretical challenge for many years. Using matrix product states, one can explicitly determine the wave functions of the one- and two-particle excitations, and, consequently, the contributions to dynamical correlations. We apply this framework to the (nonintegrable) frustrated dimerized spin-1/2 chain, a model for generic spin-Peierls systems, where low-energy quasiparticle excitations are bound states of topological solitons. The spin structure factor involving two quasiparticle scattering states is obtained in the thermodynamic limit with full momentum and frequency resolution. This allows very subtle features in the two-particle spectral function to be revealed which, we argue, could be seen, e.g., in inelastic neutron scattering of spin-Peierls compounds under a change of the external pressure.