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arXiv:2307.02407 · Physical review. B./Physical review. B

Quantum Fisher information and multipartite entanglement in spin-1 chains

Federico Dell’Anna, Sunny Pradhan, C. Degli Esposti Boschi, Elisa Ercolessi

Published Oct 16, 2023• 18 citations
arXivHTMLPDF

Abstract

In this paper, we study the ground-state quantum Fisher information (QFI) in one-dimensional spin-1 models, as witness to multipartite entanglement. The models addressed are the bilinear-biquadratic model, the most general isotropic $\text{SU}(2)$-invariant spin-1 chain, and the $XXZ$ spin-1 chain, both with nearest-neighbor interactions and open boundary conditions. We show that the scaling of the QFI of strictly nonlocal observables can be used for characterizing the phase diagrams and, in particular, for studying topological phases, where it scales maximally. Analyzing its behavior at the critical phases, we are also able to recover the scaling dimensions of the order parameters, both for local and string observables. The numerical results have been obtained by exploiting the density-matrix renormalization-group algorithm and tensor network techniques.

DOI
10.1103/physrevb.108.144414

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In this paper, we study the ground-state quantum Fisher information (QFI) in one-dimensional spin-1 models, as witness to multipartite entanglement. The models addressed are the bilinear-biquadratic model, the most general isotropic SU(2)\text{SU}(2)SU(2)-invariant spin-1 chain, and the XXZXXZXXZ spin-1 chain, both with nearest-neighbor interactions and open boundary conditions. We show that the scaling of the QFI of strictly nonlocal observables can be used for characterizing the phase diagrams and, in particular, for studying topological phases, where it scales maximally. Analyzing its behavior at the critical phases, we are also able to recover the scaling dimensions of the order parameters, both for local and string observables. The numerical results have been obtained by exploiting the density-matrix renormalization-group algorithm and tensor network techniques.
This study investigates quantum Fisher information (QFI) with respect to a class of nonlocal operators $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}$ in one-dimensional quantum chains. Based upon the triangular matrix-product-operator theory, we establish a universal scaling behavior of QFI with respect to $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{O}$ for general infinite matrix-product states (iMPSs): $\mathcal{F}\ensuremath{\sim}{\ensuremath{\varrho}}_{2}{N}^{2}+{\ensuremath{\varrho}}_{1}N+{\ensuremath{\varrho}}_{0}$, where the coefficients ${\ensuremath{\varrho}}_{i}$ can be efficiently identified by iMPS algorithms. Applied to a spin-1 Bilinear-Biquadratic model and an XXZ model, the scaling coefficients ${{\ensuremath{\varrho}}_{i}}$ distinguish various phases through distinct scaling behaviors: For instance, the topologically ordered Haldane phase exhibits characteristic quadratic scaling (${\ensuremath{\varrho}}_{2}\ensuremath{\ne}0$), while the trimer and dimer phases are featured with linear scaling (${\ensuremath{\varrho}}_{1}\ensuremath{\ne}0$). Notably, since the QFI scaling behavior undergoes fundamental changes between distinct phases, these coefficients provide signals at first-order, second-order, and Berezinskii-Kosterlitz-Thouless type quantum phase transitions in these models. Furthermore, finite bond-dimension (DDD) scaling analysis reveals a dichotomy. For gapped phases, the coefficients converge rapidly as DDD increases. Conversely, for critical systems, the coefficients exhibit DDD-scaling effects that depend on the specific model. By bypassing finite-size fittings with limited NNN, our approach accesses the thermodynamic-limit physics for QFI with respect to nonlocal operators, offering new avenues for characterizing exotic quantum phases in one-dimensional quantum systems.
We demonstrate that multipartite entanglement is able to characterize one-dimensional topological order, which is witnessed by the scaling behavior of the quantum Fisher information of the ground state with respect to the spin operators defined in the original and dual lattices. We investigate an exactly-solvable long-range extended spin-12\frac{1}{2}21​ Ising model with nontrivial Z\mathbf{Z}Z-symmetry topological order identified equivalently by winding numbers and paired Majorana zero modes at each end. For topologically ordered phases with high winding numbers, we can define a Z\mathbf{Z}Z topological invariant by the scaling coefficient of the dual quantum Fisher information density. Containing richer properties and more complex structures than bipartite entanglement, the dual multipartite entanglement of the topologically ordered state has promising applications in robust quantum computation and quantum metrology, and could be generalized to higher-dimensional systems.
Abstract The theory of multipartite entanglement provides a robust framework for understanding complex phenomena in quantum many-body systems, while multipartite entanglement also serves as a key resource in numerous quantum information processing tasks. Using density matrix renormalization group calculations, we study the multipartite entanglement in one-dimensional quantum long-range ferromagnetic and antiferromagnetic Ising chains with algebraically decaying interactions 1/ r α , respectively. To quantitatively probe the system’s multipartite entanglement structure, we employ the quantum Fisher information (QFI). Our results reveal two key behaviors: as the system size increases, the QFI rises, while the multipartite entanglement density decreases and converges toward a constant value. The latter trend points to a fundamental constraint on entanglement distribution in the thermodynamic limit. We further discuss the effects of the magnetic field and the exponent α on the QFI. We find that the peak values of the QFI increase with increasing α . Additionally, the discontinuity in the QFI can be regarded as a powerful tool for characterizing quantum phase transitions in the systems.
We demonstrate that multipartite entanglement is able to characterize one-dimensional symmetry-protected topological order, which is witnessed by the scaling behavior of the quantum Fisher information of the ground state with respect to the spin operators defined in the dual lattice. We investigate an extended Kitaev chain with a Z symmetry identified equivalently by winding numbers and paired Majorana zero modes at each end. The topological phases with high winding numbers are detected by the scaling coefficient of the quantum Fisher information density with respect to generators in different dual lattices. Containing richer properties and more complex structures than bipartite entanglement, the dual multipartite entanglement of the topological state has promising applications in robust quantum computation and quantum metrology, and can be generalized to identify topological order in the Kitaev honeycomb model.
Identifying and characterizing quantum phases of matter in the presence of long range correlations and/or spatial disorder is, generally, a challenging and relevant task. Here, we study a generalization of the Kiteav chain with variable-range pairing and different site-dependence of the chemical potential, addressing commensurable and incommensurable modulations as well as Anderson disorder. In particular, we analyze multipartite entanglement (ME) in the ground state of the dirty topological wires by studying the scaling of the quantum Fisher information (QFI) with the system's size. For nearest-neighbour pairing the Heisenberg scaling of the QFI is found in one-to-one correspondence with topological phases hosting Majorana modes. For finite-range pairing, we recognize long-range phases by the super-extensive scaling of the QFI and characterize complex lobe-structured phase diagrams. Overall, we observe that ME is robust against finite strengths of spatial inhomogeneity. This work contributes to establish ME as a central quantity to study intriguing aspects of topological systems.
Characterizing entanglement in quantum materials through experimentally accessible observables remains a central challenge. Quantum Fisher information associated with sums of one-site spin operators provides one route by linking entanglement to measurable magnetic response functions. Extending this approach to local composite observables requires new entanglement bounds. Here we establish a rigorous bound for the total nearest-neighbour vector chirality of periodic spin-1/21/21/2 chains with an even number of sites. The quantum Fisher information per spin cannot exceed k+1k+1k+1 for any kkk-producible state, so exceeding this bound certifies entanglement depth of at least k+1k+1k+1. Combining infinite density-matrix renormalization group and thermal pure quantum-state calculations, we demonstrate multipartite entanglement encoded in vector-chiral correlations at zero and finite temperatures in an extended J1J_1J1​-J2J_2J2​ XXZ chain. The spin-current coupling between vector chirality and electric polarization offers, in principle, access to this witness through dielectric loss spectroscopy in multiferroic magnets, extending entanglement diagnostics beyond dipolar spin correlations.