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arXiv:2301.04615

Estimating entanglement in 2D Heisenberg model in the strong rung-coupling limit

Chandrima B. Pushpan, K. J. Harikrishnan, Prithvi Narayan, Pal, Amit Kumar

Published Jan 11, 2023• 0 citations
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Abstract

In this paper, we calculate entanglement in the isotropic Heisenberg model in a magnetic field on a two-dimensional rectangular zig-zag lattice in the strong rung-coupling limit, using the one-dimensional XXZ model as a proxy. Focusing on the leading order in perturbation, for arbitrary size of the lattice, we show how the one-dimensional effective description emerges. We point out specific states in the low-energy sector of the two-dimensional model that are well-approximated by the one-dimensional spin-1/2 XXZ model. We propose a systematic approach for mapping matrix-elements of operators defined on the two-dimensional model to their low-energy counterparts on the one-dimensional XXZ model. We also show that partial trace-based description of entanglement in the two-dimensional model can be satisfactorily approximated using the one-dimensional XXZ model as a substitute. We further show numerically that the one-dimensional XXZ model performs well in estimating entanglement quantified using a measurement-based approach in the two-dimensional model for specific choices of measured Hermitian operators.

DOI
10.48550/arxiv.2301.04615

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In this paper, we calculate entanglement in the isotropic Heisenberg model in a magnetic field on a two-dimensional rectangular zig-zag lattice in the strong rung-coupling limit, using the one-dimensional XXZ model as a proxy. Focusing on the leading order in perturbation, for arbitrary size of the lattice, we show how the one-dimensional effective description emerges. We point out specific states in the low-energy sector of the two-dimensional model that are well-approximated by the one-dimensional spin-1/2 XXZ model. We propose a systematic approach for mapping matrix-elements of operators defined on the two-dimensional model to their low-energy counterparts on the one-dimensional XXZ model. We also show that partial trace-based description of entanglement in the two-dimensional model can be satisfactorily approximated using the one-dimensional XXZ model as a substitute. We further show numerically that the one-dimensional XXZ model performs well in estimating entanglement quantified using a measurement-based approach in the two-dimensional model for specific choices of measured Hermitian operators.
We consider the isotropic Heisenberg model in a magnetic field in the strong-rung-coupling limit on a two-dimensional (2D) rectangular zig-zag lattice of arbitrary size, and determine the one-dimensional (1D) effective model representing the low-energy manifold of the 2D model up to second order in perturbation theory. We consider a number of Hermitian operators defined on the Hilbert space of the 2D model, and systematically work out their action on the low-energy manifold, which are operators on the Hilbert space of the 1D effective model. For a class of operators among them, we demonstrate that the expectation values computed in the low-energy manifold of the 2D model can be mimicked by the expectation values of the corresponding operators in the 1D effective model even beyond the perturbation regime of the system parameters. We further argue that quantitatively estimating partial trace-based measures of entanglement in the 2D model may be done in the same fashion only in the perturbation regime. Our results and approach are expected to be useful in investigating observables and entanglement in the 2D models with large system sizes due to the advantage of using the effective 1D model with a smaller Hilbert space as a proxy.
We analyze the properties of low-energy bound states in the transverse-field Ising model and in the XXZ model on the square lattice. To this end, we develop an optimized implementation of perturbative continuous unitary transformations. The Ising model is studied in the small-field limit which is found to be a special case of the toric code model in a magnetic field. To analyze the XXZ model, we perform a perturbative expansion about the Ising limit in order to discuss the fate of the elementary magnon excitations when approaching the Heisenberg point.
We investigate the dynamical transfer of bipartite entanglement through a two-leg spin-1/2 ladder governed by the anisotropic Heisenberg (XXZ-type) model with a selective magnetic field applied exclusively to the mediating rungs. Starting from a maximally entangled initial rung pair, we demonstrate high-fidelity entanglement transfer to the terminal pair (F_max = 0.9998 for N = 3 rung pairs), with the intermediate rungs remaining effectively disentangled throughout. The dynamics is governed by two independent timescales: a fast carrier oscillation at frequency omega_fast = 2*sqrt(1 + 4d^2) J (set by local rung physics, field-independent) and a slow transfer envelope with period T_slow = 2.37 h/J^2 (set by virtual inter-rung coupling, field-dependent). The effective inter-rung coupling J_eff = alpha(d,g) J^2/h is derived via second-order perturbation theory through two parallel virtual paths. We systematically study the effects of magnetic field strength, Hamiltonian anisotropy, and initial state on transfer quality, establish a global parameter space map of the fidelity, and demonstrate robustness under uncorrelated coupling disorder (mean F_max > 0.998 for delta <= 10%). All results are obtained by exact diagonalisation for systems of up to N = 5 rung pairs; extension to larger systems requires tensor-network methods such as DMRG. Compared to one-dimensional chain proposals, the ladder geometry enables a spatially selective control mechanism that suppresses intermediate entanglement while preserving coherent transfer, providing a distinct route to engineered quantum channels.
We present a detailed investigation of the XXZ Heisenberg model for spin-1/21/21/2 and spin-1 systems on square and honeycomb lattices. Utilizing the density-matrix renormalization group method, complemented by spiral boundary conditions (SBCs) for mapping two-dimensional (2D) clusters onto one-dimensional chains, we meticulously explore the evolution of staggered magnetization and spin gaps across a broad spectrum of easy-axis anisotropies. Our study reveals that despite the lower site coordination number of the honeycomb lattice, which intuitively suggests increased quantum fluctuations in its N\'eel phase compared to the square lattice, the staggered magnetization in the honeycomb structure exhibits only a marginal reduction. Furthermore, our analysis demonstrates that the dependence of staggered magnetization on the XXZ anisotropy $\mathrm{\ensuremath{\Delta}}$, except in close proximity to $\mathrm{\ensuremath{\Delta}}=1$, aligns with series expansion predictions up to the 12th order. Notably, for the S=1/2S=1/2S=1/2 honeycomb lattice, deviations from the 10th-order series expansion predictions near the isotropic Heisenberg limit emphasize the critical influence of quantum fluctuations on the spin excitation in its N\'eel state. Additionally, our findings are numerically consistent with the singular behavior of the spin gap near the isotropic Heisenberg limit as forecasted by spin-wave theory. The successful implementation of SBCs marks a methodological advancement, streamlining the computational complexity involved in analyzing 2D models and paving the way for more precise determinations of physical properties in complex lattice systems.
We analyse the low-temperature behaviour of the classical isotropic ferromagnetic Heisenberg model on a two-dimensional square lattice of finite size. Presence of a residual magnetization in a finite-size system enables us to use a low-temperature approximation, which is however more restricting than the usual spin-wave approximation known to give reliable results for the XY model at low temperatures T . For the system considered, we find that the spin–spin correlation function decays as 1/ r η( T ) for large separations r bringing about the presence of a quasi-long-range ordering. We give analytic estimates for the exponent η( T ) in different regimes and support our findings by Monte Carlo simulations of the model on lattices of different sizes at different temperatures.
In this paper, we investigate spin entanglement in the XXZXXZXXZ model defined on a ddd-dimensional bipartite lattice. The concurrence, a measure of the entanglement between two spins, is analyzed. We prove rigorously that the ground-state concurrence reaches maximum at the isotropic point. For dimensionality $d\ensuremath{\geqslant}2$, the concurrence develops a cusp at the isotropic point and we attribute it to the existence of magnetic long-range order.